From 3bc0e04c0e7b1bafce0b6cf004d648b1a20ceec8 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Thu, 26 Jul 2018 15:29:54 +0200 Subject: [PATCH 01/20] Very first outline of linear elastoplastic problem --- Linear_Elastoplasticity/CMakeLists.txt | 39 + .../Linear_elastoplasticity.cc | 959 ++++++++++++++++++ Linear_Elastoplasticity/parameters.prm | 57 ++ 3 files changed, 1055 insertions(+) create mode 100644 Linear_Elastoplasticity/CMakeLists.txt create mode 100644 Linear_Elastoplasticity/Linear_elastoplasticity.cc create mode 100644 Linear_Elastoplasticity/parameters.prm diff --git a/Linear_Elastoplasticity/CMakeLists.txt b/Linear_Elastoplasticity/CMakeLists.txt new file mode 100644 index 00000000..9e8dd932 --- /dev/null +++ b/Linear_Elastoplasticity/CMakeLists.txt @@ -0,0 +1,39 @@ +## +# CMake script for the step-8 tutorial program: +## + +# Set the name of the project and target: +SET(TARGET "Linear_elastoplasticity") + +# Declare all source files the target consists of. Here, this is only +# the one step-X.cc file, but as you expand your project you may wish +# to add other source files as well. If your project becomes much larger, +# you may want to either replace the following statement by something like +# FILE(GLOB_RECURSE TARGET_SRC "source/*.cc") +# FILE(GLOB_RECURSE TARGET_INC "include/*.h") +# SET(TARGET_SRC ${TARGET_SRC} ${TARGET_INC}) +# or switch altogether to the large project CMakeLists.txt file discussed +# in the "CMake in user projects" page accessible from the "User info" +# page of the documentation. +SET(TARGET_SRC + ${TARGET}.cc + ) + +# Usually, you will not need to modify anything beyond this point... + +CMAKE_MINIMUM_REQUIRED(VERSION 2.8.8) + +FIND_PACKAGE(deal.II 8.5 QUIET + HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR} + ) +IF(NOT ${deal.II_FOUND}) + MESSAGE(FATAL_ERROR "\n" + "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n" + "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n" + "or set an environment variable \"DEAL_II_DIR\" that contains this path." + ) +ENDIF() + +DEAL_II_INITIALIZE_CACHED_VARIABLES() +PROJECT(${TARGET}) +DEAL_II_INVOKE_AUTOPILOT() diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/Linear_elastoplasticity.cc new file mode 100644 index 00000000..56dfad65 --- /dev/null +++ b/Linear_Elastoplasticity/Linear_elastoplasticity.cc @@ -0,0 +1,959 @@ +/* --------------------------------------------------------------------- + * Copyright (C) 2018 by the deal.II authors and + * Jean-Paul Pelteret and Tim Hamann + * + * This file is part of the deal.II library. + * + * The deal.II library is free software; you can use it, redistribute + * it, and/or modify it under the terms of the GNU Lesser General + * Public License as published by the Free Software Foundation; either + * version 2.1 of the License, or (at your option) any later version. + * The full text of the license can be found in the file LICENSE at + * the top level of the deal.II distribution. + * + * --------------------------------------------------------------------- + */ + +/* + * Authors: Jean-Paul Pelteret + * University of Erlangen-Nuremberg, 2018 + */ + +// @sect3{Include files} + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include +#include +#include + +namespace LMM +{ + using namespace dealii; + +// @sect3{Run-time parameters} +// +// There are several parameters that can be set in the code so we set up a +// ParameterHandler object to read in the choices at run-time. + namespace Parameters + { +// @sect4{Finite Element system} + +// Here we specify the polynomial order used to approximate the solution. +// The quadrature order should be adjusted accordingly. + struct FESystem + { + unsigned int poly_degree; + unsigned int quad_order; + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + void FESystem::declare_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Finite element system"); + { + prm.declare_entry("Polynomial degree", "1", + Patterns::Integer(0), + "Displacement system polynomial order"); + + prm.declare_entry("Quadrature order", "2", + Patterns::Integer(0), + "Gauss quadrature order"); + } + prm.leave_subsection(); + } + + void FESystem::parse_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Finite element system"); + { + poly_degree = prm.get_integer("Polynomial degree"); + quad_order = prm.get_integer("Quadrature order"); + } + prm.leave_subsection(); + } + + // @sect4{Boundary conditions} + + // The boundary conditions to be applied to the problem. + + struct BoundaryConditions + { + double final_stretch; + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + void BoundaryConditions::declare_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Boundary conditions"); + { + prm.declare_entry("Final stretch", "2.0", + Patterns::Double(0), + "The amount by which the specimen is to be stretched"); + } + prm.leave_subsection(); + } + + void BoundaryConditions::parse_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Boundary conditions"); + { + final_stretch = prm.get_double("Final stretch"); + } + prm.leave_subsection(); + } + +// @sect4{Geometry} + +// Make adjustments to the geometry of the notched specimen. + + struct Geometry + { + double length; + double radius; + double notch_length; + double notch_radius; + double scale; + unsigned int n_global_refinement_steps; + unsigned int n_local_refinement_steps; + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + void Geometry::declare_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Geometry"); + { + prm.declare_entry("Length", "100", + Patterns::Double(0), + "Overall length of the specimen"); + + prm.declare_entry("Radius", "10", + Patterns::Double(0), + "Overall radius of the specimen"); + + prm.declare_entry("Notch length", "10", + Patterns::Double(0), + "Overall length of the notch in the specimen"); + + prm.declare_entry("Notch radius", "1", + Patterns::Double(0), + "Overall radius of the notch in the specimen"); + + prm.declare_entry("Grid scale", "1e-3", + Patterns::Double(0), + "Global grid scaling factor"); + + prm.declare_entry("Global refinement steps", "1", + Patterns::Integer(0), + "Number of global refinement steps"); + + prm.declare_entry("Local refinement steps", "1", + Patterns::Integer(0), + "Number of initial local refinement cycles"); + } + prm.leave_subsection(); + } + + void Geometry::parse_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Geometry"); + { + length = prm.get_double("Length"); + radius = prm.get_double("Radius"); + notch_length = prm.get_double("Notch length"); + notch_radius = prm.get_double("Notch radius"); + scale = prm.get_double("Grid scale"); + n_global_refinement_steps = prm.get_integer("Global refinement steps"); + n_local_refinement_steps = prm.get_integer("Local refinement steps"); + } + prm.leave_subsection(); + + AssertThrow(length > notch_length, + ExcMessage("Cannot create geometry with given dimensions")); + AssertThrow(radius > notch_radius, + ExcMessage("Cannot create geometry with given dimensions")); + } + + // @sect4{Material properties} + + // Make adjustments to the material of the notched specimen. + + struct MaterialProperties + { + double E_e; // Young's modulus + double nu_e; // Poisson ratio + + double mu_e; // Shear modulus + double lambda_e; // Lame parameter + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + void MaterialProperties::declare_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Material properties"); + { + prm.declare_entry("Young's modulus", "100", + Patterns::Double(0), + "Young's modulus of the specimen"); + + prm.declare_entry("Poisson ratio", "10", + Patterns::Double(0), + "Poisson ratio of the specimen"); + } + prm.leave_subsection(); + } + + void MaterialProperties::parse_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Material properties"); + { + E_e = prm.get_double("Young's modulus"); + nu_e = prm.get_double("Poisson ratio"); + } + prm.leave_subsection(); + + mu_e = E_e/(2.0*(1.0 + nu_e)); + lambda_e = 2.0*mu_e*nu_e/(1.0 - 2.0*nu_e); + } + + + +// @sect4{Time} + +// Set the timestep size $ \varDelta t $ and the simulation end-time. + struct Time + { + double delta_t; + double end_time; + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + void Time::declare_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Time"); + { + prm.declare_entry("End time", "3", + Patterns::Double(0), + "End time"); + + prm.declare_entry("Time step size", "0.1", + Patterns::Double(0), + "Time step size"); + } + prm.leave_subsection(); + } + + void Time::parse_parameters(ParameterHandler &prm) + { + prm.enter_subsection("Time"); + { + end_time = prm.get_double("End time"); + delta_t = prm.get_double("Time step size"); + } + prm.leave_subsection(); + } + +// @sect4{All parameters} + +// Finally we consolidate all of the above structures into a single container +// that holds all of our run-time selections. + struct AllParameters : public FESystem, + public BoundaryConditions, + public Geometry, + public MaterialProperties, + public Time + { + AllParameters(const std::string &input_file); + + static void + declare_parameters(ParameterHandler &prm); + + void + parse_parameters(ParameterHandler &prm); + }; + + AllParameters::AllParameters(const std::string &input_file) + { + ParameterHandler prm; + declare_parameters(prm); + prm.parse_input(input_file); + parse_parameters(prm); + } + + void AllParameters::declare_parameters(ParameterHandler &prm) + { + FESystem::declare_parameters(prm); + BoundaryConditions::declare_parameters(prm); + Geometry::declare_parameters(prm); + MaterialProperties::declare_parameters(prm); + Time::declare_parameters(prm); + } + + void AllParameters::parse_parameters(ParameterHandler &prm) + { + FESystem::parse_parameters(prm); + BoundaryConditions::parse_parameters(prm); + Geometry::parse_parameters(prm); + MaterialProperties::parse_parameters(prm); + Time::parse_parameters(prm); + } + } + + // @sect3{Time} + + class Time + { + public: + Time (const double time_end, + const double delta_t) + : + timestep(0), + time_current(0.0), + time_end(time_end), + delta_t(delta_t) + {} + virtual ~Time() + {} + double current() const + { + return time_current; + } + double end() const + { + return time_end; + } + double get_delta_t() const + { + return delta_t; + } + unsigned int get_timestep() const + { + return timestep; + } + void increment() + { + time_current += delta_t; + ++timestep; + } + private: + unsigned int timestep; + double time_current; + const double time_end; + const double delta_t; +}; + + + + template + class Material_Linear_Elastoplastic + { + public: + Material_Linear_Elastoplastic(const double lambda_e, + const double mu_e, +// const double mu_v, +// const double tau_v, + const Time &time) + : + lambda_e(lambda_e), + mu_e(mu_e), +// mu_v(mu_v), +// tau_v(tau_v), + time(time) +// Q_n_t(Physics::Elasticity::StandardTensors::I), +// Q_t1(Physics::Elasticity::StandardTensors::I) + { +// Assert(kappa > 0, ExcInternalError()); + } + ~Material_Linear_Elastoplastic() + {} + + SymmetricTensor<2,dim> + get_sigma(const SymmetricTensor<2, dim> &epsilon) const + { + return get_K(epsilon)*epsilon; + } + SymmetricTensor<4,dim> get_K(const SymmetricTensor<2, dim> &epsilon) const + { + (void)epsilon; + static const SymmetricTensor<2,dim> I = unit_symmetric_tensor(); + + SymmetricTensor<4,dim> K; + for (unsigned int i=0; i < dim; ++i) + for (unsigned int j=i; j < dim; ++j) + for (unsigned int k=0; k < dim; ++k) + for (unsigned int l=k; l < dim; ++l) + { + // Matrix contribution + K[i][j][k][l] = lambda_e * I[i][j]*I[k][l] + + mu_e * (I[i][k]*I[j][l] + I[i][l]*I[j][k]); + } + + return K; + } + void + update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) + { + (void)epsilon; +// const double det_F = determinant(F); +// const SymmetricTensor<2,dim> C_bar = std::pow(det_F, -2.0 / dim) * Physics::Elasticity::Kinematics::C(F); +// // Linder2011 eq 54 +// // Assumes first-oder backward Euler time discretisation +// Q_n_t = (1.0/(1.0 + time.get_delta_t()/tau_v))*(Q_t1 + (time.get_delta_t()/tau_v)*invert(C_bar)); + } + void + update_end_timestep() + { +// Q_t1 = Q_n_t; + } + + protected: + const double lambda_e; + const double mu_e; +// const double mu_v; +// const double tau_v; + + const Time &time; + +// SymmetricTensor<2,dim> Q_n_t; // Value of internal variable at this Newton step and timestep +// SymmetricTensor<2,dim> Q_t1; // Value of internal variable at the previous timestep + }; + template + class PointHistory + { + public: + PointHistory() + {} + virtual ~PointHistory() + {} + void + setup_lqp (const Parameters::AllParameters ¶meters, + const Time &time) + { + material.reset(new Material_Linear_Elastoplastic( + parameters.lambda_e, parameters.mu_e, +// parameters.mu_v, parameters.tau_v, + time)); + } + + SymmetricTensor<2, dim> + get_sigma(const SymmetricTensor<2, dim> &epsilon) const + { + return material->get_sigma(epsilon); + } + SymmetricTensor<4, dim> + get_K(const SymmetricTensor<2, dim> &epsilon) const + { + return material->get_K(epsilon); + } + void + update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) + { + material->update_internal_equilibrium(epsilon); + } + void + update_end_timestep() + { + material->update_end_timestep(); + } + private: + std::shared_ptr< Material_Linear_Elastoplastic > material; +}; + + + // @sect3{The LinearElastoplasticProblem class template} + + template + class LinearElastoplasticProblem + { + public: + LinearElastoplasticProblem (const std::string &input_file); + ~LinearElastoplasticProblem (); + void run (); + + private: + void make_grid (); + void setup_system (); + void setup_qph(); + void assemble_system (); + void apply_boundary_conditions (); + void solve (); + void output_results () const; + + Parameters::AllParameters parameters; + + Triangulation triangulation; + DoFHandler dof_handler; + + Time time; + + FESystem fe; + QGauss qf_cell; + QGauss qf_face; + + ConstraintMatrix hanging_node_constraints; + + SparsityPattern sparsity_pattern; + SparseMatrix system_matrix; + + Vector solution; + Vector system_rhs; + + // Local data + CellDataStorage::cell_iterator, + PointHistory > quadrature_point_history; + + const FEValuesExtractors::Vector u_fe; + static const unsigned int n_components = dim; + static const unsigned int first_u_component = 0; + }; + + // @sect4{LinearElastoplasticProblem::LinearElastoplasticProblem} + + template + LinearElastoplasticProblem::LinearElastoplasticProblem (const std::string &input_file) + : + parameters(input_file), + dof_handler (triangulation), + fe (FE_Q(parameters.poly_degree), dim), + qf_cell (parameters.quad_order), + qf_face (parameters.quad_order), + time (parameters.end_time, parameters.delta_t), + u_fe(first_u_component) + { + Assert(dim==3, ExcNotImplemented()); + } + + + // @sect4{LinearElastoplasticProblem::~LinearElastoplasticProblem} + + template + LinearElastoplasticProblem::~LinearElastoplasticProblem () + { + dof_handler.clear (); + } + + + // @sect4{LinearElastoplasticProblem::make_grid} + + template + void LinearElastoplasticProblem::make_grid () + { + Assert (dim == 3, ExcNotImplemented()); + + GridGenerator::cylinder (triangulation, parameters.radius, parameters.length/2.0); + triangulation.refine_global(parameters.n_global_refinement_steps); + GridTools::scale (parameters.scale, triangulation); + } + + // @sect4{LinearElastoplasticProblem::setup_system} + + template + void LinearElastoplasticProblem::setup_system () + { + dof_handler.distribute_dofs (fe); + hanging_node_constraints.clear (); + DoFTools::make_hanging_node_constraints (dof_handler, + hanging_node_constraints); + hanging_node_constraints.close (); + sparsity_pattern.reinit (dof_handler.n_dofs(), + dof_handler.n_dofs(), + dof_handler.max_couplings_between_dofs()); + DoFTools::make_sparsity_pattern (dof_handler, sparsity_pattern); + + hanging_node_constraints.condense (sparsity_pattern); + + sparsity_pattern.compress(); + + system_matrix.reinit (sparsity_pattern); + + solution.reinit (dof_handler.n_dofs()); + system_rhs.reinit (dof_handler.n_dofs()); + + std::cout << " Number of active cells: " + << triangulation.n_active_cells() + << std::endl; + + std::cout << " Number of degrees of freedom: " + << dof_handler.n_dofs() + << std::endl; + + // Setup quadrature point history + setup_qph(); + } + + + template + void LinearElastoplasticProblem::setup_qph() + { + std::cout << "Setting up quadrature point data..." << std::endl; + const unsigned int n_q_points_cell = qf_cell.size(); + + quadrature_point_history.clear(); + quadrature_point_history.initialize(triangulation.begin_active(), + triangulation.end(), + n_q_points_cell); + for (typename DoFHandler::active_cell_iterator + cell = dof_handler.begin_active(); + cell!=dof_handler.end(); ++cell) + { + const std::vector > > lqph = + quadrature_point_history.get_data(cell); + Assert(lqph.size() == n_q_points_cell, ExcInternalError()); + for (unsigned int q_point = 0; q_point < n_q_points_cell; ++q_point) + lqph[q_point]->setup_lqp(parameters, time); + } +} + + // @sect4{LinearElastoplasticProblem::assemble_system} + + template + void LinearElastoplasticProblem::assemble_system () + { + // Reset system + system_matrix = 0; + system_rhs = 0; + + FEValues fe_values (fe, qf_cell, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + FEFaceValues fe_face_values (fe, qf_face, + update_values | + update_quadrature_points | update_JxW_values); + + const unsigned int dofs_per_cell = fe.dofs_per_cell; + const unsigned int n_q_points_cell = qf_cell.size(); + const unsigned int n_q_points_face = qf_face.size(); + + FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); + Vector cell_rhs (dofs_per_cell); + + std::vector local_dof_indices (dofs_per_cell); + std::vector > solution_grads_u_total (qf_cell.size()); + + for (typename DoFHandler::active_cell_iterator + cell = dof_handler.begin_active(); + cell!=dof_handler.end(); ++cell) + { + cell_matrix = 0; + cell_rhs = 0; + + fe_values.reinit (cell); + fe_values[u_fe].get_function_gradients( + solution, + solution_grads_u_total); + + const std::vector > > lqph = + quadrature_point_history.get_data(cell); + + for (unsigned int q_point_cell=0; q_point_cell epsilon = Physics::Elasticity::Kinematics::epsilon(solution_grads_u_total[q_point_cell]); + const SymmetricTensor<4,dim> K = lqph[q_point_cell]->get_K(epsilon); + + for (unsigned int I=0; Iget_dof_indices (local_dof_indices); + for (unsigned int i=0; i + void LinearElastoplasticProblem::apply_boundary_conditions () + { + std::map boundary_values; + + // Full constraint on -X faces + { + ComponentMask component_mask_all (n_components, true); + component_mask_all.set(0, true); + VectorTools::interpolate_boundary_values (dof_handler, + 1, + ZeroFunction(dim), + boundary_values, + component_mask_all); + } + // Horizontal displacement on +X faces + { + ComponentMask component_mask_x (n_components, false); + component_mask_x.set(0, true); + const double total_displacement + = (parameters.final_stretch-1.0)*parameters.length + * parameters.scale + * (time.current()/time.end()); + VectorTools::interpolate_boundary_values (dof_handler, + 2, + ConstantFunction(total_displacement,dim), + boundary_values, + component_mask_x); + } + +// // Fixed point on -X face +// { +// const Point fixed_point (-parameters.half_length_x,0.0,0.0); +// std::vector fixed_dof_indices; +// bool found_point_of_interest = false; +// +// for (typename DoFHandler::active_cell_iterator +// cell = dof_handler.begin_active(), +// endc = dof_handler.end(); cell != endc; ++cell) +// { +// for (unsigned int face = 0; face < GeometryInfo::faces_per_cell; ++face) +// { +// // We know that the fixed point is on the -X Dirichlet boundary +// if (cell->face(face)->at_boundary() == true && +// cell->face(face)->boundary_id() == parameters.bid_CC_dirichlet_symm_X) +// { +// for (unsigned int face_vertex_index = 0; face_vertex_index < GeometryInfo::vertices_per_face; ++face_vertex_index) +// { +// if (cell->face(face)->vertex(face_vertex_index).distance(fixed_point) < 1e-6) +// { +// found_point_of_interest = true; +// for (unsigned int index_component = 0; index_component < dim; ++index_component) +// fixed_dof_indices.push_back(cell->face(face)->vertex_dof_index(face_vertex_index, +// index_component)); +// } +// +// if (found_point_of_interest == true) break; +// } +// } +// if (found_point_of_interest == true) break; +// } +// if (found_point_of_interest == true) break; +// } +// +// Assert(found_point_of_interest == true, ExcMessage("Didn't find point of interest")); +// AssertThrow(fixed_dof_indices.size() == dim, ExcMessage("Didn't find the correct number of DoFs to fix")); +// +// for (unsigned int i=0; i < fixed_dof_indices.size(); ++i) +// boundary_values[fixed_dof_indices[i]] = 0.0; +// } +// } + + MatrixTools::apply_boundary_values (boundary_values, + system_matrix, + solution, + system_rhs); + } + + + // @sect4{LinearElastoplasticProblem::solve} + + template + void LinearElastoplasticProblem::solve () + { + SolverControl solver_control (system_matrix.m(), 1e-12); + SolverCG<> cg (solver_control); + + PreconditionSSOR<> preconditioner; + preconditioner.initialize(system_matrix, 1.2); + + cg.solve (system_matrix, solution, system_rhs, + preconditioner); + + hanging_node_constraints.distribute (solution); + } + + + // @sect4{LinearElastoplasticProblem::output_results} + + + template + void LinearElastoplasticProblem::output_results () const + { + // Visual output: FEM results + std::string filename = "solution-"; + filename += Utilities::int_to_string(time.get_timestep(),4); + filename += ".vtk"; + std::ofstream output (filename.c_str()); + + DataOut data_out; + data_out.attach_dof_handler (dof_handler); + + std::vector + data_component_interpretation(dim, + DataComponentInterpretation::component_is_part_of_vector); + std::vector solution_name(dim, "displacement"); + + data_out.add_data_vector (solution, solution_name, + DataOut::type_dof_data, + data_component_interpretation); + data_out.build_patches (); + data_out.write_vtk (output); + } + + + + // @sect4{LinearElastoplasticProblem::run} + + template + void LinearElastoplasticProblem::run () + { + make_grid(); + setup_system (); + +// const bool do_grid_refinement = false; + time.increment(); + while (time.current() < time.end()+0.01*time.get_delta_t()) + { + std::cout + << "Timestep " << time.get_timestep() + << " @ time " << time.current() + << std::endl; + + // Next we assemble the system and enforce boundary + // conditions. + // Here we assume that the system and fibres have + // a fixed state, and we will assemble based on how + // epsilon_c will update given the current state of + // the body. + assemble_system (); + apply_boundary_conditions (); + + // Then we solve the linear system + solve (); + + // Output some values to file + output_results (); + + time.increment(); + } + } +} + +// @sect3{The main function} + +int main () +{ + try + { + dealii::deallog.depth_console (0); + const unsigned int dim = 3; + + LMM::LinearElastoplasticProblem elastoplastic_problem ("parameters.prm"); + elastoplastic_problem.run (); + } + catch (std::exception &exc) + { + std::cerr << std::endl << std::endl + << "----------------------------------------------------" + << std::endl; + std::cerr << "Exception on processing: " << std::endl + << exc.what() << std::endl + << "Aborting!" << std::endl + << "----------------------------------------------------" + << std::endl; + + return 1; + } + catch (...) + { + std::cerr << std::endl << std::endl + << "----------------------------------------------------" + << std::endl; + std::cerr << "Unknown exception!" << std::endl + << "Aborting!" << std::endl + << "----------------------------------------------------" + << std::endl; + return 1; + } + + return 0; +} diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm new file mode 100644 index 00000000..c66ec9f0 --- /dev/null +++ b/Linear_Elastoplasticity/parameters.prm @@ -0,0 +1,57 @@ +# Listing of Parameters +# --------------------- +subsection Finite element system + # Displacement system polynomial order + set Polynomial degree = 1 + + # Gauss quadrature order + set Quadrature order = 2 +end + + +subsection Boundary conditions + # The amount by which the specimen is to be stretched + set Final stretch = 2.0 +end + + +subsection Geometry + # Overall length of the specimen + set Length = 100 + + # Overall radius of the specimen + set Radius = 10 + + # Overall length of the notch in the specimen + set Notch length = 10 + + # Overall radius of the notch in the specimen + set Notch radius = 1 + + # Global grid scaling factor + set Grid scale = 1e-3 + + # Number of global refinement steps + set Global refinement steps = 1 + + # Number of initial local refinement cycles + set Local refinement steps = 1 +end + + +subsection Material properties + # Young's modulus + set Young's modulus = 100 + + # Poisson ratio + set Poisson ratio = 0.3 +end + + +subsection Time + # End time + set End time = 1.0 + + # Time step size + set Time step size = 0.1 +end From 023da9331620ef463fb02a57d97865065c2182bd Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Thu, 2 Aug 2018 17:24:54 +0200 Subject: [PATCH 02/20] Add elastoplasticity with linear isotropic hardening --- .../Linear_elastoplasticity.cc | 441 ++++++++++++------ Linear_Elastoplasticity/parameters.prm | 19 +- 2 files changed, 325 insertions(+), 135 deletions(-) diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/Linear_elastoplasticity.cc index 56dfad65..ba27cbe6 100644 --- a/Linear_Elastoplasticity/Linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/Linear_elastoplasticity.cc @@ -227,11 +227,11 @@ namespace LMM struct MaterialProperties { - double E_e; // Young's modulus - double nu_e; // Poisson ratio - + std::string mat_type; double mu_e; // Shear modulus - double lambda_e; // Lame parameter + double nu_e; // Poisson ratio + double k_p; // Isotropic hardening constant + double sigma_y_p; // Yield stress static void declare_parameters(ParameterHandler &prm); @@ -244,13 +244,25 @@ namespace LMM { prm.enter_subsection("Material properties"); { - prm.declare_entry("Young's modulus", "100", - Patterns::Double(0), - "Young's modulus of the specimen"); + prm.declare_entry("Type", "Elastic", + Patterns::Selection("Elastic|Elastoplastic (isotropic hardening)"), + "Type of material that composes the specimen"); - prm.declare_entry("Poisson ratio", "10", + prm.declare_entry("Shear modulus", "100", Patterns::Double(0), + "Shear modulus of the specimen"); + + prm.declare_entry("Poisson ratio", "0.3", + Patterns::Double(-1,0.5), "Poisson ratio of the specimen"); + + prm.declare_entry("Isotropic hardening constant", "10", + Patterns::Double(0), + "Isotropic hardening constant"); + + prm.declare_entry("Yield stress", "20", + Patterns::Double(0), + "Yield stress"); } prm.leave_subsection(); } @@ -259,13 +271,13 @@ namespace LMM { prm.enter_subsection("Material properties"); { - E_e = prm.get_double("Young's modulus"); + mat_type = prm.get("Type"); + mu_e = prm.get_double("Shear modulus"); nu_e = prm.get_double("Poisson ratio"); + k_p = prm.get_double("Isotropic hardening constant"); + sigma_y_p = prm.get_double("Yield stress"); } prm.leave_subsection(); - - mu_e = E_e/(2.0*(1.0 + nu_e)); - lambda_e = 2.0*mu_e*nu_e/(1.0 - 2.0*nu_e); } @@ -400,80 +412,269 @@ namespace LMM }; + template + class Material_Base + { + public: + Material_Base() + {} + + virtual ~Material_Base() + {} + + // Cauchy stress + virtual SymmetricTensor<2,dim> + get_sigma(const SymmetricTensor<2, dim> &epsilon) const = 0; + + // Elastic / Elastoplastic tangent + virtual SymmetricTensor<4,dim> + get_K(const SymmetricTensor<2, dim> &epsilon) const = 0; + + virtual void + update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) = 0; + + virtual void + update_end_timestep() = 0; + }; + template - class Material_Linear_Elastoplastic + class Material_Linear_Elastic : public Material_Base { public: - Material_Linear_Elastoplastic(const double lambda_e, - const double mu_e, -// const double mu_v, -// const double tau_v, + Material_Linear_Elastic(const double mu_e, + const double nu_e) + : + kappa_e((2.0 * mu_e * (1.0 + nu_e)) / (3.0 * (1.0 - 2.0 * nu_e))), + mu_e(mu_e) + { + Assert(kappa_e > 0, ExcInternalError()); + } + + virtual ~Material_Linear_Elastic() + {} + + // Cauchy stress + SymmetricTensor<2,dim> + get_sigma(const SymmetricTensor<2, dim> &epsilon) const override + { + // Assume isochoric/deviatoric split + const SymmetricTensor<2, dim> epsilon_vol = (trace(epsilon)/dim)*Physics::Elasticity::StandardTensors::I; + const SymmetricTensor<2, dim> epsilon_dev = epsilon - epsilon_vol; + + return dim*kappa_e*epsilon_vol + + 2.0*mu_e*epsilon_dev; + } + + // Elastic / Elastoplastic tangent + SymmetricTensor<4,dim> get_K(const SymmetricTensor<2, dim> &epsilon) const override + { + (void)epsilon; + + // Elastic part of tangent, assuming an isochoric/deviatoric split + const SymmetricTensor<4,dim> K_e + = (kappa_e - 2.0/dim*mu_e)*Physics::Elasticity::StandardTensors::IxI + + (2.0*mu_e)*Physics::Elasticity::StandardTensors::S; + + // Check that stress-strain relationship really is linear and + // correctly implemented + Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-9, ExcInternalError()); + + return K_e; + } + + void + update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) override + { + (void)epsilon; + } + + void + update_end_timestep() override + {} + + protected: + const double kappa_e; // buld modulus + const double mu_e; // shear modulus + }; + + + + template + class Material_Linear_Elastoplastic : public Material_Base + { + public: + Material_Linear_Elastoplastic(const double mu_e, + const double nu_e, + const double k_p, + const double sigma_y_p, const Time &time) : - lambda_e(lambda_e), + kappa_e((2.0 * mu_e * (1.0 + nu_e)) / (3.0 * (1.0 - 2.0 * nu_e))), mu_e(mu_e), -// mu_v(mu_v), -// tau_v(tau_v), - time(time) -// Q_n_t(Physics::Elasticity::StandardTensors::I), -// Q_t1(Physics::Elasticity::StandardTensors::I) + k_p (k_p), + sigma_y_p (sigma_y_p), + time(time), + phi_t(std::numeric_limits::min()), + n_t(), + delta_lambda_t(0.0), + sigma_dev_trial_norm(1.0), + epsilon_p_t(), + epsilon_p_t1(), + alpha_p_t (0.0), + alpha_p_t1 (0.0) { -// Assert(kappa > 0, ExcInternalError()); + Assert(kappa_e > 0, ExcInternalError()); } - ~Material_Linear_Elastoplastic() + + virtual ~Material_Linear_Elastoplastic() {} + // Cauchy stress SymmetricTensor<2,dim> - get_sigma(const SymmetricTensor<2, dim> &epsilon) const + get_sigma(const SymmetricTensor<2, dim> &epsilon) const override { - return get_K(epsilon)*epsilon; + return get_sigma(epsilon, epsilon_p_t); } - SymmetricTensor<4,dim> get_K(const SymmetricTensor<2, dim> &epsilon) const + + // Isotropic hardening stress + double + get_R() const + { + return get_R(alpha_p_t); + } + + // Elastic / Elastoplastic tangent + SymmetricTensor<4,dim> + get_K(const SymmetricTensor<2, dim> &epsilon) const override { (void)epsilon; - static const SymmetricTensor<2,dim> I = unit_symmetric_tensor(); - SymmetricTensor<4,dim> K; - for (unsigned int i=0; i < dim; ++i) - for (unsigned int j=i; j < dim; ++j) - for (unsigned int k=0; k < dim; ++k) - for (unsigned int l=k; l < dim; ++l) - { - // Matrix contribution - K[i][j][k][l] = lambda_e * I[i][j]*I[k][l] - + mu_e * (I[i][k]*I[j][l] + I[i][l]*I[j][k]); - } + // Elastic part of tangent, assuming an isochoric/deviatoric split + const SymmetricTensor<4,dim> K_e + = (kappa_e - 2.0/dim*mu_e)*Physics::Elasticity::StandardTensors::IxI + + (2.0*mu_e)*Physics::Elasticity::StandardTensors::S; - return K; + if (phi_t <= 0) + { + // Check that stress-strain relationship really is linear and + // correctly implemented + Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-9, ExcInternalError()); + + return K_e; + } + else + { + // Compute correction for radial projection during plastic flow + static const SymmetricTensor<4,dim> I_dev + = Physics::Elasticity::StandardTensors::S + - (1.0/dim)*Physics::Elasticity::StandardTensors::IxI; + const SymmetricTensor<4,dim> n_t_x_n_t = outer_product(n_t,n_t); + + Assert(sigma_dev_trial_norm != 0.0, ExcInternalError()); + const SymmetricTensor<4,dim> K_p_correction + = - (Utilities::fixed_power<2>(2.0*mu_e))/(2.0*mu_e + 2.0/dim*kappa_e)*n_t_x_n_t + - (Utilities::fixed_power<2>(2.0*mu_e)*delta_lambda_t)/(sigma_dev_trial_norm)*(I_dev - n_t_x_n_t); + + return K_e + K_p_correction; + } } + void - update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) + update_internal_equilibrium(const SymmetricTensor<2, dim> &epsilon) override { - (void)epsilon; -// const double det_F = determinant(F); -// const SymmetricTensor<2,dim> C_bar = std::pow(det_F, -2.0 / dim) * Physics::Elasticity::Kinematics::C(F); -// // Linder2011 eq 54 -// // Assumes first-oder backward Euler time discretisation -// Q_n_t = (1.0/(1.0 + time.get_delta_t()/tau_v))*(Q_t1 + (time.get_delta_t()/tau_v)*invert(C_bar)); + // Trial stress + const SymmetricTensor<2, dim> sigma_trial = get_sigma(epsilon,epsilon_p_t1); + // Deviatoric part of trial stress + const SymmetricTensor<2, dim> sigma_dev_trial = deviator(sigma_trial); + // Trial hardening stress + const double R_trial = get_R(alpha_p_t1); + + // Update yield function value (von Mises plasticity for linear + // isotropic hardening) + { + phi_t = sigma_dev_trial.norm() - std::sqrt(2.0/dim)*(sigma_y_p - R_trial); + + // Update internal variables based condition of + // elasticity or plastic flow + if (phi_t <= 0.0) + { + // Elastic update + epsilon_p_t = epsilon_p_t1; + alpha_p_t = alpha_p_t1; + } + else + { + // Plastic update: + // Linear isotropic hardening -> use radial return algorithm + + // Internal variables for radial return algorithm + sigma_dev_trial_norm = sigma_dev_trial.norm(); + n_t = sigma_dev_trial / sigma_dev_trial_norm; + delta_lambda_t = phi_t / (2.0*mu_e + 2.0/dim*kappa_e); + + // Update internal plastic variables + epsilon_p_t = epsilon_p_t1 + delta_lambda_t*n_t; + alpha_p_t = alpha_p_t1 + delta_lambda_t*std::sqrt(2.0/dim); + } + } } + void - update_end_timestep() + update_end_timestep() override { -// Q_t1 = Q_n_t; + // For the linear problem, it is easier to do the update + // between the previous and the current timestep in the + // call to update_internal_equilibrium(). For a nonlinear + // problem, it is more useful to employ this function to + // accept the accumulation of incremental values in the + // context of a Newton scheme. } protected: - const double lambda_e; - const double mu_e; -// const double mu_v; -// const double tau_v; + const double kappa_e; // buld modulus + const double mu_e; // shear modulus + const double k_p; // isotropic hardening modulus + const double sigma_y_p; // yield stress const Time &time; -// SymmetricTensor<2,dim> Q_n_t; // Value of internal variable at this Newton step and timestep -// SymmetricTensor<2,dim> Q_t1; // Value of internal variable at the previous timestep + // Variables related to yield function at current timestep + double phi_t; // yield function + SymmetricTensor<2,dim> n_t; // radial return direction + double delta_lambda_t;// increment in plastic multiplier + double sigma_dev_trial_norm; // norm of deviatoric part of trial stress + + // Internal variables: + // Plastic strain + SymmetricTensor<2,dim> epsilon_p_t; // current timestep + SymmetricTensor<2,dim> epsilon_p_t1; // previous timestep + // Isotropic hardening variable + double alpha_p_t; // current timestep + double alpha_p_t1; // previous timestep + + SymmetricTensor<2,dim> + get_sigma(const SymmetricTensor<2, dim> &epsilon, + const SymmetricTensor<2, dim> &epsilon_p) const + { + // Assume isochoric/deviatoric split + const SymmetricTensor<2, dim> epsilon_vol = (trace(epsilon)/dim)*Physics::Elasticity::StandardTensors::I; + const SymmetricTensor<2, dim> epsilon_dev = epsilon - epsilon_vol; + + const SymmetricTensor<2,dim> sigma + = dim*kappa_e*epsilon_vol + + 2.0*mu_e*(epsilon_dev - epsilon_p); + return sigma; + } + + double + get_R(const double alpha_p) const + { + return -k_p*alpha_p; + } }; + + template class PointHistory { @@ -483,13 +684,24 @@ namespace LMM virtual ~PointHistory() {} void - setup_lqp (const Parameters::AllParameters ¶meters, + setup_lqp (const types::material_id mat_id, + const Parameters::AllParameters ¶meters, const Time &time) { - material.reset(new Material_Linear_Elastoplastic( - parameters.lambda_e, parameters.mu_e, -// parameters.mu_v, parameters.tau_v, - time)); + if (mat_id == 0) + { + material.reset(new Material_Linear_Elastic( + parameters.mu_e, parameters.nu_e)); + } + else if (mat_id == 1) + { + material.reset(new Material_Linear_Elastoplastic( + parameters.mu_e, parameters.nu_e, + parameters.k_p, parameters.sigma_y_p, + time)); + } + else + AssertThrow(mat_id <= 1, ExcMessage("Unknown material id")); } SymmetricTensor<2, dim> @@ -513,7 +725,7 @@ namespace LMM material->update_end_timestep(); } private: - std::shared_ptr< Material_Linear_Elastoplastic > material; + std::shared_ptr< Material_Base > material; }; @@ -571,10 +783,10 @@ namespace LMM : parameters(input_file), dof_handler (triangulation), + time (parameters.end_time, parameters.delta_t), fe (FE_Q(parameters.poly_degree), dim), qf_cell (parameters.quad_order), qf_face (parameters.quad_order), - time (parameters.end_time, parameters.delta_t), u_fe(first_u_component) { Assert(dim==3, ExcNotImplemented()); @@ -600,6 +812,18 @@ namespace LMM GridGenerator::cylinder (triangulation, parameters.radius, parameters.length/2.0); triangulation.refine_global(parameters.n_global_refinement_steps); GridTools::scale (parameters.scale, triangulation); + + for (typename Triangulation::active_cell_iterator + cell = triangulation.begin_active(); + cell!=triangulation.end(); ++cell) + { + if (parameters.mat_type == "Elastic") + cell->set_material_id(0); + else if (parameters.mat_type == "Elastoplastic (isotropic hardening)") + cell->set_material_id(1); + else + AssertThrow(false, ExcMessage("Unknown material id")); + } } // @sect4{LinearElastoplasticProblem::setup_system} @@ -657,7 +881,7 @@ namespace LMM quadrature_point_history.get_data(cell); Assert(lqph.size() == n_q_points_cell, ExcInternalError()); for (unsigned int q_point = 0; q_point < n_q_points_cell; ++q_point) - lqph[q_point]->setup_lqp(parameters, time); + lqph[q_point]->setup_lqp(cell->material_id(), parameters, time); } } @@ -679,7 +903,7 @@ namespace LMM const unsigned int dofs_per_cell = fe.dofs_per_cell; const unsigned int n_q_points_cell = qf_cell.size(); - const unsigned int n_q_points_face = qf_face.size(); + // const unsigned int n_q_points_face = qf_face.size(); FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); Vector cell_rhs (dofs_per_cell); @@ -704,41 +928,37 @@ namespace LMM for (unsigned int q_point_cell=0; q_point_cell epsilon = Physics::Elasticity::Kinematics::epsilon(solution_grads_u_total[q_point_cell]); + const Tensor<2,dim> &Grad_u = solution_grads_u_total[q_point_cell]; + const SymmetricTensor<2,dim> epsilon = Physics::Elasticity::Kinematics::epsilon(Grad_u); + lqph[q_point_cell]->update_internal_equilibrium(epsilon); const SymmetricTensor<4,dim> K = lqph[q_point_cell]->get_K(epsilon); + const double JxW = fe_values.JxW(q_point_cell); + + // Precompute the shape function symmetric gradients + // (related to the variation of the small strain) + std::vector< SymmetricTensor<2,dim> > d_epsilon (dofs_per_cell); + for (unsigned int K=0; K &d_epsilon_I = d_epsilon[I]; - for (unsigned int J=0; J &d_epsilon_J = d_epsilon[J]; + + cell_matrix(I,J) + += contract3(d_epsilon_I,K,d_epsilon_J) * JxW; } } - -// for (unsigned int I=0; Iget_dof_indices (local_dof_indices); for (unsigned int i=0; i fixed_point (-parameters.half_length_x,0.0,0.0); -// std::vector fixed_dof_indices; -// bool found_point_of_interest = false; -// -// for (typename DoFHandler::active_cell_iterator -// cell = dof_handler.begin_active(), -// endc = dof_handler.end(); cell != endc; ++cell) -// { -// for (unsigned int face = 0; face < GeometryInfo::faces_per_cell; ++face) -// { -// // We know that the fixed point is on the -X Dirichlet boundary -// if (cell->face(face)->at_boundary() == true && -// cell->face(face)->boundary_id() == parameters.bid_CC_dirichlet_symm_X) -// { -// for (unsigned int face_vertex_index = 0; face_vertex_index < GeometryInfo::vertices_per_face; ++face_vertex_index) -// { -// if (cell->face(face)->vertex(face_vertex_index).distance(fixed_point) < 1e-6) -// { -// found_point_of_interest = true; -// for (unsigned int index_component = 0; index_component < dim; ++index_component) -// fixed_dof_indices.push_back(cell->face(face)->vertex_dof_index(face_vertex_index, -// index_component)); -// } -// -// if (found_point_of_interest == true) break; -// } -// } -// if (found_point_of_interest == true) break; -// } -// if (found_point_of_interest == true) break; -// } -// -// Assert(found_point_of_interest == true, ExcMessage("Didn't find point of interest")); -// AssertThrow(fixed_dof_indices.size() == dim, ExcMessage("Didn't find the correct number of DoFs to fix")); -// -// for (unsigned int i=0; i < fixed_dof_indices.size(); ++i) -// boundary_values[fixed_dof_indices[i]] = 0.0; -// } -// } - MatrixTools::apply_boundary_values (boundary_values, system_matrix, solution, @@ -888,6 +1066,7 @@ namespace LMM { make_grid(); setup_system (); + output_results (); // const bool do_grid_refinement = false; time.increment(); diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index c66ec9f0..46cd7015 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -11,7 +11,7 @@ end subsection Boundary conditions # The amount by which the specimen is to be stretched - set Final stretch = 2.0 + set Final stretch = 1.5 end @@ -32,7 +32,7 @@ subsection Geometry set Grid scale = 1e-3 # Number of global refinement steps - set Global refinement steps = 1 + set Global refinement steps = 2 # Number of initial local refinement cycles set Local refinement steps = 1 @@ -40,11 +40,22 @@ end subsection Material properties - # Young's modulus - set Young's modulus = 100 + # Type of material that composes the specimen + # Options: Elastic ; Elastoplastic (isotropic hardening) + #set Type = Elastic + set Type = Elastoplastic (isotropic hardening) + + # Shear modulus + set Shear modulus = 100 # Poisson ratio set Poisson ratio = 0.3 + + # Isotropic hardening constant + set Isotropic hardening constant = 10 + + # Yield stress + set Yield stress = 20 end From 8e417ba7ff67ab01ce541626f1b8553f3466596f Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Wed, 19 Dec 2018 16:42:25 +0200 Subject: [PATCH 03/20] Minor updates; modernisation for affine constraints --- .../Linear_elastoplasticity.cc | 19 ++++++++++++------- 1 file changed, 12 insertions(+), 7 deletions(-) diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/Linear_elastoplasticity.cc index ba27cbe6..2eb0316e 100644 --- a/Linear_Elastoplasticity/Linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/Linear_elastoplasticity.cc @@ -31,7 +31,7 @@ #include #include #include -#include +#include #include #include #include @@ -438,6 +438,9 @@ namespace LMM }; + /** + * Linear elastic material + */ template class Material_Linear_Elastic : public Material_Base { @@ -499,12 +502,14 @@ namespace LMM }; - + /** + * Linear elastoplastic material: von Mises plasticity with linear isotropic hardening + */ template - class Material_Linear_Elastoplastic : public Material_Base + class Material_Linear_Elastoplastic_Isotropic_Hardening : public Material_Base { public: - Material_Linear_Elastoplastic(const double mu_e, + Material_Linear_Elastoplastic_Isotropic_Hardening(const double mu_e, const double nu_e, const double k_p, const double sigma_y_p, @@ -527,7 +532,7 @@ namespace LMM Assert(kappa_e > 0, ExcInternalError()); } - virtual ~Material_Linear_Elastoplastic() + virtual ~Material_Linear_Elastoplastic_Isotropic_Hardening() {} // Cauchy stress @@ -695,7 +700,7 @@ namespace LMM } else if (mat_id == 1) { - material.reset(new Material_Linear_Elastoplastic( + material.reset(new Material_Linear_Elastoplastic_Isotropic_Hardening( parameters.mu_e, parameters.nu_e, parameters.k_p, parameters.sigma_y_p, time)); @@ -759,7 +764,7 @@ namespace LMM QGauss qf_cell; QGauss qf_face; - ConstraintMatrix hanging_node_constraints; + AffineConstraints hanging_node_constraints; SparsityPattern sparsity_pattern; SparseMatrix system_matrix; From 8df06254122a3301b9d57c6407a5c16dccda6851 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Wed, 19 Dec 2018 18:17:01 +0200 Subject: [PATCH 04/20] Outline of notched cylinder geometry --- .../Linear_elastoplasticity.cc | 163 +++++++++++++++--- Linear_Elastoplasticity/parameters.prm | 4 +- 2 files changed, 142 insertions(+), 25 deletions(-) diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/Linear_elastoplasticity.cc index 2eb0316e..49a48ee9 100644 --- a/Linear_Elastoplasticity/Linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/Linear_elastoplasticity.cc @@ -151,6 +151,7 @@ namespace LMM struct Geometry { + std::string geometry_type; double length; double radius; double notch_length; @@ -170,6 +171,10 @@ namespace LMM { prm.enter_subsection("Geometry"); { + prm.declare_entry("Geometry type", "Notched cylinder", + Patterns::Selection("Notched cylinder|Notched tensile specimen"), + "The geometry to be modelled"); + prm.declare_entry("Length", "100", Patterns::Double(0), "Overall length of the specimen"); @@ -205,6 +210,7 @@ namespace LMM { prm.enter_subsection("Geometry"); { + geometry_type = prm.get("Geometry type"); length = prm.get_double("Length"); radius = prm.get_double("Radius"); notch_length = prm.get_double("Notch length"); @@ -814,9 +820,101 @@ namespace LMM { Assert (dim == 3, ExcNotImplemented()); - GridGenerator::cylinder (triangulation, parameters.radius, parameters.length/2.0); - triangulation.refine_global(parameters.n_global_refinement_steps); - GridTools::scale (parameters.scale, triangulation); + if (parameters.geometry_type == "Notched cylinder") + { + Triangulation<2> tria_cross_section; + { + const Point<2> centre; + const double notch_radius_inner = parameters.notch_radius - (parameters.radius-parameters.notch_radius); + Triangulation<2> tria_inner_1; + GridGenerator::hyper_ball (tria_inner_1,centre, + notch_radius_inner,false); + + Triangulation<2> tria_inner_2; + GridGenerator::hyper_shell (tria_inner_2,centre, + notch_radius_inner, + parameters.notch_radius, + 4); + GridTools::rotate(M_PI/4,tria_inner_2); + + Triangulation<2> tria_outer; + GridGenerator::hyper_shell (tria_outer,centre, + parameters.notch_radius, + parameters.radius, + 4); + GridTools::rotate(M_PI/4,tria_outer); + + Triangulation<2> tria_inner; + GridGenerator::merge_triangulations(tria_inner_1,tria_inner_2,tria_inner); + GridGenerator::merge_triangulations(tria_inner,tria_outer,tria_cross_section); + } + + Triangulation tria_unnotched; + { + std::vector slice_coordinates; + slice_coordinates.push_back(0.0); + slice_coordinates.push_back(parameters.notch_length); + slice_coordinates.push_back(2.0*parameters.notch_length); + slice_coordinates.push_back(parameters.length); + GridGenerator::extrude_triangulation (tria_cross_section, + slice_coordinates, + tria_unnotched); + GridTools::rotate(M_PI/2,1,tria_unnotched); + } + + // Remove all cells within the notch length and greater than + // the notch radius + std::set::active_cell_iterator> cells_to_remove; + for (typename Triangulation::active_cell_iterator + cell = tria_unnotched.begin_active(); + cell!=tria_unnotched.end(); ++cell) + { + if (cell->center()[0] < parameters.notch_length) + { + // Outer layer of cells correspond to the notch to be removed + for (unsigned int face=0; face < GeometryInfo::faces_per_cell; ++face) + if (cell->face(face)->at_boundary() == true && + std::abs(cell->face(face)->center()[0] - parameters.notch_length/2.0) < 1e-12) + cells_to_remove.insert(cell); + + // Additional verification + Point rad = cell->center(); + rad[0] = 0.0; + const double radius = rad.norm(); + if (radius > parameters.notch_radius) + cells_to_remove.insert(cell); + } + } + + AssertThrow(cells_to_remove.empty() == false, ExcMessage("Found no cells to remove.")); + GridGenerator::create_triangulation_with_removed_cells(tria_unnotched,cells_to_remove,triangulation); + + // Set boundary and manifold IDs + for (typename Triangulation::active_cell_iterator + cell = triangulation.begin_active(); + cell!=triangulation.end(); ++cell) + { + for (unsigned int face=0; face < GeometryInfo::faces_per_cell; ++face) + { + if (cell->face(face)->at_boundary() == true) + { + if (std::abs(cell->face(face)->center()[0] - 0.0) < 1e-6) + cell->face(face)->set_boundary_id(1); + else if (std::abs(cell->face(face)->center()[0] - parameters.length) < 1e-6) + cell->face(face)->set_boundary_id(2); + } + } + } + + triangulation.refine_global(1); + GridTools::scale (parameters.scale, triangulation); + } + else if (parameters.geometry_type == "Notched tensile specimen") + { + AssertThrow(false, ExcNotImplemented()); + } + else + AssertThrow(false, ExcMessage("Unknown geometry")); for (typename Triangulation::active_cell_iterator cell = triangulation.begin_active(); @@ -985,30 +1083,49 @@ namespace LMM { std::map boundary_values; - // Full constraint on -X faces + if (parameters.geometry_type == "Notched cylinder") { - ComponentMask component_mask_all (n_components, true); - component_mask_all.set(0, true); - VectorTools::interpolate_boundary_values (dof_handler, - 1, - ZeroFunction(dim), - boundary_values, - component_mask_all); + // No X displacement on constraint on -X faces + { + ComponentMask component_mask_x (n_components, false); + component_mask_x.set(0, true); + VectorTools::interpolate_boundary_values (dof_handler, + 1, + ZeroFunction(dim), + boundary_values, + component_mask_x); + } + // Prescribed horizontal displacement on +X faces + { + ComponentMask component_mask_x (n_components, false); + component_mask_x.set(0, true); + const double total_displacement + = (parameters.final_stretch-1.0)*parameters.length + * parameters.scale + * (time.current()/time.end()); + VectorTools::interpolate_boundary_values (dof_handler, + 2, + ConstantFunction(total_displacement,dim), + boundary_values, + component_mask_x); + } + // No radial displacement of +X faces + { + ComponentMask component_mask_yz (n_components, true); + component_mask_yz.set(0, false); + VectorTools::interpolate_boundary_values (dof_handler, + 2, + ZeroFunction(dim), + boundary_values, + component_mask_yz); + } } - // Horizontal displacement on +X faces + else if (parameters.geometry_type == "Notched tensile specimen") { - ComponentMask component_mask_x (n_components, false); - component_mask_x.set(0, true); - const double total_displacement - = (parameters.final_stretch-1.0)*parameters.length - * parameters.scale - * (time.current()/time.end()); - VectorTools::interpolate_boundary_values (dof_handler, - 2, - ConstantFunction(total_displacement,dim), - boundary_values, - component_mask_x); + AssertThrow(false, ExcNotImplemented()); } + else + AssertThrow(false, ExcMessage("Unknown geometry")); MatrixTools::apply_boundary_values (boundary_values, system_matrix, diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index 46cd7015..d91c413f 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -23,10 +23,10 @@ subsection Geometry set Radius = 10 # Overall length of the notch in the specimen - set Notch length = 10 + set Notch length = 1 # Overall radius of the notch in the specimen - set Notch radius = 1 + set Notch radius = 9 # Global grid scaling factor set Grid scale = 1e-3 From f5c8d8307b7a32b79f2b6b8f1c0432ed36537dd1 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Thu, 20 Dec 2018 12:24:51 +0200 Subject: [PATCH 05/20] Notched cylinder geometry now more precise --- .../Linear_elastoplasticity.cc | 44 +++++++++++++++++-- 1 file changed, 41 insertions(+), 3 deletions(-) diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/Linear_elastoplasticity.cc index 49a48ee9..9cc8e7ab 100644 --- a/Linear_Elastoplasticity/Linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/Linear_elastoplasticity.cc @@ -1,6 +1,6 @@ /* --------------------------------------------------------------------- * Copyright (C) 2018 by the deal.II authors and - * Jean-Paul Pelteret and Tim Hamann + * Jean-Paul Pelteret * * This file is part of the deal.II library. * @@ -38,6 +38,7 @@ #include #include #include +#include #include #include #include @@ -822,10 +823,11 @@ namespace LMM if (parameters.geometry_type == "Notched cylinder") { + const double notch_radius_inner = parameters.notch_radius - (parameters.radius-parameters.notch_radius); + Triangulation<2> tria_cross_section; { const Point<2> centre; - const double notch_radius_inner = parameters.notch_radius - (parameters.radius-parameters.notch_radius); Triangulation<2> tria_inner_1; GridGenerator::hyper_ball (tria_inner_1,centre, notch_radius_inner,false); @@ -855,6 +857,19 @@ namespace LMM slice_coordinates.push_back(0.0); slice_coordinates.push_back(parameters.notch_length); slice_coordinates.push_back(2.0*parameters.notch_length); + // Biased slices with function taken from GridGenerator::concentric_hyper_shells + const double bias_first_slice = 2.0*parameters.notch_length; + const double bias_last_slice = parameters.length; + const double skewness = 2.0; + const unsigned int n_slices = 5; + for (unsigned int s=0; s(n_slices)))) / std::tanh(skewness); + AssertThrow(f >= 0.0 && f <= 1.0, ExcMessage("Bias function not in correct range.")); + const double r = bias_first_slice + (bias_last_slice-bias_first_slice)*f; + AssertThrow(r >= bias_first_slice && r <= bias_last_slice, ExcMessage("New slice location not in correct range.")); + slice_coordinates.push_back(r); + } slice_coordinates.push_back(parameters.length); GridGenerator::extrude_triangulation (tria_cross_section, slice_coordinates, @@ -890,6 +905,19 @@ namespace LMM GridGenerator::create_triangulation_with_removed_cells(tria_unnotched,cells_to_remove,triangulation); // Set boundary and manifold IDs + triangulation.set_all_manifold_ids(2); // TFI manifold + + const double radius_for_cyl_cells = notch_radius_inner*std::cos(M_PI/4); + for (typename Triangulation::active_cell_iterator + cell = triangulation.begin_active(); + cell!=triangulation.end(); ++cell) + { + Point rad = cell->center(); + rad[0] = 0.0; + const double radius = rad.norm(); + if (radius > radius_for_cyl_cells) + cell->set_all_manifold_ids(1); // Cylindrical manifold + } for (typename Triangulation::active_cell_iterator cell = triangulation.begin_active(); cell!=triangulation.end(); ++cell) @@ -902,11 +930,21 @@ namespace LMM cell->face(face)->set_boundary_id(1); else if (std::abs(cell->face(face)->center()[0] - parameters.length) < 1e-6) cell->face(face)->set_boundary_id(2); + else + cell->face(face)->set_all_manifold_ids(1); // Cylindrical manifold } } } - triangulation.refine_global(1); + CylindricalManifold cylindrical_manifold (0 /*axis*/); + CylindricalManifold cylindrical_manifold_tfi (0 /*axis*/); + TransfiniteInterpolationManifold tfi_manifold; + + triangulation.set_manifold (1, cylindrical_manifold); + tfi_manifold.initialize(triangulation); + triangulation.set_manifold (2, tfi_manifold); + + triangulation.refine_global(parameters.n_global_refinement_steps); GridTools::scale (parameters.scale, triangulation); } else if (parameters.geometry_type == "Notched tensile specimen") From 2c007fc90d8520c919b1955b8f2eed64386f60d7 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Thu, 20 Dec 2018 17:13:26 +0200 Subject: [PATCH 06/20] Rename file --- .../{Linear_elastoplasticity.cc => linear_elastoplasticity.cc} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename Linear_Elastoplasticity/{Linear_elastoplasticity.cc => linear_elastoplasticity.cc} (100%) diff --git a/Linear_Elastoplasticity/Linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc similarity index 100% rename from Linear_Elastoplasticity/Linear_elastoplasticity.cc rename to Linear_Elastoplasticity/linear_elastoplasticity.cc From a0024695dcea7b541acd05a965cfbd9c94ae85c5 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Thu, 20 Dec 2018 18:27:50 +0200 Subject: [PATCH 07/20] Added initial version of notes on elastoplasticity --- .../Figures/Sec_3_5_1-Radial_projection.png | Bin 0 -> 270530 bytes ...3_5_2-General_closest_point_projection.png | Bin 0 -> 194592 bytes .../theory/theory-linear_elastoplasticity.tex | 810 ++++++++++++++++++ 3 files changed, 810 insertions(+) create mode 100644 Linear_Elastoplasticity/doc/theory/Figures/Sec_3_5_1-Radial_projection.png create mode 100644 Linear_Elastoplasticity/doc/theory/Figures/Sec_3_5_2-General_closest_point_projection.png create mode 100644 Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex diff --git a/Linear_Elastoplasticity/doc/theory/Figures/Sec_3_5_1-Radial_projection.png b/Linear_Elastoplasticity/doc/theory/Figures/Sec_3_5_1-Radial_projection.png new file mode 100644 index 0000000000000000000000000000000000000000..f6e94e99c1ee38c0b7780c94c7671596bba723f8 GIT binary patch literal 270530 zcmb@tWmFtnwa0#x#A-E>NEqIXJhJD^S-yZLt zf0t2=E~;zIxu)A9T180)6^Q@|0s;b6PF7M40sG8(u?Q4!1yC4FQn}07+NOBlaPEGd9S*9wP;!co2X0#gal#n+bs;^#b~na9)|i5LlgqZ0s@k@ad`!5`r-lrVR`cG<>Q5=C(&-joNzP! z@|Qk_$dQ{6f|6*LQi)ptYqu0)`^Y|+6&^yIX5skTH;kxD{iyWtBr*ueM!ZVdh6p?w zxrXvVYWjXcGg63A4IKF_sGvMvVUEmvsJ&4{vX&oi-hNSDq9?;jmM&yx5#DN6zlfrK zRs#rW{Bdc1IXsv?djyiN{$M5*^9xwfBCDM@RunYzodc9-L$j<2^uuWH_)+I zHjRj5K1$5jr;LOes)M*2E0{276n9<`163k(3ctLvS#K2txozR5BisiV&?`Yv{m{Hm z+hT=C)iXna`tv7@bo?&mPF~dPy$!Pn1*Zk))VPM@Hs0p&x7&E4Cb?e^u;M7%1RpqA 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+\author{Jean-Paul Pelteret} + +\begin{filecontents}{\jobname.bib} +@Misc{Mergheim2018a, + author = {Julia Mergheim and Jean-Paul Pelteret and Ester Comellas}, + title = {Lecture notes: Material Modelling and Simulation}, + year = {2018}, + publisher = {Friedrich--Alexander Uniersit\"{a}t Erlangen--N\"{u}rnberg} +} +@Book{Simo2006a, + title = {Computational Inelasticity}, + publisher = {Springer Science \& Business Media}, + year = {2000}, + author = {Simo, J. C. and Hughes, Thomas J. R.}, + editor = {Marsden, J. E. and Sirovich, L. and Wiggins, S.}, + volume = {7}, + series = {Interdisciplinary Applied Mathematics}, + isbn = {978-0-387-97520-7}, + doi = {10.1007/b98904}, +} +\end{filecontents} + +\begin{document} + +\maketitle + +\begin{abstract} +An introduction to the theory applied for elastoplasticity. +\end{abstract} + +\section{Governing equations for quasi-static linear elasticity} +The strong statement of the balance of linear momentum reads +\begin{gather} +\nabla \cdot \boldsymbol{\sigma} + \mathbf{b} + = \mathbf{0} +\quad \text{on} \quad \Omega \quad , +\end{gather} +where $\nabla = \frac{\partial}{\partial x}$ is a differential operator, +$\boldsymbol{\sigma}$ is the Cauchy stress tensor and +$\mathbf{b} = \rho \mathbf{g}$ is the body force density vector. +This is expressed in index notation as +\begin{gather} +\frac{\partial \sigma_{ij}}{\partial x_{j}} + b_{i} + = 0 +\quad \text{on} \quad \Omega \quad . +\end{gather} +Pre-multiplying the above by test function $\delta \mathbf{v}$ and integrating over the domain $\Omega$ renders +\begin{gather} +- \int\limits_{\Omega} \delta v_{i} \, \frac{\partial \sigma_{ij}}{\partial x_{j}} \, dv + = \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv +\end{gather} +that, by using the product rule for derivatives (i.e. integration by parts), becomes +\begin{gather} +\int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \sigma_{ij} \, dv +- \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \delta v_{i} \, \sigma_{ij} \right] \, dv + = \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv +\quad . +\end{gather} +Finally, by applying divergence theorem to the second term in the above, we attain the weak form of the balance of linear momentum +\begin{gather} +\int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \sigma_{ij} \, dv + = \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv + + \int\limits_{\partial\Omega} \delta v_{i} \, \underbrace{\sigma_{ij} \, n_{j}}_{\bar{t}_{i}} \, da +\quad , +\label{equ: Weak form of the balance of linear momentum} +\end{gather} +wherein $\mathbf{n}$ represents the outward facing normal on $\partial\Omega$, the boundary of the domain, and $\bar{\mathbf{t}}$ the prescribed traction on the Neumann boundary. + +\clearpage +\section{General framework for elastoplasticity} +\begin{easylist}[itemize] +# Kinematics +## Small strain tensor +\begin{gather} +\boldsymbol{\varepsilon} + = \frac{1}{2} \left[ \nabla \mathbf{u} + \left[ \nabla \mathbf{u} \right]^{T} \right] + = \nabla\mathbf{u}^{sym} +\end{gather} +## Additive volumetric-isochoric split of the strain +\begin{gather} +\boldsymbol{\varepsilon} + = \boldsymbol{\varepsilon}^{vol} + \boldsymbol{\varepsilon}^{dev} +\quad ; \quad +\boldsymbol{\varepsilon}^{vol} = \dfrac{1}{3} \textrm{ tr}(\boldsymbol{\varepsilon}) \mathbf{I} +\end{gather} +## Additive split of strain into elastic and plastic contributions +\begin{gather*} +\boldsymbol{\varepsilon} + = \boldsymbol{\varepsilon}^{e} + \boldsymbol{\varepsilon}^{p} +\end{gather*} +# General free energy function +\begin{gather*} +\psi = \psi (\boldsymbol{\varepsilon}^{e}, \alpha, \boldsymbol{\beta}) +\end{gather*} +## Two new internal variables: +### $\alpha$ describes the relative increase of the elastic region (isotropic hardening variable) +### $\boldsymbol{\beta}$ describes the kinematic hardening (a rank-2 tensor; related to the centre point of the elastic region and introduces anisotropy due to plastic flow) +# Principle of irreversibility: +## Dissipation inequality +\begin{align*} +\mathcal{D} + &= \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}} - \dot\psi (\boldsymbol{\varepsilon}^{e}, \alpha, \boldsymbol{\beta}) \\ + &= \boldsymbol{\sigma} : \left[ \dot{\boldsymbol{\varepsilon}}^{e} + \dot{\boldsymbol{\varepsilon}}^{p} \right] + - \frac{\partial \psi}{\partial \boldsymbol{\varepsilon}^{e}} : \dot{\boldsymbol{\varepsilon}^{e}} + \underbrace{- \frac{\partial \psi}{\partial \alpha}}_{R} \dot{\alpha} + \underbrace{- \frac{\partial \psi}{\partial \boldsymbol{\beta}}}_{\boldsymbol{B}} : \dot{\boldsymbol{\beta}} \\ + &= \left[ \boldsymbol{\sigma} - \frac{\partial \psi}{\partial \boldsymbol{\varepsilon}^{e}} \right] : \dot{\boldsymbol{\varepsilon}^{e}} + + \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}}^{p} + + R \dot{\alpha} + + \boldsymbol{B} : \dot{\boldsymbol{\beta}} + \\ + &\geq 0 +\end{align*} +### Hardening stress $R$ +### Back stress $\boldsymbol{B}$ +## Definition of Cauchy stress +\begin{gather} +\boldsymbol{\sigma} + := \frac{\partial \psi}{\partial \boldsymbol{\varepsilon}^{e}} +\end{gather} +## Reduced dissipation inequality (using definition of Cauchy stress) +\begin{gather*} +\mathcal{D}^{red} + = \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}}^{p} + + R \dot{\alpha} + + \boldsymbol{B} : \dot{\boldsymbol{\beta}} + \geq 0 +\end{gather*} +# Evolution equations derived by the postulate of maximum dissipation [note: this ensures that hardening occurs when accommodated by the model, thus dissipating more energy than perfect plastic flow] +## Elastic region is restricted by yield function $\Phi \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right)$: +\begin{gather} +\mathbb{E} := \left\lbrace \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \vert \Phi \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \leq 0 \right\rbrace +\end{gather} +## Conditions: +### Elastic state: $\Phi < 0$ +### Plastic state: $\Phi = 0$ +# Define Lagrange multiplier problem to maximise the dissipation, i.e. +\begin{gather*} +\textnormal{maximise } \mathcal{D}^{red} \textnormal{ subject to } \Phi \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \leq 0 \\ +\mathcal{L} \left( \boldsymbol{\sigma},R,\boldsymbol{B}, \dot{\lambda} \right) + = -\mathcal{D}^{red} + \dot{\lambda} \Phi + \quad \rightarrow \quad \textnormal{stationary} +\end{gather*} +### $\dot{\lambda}$ is known as the consistency parameter +## Taking directional derivatives, we can identify the following evolution equations for the internal variables: +\begin{align*} +\frac{\partial \mathcal{L}}{\partial \boldsymbol{\sigma}} = \boldsymbol{0} +\quad &\Rightarrow \quad +\dot{\boldsymbol{\varepsilon}}^{p} = \dot{\lambda} \frac{\partial \Psi}{\partial \boldsymbol{\sigma}} +\quad \textnormal{(flow rule)} +\\ +\frac{\partial \mathcal{L}}{\partial R} = 0 +\quad &\Rightarrow \quad +\dot{\alpha} = \dot{\lambda} \frac{\partial \Psi}{\partial R} +\quad \textnormal{(evolution equation for isotropic hardening)} +\\ +\frac{\partial \mathcal{L}}{\partial \boldsymbol{B}} = \boldsymbol{0} +\quad &\Rightarrow \quad +\dot{\boldsymbol{\beta}} = \dot{\lambda} \frac{\partial \Psi}{\partial \boldsymbol{B}} +\quad \textnormal{(evolution equation for kinematic hardening)} +\\ +%\frac{\partial \mathcal{L}}{\partial \dot{\lambda}} +% = \\ +\end{align*} +% http://www.engr.colostate.edu/~thompson/hPage/CourseMat/Courses/CE696DV/plasticity.pdf +%https://www.ethz.ch/content/dam/ethz/special-interest/baug/ifb/ifb-dam/homepage-IfB/Education/msc_courses/msc-mechanics-building-materials/documents/plastizitaet.pdf +% p 36 +## Loading / unloading condition: +\begin{align*} +\frac{\partial \Psi}{\partial \boldsymbol{\sigma}} \dot{\boldsymbol{\sigma}} +\begin{cases} +< 0 \quad & \textnormal{Unloading (elastic)} \\ += 0 \quad & \textnormal{Loading (plastic flow)} \\ +> 0 \quad & \textnormal{Loading (plastic hardening)} +\end{cases} +%\textnormal{Loading (plastic hardening)} &\frac{\partial \Psi}{\partial \boldsymbol{\sigma}} \dot{\boldsymbol{\sigma}} > 0 \\ +%\textnormal{Loading (plastic flow): } &\frac{\partial \Psi}{\partial \boldsymbol{\sigma}} \dot{\boldsymbol{\sigma}} = 0 \\ +%\textnormal{Unloading: } &\frac{\partial \Psi}{\partial \boldsymbol{\sigma}} \dot{\boldsymbol{\sigma}} < 0 +\end{align*} +### The loading-unloading conditions can be shown to be equivalent to the Karush-Kuhn-Tucker (KKT) conditions (an important statement in optimisation theory): +% https://en.wikipedia.org/wiki/Flow_plasticity_theory +% Simo, Computational elasticity, p77 +\begin{gather} +\dot{\lambda} \geq 0 +\quad;\quad +\Phi \leq 0 +\quad;\quad +\dot{\lambda} \Phi = 0 +\end{gather} +#### Situations and definitions for any $\left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \in \mathbb{E}$ +\begin{align*} +\Phi < 0 + \quad &\Rightarrow \quad + \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \in \textnormal{int}(\mathbb{E}) + \quad \Rightarrow \quad + \dot{\lambda} = 0 \quad \textnormal{(Elastic response)} \\ +\Phi = 0 + \quad &\Rightarrow \quad + \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) \in \partial\mathbb{E} + \quad \Rightarrow \quad + \begin{cases} + \dot{\Phi} < 0 \quad \Rightarrow \quad \dot{\lambda} = 0 \quad \textnormal{(Elastic unloading)} \\ + \dot{\Phi} = 0 \quad \textnormal{and} \quad \dot{\lambda} = 0 \quad \textnormal{(Neutral loading)} \\ + \dot{\Phi} = 0 \quad \textnormal{and} \quad \dot{\lambda} > 0 \quad \textnormal{(Plastic loading)} + \end{cases} +\end{align*} +## Hardening / softening condition: +\begin{align*} +\frac{\partial \Psi}{\partial \alpha} \dot{\alpha} +\begin{cases} +< 0 \quad & \textnormal{Softening} \\ +> 0 \quad & \textnormal{Hardening} +\end{cases} +%\textnormal{Softening: } &\frac{\partial \Psi}{\partial \alpha} \dot{\alpha} > 0 \\ +%\textnormal{Hardening: } &\frac{\partial \Psi}{\partial \alpha} \dot{\alpha} < 0 +\end{align*} +# Example of Von-Mises yield criterion +## Convex yield surface defined by +\begin{gather*} +\Phi \left( \boldsymbol{\sigma} \right) + = \sqrt{\frac{3}{2}} \Vert \boldsymbol{\sigma}^{dev} \Vert - \sigma_{y} + \leq 0 +\end{gather*} +### Cylindrical yield-surface as observed in the principal stress space +### Norm of deviatoric stress contribution: +\begin{gather*} +\Vert \boldsymbol{\sigma}^{dev} \Vert + = \sqrt{\boldsymbol{\sigma}^{dev} : \boldsymbol{\sigma}^{dev}} +\end{gather*} +\end{easylist} + +\clearpage +\section{Constitutive law: Hyperelasticity} + +\begin{easylist}[itemize] +# Hyperelasticity +## Dissipation (in-)equality +\begin{gather} +\mathcal{D} + = \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}} - \dot\psi \left( \boldsymbol{\varepsilon} \right) + = 0 +\quad \Rightarrow \quad +\boldsymbol{\sigma} + = \frac{\partial \psi \left( \boldsymbol{\varepsilon} \right)}{\partial \boldsymbol{\varepsilon}} +\end{gather} +## Free/Strain energy function (Hooke's law) +\begin{gather} +\psi(\boldsymbol{\varepsilon}) + = \dfrac{\lambda}{2} \left[\textrm{tr}(\boldsymbol{\varepsilon})\right]^2 + + \mu \textrm{ tr}(\boldsymbol{\varepsilon}^2) +\end{gather} +## Cauchy stress +\begin{gather} +\boldsymbol{\sigma}(\boldsymbol{\varepsilon}) = \dfrac{\partial \psi}{\partial \boldsymbol{\varepsilon}} = \lambda \textrm{ tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2\mu\boldsymbol{\varepsilon} +\end{gather} +## Elastic tangent +\begin{gather} +\mathbb{C} = \dfrac{{\partial}^2 \psi}{\partial \boldsymbol{\varepsilon}^2}: \dot{\boldsymbol{\varepsilon}} = \lambda \mathbf{I} \otimes \mathbf{I} + 2\mu\;\mathbb{I}^{sym} +\end{gather} +### with the linear relationship +\begin{gather} +\boldsymbol{\sigma} + = \boldsymbol{\mathbb{C}} : \boldsymbol{\varepsilon} +\end{gather} +\end{easylist} + +\clearpage +\section{Constitutive law: Elastoplasticity} + +The linear elastoplastic constitutive laws and framework described here are described in detail by \citep{Simo2006a,Mergheim2018a}. + +\subsection{Associated Von Mises elastoplasticity with linear isotropic and kinematic hardening} +\begin{easylist} +# Kinematics +\begin{gather*} +\varepsilon + = \varepsilon^{e} + \varepsilon^{p} +\end{gather*} +# Free energy (additive decomposition into elastic and plastic parts) +\begin{align*} +\psi (\boldsymbol{\varepsilon}^{e}, \alpha, \boldsymbol{\beta}) + &= \psi^{e} (\boldsymbol{\varepsilon}^{e}) + + \psi^{p} (\alpha, \boldsymbol{\beta}) \\ +\psi^{e} (\boldsymbol{\varepsilon}^{e}) + &= \frac{\lambda}{2} \left[ \textnormal{tr} (\boldsymbol{\varepsilon}^{e}) \right]^{2} + + \mu \textnormal{tr} \left( \left[\boldsymbol{\varepsilon}^{e} \right]^{2} \right) \\ +\psi^{p} (\alpha, \boldsymbol{\beta}) + &= \frac{1}{2} k r \alpha^{2} + \frac{1}{2} k \left[ 1 - r \right] \Vert \boldsymbol{\beta} \Vert^{2} + \quad , \quad r \in \left[ 0,1 \right] +\end{align*} +# Stresses +\begin{align*} +\boldsymbol{\sigma} + &= \dfrac{\partial \psi}{\partial \boldsymbol{\varepsilon^{e}}} + = \lambda \textrm{ tr}(\boldsymbol{\varepsilon}^{e}) \mathbf{1} + 2\mu\boldsymbol{\varepsilon}^{e} \\ +R + &= - \dfrac{\partial \psi}{\partial \alpha} + = - k r \alpha \\ +\boldsymbol{B} + &= - \dfrac{\partial \psi}{\partial \boldsymbol{\beta}} + = -k \left[ 1 - r \right] \boldsymbol{\beta} +\end{align*} +# Yield function (Von Mises) +\begin{gather*} +\Phi \left( \boldsymbol{\sigma},R,\boldsymbol{B} \right) + = \Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert - \sqrt{\frac{2}{3}} \left[ \sigma_{y} - R \right] + \leq 0 +\end{gather*} +# Evolution equations +\begin{align*} +\dot{\boldsymbol{\varepsilon}}^{p} + &= \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} + = \dot{\lambda} \dfrac{\boldsymbol{\sigma}^{dev} + \boldsymbol{B}}{\Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert} + = \dot{\lambda} \boldsymbol{n} +\\ +\dot{\alpha} + &= \dot{\lambda} \frac{\partial \Phi}{\partial R} + = \sqrt{\frac{2}{3}} \dot{\lambda} +\\ +\dot{\boldsymbol{\beta}} + &= \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{B}} + = \dot{\lambda} \dfrac{\boldsymbol{\sigma}^{dev} + \boldsymbol{B}}{\Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert} + = \dot{\lambda} \boldsymbol{n} +\end{align*} +## Must satisfy KTT conditions +\begin{gather*} +\dot{\lambda} \geq 0 +\quad;\quad +\Phi \leq 0 +\quad;\quad +\dot{\lambda} \Phi = 0 +\end{gather*} +# Principle of irreversibility +\begin{align*} +\mathcal{D}^{red} + &= \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}}^{p} + + R \dot{\alpha} + + \boldsymbol{B} : \dot{\boldsymbol{\beta}} \\ + &= \boldsymbol{\sigma} : \dot{\lambda} \boldsymbol{n} + + R \sqrt{\frac{2}{3}} \dot{\lambda} + + \boldsymbol{B} : \dot{\lambda} \boldsymbol{n} \\ + &= \dot{\lambda} \left[ \left[ + \boldsymbol{\sigma} + \boldsymbol{B} \right] : \boldsymbol{n} + + R \sqrt{\frac{2}{3}} + \right] \\ + &= \dot{\lambda} \left[ \left[ + \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \right] : \dfrac{\boldsymbol{\sigma}^{dev} + \boldsymbol{B}}{\Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert} + + R \sqrt{\frac{2}{3}} + \right] \\ + &= \dot{\lambda} \left[ \ + \Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert + + R \sqrt{\frac{2}{3}} + \right] \\ + &\geq 0 +\end{align*} +## If $\Phi < 0 \quad \Rightarrow \quad \dot{\lambda} = 0$ +## If $\Phi = 0 \quad \Rightarrow \quad \dot{\lambda} > 0$ + and $\Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert + R \sqrt{\dfrac{2}{3}} = \sqrt{\dfrac{2}{3}} \sigma_{y} \geq 0$ +# Consistency condition (evolution of yield surface) +\begin{align*} +\dot{\Phi} + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dot{\boldsymbol{\sigma}} + + \frac{\partial \Phi}{\partial R} \dot{R} + + \frac{\partial \Phi}{\partial \boldsymbol{B}} : \dot{\boldsymbol{B}} \\ + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \underbrace{ \left[ \dot{\boldsymbol{\varepsilon}} - \dot{\boldsymbol{\varepsilon^{p}}} \right]}_{\dot{\boldsymbol{\varepsilon^{e}}}} + - \frac{\partial \Phi}{\partial R} \dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} \dot{\alpha} + - \frac{\partial \Phi}{\partial \boldsymbol{B}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} : \dot{\boldsymbol{\beta}} \\ +% &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dot{\boldsymbol{\varepsilon}} +% - \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dot{\boldsymbol{\varepsilon^{p}}} +% - \frac{\partial \Phi}{\partial R} \dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} \dot{\alpha} +% - \frac{\partial \Phi}{\partial \boldsymbol{B}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} : \dot{\boldsymbol{\beta}} \\ + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dot{\boldsymbol{\varepsilon}} + - \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} \dot{\lambda} + - \frac{\partial \Phi}{\partial R} \dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} \frac{\partial \Phi}{\partial R} \dot{\lambda} + - \frac{\partial \Phi}{\partial \boldsymbol{B}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} : \frac{\partial \Phi}{\partial \boldsymbol{B}} \dot{\lambda} \\ + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dot{\boldsymbol{\varepsilon}} + - \dot{\lambda} \left[ + \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} + + \frac{\partial \Phi}{\partial R} \dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} \frac{\partial \Phi}{\partial R} + + \frac{\partial \Phi}{\partial \boldsymbol{B}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} : \frac{\partial \Phi}{\partial \boldsymbol{B}} + \right] \\ + &= 0 \quad \textnormal{(stay on yield surface during plastic deformation)} +\end{align*} +\begin{align*} +\Rightarrow \quad \dot{\lambda} + = \dfrac { \dfrac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dot{\boldsymbol{\varepsilon}} }{ \dfrac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} : \dfrac{\partial \Phi}{\partial \boldsymbol{\sigma}} + + \dfrac{\partial \Phi}{\partial R} \dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} \dfrac{\partial \Phi}{\partial R} + + \dfrac{\partial \Phi}{\partial \boldsymbol{B}} : \dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} : \dfrac{\partial \Phi}{\partial \boldsymbol{B}} } +\end{align*} +with +\begin{align*} +\dfrac{\partial^{2} \psi}{\partial \boldsymbol{\varepsilon^{e}} \otimes \partial \boldsymbol{\varepsilon^{e}}} + &= \mathbb{C} \\ +\dfrac{\partial^{2} \psi}{\partial \alpha \partial \alpha} + &= -k r \\ +\dfrac{\partial^{2} \psi}{\partial \boldsymbol{\beta} \otimes \partial \boldsymbol{\beta}} + &= -k \left[ 1-r \right] \mathbb{I}^{sym} \\ +\dfrac{\partial \Phi}{\partial \boldsymbol{\sigma}} + &= \dfrac{\partial \Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert}{\partial \left( \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \right)} : \dfrac{\partial \left( \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \right)}{\partial \boldsymbol{\sigma}} + = \dfrac{\boldsymbol{\sigma}^{dev} + \boldsymbol{B}}{\Vert \boldsymbol{\sigma}^{dev} + \boldsymbol{B} \Vert} : \mathbb{I}^{sym} + = \boldsymbol{n} \\ +\dfrac{\partial \Phi}{\partial R} + &= \sqrt{\frac{2}{3}} \\ +\dfrac{\partial \Phi}{\partial \boldsymbol{B}} + &= \boldsymbol{n} +\end{align*} +Then +\begin{align*} +\Rightarrow \quad \dot{\lambda} + &= \dfrac { \boldsymbol{n} : \mathbb{C} : \dot{\boldsymbol{\varepsilon}} }{ \boldsymbol{n} : \mathbb{C} : \boldsymbol{n} + + \sqrt{\frac{2}{3}} \left[ -k r \right] \sqrt{\frac{2}{3}} + + \boldsymbol{n} : \left[ -k \left[ 1-r \right] \mathbb{I}^{sym} \right] : \boldsymbol{n} } \\ + &= \dfrac { \boldsymbol{n} : \mathbb{C} : \dot{\boldsymbol{\varepsilon}} }{ \boldsymbol{n} : \mathbb{C} : \boldsymbol{n} + - \frac{2}{3} k r + -k \left[ 1-r \right] } \\ + &= \dfrac { \boldsymbol{n} : \mathbb{C} : \dot{\boldsymbol{\varepsilon}} }{ \boldsymbol{n} : \mathbb{C} : \boldsymbol{n} + + k \left[\frac{1}{3} r - 1 \right] } \\ +\end{align*} +\end{easylist} + +\subsection{Elasto-plastic tangent modulus (continuous setting)} +\begin{easylist} +# Goal +\begin{gather*} +\dot{\boldsymbol{\sigma}} + = \boldsymbol{\mathcal{C}}^{ep} : \dot{\boldsymbol{\varepsilon}} +\end{gather*} +# Stress rate +\begin{gather*} +\boldsymbol{\sigma} + = \frac{\partial \Psi}{\partial \boldsymbol{\varepsilon}^{e}} \\ +\Rightarrow \quad +\dot{\boldsymbol{\sigma}} + = \frac{\partial^{2} \Psi}{\partial \boldsymbol{\varepsilon}^{e} \otimes \partial \boldsymbol{\varepsilon}^{e}} : \dot{\boldsymbol{\varepsilon}}^{e} + = \boldsymbol{\mathcal{C}} : \left[ \dot{\boldsymbol{\varepsilon}} - \dot{\boldsymbol{\varepsilon}}^{p} \right] + = \boldsymbol{\mathcal{C}} : \left[ \dot{\boldsymbol{\varepsilon}} - \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} \right] +\end{gather*} +# Remember: Plastic potential $\Phi \left( \boldsymbol{\sigma},\boldsymbol{S} \right)$ with evolution laws +$ +\dot{\boldsymbol{\varepsilon}}^{p} + = \dot{\lambda} \dfrac{\partial \Phi}{\partial \boldsymbol{\sigma}} +$, $ +\dot{\boldsymbol{d}} + = \dot{\lambda} \dfrac{\partial \Phi}{\partial \boldsymbol{S}} +$ +## Definition and derivatives of stress-like terms $\boldsymbol{S} = \left[ R,\boldsymbol{B} \right]$ with internal variables $\boldsymbol{d} = \left[ \alpha,\boldsymbol{\beta} \right]$ +\begin{gather*} +\boldsymbol{S} + = - \frac{\partial \Psi}{\partial \boldsymbol{d}} \\ +\dot{\boldsymbol{S}} + = - \frac{\partial^{2} \Psi}{\partial \boldsymbol{d} \otimes \partial \boldsymbol{d}} \circ \dot{\boldsymbol{d}} + = - \frac{\partial^{2} \Psi}{\partial \boldsymbol{d} \otimes \partial \boldsymbol{d}} \circ \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{S}} \\ +\end{gather*} +# Use consistency condition to compute $\dot{\lambda}$: +\begin{gather*} +\dot{\lambda} \dot{\Phi} = 0 +\quad \Rightarrow \quad +\dot{\Phi} = 0 \quad \textnormal{for plastic loading} +\end{gather*} +\begin{align*} +\dot{\Phi} + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \dot{\boldsymbol{\sigma}} + + \frac{\partial \Phi}{\partial \boldsymbol{S}} \circ \dot{\boldsymbol{S}} \\ + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \boldsymbol{\mathcal{C}} : \left[ \dot{\boldsymbol{\varepsilon}} - \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} \right] + - \frac{\partial \Phi}{\partial \boldsymbol{S}} \circ \frac{\partial^{2} \Psi}{\partial \boldsymbol{d} \otimes \partial \boldsymbol{d}} \circ \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{S}} \\ + &= 0 \\ +\Rightarrow \quad +\dot{\lambda} + &= \frac{1}{D} \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \boldsymbol{\mathcal{C}} : \dot{\boldsymbol{\varepsilon}} +\quad \textnormal{with} \\ +D + &= \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \boldsymbol{\mathcal{C}} : \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} + + \frac{\partial \Phi}{\partial \boldsymbol{S}} \circ \frac{\partial^{2} \Psi}{\partial \boldsymbol{d} \otimes \partial \boldsymbol{d}} \circ \frac{\partial \Phi}{\partial \boldsymbol{S}} +\end{align*} +# Elasto-plastic tangent +\begin{gather*} +\dot{\boldsymbol{\sigma}} + = \boldsymbol{\mathcal{C}}^{ep} : \dot{\boldsymbol{\varepsilon}} + = \underbrace{\left[ \boldsymbol{\mathcal{C}} - \frac{1}{D} \left[ \boldsymbol{\mathcal{C}} : \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} \right] \otimes \left[ \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \boldsymbol{\mathcal{C}} \right] \right]}_{\boldsymbol{\mathcal{C}}^{ep} \textnormal{ (symmetric)}} : \dot{\boldsymbol{\varepsilon}} +\end{gather*} +# Result for isotropic hardening +\begin{align*} +\boldsymbol{\mathcal{C}}^{ep, dev} + &= 2 \mu \boldsymbol{\mathcal{I}}^{dev} - \frac{\left[ 2\mu \right]^{2}}{2\mu + \frac{2}{3}k} \boldsymbol{n} \otimes \boldsymbol{n} \\ +\boldsymbol{\mathcal{C}}^{ep, vol} + &= \kappa \boldsymbol{1} \otimes \boldsymbol{1} +\end{align*} +with +\begin{gather*} +\boldsymbol{n} + = \frac{\boldsymbol{\sigma}^{dev}}{\Vert \boldsymbol{\sigma}^{dev} \Vert} +\end{gather*} +\end{easylist} + +\section{Integration algorithms for elasto-plasticity} +\subsection{Radial return algorithm for von Mises plasticity with linear isotropic hardening} +\begin{easylist} +# Constitutive setting +\begin{gather*} +\boldsymbol{\varepsilon} + = \boldsymbol{\varepsilon}^{e} + \boldsymbol{\varepsilon}^{p} + = \boldsymbol{\varepsilon}^{vol} + \boldsymbol{\varepsilon}^{dev} +\\ +\psi + = \frac{1}{2} \kappa \left[ \textnormal{tr} \left( \boldsymbol{\varepsilon}^{vol} \right) \right]^{2} + + \mu \textnormal{tr} \left( \left[\boldsymbol{\varepsilon}^{e, dev} \right]^{2} \right) + + \frac{1}{2} k \alpha^{2} +\\ +\Phi \left( \boldsymbol{\sigma}, R \right) + = \Vert \boldsymbol{\sigma}^{dev} \Vert - \sqrt{\frac{2}{3}} \left[ \sigma_{y} - R \right] + \leq 0 +\\ +\dot{\boldsymbol{\varepsilon}}^{p} + = \dot{\lambda} \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} +% = \dot{\lambda} \dfrac{\boldsymbol{\sigma}^{dev}}{\Vert \boldsymbol{\sigma}^{dev} \Vert} + = \dot{\lambda} \boldsymbol{n} +\quad , \quad +\boldsymbol{n} + = \dfrac{\boldsymbol{\sigma}^{dev}}{\Vert \boldsymbol{\sigma}^{dev} \Vert} +\\ +\dot{\alpha} + = \dot{\lambda} \frac{\partial \Phi}{\partial R} + = \sqrt{\frac{2}{3}} \dot{\lambda} +\end{gather*} +# Apply backward Euler method to evolution equations +\begin{gather*} +\boldsymbol{\varepsilon}^{p}_{n+1} + = \boldsymbol{\varepsilon}^{p}_{n} + + \Delta t \, \dot{\boldsymbol{\varepsilon}}^{p}_{n+1} + = \boldsymbol{\varepsilon}^{p}_{n} + + \Delta t \, \dot{\lambda}_{n+1} \boldsymbol{n}_{n+1} +\\ +\alpha_{n+1} + = \alpha_{n} + + \Delta t \, \dot{\alpha}_{n+1} + = \alpha_{n} + + \Delta t \, \dot{\lambda}_{n+1} \sqrt{\frac{2}{3}} +\end{gather*} +# Deviatoric stress +\begin{align*} +\boldsymbol{\sigma}^{dev}_{n+1} + & = 2 \mu \boldsymbol{\varepsilon}^{e, dev}_{n+1} + = 2 \mu \left[ \boldsymbol{\varepsilon}^{dev}_{n+1} - \boldsymbol{\varepsilon}^{p, dev}_{n+1} \right] \\ + &= 2 \mu \boldsymbol{\varepsilon}^{dev}_{n+1} + - 2 \mu \left[ \boldsymbol{\varepsilon}^{p}_{n} + + \Delta t \, \dot{\lambda}_{n+1} \boldsymbol{n}_{n+1} \right] \\ +% &= \underbrace{2 \mu \left[ \boldsymbol{\varepsilon}^{dev} - \boldsymbol{\varepsilon}^{p}_{n} \right]}_{\boldsymbol{\sigma}^{dev, trial}_{n+1}} +% - 2 \mu \Delta t \, \dot{\lambda}_{n+1} \boldsymbol{n}_{n+1} \\ + &= \underbrace{2 \mu \left[ \boldsymbol{\varepsilon}^{dev}_{n+1} - \boldsymbol{\varepsilon}^{p}_{n} \right]}_{\boldsymbol{\sigma}^{dev, trial}_{n+1}} + - 2 \mu \Delta \lambda_{n+1} \dfrac{\boldsymbol{\sigma}^{dev}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} +\quad (\textnormal{Note: } \Delta t \, \dot{\lambda}_{n+1} = \Delta \lambda_{n+1}) +%\\ +%\Rightarrow \quad +%\boldsymbol{\sigma}^{dev, trial}_{n+1} +% &= \left[ 1 + \dfrac{2 \mu \Delta \lambda_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} \right] \boldsymbol{\sigma}^{dev}_{n+1} +\\ +\Rightarrow \quad +\boldsymbol{\sigma}^{dev}_{n+1} + &= \left[ 1 + \dfrac{2 \mu \Delta \lambda_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} \right]^{-1} \boldsymbol{\sigma}^{dev, trial}_{n+1} +\end{align*} +\clearpage +### The final stress state is obtained by projecting the trial stress state onto the current yield surface. +\begin{figure}[!htb] +\centering +\includegraphics[width=0.7\linewidth]{Figures/Sec_3_5_1-Radial_projection} +\caption{Radial projection \citep{Mergheim2018a}} +\label{fig:sec351-radialprojection} +\end{figure} +## Still need to determine $\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert$. Use coaxiality of the stresses: +\begin{gather*} +\dfrac{\boldsymbol{\sigma}^{dev, trial}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} + = \dfrac{\boldsymbol{\sigma}^{dev}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} + = \boldsymbol{n}_{n+1} +\end{gather*} +### Contract both sides of previous equation by $\boldsymbol{n}_{n+1}$ and use definition of tensor norm $\Vert \boldsymbol{\sigma} \Vert$: +\begin{gather*} +\boldsymbol{\sigma}^{dev}_{n+1} : \dfrac{\boldsymbol{\sigma}^{dev}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} + = \boldsymbol{\sigma}^{dev, trial}_{n+1} : \dfrac{\boldsymbol{\sigma}^{dev, trial}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} + - 2 \mu \Delta \lambda_{n+1} \dfrac{\boldsymbol{\sigma}^{dev}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} : \dfrac{\boldsymbol{\sigma}^{dev}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert} \\ +\Rightarrow \quad +\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert + = \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert + - 2 \mu \Delta \lambda_{n+1} +\end{gather*} +## Yield function (plastic flow: $\Phi = 0$) +\begin{gather*} +\Phi_{n+1} + = \Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert - \sqrt{\frac{2}{3}} \left[ \sigma_{y} + \underbrace{k \alpha_{n+1}}_{- R_{n+1}} \right] + = 0 \\ +\Rightarrow \quad +\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert + = \sqrt{\frac{2}{3}} \left[ \sigma_{y} + k \left[ \alpha_{n} + \Delta \lambda_{n+1} \sqrt{\frac{2}{3}} \right] \right] +\end{gather*} +## Plastic multiplier update computed by equating the two above equations +\begin{gather*} +\Vert \boldsymbol{\sigma}^{dev}_{n+1} \Vert + = \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert + - 2 \mu \Delta \lambda_{n+1} + = \sqrt{\frac{2}{3}} \left[ \sigma_{y} + k \left[ \alpha_{n} + \Delta \lambda_{n+1} \sqrt{\frac{2}{3}} \right] \right] \\ +\Rightarrow \quad +\underbrace{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert - \sqrt{\frac{2}{3}} \left[ \sigma_{y} + k \alpha_{n} \right]}_{\Phi^{trial}_{n+1}} + = \Delta \lambda_{n+1} \left[ 2 \mu + \frac{2}{3} k \right] \\ +\Rightarrow \quad +\Delta \lambda_{n+1} + = \frac{\Phi^{trial}_{n+1}}{2 \mu + \frac{2}{3} k} +\end{gather*} +# Remarks +## Within ideal von Mises plasticity or with linear hardening, $\dot{\lambda}_{n+1}$ can be computed directly. + Otherwise, $\Phi_{n+1}$ has to be solver iteratively for $\dot{\lambda}_{n+1}$ (e.g. using Newton's method) +## For general yield functions, the closest point projection is the extension of the radial return algorithm. +%# Algorithm +%## Show slides +\end{easylist} + +\subsection{General closest-point projection (for general plasticity and hardening laws)} +\begin{easylist} +# Difference between radial and closest point projection +\begin{figure}[!htb] +\centering +\begin{subfigure}[t]{0.45\textwidth} +\includegraphics[width=0.8\linewidth]{Figures/Sec_3_5_1-Radial_projection} +\caption{Radial projection \citep{Mergheim2018a}} +\label{fig:sec351-radialprojection} +\end{subfigure} +\quad +\begin{subfigure}[t]{0.45\textwidth} +\includegraphics[width=0.8\linewidth]{Figures/Sec_3_5_2-General_closest_point_projection} +\caption{Closest-point projection \citep{Simo2006a}} +\label{fig:sec352-generalclosestpointprojection} +\end{subfigure} +\end{figure} +\end{easylist} + +\subsection{Consistent elastoplastic tangent modulus for von Mises plasticity with only linear isotropic hardening (time discrete setting)} +\begin{easylist} +# Note: The definition of the consistent elastoplastic tangent depends on the algorithms used to update stresses, internal variables +# Goal: Tangent to compute +\begin{gather*} + \boldsymbol{\mathcal{C}}^{ep} + = \frac{\partial \boldsymbol{\sigma}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \boldsymbol{\mathcal{C}}^{ep, vol} + \boldsymbol{\mathcal{C}}^{ep, dev} +\end{gather*} +# Stress tensors +\begin{gather*} +\boldsymbol{\sigma}_{n+1} + = \boldsymbol{\sigma}^{vol}_{n+1} + \boldsymbol{\sigma}^{dev}_{n+1} \\ +\boldsymbol{\sigma}^{vol}_{n+1} + = \kappa \boldsymbol{\varepsilon}^{vol}_{n+1} \\ +\boldsymbol{\sigma}^{dev}_{n+1} + = \boldsymbol{\sigma}^{dev, trial}_{n+1} + - 2 \mu \Delta \lambda_{n+1} \boldsymbol{n}_{n+1} \\ +\boldsymbol{\sigma}^{dev, trial}_{n+1} + = 2 \mu \left[ \boldsymbol{\varepsilon}^{dev}_{n+1} - \boldsymbol{\varepsilon}^{p}_{n} \right] +\end{gather*} +# Derivatives (for tangent stiffness contributions for trial solution) +\begin{gather*} +\boldsymbol{\mathcal{C}}^{ep, vol} + = \frac{\partial \boldsymbol{\sigma}^{vol}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \kappa \boldsymbol{1} \otimes \boldsymbol{1} \\ +\boldsymbol{\mathcal{C}}^{ep, dev, trial} + = \frac{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = 2 \mu \boldsymbol{\mathcal{I}}^{dev} +\end{gather*} +# Derivatives (for tangent stiffness contributions when plastic flow) +## Remember: +\begin{gather*} +\Delta \lambda_{n+1} + = \frac{\Phi^{trial}_{n+1}}{2 \mu + \frac{2}{3} k} + = \frac{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert - \sqrt{\frac{2}{3}} \left[ \sigma_{y} - R_{n} \right]}{2 \mu + \frac{2}{3} k} +\quad \textnormal{(Note: $R_{n}$ is fixed)} \\ +\boldsymbol{n}_{n+1} + = \dfrac{\boldsymbol{\sigma}^{dev, trial}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} +\end{gather*} +## Therefore: +\begin{gather*} +\frac{\partial \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \frac{\partial \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}} : \frac{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \underbrace{\frac{\boldsymbol{\sigma}^{dev, trial}_{n+1}}{\Vert\boldsymbol{\sigma}^{dev, trial}_{n+1}\Vert}}_{\boldsymbol{n}_{n+1}} : 2 \mu \boldsymbol{\mathcal{I}}^{dev} + = 2 \mu \boldsymbol{n}_{n+1} +\\ +\frac{\partial \Delta \lambda_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \frac{1}{{2 \mu + \frac{2}{3} k}} \frac{\partial \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \frac{2 \mu}{{2 \mu + \frac{2}{3} k}} \boldsymbol{n}_{n+1} \\ +\end{gather*} +\begin{align*} +\frac{\partial \boldsymbol{n}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} +% &= 2 \mu \boldsymbol{\mathcal{I}}^{dev}_{n+1} - \frac{2\mu}{2\mu + \frac{2}{3}k} \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \quad \textnormal{(from previous section)} \\ + &\equiv \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert^{-1} \frac{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + + \boldsymbol{\sigma}^{dev, trial}_{n+1} \otimes \frac{\partial \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert^{-1}}{\partial \boldsymbol{\varepsilon}_{n+1}} \\ + &= \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert^{-1} 2 \mu \boldsymbol{\mathcal{I}}^{dev} + - \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert^{-2} \boldsymbol{\sigma}^{dev, trial}_{n+1} \otimes \frac{\partial \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}{\partial \boldsymbol{\varepsilon}_{n+1}} \\ + &= \Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert^{-1} \left[ 2 \mu \boldsymbol{\mathcal{I}}^{dev} + - \frac{\boldsymbol{\sigma}^{dev, trial}_{n+1}}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} \otimes 2 \mu \boldsymbol{n}_{n+1} \right] \\ + &= \frac{2\mu}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} \left[ \boldsymbol{\mathcal{I}}^{dev} + - \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \right] +\end{align*} +## Resulting deviatoric part of the elasto-plastic tangent +\begin{align*} +\boldsymbol{\mathcal{C}}^{ep, dev} + &= \frac{\partial \boldsymbol{\sigma}^{dev}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + = \frac{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + - 2 \mu \left[ \boldsymbol{n}_{n+1} \otimes \frac{\partial \Delta \lambda_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} + + \Delta \lambda_{n+1} \frac{\partial \boldsymbol{n}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} \right] +\\ + &= 2 \mu \boldsymbol{\mathcal{I}}^{dev} + - 2 \mu \left[ \boldsymbol{n}_{n+1} \otimes \left[ \frac{2 \mu}{{2 \mu + \frac{2}{3} k}} \boldsymbol{n}_{n+1} \right] + \right.\\ &\left. + + \Delta \lambda_{n+1} \left[ \frac{2\mu}{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert} \left[ \boldsymbol{\mathcal{I}}^{dev} - \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \right] \right] \right] +\\ + &= 2 \mu \boldsymbol{\mathcal{I}}^{dev} + - \frac{ \left[2 \mu \right]^{2}}{2 \mu + \frac{2}{3} k} \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \\ + & + \left[\Delta \lambda_{n+1} \frac{\left[ 2 \mu \right]^{2}}{{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}} \left[ \boldsymbol{\mathcal{I}}^{dev}_{n+1} - \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \right] \right] +\end{align*} +## Compared to continuous case, the tangents are not equal but they converge to the same expression when $\Delta \lambda_{n+1} \rightarrow 0$ +\end{easylist} + +\clearpage +\section{Finite element discretisation +\label{sec: FE discretisation} +} +For clarity, it is worth mentioning up front that in this \namecref{sec: FE discretisation} we will denote the general elasto-plastic tangent by $\boldsymbol{\mathbb{C}}^{ep}$; +so, in the elastic region $\boldsymbol{\mathbb{C}}^{ep} \equiv \boldsymbol{\mathbb{C}}$ while in the plastic region $\boldsymbol{\mathbb{C}}^{ep}$ represents the consistent tangent. + +Combining \cref{equ: Weak form of the balance of linear momentum,equ: Full expansion of constitutive laws} renders the complete expression of the balance of linear momentum, with accommodation of the muscle fibre model, namely +\begin{gather} +\int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \left[ \boldsymbol{\mathbb{C}}^{ep} \right]_{ijkl} \varepsilon_{kl} \, dv + = \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv + + \int\limits_{\partial\Omega} \delta v_{i} \, \underbrace{\sigma_{ij} \, n_{j}}_{\bar{t}_{i}} \, da +\quad , +\label{equ: Weak form of the balance of linear momentum: Muscle model} +\end{gather} + +We discretise the trial solution and test function using finite element shape functions (ansatz) +\begin{gather} +\mathbf{u} \left( \mathbf{x} \right) + \approx \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) u^{I} +\quad , \quad +\mathbf{v} \left( \mathbf{x} \right) + \approx \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) v^{I} +\end{gather} +where $\mathbf{N}^{I} \left( \mathbf{x} \right)$ is the (position-dependent) vector-valued finite element shape function corresponding to the $I^{\textnormal{th}}$ degree-of-freedom, and $u^{I}, v^{I}$ are coefficients of the solution and trial function. +In \texttt{deal.II} nomenclature, the shape function is computed from a scalar base shape function and some expansion into higher-dimensional space by +\begin{gather} +\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) + = N^{I} \left( \mathbf{x} \right) \mathbf{e}_{\textnormal{comp}(I)} +\end{gather} +where $N^{I}$ is a scalar shape function and $\mathbf{e}_{\textnormal{comp}(I)}$ is the basis direction associated with the $I^{\textnormal{th}}$ degree-of-freedom. +Therefore, the $j^{\textnormal{th}}$ local component of shape function $\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right)$ is given by +\begin{gather} +\left[\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right)\right]_{j} + = N^{I} \left( \mathbf{x} \right) \left[ \mathbf{e}_{\textnormal{comp}(I)} \right]_{j} + = N^{I} \left( \mathbf{x} \right) \delta_{\textnormal{comp}(I) j} +\quad . +\end{gather} +where $\delta_{ij}$ is the Kronecker delta. +Note that in this instance we use the same ansatz for the test and trial spaces, and the $0 \leq \textnormal{comp}(I), j < \textnormal{spacedim}$. + +We now use these shape functions to discretise the weak expression for the balance of linear momentum. +Starting on the right-hand side of \cref{equ: Weak form of the balance of linear momentum: Muscle model}, the body force and traction contributions are computed by +\begin{align} +\int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv + &= \int\limits_{\Omega} \left[ \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, b_{i} \, dv + = \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} \left[ \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \right]_{i} \, b_{i} \, dv \notag\\ + &= \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \left( \mathbf{x} \right) \delta_{\textnormal{comp}(I) i} \, b_{i} \, dv + = \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \, b_{\textnormal{comp}(I)} \, dv +\label{equ: Discretisation: Body force} +\\ +\int\limits_{\Omega} \delta v_{i} \, t_{i} \, dv + &= \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \, t_{\textnormal{comp}(I)} \, dv +\label{equ: Discretisation: Traction} +\quad . +\end{align} +The last component of \cref{equ: Weak form of the balance of linear momentum: Muscle model} that we wish to express in discrete form is the left-hand side of the equation. +Before we do, we observe that using the minor symmetry of the material stiffness tensor we can re-express the contraction of it and the small strain tensor as +\begin{gather} +\boldsymbol{\mathbb{C}} : \boldsymbol{\varepsilon} + = \boldsymbol{\mathbb{C}} : \frac{1}{2} \left[ \nabla \mathbf{u} + \left[ \nabla \mathbf{u} \right]^{T} \right] + \equiv \boldsymbol{\mathbb{C}} : \nabla \mathbf{u} +\end{gather} +Therefore, this contribution written in discrete form is +\begin{align} +&\int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \left[ \boldsymbol{\mathbb{C}}^{ep} \right]_{ijkl} \varepsilon_{kl} \, dv + \equiv \int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial \delta u_{k}}{\partial x_{l}} \, dv \notag\\ + &\equiv \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \sum\limits_{J} \boldsymbol{\varPhi}^{J} \left( \mathbf{x} \right) \delta u^{J} \right]_{k} \, dv \notag\\ + &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \boldsymbol{\varPhi}^{J} \left( \mathbf{x} \right) \right]_{k} \, dv \right] \delta u^{J} \notag\\ + &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial N^{I} \left( \mathbf{x} \right)}{\partial x_{j}} \delta_{\textnormal{comp}(I) i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial N^{J} \left( \mathbf{x} \right)}{\partial x_{l}} \delta_{\textnormal{comp}(J) k} \, dv \right] \delta u^{J} \notag\\ + &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial N^{I} \left( \mathbf{x} \right)}{\partial x_{j}} \, \mathbb{C}^{ep}_{\textnormal{comp}(I) \,j \, \textnormal{comp}(J) \, l} \, \frac{\partial N^{J} \left( \mathbf{x} \right)}{\partial x_{l}} \, dv \right] \delta u^{J} +\label{equ: Discretisation: Material tangent} +\quad . +\end{align} + +\Cref{equ: Discretisation: Body force,equ: Discretisation: Traction,equ: Discretisation: Stress history,equ: Discretisation: Material tangent} are collectively used to develop the system of linear equations that are solved at each time step. + +\bibliographystyle{plain} +\bibliography{\jobname} + +\end{document} From 378a3c729bf1c59ea63549d0bb754882ce56002f Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Sat, 29 Dec 2018 18:34:02 +0200 Subject: [PATCH 08/20] Muchos updates --- Linear_Elastoplasticity/CMakeLists.txt | 2 +- .../linear_elastoplasticity.cc | 250 ++++++++--- .../mesh/tensile_specimen.inp | 422 ++++++++++++++++++ Linear_Elastoplasticity/parameters.prm | 32 +- 4 files changed, 622 insertions(+), 84 deletions(-) create mode 100755 Linear_Elastoplasticity/mesh/tensile_specimen.inp diff --git a/Linear_Elastoplasticity/CMakeLists.txt b/Linear_Elastoplasticity/CMakeLists.txt index 9e8dd932..8aa14a3b 100644 --- a/Linear_Elastoplasticity/CMakeLists.txt +++ b/Linear_Elastoplasticity/CMakeLists.txt @@ -3,7 +3,7 @@ ## # Set the name of the project and target: -SET(TARGET "Linear_elastoplasticity") +SET(TARGET "linear_elastoplasticity") # Declare all source files the target consists of. Here, this is only # the one step-X.cc file, but as you expand your project you may wish diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index 9cc8e7ab..f9e4b90b 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -37,6 +37,7 @@ #include #include #include +#include #include #include #include @@ -153,14 +154,27 @@ namespace LMM struct Geometry { std::string geometry_type; - double length; - double radius; - double notch_length; - double notch_radius; double scale; unsigned int n_global_refinement_steps; unsigned int n_local_refinement_steps; + struct NotchedCylinder + { + double length; + double radius; + double notch_length; + double notch_radius; + } notched_cylinder; + + struct TensileSpecimen + { + const double length = 35.0/2; + + const Tensor<1,3> manifold_direction = Tensor<1,3>({0,0,1}); + const Point<3> centre_manifold_id_10 = {6.0,4.0,0.0}; + const Point<3> centre_manifold_id_11 = {10.472135955,0.0,0.0}; + } tensile_specimen; + static void declare_parameters(ParameterHandler &prm); @@ -173,9 +187,23 @@ namespace LMM prm.enter_subsection("Geometry"); { prm.declare_entry("Geometry type", "Notched cylinder", - Patterns::Selection("Notched cylinder|Notched tensile specimen"), + Patterns::Selection("Notched cylinder|Tensile specimen|Notched tensile specimen"), "The geometry to be modelled"); + prm.declare_entry("Grid scale", "1e-3", + Patterns::Double(0), + "Global grid scaling factor"); + + prm.declare_entry("Global refinement steps", "1", + Patterns::Integer(0), + "Number of global refinement steps"); + + prm.declare_entry("Local refinement steps", "1", + Patterns::Integer(0), + "Number of initial local refinement cycles"); + + prm.enter_subsection("Notched cylinder"); + { prm.declare_entry("Length", "100", Patterns::Double(0), "Overall length of the specimen"); @@ -191,18 +219,8 @@ namespace LMM prm.declare_entry("Notch radius", "1", Patterns::Double(0), "Overall radius of the notch in the specimen"); - - prm.declare_entry("Grid scale", "1e-3", - Patterns::Double(0), - "Global grid scaling factor"); - - prm.declare_entry("Global refinement steps", "1", - Patterns::Integer(0), - "Number of global refinement steps"); - - prm.declare_entry("Local refinement steps", "1", - Patterns::Integer(0), - "Number of initial local refinement cycles"); + } + prm.leave_subsection(); } prm.leave_subsection(); } @@ -212,19 +230,24 @@ namespace LMM prm.enter_subsection("Geometry"); { geometry_type = prm.get("Geometry type"); - length = prm.get_double("Length"); - radius = prm.get_double("Radius"); - notch_length = prm.get_double("Notch length"); - notch_radius = prm.get_double("Notch radius"); scale = prm.get_double("Grid scale"); n_global_refinement_steps = prm.get_integer("Global refinement steps"); n_local_refinement_steps = prm.get_integer("Local refinement steps"); + + prm.enter_subsection("Notched cylinder"); + { + notched_cylinder.length = prm.get_double("Length"); + notched_cylinder.radius = prm.get_double("Radius"); + notched_cylinder.notch_length = prm.get_double("Notch length"); + notched_cylinder.notch_radius = prm.get_double("Notch radius"); + } + prm.leave_subsection(); } prm.leave_subsection(); - AssertThrow(length > notch_length, + AssertThrow(notched_cylinder.length > notched_cylinder.notch_length, ExcMessage("Cannot create geometry with given dimensions")); - AssertThrow(radius > notch_radius, + AssertThrow(notched_cylinder.radius > notched_cylinder.notch_radius, ExcMessage("Cannot create geometry with given dimensions")); } @@ -643,6 +666,15 @@ namespace LMM // context of a Newton scheme. } + // --- Post-processing --- + + // Plastic variables + double + get_alpha_p() + { + return alpha_p_t; + } + protected: const double kappa_e; // buld modulus const double mu_e; // shear modulus @@ -755,8 +787,8 @@ namespace LMM void make_grid (); void setup_system (); void setup_qph(); + void make_constraints (); void assemble_system (); - void apply_boundary_conditions (); void solve (); void output_results () const; @@ -772,6 +804,7 @@ namespace LMM QGauss qf_face; AffineConstraints hanging_node_constraints; + AffineConstraints constraints; SparsityPattern sparsity_pattern; SparseMatrix system_matrix; @@ -823,7 +856,7 @@ namespace LMM if (parameters.geometry_type == "Notched cylinder") { - const double notch_radius_inner = parameters.notch_radius - (parameters.radius-parameters.notch_radius); + const double notch_radius_inner = parameters.notched_cylinder.notch_radius - (parameters.notched_cylinder.radius-parameters.notched_cylinder.notch_radius); Triangulation<2> tria_cross_section; { @@ -835,14 +868,14 @@ namespace LMM Triangulation<2> tria_inner_2; GridGenerator::hyper_shell (tria_inner_2,centre, notch_radius_inner, - parameters.notch_radius, + parameters.notched_cylinder.notch_radius, 4); GridTools::rotate(M_PI/4,tria_inner_2); Triangulation<2> tria_outer; GridGenerator::hyper_shell (tria_outer,centre, - parameters.notch_radius, - parameters.radius, + parameters.notched_cylinder.notch_radius, + parameters.notched_cylinder.radius, 4); GridTools::rotate(M_PI/4,tria_outer); @@ -855,11 +888,11 @@ namespace LMM { std::vector slice_coordinates; slice_coordinates.push_back(0.0); - slice_coordinates.push_back(parameters.notch_length); - slice_coordinates.push_back(2.0*parameters.notch_length); + slice_coordinates.push_back(parameters.notched_cylinder.notch_length); + slice_coordinates.push_back(2.0*parameters.notched_cylinder.notch_length); // Biased slices with function taken from GridGenerator::concentric_hyper_shells - const double bias_first_slice = 2.0*parameters.notch_length; - const double bias_last_slice = parameters.length; + const double bias_first_slice = 2.0*parameters.notched_cylinder.notch_length; + const double bias_last_slice = parameters.notched_cylinder.length; const double skewness = 2.0; const unsigned int n_slices = 5; for (unsigned int s=0; s= bias_first_slice && r <= bias_last_slice, ExcMessage("New slice location not in correct range.")); slice_coordinates.push_back(r); } - slice_coordinates.push_back(parameters.length); + slice_coordinates.push_back(parameters.notched_cylinder.length); GridGenerator::extrude_triangulation (tria_cross_section, slice_coordinates, tria_unnotched); GridTools::rotate(M_PI/2,1,tria_unnotched); } - // Remove all cells within the notch length and greater than - // the notch radius std::set::active_cell_iterator> cells_to_remove; for (typename Triangulation::active_cell_iterator cell = tria_unnotched.begin_active(); cell!=tria_unnotched.end(); ++cell) { - if (cell->center()[0] < parameters.notch_length) + // Remove all cells within the notch length and greater than + // the notch radius. + if (cell->center()[0] < parameters.notched_cylinder.notch_length) { // Outer layer of cells correspond to the notch to be removed for (unsigned int face=0; face < GeometryInfo::faces_per_cell; ++face) if (cell->face(face)->at_boundary() == true && - std::abs(cell->face(face)->center()[0] - parameters.notch_length/2.0) < 1e-12) + std::abs(cell->face(face)->center()[0] - parameters.notched_cylinder.notch_length/2.0) < 1e-12) cells_to_remove.insert(cell); // Additional verification Point rad = cell->center(); rad[0] = 0.0; const double radius = rad.norm(); - if (radius > parameters.notch_radius) + if (radius > parameters.notched_cylinder.notch_radius) cells_to_remove.insert(cell); } } @@ -928,7 +961,7 @@ namespace LMM { if (std::abs(cell->face(face)->center()[0] - 0.0) < 1e-6) cell->face(face)->set_boundary_id(1); - else if (std::abs(cell->face(face)->center()[0] - parameters.length) < 1e-6) + else if (std::abs(cell->face(face)->center()[0] - parameters.notched_cylinder.length) < 1e-6) cell->face(face)->set_boundary_id(2); else cell->face(face)->set_all_manifold_ids(1); // Cylindrical manifold @@ -947,6 +980,39 @@ namespace LMM triangulation.refine_global(parameters.n_global_refinement_steps); GridTools::scale (parameters.scale, triangulation); } + else if (parameters.geometry_type == "Tensile specimen") + { + const std::string filename = "tensile_specimen.inp"; + std::ifstream input (filename.c_str()); + + GridIn gridin; + gridin.attach_triangulation(triangulation); + gridin.read_abaqus(input, false); + +// GridTools::copy_boundary_to_manifold_id(triangulation,false); + for (typename Triangulation::active_cell_iterator + cell = triangulation.begin_active(); + cell!=triangulation.end(); ++cell) + { + for (unsigned int face=0; face < GeometryInfo::faces_per_cell; ++face) + { + if (cell->face(face)->at_boundary() == true) + { + if (cell->face(face)->boundary_id() == 10 || cell->face(face)->boundary_id() == 11) + cell->face(face)->set_all_manifold_ids(cell->face(face)->boundary_id()); + } + } + } + + GridTools::scale (parameters.scale, triangulation); + + CylindricalManifold cylindrical_manifold_id_10 (parameters.tensile_specimen.manifold_direction,parameters.tensile_specimen.centre_manifold_id_10*parameters.scale); + CylindricalManifold cylindrical_manifold_id_11 (parameters.tensile_specimen.manifold_direction,parameters.tensile_specimen.centre_manifold_id_11*parameters.scale); + triangulation.set_manifold (10, cylindrical_manifold_id_10); + triangulation.set_manifold (11, cylindrical_manifold_id_11); + + triangulation.refine_global(parameters.n_global_refinement_steps); + } else if (parameters.geometry_type == "Notched tensile specimen") { AssertThrow(false, ExcNotImplemented()); @@ -1101,28 +1167,32 @@ namespace LMM cell_matrix(J,I) = cell_matrix(I,J); cell->get_dof_indices (local_dof_indices); - for (unsigned int i=0; i - void LinearElastoplasticProblem::apply_boundary_conditions () + void LinearElastoplasticProblem::make_constraints () { - std::map boundary_values; + constraints.clear(); if (parameters.geometry_type == "Notched cylinder") { + // Fully constraints (pinned) vertex at centre of notched (-X) face + { + Point pinned_point; + typename DoFHandler::active_cell_iterator cell = + dof_handler.begin_active(), endc = dof_handler.end(); + for (; cell != endc; ++cell) + { + for (unsigned int v=0; v::vertices_per_cell; ++v) + if (cell->vertex(v).distance(pinned_point) < 1e-9) + { + for (unsigned int d=0; dvertex_dof_index(v,d)); + } + } + } // No X displacement on constraint on -X faces { ComponentMask component_mask_x (n_components, false); @@ -1130,7 +1200,7 @@ namespace LMM VectorTools::interpolate_boundary_values (dof_handler, 1, ZeroFunction(dim), - boundary_values, + constraints, component_mask_x); } // Prescribed horizontal displacement on +X faces @@ -1138,23 +1208,70 @@ namespace LMM ComponentMask component_mask_x (n_components, false); component_mask_x.set(0, true); const double total_displacement - = (parameters.final_stretch-1.0)*parameters.length + = (parameters.final_stretch-1.0)*parameters.notched_cylinder.length * parameters.scale * (time.current()/time.end()); VectorTools::interpolate_boundary_values (dof_handler, 2, ConstantFunction(total_displacement,dim), - boundary_values, + constraints, + component_mask_x); + } + } + else if (parameters.geometry_type == "Tensile specimen") + { + // No X displacement on constraint on -X faces + { + ComponentMask component_mask_x (n_components, false); + component_mask_x.set(0, true); + VectorTools::interpolate_boundary_values (dof_handler, + 1, + ZeroFunction(dim), + constraints, + component_mask_x); + } + // No Y displacement on constraint on -Y faces + { + ComponentMask component_mask_y (n_components, false); + component_mask_y.set(1, true); + VectorTools::interpolate_boundary_values (dof_handler, + 3, + ZeroFunction(dim), + constraints, + component_mask_y); + } + // No Z displacement on constraint on -Z faces + { + ComponentMask component_mask_z (n_components, false); + component_mask_z.set(2, true); + VectorTools::interpolate_boundary_values (dof_handler, + 5, + ZeroFunction(dim), + constraints, + component_mask_z); + } + // Prescribed horizontal displacement on clamped surface + { + ComponentMask component_mask_x (n_components, false); + component_mask_x.set(0, true); + const double total_displacement + = (parameters.final_stretch-1.0)*parameters.tensile_specimen.length + * parameters.scale + * (time.current()/time.end()); + VectorTools::interpolate_boundary_values (dof_handler, + 6, + ConstantFunction(total_displacement,dim), + constraints, component_mask_x); } - // No radial displacement of +X faces + // No out-of-plane displacement on clamped surface { ComponentMask component_mask_yz (n_components, true); component_mask_yz.set(0, false); VectorTools::interpolate_boundary_values (dof_handler, - 2, + 6, ZeroFunction(dim), - boundary_values, + constraints, component_mask_yz); } } @@ -1165,10 +1282,8 @@ namespace LMM else AssertThrow(false, ExcMessage("Unknown geometry")); - MatrixTools::apply_boundary_values (boundary_values, - system_matrix, - solution, - system_rhs); + constraints.merge(hanging_node_constraints, AffineConstraints::right_object_wins); + constraints.close(); } @@ -1186,7 +1301,7 @@ namespace LMM cg.solve (system_matrix, solution, system_rhs, preconditioner); - hanging_node_constraints.distribute (solution); + constraints.distribute(solution); } @@ -1228,7 +1343,6 @@ namespace LMM setup_system (); output_results (); -// const bool do_grid_refinement = false; time.increment(); while (time.current() < time.end()+0.01*time.get_delta_t()) { @@ -1239,12 +1353,8 @@ namespace LMM // Next we assemble the system and enforce boundary // conditions. - // Here we assume that the system and fibres have - // a fixed state, and we will assemble based on how - // epsilon_c will update given the current state of - // the body. + make_constraints(); assemble_system (); - apply_boundary_conditions (); // Then we solve the linear system solve (); diff --git a/Linear_Elastoplasticity/mesh/tensile_specimen.inp b/Linear_Elastoplasticity/mesh/tensile_specimen.inp new file mode 100755 index 00000000..8f037c58 --- /dev/null +++ b/Linear_Elastoplasticity/mesh/tensile_specimen.inp @@ -0,0 +1,422 @@ +*HEADING +cubit(ems/elasticity/tensile_specimen/mesh/tensile_specimen.inp): 10/23/2012: 16 +version: 13.2 +** +********************************** P A R T S ********************************** +*PART, NAME=Part-Default +** +********************************** N O D E S ********************************** +*NODE, NSET=ALLNODES + 1, 0.000000e+00, 1.000000e+00, 0.000000e+00 + 2, 0.000000e+00, 5.000000e-01, 0.000000e+00 + 3, 5.000000e-01, 5.000000e-01, 0.000000e+00 + 4, 5.000000e-01, 1.000000e+00, 0.000000e+00 + 5, 0.000000e+00, 1.000000e+00, 2.500000e-01 + 6, 0.000000e+00, 5.000000e-01, 2.500000e-01 + 7, 5.000000e-01, 5.000000e-01, 2.500000e-01 + 8, 5.000000e-01, 1.000000e+00, 2.500000e-01 + 9, 0.000000e+00, 0.000000e+00, 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ELSET=SS5_S5 + 43, 44, 47, 48, 53, 54, 55, 56, 61, 62, + 63, 64, 75, 76, 79, 80, 85, 86, 87, 88, + 93, 94, 95, 96 +*SURFACE, NAME=SS5 +SS5_S1, S1 +SS5_S5, S5 +*ELSET, ELSET=SS6_S3 + 81, 82, 83, 84, 89, 90, 91, 92 +*SURFACE, NAME=SS6 +SS6_S3, S3 +*ELSET, ELSET=SS10_S2 + 57, 58, 59, 60, 61, 62, 63, 64 +*SURFACE, NAME=SS10 +SS10_S2, S2 +*ELSET, ELSET=SS11_S4 + 66, 68, 70, 72 +*SURFACE, NAME=SS11 +SS11_S4, S4 +** +********************************** P R O P E R T I E S ************************ +*SOLID SECTION, ELSET=EB1, MATERIAL=Default-Steel +** +*END PART +** +** +** +********************************** E N D P A R T S ********************************** +** +** +********************************** A S S E M B L Y ************************************ +** +*ASSEMBLY, NAME=ASSEMBLY1 +** +*INSTANCE, NAME=Part-Default_1, PART=Part-Default +*END INSTANCE +** +*END ASSEMBLY +** +** +** +*MATERIAL, NAME = Default-Steel +*ELASTIC, TYPE=ISOTROPIC +2.068000e+05, 2.900000e-01 +*DENSITY +7.000000e-06 +*CONDUCTIVITY,TYPE=ISO +4.500000e-02 +*SPECIFIC HEAT +5.000000e+02 +** +** +************************************** H I S T O R Y ************************************* +** +*PREPRINT +** +**************************************** S T E P 1 *************************************** +*STEP,INC=100,NAME=Default Set +** +*STATIC +1, 1, 1e-05, 1 +** +** +** +** +*END STEP diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index d91c413f..eb9d2b50 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -11,31 +11,37 @@ end subsection Boundary conditions # The amount by which the specimen is to be stretched - set Final stretch = 1.5 + set Final stretch = 1.2 end subsection Geometry - # Overall length of the specimen - set Length = 100 - - # Overall radius of the specimen - set Radius = 10 - - # Overall length of the notch in the specimen - set Notch length = 1 - - # Overall radius of the notch in the specimen - set Notch radius = 9 + # Geometry to simulate + # Options: Notched cylinder ; Tensile specimen ; Notched tensile specimen + set Geometry type = Tensile specimen # Global grid scaling factor set Grid scale = 1e-3 # Number of global refinement steps - set Global refinement steps = 2 + set Global refinement steps = 1 # Number of initial local refinement cycles set Local refinement steps = 1 + + subsection Notched cylinder + # Overall length of the specimen + set Length = 100 + + # Overall radius of the specimen + set Radius = 10 + + # Overall length of the notch in the specimen + set Notch length = 1 + + # Overall radius of the notch in the specimen + set Notch radius = 9 +end end From 3648f29844b0dace3a5cac969f32496c3e0adf91 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 22 Jan 2019 14:41:14 +0100 Subject: [PATCH 09/20] TMP: Start to work on adding a data post-processor --- .../linear_elastoplasticity.cc | 46 +++++++++++++++++++ 1 file changed, 46 insertions(+) diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index f9e4b90b..eecfdf7a 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -52,6 +52,7 @@ #include #include #include +#include #include #include #include @@ -1004,6 +1005,8 @@ namespace LMM } } + // The triangulation scaling needs to happen before the manifolds + // with a non-trivial centre point are defined / applied. GridTools::scale (parameters.scale, triangulation); CylindricalManifold cylindrical_manifold_id_10 (parameters.tensile_specimen.manifold_direction,parameters.tensile_specimen.centre_manifold_id_10*parameters.scale); @@ -1307,6 +1310,47 @@ namespace LMM // @sect4{LinearElastoplasticProblem::output_results} + template + class ComputeIntensity : public DataPostprocessorScalar + { + public: + ComputeIntensity(); + virtual ~ComputeIntensity(); + virtual void evaluate_vector_field( + const DataPostprocessorInputs::Vector &inputs, + std::vector> &computed_quantities) const override; + }; + + template + ComputeIntensity::ComputeIntensity() + : DataPostprocessorScalar("Intensity", update_values) + {} + + template + ComputeIntensity::~ComputeIntensity() + {} + + template + void ComputeIntensity::evaluate_vector_field( + const DataPostprocessorInputs::Vector &inputs, + std::vector> & computed_quantities) const + { + Assert(computed_quantities.size() == inputs.solution_values.size(), + ExcDimensionMismatch(computed_quantities.size(), + inputs.solution_values.size())); + + for (unsigned int i = 0; i < computed_quantities.size(); i++) + { + Assert(computed_quantities[i].size() == 1, + ExcDimensionMismatch(computed_quantities[i].size(), 1)); + Assert(inputs.solution_values[i].size() == 2, + ExcDimensionMismatch(inputs.solution_values[i].size(), 2)); + computed_quantities[i](0) = std::sqrt( + inputs.solution_values[i](0) * inputs.solution_values[i](0) + + inputs.solution_values[i](1) * inputs.solution_values[i](1)); + } + } + template void LinearElastoplasticProblem::output_results () const @@ -1317,6 +1361,7 @@ namespace LMM filename += ".vtk"; std::ofstream output (filename.c_str()); + ComputeIntensity intensities; DataOut data_out; data_out.attach_dof_handler (dof_handler); @@ -1328,6 +1373,7 @@ namespace LMM data_out.add_data_vector (solution, solution_name, DataOut::type_dof_data, data_component_interpretation); + data_out.add_data_vector(solution, intensities); data_out.build_patches (); data_out.write_vtk (output); } From a605fe92433ae2a486ab7c7f18a3e521f32bda5a Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Sun, 30 Jun 2019 11:01:33 +0200 Subject: [PATCH 10/20] Update parameters file --- Linear_Elastoplasticity/parameters.prm | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index eb9d2b50..80701cfc 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -18,7 +18,7 @@ end subsection Geometry # Geometry to simulate # Options: Notched cylinder ; Tensile specimen ; Notched tensile specimen - set Geometry type = Tensile specimen + set Geometry type = Notched cylinder # Global grid scaling factor set Grid scale = 1e-3 @@ -41,14 +41,13 @@ subsection Geometry # Overall radius of the notch in the specimen set Notch radius = 9 -end + end end subsection Material properties # Type of material that composes the specimen # Options: Elastic ; Elastoplastic (isotropic hardening) - #set Type = Elastic set Type = Elastoplastic (isotropic hardening) # Shear modulus From 26c2cb9d4a920d47f47daf5fa0851bb43e269d20 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Sun, 30 Jun 2019 12:03:45 +0200 Subject: [PATCH 11/20] Added grid refinement --- .../linear_elastoplasticity.cc | 224 ++++++++++++++---- 1 file changed, 174 insertions(+), 50 deletions(-) diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index eecfdf7a..edbdbb63 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -24,11 +24,13 @@ #include #include #include +#include #include #include #include #include #include +#include #include #include #include @@ -528,7 +530,7 @@ namespace LMM {} protected: - const double kappa_e; // buld modulus + const double kappa_e; // bulk modulus const double mu_e; // shear modulus }; @@ -659,19 +661,17 @@ namespace LMM void update_end_timestep() override { - // For the linear problem, it is easier to do the update - // between the previous and the current timestep in the - // call to update_internal_equilibrium(). For a nonlinear - // problem, it is more useful to employ this function to - // accept the accumulation of incremental values in the - // context of a Newton scheme. + // Record history of plastic variables + epsilon_p_t1 = epsilon_p_t; + alpha_p_t1 = alpha_p_t; } // --- Post-processing --- - // Plastic variables + // Plastic variables: + // Isotropic hardening variable double - get_alpha_p() + get_alpha_p() const { return alpha_p_t; } @@ -769,6 +769,12 @@ namespace LMM { material->update_end_timestep(); } + const Material_Base * const + get_material() const + { + Assert(material, ExcInternalError()); + return &(*material); + } private: std::shared_ptr< Material_Base > material; }; @@ -786,12 +792,14 @@ namespace LMM private: void make_grid (); + void refine_grid (); void setup_system (); void setup_qph(); void make_constraints (); void assemble_system (); void solve (); void output_results () const; + void update_end_timestep (); Parameters::AllParameters parameters; @@ -814,7 +822,7 @@ namespace LMM Vector system_rhs; // Local data - CellDataStorage::cell_iterator, + CellDataStorage::active_cell_iterator, PointHistory > quadrature_point_history; const FEValuesExtractors::Vector u_fe; @@ -1036,6 +1044,25 @@ namespace LMM } } + // @sect4{LinearElastoplasticProblem::refine_grid} + + template + void LinearElastoplasticProblem::refine_grid () + { + Vector estimated_error_per_cell(triangulation.n_active_cells()); + KellyErrorEstimator::estimate( + dof_handler, + QGauss(fe.degree + 1), + std::map *>(), + solution, + estimated_error_per_cell); + GridRefinement::refine_and_coarsen_fixed_number(triangulation, + estimated_error_per_cell, + 0.3, + 0.03); + triangulation.execute_coarsening_and_refinement(); + } + // @sect4{LinearElastoplasticProblem::setup_system} template @@ -1046,17 +1073,15 @@ namespace LMM DoFTools::make_hanging_node_constraints (dof_handler, hanging_node_constraints); hanging_node_constraints.close (); - sparsity_pattern.reinit (dof_handler.n_dofs(), - dof_handler.n_dofs(), - dof_handler.max_couplings_between_dofs()); - DoFTools::make_sparsity_pattern (dof_handler, sparsity_pattern); - hanging_node_constraints.condense (sparsity_pattern); - - sparsity_pattern.compress(); + DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs()); + DoFTools::make_sparsity_pattern(dof_handler, + dsp, + hanging_node_constraints, + /*keep_constrained_dofs = */ true); + sparsity_pattern.copy_from(dsp); system_matrix.reinit (sparsity_pattern); - solution.reinit (dof_handler.n_dofs()); system_rhs.reinit (dof_handler.n_dofs()); @@ -1308,47 +1333,118 @@ namespace LMM } + template + void LinearElastoplasticProblem::update_end_timestep () + { + const unsigned int n_q_points_cell = qf_cell.size(); + + for (typename DoFHandler::active_cell_iterator + cell = dof_handler.begin_active(); + cell!=dof_handler.end(); ++cell) + { + const std::vector > > lqph = + quadrature_point_history.get_data(cell); + + for (unsigned int q_point_cell=0; q_point_cellupdate_end_timestep(); + } + } + + // @sect4{LinearElastoplasticProblem::output_results} template - class ComputeIntensity : public DataPostprocessorScalar + class PostProcessIsotropicHardening : public DataPostprocessorScalar { public: - ComputeIntensity(); - virtual ~ComputeIntensity(); + PostProcessIsotropicHardening(const CellDataStorage::active_cell_iterator, + PointHistory > &quadrature_point_history, + const QGauss &qf_cell); + virtual ~PostProcessIsotropicHardening() = default; virtual void evaluate_vector_field( const DataPostprocessorInputs::Vector &inputs, std::vector> &computed_quantities) const override; - }; - template - ComputeIntensity::ComputeIntensity() - : DataPostprocessorScalar("Intensity", update_values) - {} + private: + const CellDataStorage::active_cell_iterator, + PointHistory > &quadrature_point_history; + + FullMatrix projection_matrix_qpoint_to_support_point; + }; template - ComputeIntensity::~ComputeIntensity() - {} + PostProcessIsotropicHardening::PostProcessIsotropicHardening( + const CellDataStorage::active_cell_iterator, + PointHistory > &quadrature_point_history, + const QGauss &quadrature_cell) + : DataPostprocessorScalar("isotropic_hardening", update_values), + quadrature_point_history (quadrature_point_history) + { + const FE_Q fe_projection (1); + const QTrapez quadrature_projection; + projection_matrix_qpoint_to_support_point = FullMatrix ( + quadrature_projection.size(), + quadrature_cell.size()); + + FETools::compute_projection_from_quadrature_points_matrix + (fe_projection, + quadrature_projection, + quadrature_cell, + projection_matrix_qpoint_to_support_point); + } template - void ComputeIntensity::evaluate_vector_field( + void PostProcessIsotropicHardening::evaluate_vector_field( const DataPostprocessorInputs::Vector &inputs, std::vector> & computed_quantities) const { Assert(computed_quantities.size() == inputs.solution_values.size(), ExcDimensionMismatch(computed_quantities.size(), - inputs.solution_values.size())); + inputs.solution_values.size())); - for (unsigned int i = 0; i < computed_quantities.size(); i++) - { - Assert(computed_quantities[i].size() == 1, - ExcDimensionMismatch(computed_quantities[i].size(), 1)); - Assert(inputs.solution_values[i].size() == 2, - ExcDimensionMismatch(inputs.solution_values[i].size(), 2)); - computed_quantities[i](0) = std::sqrt( - inputs.solution_values[i](0) * inputs.solution_values[i](0) + - inputs.solution_values[i](1) * inputs.solution_values[i](1)); - } + const typename DoFHandler::active_cell_iterator cell + = inputs.template get_cell>(); + + // number of support points (nodes) to project to + const unsigned int n_support_points = projection_matrix_qpoint_to_support_point.n_rows(); + // number of quadrature points to project from + const unsigned int n_quad_points = projection_matrix_qpoint_to_support_point.n_cols(); + + // component projected to the nodes + Vector component_at_node(n_support_points); + // component at the quadrature point + Vector component_at_qp(n_quad_points); + + const std::vector > > lqph = + quadrature_point_history.get_data(cell); + Assert(lqph.size() == n_quad_points, ExcDimensionMismatch(lqph.size(), n_quad_points)); + + // populate the vector of components at the qps + for (unsigned int q_point = 0; q_point < n_quad_points; ++q_point) + { + const Material_Base * const material = lqph[q_point]->get_material(); + if (const Material_Linear_Elastoplastic_Isotropic_Hardening* const material_ep + = dynamic_cast* const>(material)) + component_at_qp(q_point) = material_ep->get_alpha_p(); + else + component_at_qp(q_point) = 0.0; + } + + // project from the qps -> nodes + // component_at_node = projection_matrix_u * component_at_qp + projection_matrix_qpoint_to_support_point.vmult(component_at_node, component_at_qp); + + Assert(computed_quantities.size() == n_support_points, + ExcDimensionMismatch(computed_quantities.size(), + n_support_points)); + for (unsigned int i = 0; i < n_support_points; i++) + { + Assert(inputs.solution_values[i].size() == dim, + ExcDimensionMismatch(inputs.solution_values[i].size(), dim)); + (void)inputs; + + computed_quantities[i](0) = component_at_node(i); + } } @@ -1361,7 +1457,7 @@ namespace LMM filename += ".vtk"; std::ofstream output (filename.c_str()); - ComputeIntensity intensities; + PostProcessIsotropicHardening pp_isotropic_hardening (quadrature_point_history, qf_cell); DataOut data_out; data_out.attach_dof_handler (dof_handler); @@ -1373,8 +1469,12 @@ namespace LMM data_out.add_data_vector (solution, solution_name, DataOut::type_dof_data, data_component_interpretation); - data_out.add_data_vector(solution, intensities); - data_out.build_patches (); + data_out.add_data_vector(solution, pp_isotropic_hardening); + + DataOutBase::VtkFlags flags; + flags.write_higher_order_cells = true; + data_out.set_flags (flags); + data_out.build_patches (StaticMappingQ1::mapping, fe.degree, DataOut::curved_inner_cells); data_out.write_vtk (output); } @@ -1389,7 +1489,20 @@ namespace LMM setup_system (); output_results (); + update_end_timestep(); time.increment(); + + auto solve_timestep = [&]() + { + // Next we assemble the system and enforce boundary + // conditions. + make_constraints(); + assemble_system (); + + // Then we solve the linear system + solve (); + }; + while (time.current() < time.end()+0.01*time.get_delta_t()) { std::cout @@ -1397,17 +1510,26 @@ namespace LMM << " @ time " << time.current() << std::endl; - // Next we assemble the system and enforce boundary - // conditions. - make_constraints(); - assemble_system (); + // Compute the solution at the current timestep + solve_timestep(); - // Then we solve the linear system - solve (); + // Perform local refinement + if (time.get_timestep() == 1) + for (unsigned int cycle = 0; cycle < parameters.n_local_refinement_steps; ++cycle) + { + std::cout + << "Executing refinement cycle " << cycle + << " of " << parameters.n_local_refinement_steps + << "..." << std::endl; + refine_grid(); + setup_system(); + solve_timestep(); + } // Output some values to file output_results (); + update_end_timestep(); time.increment(); } } @@ -1415,13 +1537,15 @@ namespace LMM // @sect3{The main function} -int main () +int main (int argc, char *argv[]) { try { dealii::deallog.depth_console (0); const unsigned int dim = 3; + dealii::Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1); + LMM::LinearElastoplasticProblem elastoplastic_problem ("parameters.prm"); elastoplastic_problem.run (); } From b189e4c5b3a2532d321bf1dfd253de556aaa7bda Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Sun, 30 Jun 2019 12:04:35 +0200 Subject: [PATCH 12/20] Fixed title of theory tex file --- .../doc/theory/theory-linear_elastoplasticity.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex index a2f492d7..139d4bae 100644 --- a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex +++ b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex @@ -10,7 +10,7 @@ \usepackage{filecontents} \usepackage[sort]{natbib} -\title{Theory: Linear elastic active muscle model} +\title{Theory: Linear elastoplasicity} \author{Jean-Paul Pelteret} \begin{filecontents}{\jobname.bib} From 2ab9f89a17943bbde4adc90ac4e8d86a5abbc41c Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Sun, 30 Jun 2019 12:05:29 +0200 Subject: [PATCH 13/20] Add README.md --- Linear_Elastoplasticity/README.md | 4 ++++ 1 file changed, 4 insertions(+) create mode 100644 Linear_Elastoplasticity/README.md diff --git a/Linear_Elastoplasticity/README.md b/Linear_Elastoplasticity/README.md new file mode 100644 index 00000000..679d241c --- /dev/null +++ b/Linear_Elastoplasticity/README.md @@ -0,0 +1,4 @@ +## Overview +Linear elasticity with linear isotropic hardening. +Solved on a notched cylindrical specimen or a tensile specimen. + From e6d346bf2ac2d264294ce545173dafa8f72eedb8 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 26 May 2020 07:19:51 +0200 Subject: [PATCH 14/20] Updates --- Linear_Elastoplasticity/CMakeLists.txt | 2 +- .../theory/theory-linear_elastoplasticity.bib | 21 +++++ .../theory/theory-linear_elastoplasticity.tex | 67 +++++++-------- .../linear_elastoplasticity.cc | 86 +++++++++++-------- Linear_Elastoplasticity/parameters.prm | 2 +- 5 files changed, 104 insertions(+), 74 deletions(-) create mode 100644 Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.bib diff --git a/Linear_Elastoplasticity/CMakeLists.txt b/Linear_Elastoplasticity/CMakeLists.txt index 8aa14a3b..44168d3d 100644 --- a/Linear_Elastoplasticity/CMakeLists.txt +++ b/Linear_Elastoplasticity/CMakeLists.txt @@ -23,7 +23,7 @@ SET(TARGET_SRC CMAKE_MINIMUM_REQUIRED(VERSION 2.8.8) -FIND_PACKAGE(deal.II 8.5 QUIET +FIND_PACKAGE(deal.II 9.1 QUIET HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR} ) IF(NOT ${deal.II_FOUND}) diff --git a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.bib b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.bib new file mode 100644 index 00000000..024e7d4b --- /dev/null +++ b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.bib @@ -0,0 +1,21 @@ +%% LaTeX2e file `theory-linear_elastoplasticity.bib' +%% generated by the `filecontents' environment +%% from source `theory-linear_elastoplasticity' on 2020/05/25. +%% +@Misc{Mergheim2018a, + author = {Julia Mergheim and Jean-Paul Pelteret and Ester Comellas}, + title = {Lecture notes: Material Modelling and Simulation}, + year = {2018}, + publisher = {Friedrich--Alexander Uniersit\"{a}t Erlangen--N\"{u}rnberg} +} +@Book{Simo2006a, + title = {Computational Inelasticity}, + publisher = {Springer Science \& Business Media}, + year = {2000}, + author = {Simo, J. C. and Hughes, Thomas J. R.}, + editor = {Marsden, J. E. and Sirovich, L. and Wiggins, S.}, + volume = {7}, + series = {Interdisciplinary Applied Mathematics}, + isbn = {978-0-387-97520-7}, + doi = {10.1007/b98904}, +} diff --git a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex index 139d4bae..bea7a5d1 100644 --- a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex +++ b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.tex @@ -243,10 +243,10 @@ \section{General framework for elastoplasticity} \end{easylist} \clearpage -\section{Constitutive law: Hyperelasticity} +\section{Constitutive law: Linear elasticity} \begin{easylist}[itemize] -# Hyperelasticity +# Linear elasticity (a special case of hyperelasticity) ## Dissipation (in-)equality \begin{gather} \mathcal{D} @@ -492,12 +492,15 @@ \subsection{Elasto-plastic tangent modulus (continuous setting)} = \underbrace{\left[ \boldsymbol{\mathcal{C}} - \frac{1}{D} \left[ \boldsymbol{\mathcal{C}} : \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} \right] \otimes \left[ \frac{\partial \Phi}{\partial \boldsymbol{\sigma}} : \boldsymbol{\mathcal{C}} \right] \right]}_{\boldsymbol{\mathcal{C}}^{ep} \textnormal{ (symmetric)}} : \dot{\boldsymbol{\varepsilon}} \end{gather*} # Result for isotropic hardening -\begin{align*} +\begin{subequations} +\begin{align} \boldsymbol{\mathcal{C}}^{ep, dev} &= 2 \mu \boldsymbol{\mathcal{I}}^{dev} - \frac{\left[ 2\mu \right]^{2}}{2\mu + \frac{2}{3}k} \boldsymbol{n} \otimes \boldsymbol{n} \\ \boldsymbol{\mathcal{C}}^{ep, vol} &= \kappa \boldsymbol{1} \otimes \boldsymbol{1} -\end{align*} + \label{equ: Constutitve law: Algorithmically correct tangent: Volumetric} +\end{align} +\end{subequations} with \begin{gather*} \boldsymbol{n} @@ -648,11 +651,12 @@ \subsection{Consistent elastoplastic tangent modulus for von Mises plasticity wi \begin{easylist} # Note: The definition of the consistent elastoplastic tangent depends on the algorithms used to update stresses, internal variables # Goal: Tangent to compute -\begin{gather*} +\begin{gather} \boldsymbol{\mathcal{C}}^{ep} = \frac{\partial \boldsymbol{\sigma}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} = \boldsymbol{\mathcal{C}}^{ep, vol} + \boldsymbol{\mathcal{C}}^{ep, dev} -\end{gather*} +\label{equ: Constutitve law: Algorithmically correct tangent} +\end{gather} # Stress tensors \begin{gather*} \boldsymbol{\sigma}_{n+1} @@ -708,7 +712,7 @@ \subsection{Consistent elastoplastic tangent modulus for von Mises plasticity wi - \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \right] \end{align*} ## Resulting deviatoric part of the elasto-plastic tangent -\begin{align*} +\begin{align} \boldsymbol{\mathcal{C}}^{ep, dev} &= \frac{\partial \boldsymbol{\sigma}^{dev}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} = \frac{\partial \boldsymbol{\sigma}^{dev, trial}_{n+1}}{\partial \boldsymbol{\varepsilon}_{n+1}} @@ -723,7 +727,8 @@ \subsection{Consistent elastoplastic tangent modulus for von Mises plasticity wi &= 2 \mu \boldsymbol{\mathcal{I}}^{dev} - \frac{ \left[2 \mu \right]^{2}}{2 \mu + \frac{2}{3} k} \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \\ & + \left[\Delta \lambda_{n+1} \frac{\left[ 2 \mu \right]^{2}}{{\Vert \boldsymbol{\sigma}^{dev, trial}_{n+1} \Vert}} \left[ \boldsymbol{\mathcal{I}}^{dev}_{n+1} - \boldsymbol{n}_{n+1} \otimes \boldsymbol{n}_{n+1} \right] \right] -\end{align*} + \label{equ: Constutitve law: Algorithmically correct tangent: Deviatoric} +\end{align} ## Compared to continuous case, the tangents are not equal but they converge to the same expression when $\Delta \lambda_{n+1} \rightarrow 0$ \end{easylist} @@ -734,7 +739,7 @@ \section{Finite element discretisation For clarity, it is worth mentioning up front that in this \namecref{sec: FE discretisation} we will denote the general elasto-plastic tangent by $\boldsymbol{\mathbb{C}}^{ep}$; so, in the elastic region $\boldsymbol{\mathbb{C}}^{ep} \equiv \boldsymbol{\mathbb{C}}$ while in the plastic region $\boldsymbol{\mathbb{C}}^{ep}$ represents the consistent tangent. -Combining \cref{equ: Weak form of the balance of linear momentum,equ: Full expansion of constitutive laws} renders the complete expression of the balance of linear momentum, with accommodation of the muscle fibre model, namely +Combining \cref{equ: Weak form of the balance of linear momentum,equ: Constutitve law: Algorithmically correct tangent} (specifically, \cref{equ: Constutitve law: Algorithmically correct tangent: Volumetric,equ: Constutitve law: Algorithmically correct tangent: Deviatoric}) renders the complete expression of the balance of linear momentum, namely \begin{gather} \int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \left[ \boldsymbol{\mathbb{C}}^{ep} \right]_{ijkl} \varepsilon_{kl} \, dv = \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv @@ -743,43 +748,26 @@ \section{Finite element discretisation \label{equ: Weak form of the balance of linear momentum: Muscle model} \end{gather} -We discretise the trial solution and test function using finite element shape functions (ansatz) +We discretise the trial solution and test function using vector-valued finite element shape functions (ansatz) \begin{gather} \mathbf{u} \left( \mathbf{x} \right) - \approx \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) u^{I} + \approx \sum\limits_{I} \mathbf{N}^{I} \left( \mathbf{x} \right) u^{I} \quad , \quad \mathbf{v} \left( \mathbf{x} \right) - \approx \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) v^{I} + \approx \sum\limits_{I} \mathbf{N}^{I} \left( \mathbf{x} \right) v^{I} \end{gather} where $\mathbf{N}^{I} \left( \mathbf{x} \right)$ is the (position-dependent) vector-valued finite element shape function corresponding to the $I^{\textnormal{th}}$ degree-of-freedom, and $u^{I}, v^{I}$ are coefficients of the solution and trial function. -In \texttt{deal.II} nomenclature, the shape function is computed from a scalar base shape function and some expansion into higher-dimensional space by -\begin{gather} -\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) - = N^{I} \left( \mathbf{x} \right) \mathbf{e}_{\textnormal{comp}(I)} -\end{gather} -where $N^{I}$ is a scalar shape function and $\mathbf{e}_{\textnormal{comp}(I)}$ is the basis direction associated with the $I^{\textnormal{th}}$ degree-of-freedom. -Therefore, the $j^{\textnormal{th}}$ local component of shape function $\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right)$ is given by -\begin{gather} -\left[\boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right)\right]_{j} - = N^{I} \left( \mathbf{x} \right) \left[ \mathbf{e}_{\textnormal{comp}(I)} \right]_{j} - = N^{I} \left( \mathbf{x} \right) \delta_{\textnormal{comp}(I) j} -\quad . -\end{gather} -where $\delta_{ij}$ is the Kronecker delta. -Note that in this instance we use the same ansatz for the test and trial spaces, and the $0 \leq \textnormal{comp}(I), j < \textnormal{spacedim}$. We now use these shape functions to discretise the weak expression for the balance of linear momentum. Starting on the right-hand side of \cref{equ: Weak form of the balance of linear momentum: Muscle model}, the body force and traction contributions are computed by \begin{align} \int\limits_{\Omega} \delta v_{i} \, b_{i} \, dv - &= \int\limits_{\Omega} \left[ \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, b_{i} \, dv - = \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} \left[ \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \right]_{i} \, b_{i} \, dv \notag\\ - &= \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \left( \mathbf{x} \right) \delta_{\textnormal{comp}(I) i} \, b_{i} \, dv - = \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \, b_{\textnormal{comp}(I)} \, dv + &= \int\limits_{\Omega} \left[ \sum\limits_{I} \mathbf{N}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, b_{i} \, dv + = \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} \mathbf{N}^{I}_{i} \, b_{i} \, dv \label{equ: Discretisation: Body force} \\ \int\limits_{\Omega} \delta v_{i} \, t_{i} \, dv - &= \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} N^{I} \, t_{\textnormal{comp}(I)} \, dv + &= \sum\limits_{I} \delta v^{I} \int\limits_{\Omega} \mathbf{N}^{I}_{i} \, t_{i} \, dv \label{equ: Discretisation: Traction} \quad . \end{align} @@ -790,19 +778,24 @@ \section{Finite element discretisation = \boldsymbol{\mathbb{C}} : \frac{1}{2} \left[ \nabla \mathbf{u} + \left[ \nabla \mathbf{u} \right]^{T} \right] \equiv \boldsymbol{\mathbb{C}} : \nabla \mathbf{u} \end{gather} +from which we can similarly deduce that +\begin{gather} +\nabla \mathbf{u} : \boldsymbol{\mathbb{C}} + \equiv \boldsymbol{\varepsilon} : \boldsymbol{\mathbb{C}} +\end{gather} Therefore, this contribution written in discrete form is \begin{align} &\int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \left[ \boldsymbol{\mathbb{C}}^{ep} \right]_{ijkl} \varepsilon_{kl} \, dv \equiv \int\limits_{\Omega} \frac{\partial \delta v_{i}}{\partial x_{j}} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial \delta u_{k}}{\partial x_{l}} \, dv \notag\\ - &\equiv \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \sum\limits_{I} \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \sum\limits_{J} \boldsymbol{\varPhi}^{J} \left( \mathbf{x} \right) \delta u^{J} \right]_{k} \, dv \notag\\ - &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \boldsymbol{\varPhi}^{I} \left( \mathbf{x} \right) \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \boldsymbol{\varPhi}^{J} \left( \mathbf{x} \right) \right]_{k} \, dv \right] \delta u^{J} \notag\\ - &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial N^{I} \left( \mathbf{x} \right)}{\partial x_{j}} \delta_{\textnormal{comp}(I) i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial N^{J} \left( \mathbf{x} \right)}{\partial x_{l}} \delta_{\textnormal{comp}(J) k} \, dv \right] \delta u^{J} \notag\\ - &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial N^{I} \left( \mathbf{x} \right)}{\partial x_{j}} \, \mathbb{C}^{ep}_{\textnormal{comp}(I) \,j \, \textnormal{comp}(J) \, l} \, \frac{\partial N^{J} \left( \mathbf{x} \right)}{\partial x_{l}} \, dv \right] \delta u^{J} + &= \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \sum\limits_{I} \mathbf{N}^{I} \left( \mathbf{x} \right) \delta v^{I} \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \sum\limits_{J} \mathbf{N}^{J} \left( \mathbf{x} \right) \delta u^{J} \right]_{k} \, dv \notag\\ + &= \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \frac{\partial}{\partial x_{j}} \left[ \mathbf{N}^{I} \left( \mathbf{x} \right) \right]_{i} \, \mathbb{C}^{ep}_{ijkl} \, \frac{\partial}{\partial x_{l}} \left[ \mathbf{N}^{J} \left( \mathbf{x} \right) \right]_{k} \, dv \right] \delta u^{J} \notag\\ + &= \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \left[ \mathbf{N}^{I} \left( \mathbf{x} \right) \right]_{i,j} \, \mathbb{C}^{ep}_{ijkl} \, \left[ \mathbf{N}^{J} \left( \mathbf{x} \right) \right]_{k,l} \, dv \right] \delta u^{J} \notag\\ + &\equiv \sum\limits_{I,J} \delta v^{I} \left[ \int\limits_{\Omega} \left[\left[ \mathbf{N}^{I} \left( \mathbf{x} \right) \right]_{i,j} \right]^{S} \, \mathbb{C}^{ep}_{ijkl} \, \left[\left[ \mathbf{N}^{J} \left( \mathbf{x} \right) \right]_{k,l}\right]^{S} \, dv \right] \delta u^{J} \label{equ: Discretisation: Material tangent} \quad . \end{align} -\Cref{equ: Discretisation: Body force,equ: Discretisation: Traction,equ: Discretisation: Stress history,equ: Discretisation: Material tangent} are collectively used to develop the system of linear equations that are solved at each time step. +\Cref{equ: Discretisation: Body force,equ: Discretisation: Traction,equ: Discretisation: Material tangent} are collectively used to develop the system of linear equations that are solved at each time step. \bibliographystyle{plain} \bibliography{\jobname} diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index edbdbb63..b3896b82 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -19,6 +19,7 @@ * University of Erlangen-Nuremberg, 2018 */ + // @sect3{Include files} #include @@ -64,16 +65,18 @@ #include #include -namespace LMM +namespace LinearElastoplasticity { using namespace dealii; + // @sect3{Run-time parameters} // // There are several parameters that can be set in the code so we set up a // ParameterHandler object to read in the choices at run-time. namespace Parameters { + // @sect4{Finite Element system} // Here we specify the polynomial order used to approximate the solution. @@ -115,9 +118,10 @@ namespace LMM prm.leave_subsection(); } - // @sect4{Boundary conditions} - // The boundary conditions to be applied to the problem. +// @sect4{Boundary conditions} + +// The boundary conditions to be applied to the problem. struct BoundaryConditions { @@ -150,6 +154,7 @@ namespace LMM prm.leave_subsection(); } + // @sect4{Geometry} // Make adjustments to the geometry of the notched specimen. @@ -190,7 +195,7 @@ namespace LMM prm.enter_subsection("Geometry"); { prm.declare_entry("Geometry type", "Notched cylinder", - Patterns::Selection("Notched cylinder|Tensile specimen|Notched tensile specimen"), + Patterns::Selection("Notched cylinder|Tensile specimen"), "The geometry to be modelled"); prm.declare_entry("Grid scale", "1e-3", @@ -254,9 +259,10 @@ namespace LMM ExcMessage("Cannot create geometry with given dimensions")); } - // @sect4{Material properties} - // Make adjustments to the material of the notched specimen. +// @sect4{Material properties} + +// Make adjustments to the material of the notched specimen. struct MaterialProperties { @@ -314,7 +320,6 @@ namespace LMM } - // @sect4{Time} // Set the timestep size $ \varDelta t $ and the simulation end-time. @@ -355,6 +360,7 @@ namespace LMM prm.leave_subsection(); } + // @sect4{All parameters} // Finally we consolidate all of the above structures into a single container @@ -401,7 +407,8 @@ namespace LMM } } - // @sect3{Time} + +// @sect3{Time incrementation} class Time { @@ -445,6 +452,8 @@ namespace LMM }; +// @sect3{Constitutive laws} + template class Material_Base { @@ -720,6 +729,8 @@ namespace LMM }; +// @sect3{Quadrature point data} + template class PointHistory { @@ -780,7 +791,7 @@ namespace LMM }; - // @sect3{The LinearElastoplasticProblem class template} +// @sect3{The LinearElastoplasticProblem class template} template class LinearElastoplasticProblem @@ -830,7 +841,8 @@ namespace LMM static const unsigned int first_u_component = 0; }; - // @sect4{LinearElastoplasticProblem::LinearElastoplasticProblem} + +// @sect4{LinearElastoplasticProblem::LinearElastoplasticProblem} template LinearElastoplasticProblem::LinearElastoplasticProblem (const std::string &input_file) @@ -847,7 +859,7 @@ namespace LMM } - // @sect4{LinearElastoplasticProblem::~LinearElastoplasticProblem} +// @sect4{LinearElastoplasticProblem::~LinearElastoplasticProblem} template LinearElastoplasticProblem::~LinearElastoplasticProblem () @@ -856,7 +868,7 @@ namespace LMM } - // @sect4{LinearElastoplasticProblem::make_grid} +// @sect4{LinearElastoplasticProblem::make_grid} template void LinearElastoplasticProblem::make_grid () @@ -958,7 +970,9 @@ namespace LMM rad[0] = 0.0; const double radius = rad.norm(); if (radius > radius_for_cyl_cells) + { cell->set_all_manifold_ids(1); // Cylindrical manifold + } } for (typename Triangulation::active_cell_iterator cell = triangulation.begin_active(); @@ -1024,10 +1038,6 @@ namespace LMM triangulation.refine_global(parameters.n_global_refinement_steps); } - else if (parameters.geometry_type == "Notched tensile specimen") - { - AssertThrow(false, ExcNotImplemented()); - } else AssertThrow(false, ExcMessage("Unknown geometry")); @@ -1044,7 +1054,8 @@ namespace LMM } } - // @sect4{LinearElastoplasticProblem::refine_grid} + +// @sect4{LinearElastoplasticProblem::refine_grid} template void LinearElastoplasticProblem::refine_grid () @@ -1063,7 +1074,8 @@ namespace LMM triangulation.execute_coarsening_and_refinement(); } - // @sect4{LinearElastoplasticProblem::setup_system} + +// @sect4{LinearElastoplasticProblem::setup_system} template void LinearElastoplasticProblem::setup_system () @@ -1098,6 +1110,8 @@ namespace LMM } +// @sect4{LinearElastoplasticProblem::setup_qph} + template void LinearElastoplasticProblem::setup_qph() { @@ -1125,20 +1139,15 @@ namespace LMM template void LinearElastoplasticProblem::assemble_system () { - // Reset system system_matrix = 0; system_rhs = 0; FEValues fe_values (fe, qf_cell, update_values | update_gradients | update_quadrature_points | update_JxW_values); - FEFaceValues fe_face_values (fe, qf_face, - update_values | - update_quadrature_points | update_JxW_values); const unsigned int dofs_per_cell = fe.dofs_per_cell; const unsigned int n_q_points_cell = qf_cell.size(); - // const unsigned int n_q_points_face = qf_face.size(); FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); Vector cell_rhs (dofs_per_cell); @@ -1199,6 +1208,9 @@ namespace LMM } } + +// @sect4{LinearElastoplasticProblem::make_constraints} + template void LinearElastoplasticProblem::make_constraints () { @@ -1303,10 +1315,6 @@ namespace LMM component_mask_yz); } } - else if (parameters.geometry_type == "Notched tensile specimen") - { - AssertThrow(false, ExcNotImplemented()); - } else AssertThrow(false, ExcMessage("Unknown geometry")); @@ -1315,7 +1323,7 @@ namespace LMM } - // @sect4{LinearElastoplasticProblem::solve} +// @sect4{LinearElastoplasticProblem::solve} template void LinearElastoplasticProblem::solve () @@ -1333,6 +1341,8 @@ namespace LMM } +// @sect4{LinearElastoplasticProblem::update_end_timestep} + template void LinearElastoplasticProblem::update_end_timestep () { @@ -1372,6 +1382,7 @@ namespace LMM FullMatrix projection_matrix_qpoint_to_support_point; }; + template PostProcessIsotropicHardening::PostProcessIsotropicHardening( const CellDataStorage::active_cell_iterator, @@ -1393,6 +1404,7 @@ namespace LMM projection_matrix_qpoint_to_support_point); } + template void PostProcessIsotropicHardening::evaluate_vector_field( const DataPostprocessorInputs::Vector &inputs, @@ -1451,10 +1463,9 @@ namespace LMM template void LinearElastoplasticProblem::output_results () const { - // Visual output: FEM results std::string filename = "solution-"; filename += Utilities::int_to_string(time.get_timestep(),4); - filename += ".vtk"; + filename += ".vtu"; std::ofstream output (filename.c_str()); PostProcessIsotropicHardening pp_isotropic_hardening (quadrature_point_history, qf_cell); @@ -1474,10 +1485,14 @@ namespace LMM DataOutBase::VtkFlags flags; flags.write_higher_order_cells = true; data_out.set_flags (flags); - data_out.build_patches (StaticMappingQ1::mapping, fe.degree, DataOut::curved_inner_cells); - data_out.write_vtk (output); - } + data_out.build_patches (StaticMappingQ1::mapping, std::max(fe.degree,2u), DataOut::curved_inner_cells); + data_out.write_vtu (output); + static std::vector< std::pair< double, std::string >> times_and_names; + times_and_names.emplace_back (time.current(), filename); + std::ofstream pvd_output ("solution.pvd"); + DataOutBase::write_pvd_record (pvd_output, times_and_names); + } // @sect4{LinearElastoplasticProblem::run} @@ -1533,7 +1548,8 @@ namespace LMM time.increment(); } } -} +} // namespace LinearElastoplasticity + // @sect3{The main function} @@ -1546,7 +1562,7 @@ int main (int argc, char *argv[]) dealii::Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1); - LMM::LinearElastoplasticProblem elastoplastic_problem ("parameters.prm"); + LinearElastoplasticity::LinearElastoplasticProblem elastoplastic_problem ("parameters.prm"); elastoplastic_problem.run (); } catch (std::exception &exc) diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index 80701cfc..e8a4e8c6 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -17,7 +17,7 @@ end subsection Geometry # Geometry to simulate - # Options: Notched cylinder ; Tensile specimen ; Notched tensile specimen + # Options: Notched cylinder ; Tensile specimen set Geometry type = Notched cylinder # Global grid scaling factor From a3806739d62b7c4a98f0354be35bcb5cb5fb897f Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 26 May 2020 22:26:31 +0200 Subject: [PATCH 15/20] Fix bug with PP output --- Linear_Elastoplasticity/linear_elastoplasticity.cc | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index b3896b82..38d5a941 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -1485,7 +1485,7 @@ namespace LinearElastoplasticity DataOutBase::VtkFlags flags; flags.write_higher_order_cells = true; data_out.set_flags (flags); - data_out.build_patches (StaticMappingQ1::mapping, std::max(fe.degree,2u), DataOut::curved_inner_cells); + data_out.build_patches (StaticMappingQ1::mapping, fe.degree, DataOut::curved_inner_cells); data_out.write_vtu (output); static std::vector< std::pair< double, std::string >> times_and_names; @@ -1560,7 +1560,7 @@ int main (int argc, char *argv[]) dealii::deallog.depth_console (0); const unsigned int dim = 3; - dealii::Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1); + dealii::Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv); LinearElastoplasticity::LinearElastoplasticProblem elastoplastic_problem ("parameters.prm"); elastoplastic_problem.run (); From c2485a46d1b6b559e06ec8c97f489ee5012a386c Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 26 May 2020 22:51:17 +0200 Subject: [PATCH 16/20] Use mesh loop for assembly --- .../linear_elastoplasticity.cc | 129 ++++++++++-------- Linear_Elastoplasticity/parameters.prm | 2 +- 2 files changed, 75 insertions(+), 56 deletions(-) diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index 38d5a941..f74e56ca 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -52,6 +52,9 @@ #include #include #include +#include +#include +#include #include #include #include @@ -1142,70 +1145,86 @@ namespace LinearElastoplasticity system_matrix = 0; system_rhs = 0; - FEValues fe_values (fe, qf_cell, - update_values | update_gradients | - update_quadrature_points | update_JxW_values); + using ScratchData = MeshWorker::ScratchData; + using CopyData = MeshWorker::CopyData<1, 1, 1>; + using CellIteratorType = decltype(dof_handler.begin_active()); + + const ScratchData sample_scratch_data (fe, qf_cell, + update_values | update_gradients | + update_quadrature_points | update_JxW_values); + CopyData sample_copy_data (fe.dofs_per_cell); + + auto cell_worker = [this] (const CellIteratorType &cell, + ScratchData &scratch_data, + CopyData ©_data) + { + const FEValues &fe_values = scratch_data.reinit(cell); + FullMatrix &cell_matrix = copy_data.matrices[0]; - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_q_points_cell = qf_cell.size(); + std::vector &local_dof_indices = copy_data.local_dof_indices[0]; + cell->get_dof_indices(local_dof_indices); - FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); - Vector cell_rhs (dofs_per_cell); + const unsigned int dofs_per_cell = fe_values.dofs_per_cell; + const unsigned int n_q_points_cell = fe_values.n_quadrature_points; - std::vector local_dof_indices (dofs_per_cell); - std::vector > solution_grads_u_total (qf_cell.size()); + std::vector > solution_grads_u_total (n_q_points_cell); + fe_values[u_fe].get_function_gradients( + solution, + solution_grads_u_total); - for (typename DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(); - cell!=dof_handler.end(); ++cell) - { - cell_matrix = 0; - cell_rhs = 0; + const std::vector > > lqph = + quadrature_point_history.get_data(cell); - fe_values.reinit (cell); - fe_values[u_fe].get_function_gradients( - solution, - solution_grads_u_total); + for (unsigned int q_point_cell=0; q_point_cell &Grad_u = solution_grads_u_total[q_point_cell]; + const SymmetricTensor<2,dim> epsilon = Physics::Elasticity::Kinematics::epsilon(Grad_u); + lqph[q_point_cell]->update_internal_equilibrium(epsilon); + const SymmetricTensor<4,dim> K = lqph[q_point_cell]->get_K(epsilon); - const std::vector > > lqph = - quadrature_point_history.get_data(cell); + const double JxW = fe_values.JxW(q_point_cell); - for (unsigned int q_point_cell=0; q_point_cell &Grad_u = solution_grads_u_total[q_point_cell]; - const SymmetricTensor<2,dim> epsilon = Physics::Elasticity::Kinematics::epsilon(Grad_u); - lqph[q_point_cell]->update_internal_equilibrium(epsilon); - const SymmetricTensor<4,dim> K = lqph[q_point_cell]->get_K(epsilon); - - const double JxW = fe_values.JxW(q_point_cell); - - // Precompute the shape function symmetric gradients - // (related to the variation of the small strain) - std::vector< SymmetricTensor<2,dim> > d_epsilon (dofs_per_cell); - for (unsigned int K=0; K &d_epsilon_I = d_epsilon[I]; - - for (unsigned int J=I; J &d_epsilon_J = d_epsilon[J]; - - cell_matrix(I,J) - += contract3(d_epsilon_I,K,d_epsilon_J) * JxW; - } - } - } + // Precompute the shape function symmetric gradients + // (related to the variation of the small strain) + std::vector< SymmetricTensor<2,dim> > d_epsilon (dofs_per_cell); + for (unsigned int K=0; K &d_epsilon_I = d_epsilon[I]; - cell->get_dof_indices (local_dof_indices); - constraints.distribute_local_to_global(cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs); - } + for (unsigned int J=I; J &d_epsilon_J = d_epsilon[J]; + + cell_matrix(I,J) + += contract3(d_epsilon_I,K,d_epsilon_J) * JxW; + } + } + } + + for (unsigned int I=0; I &cell_matrix = copy_data.matrices[0]; + const Vector &cell_rhs = copy_data.vectors[0]; + const std::vector &local_dof_indices = copy_data.local_dof_indices[0]; + + constraints.distribute_local_to_global(cell_matrix, cell_rhs, + local_dof_indices, + system_matrix, system_rhs); + }; + + MeshWorker::mesh_loop(dof_handler.active_cell_iterators(), + cell_worker, copier, + sample_scratch_data, + sample_copy_data, + MeshWorker::assemble_own_cells); } diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index e8a4e8c6..3dc2c18a 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -24,7 +24,7 @@ subsection Geometry set Grid scale = 1e-3 # Number of global refinement steps - set Global refinement steps = 1 + set Global refinement steps = 2 # Number of initial local refinement cycles set Local refinement steps = 1 From 2d6a15d666c5d3ef26125a6013d40b675b9b9ccd Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 26 May 2020 22:55:41 +0200 Subject: [PATCH 17/20] Change default simulation; fix tols --- Linear_Elastoplasticity/linear_elastoplasticity.cc | 6 +++--- Linear_Elastoplasticity/parameters.prm | 2 +- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/Linear_Elastoplasticity/linear_elastoplasticity.cc b/Linear_Elastoplasticity/linear_elastoplasticity.cc index f74e56ca..38efc322 100644 --- a/Linear_Elastoplasticity/linear_elastoplasticity.cc +++ b/Linear_Elastoplasticity/linear_elastoplasticity.cc @@ -526,7 +526,7 @@ namespace LinearElastoplasticity // Check that stress-strain relationship really is linear and // correctly implemented - Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-9, ExcInternalError()); + Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-6, ExcInternalError()); return K_e; } @@ -609,7 +609,7 @@ namespace LinearElastoplasticity { // Check that stress-strain relationship really is linear and // correctly implemented - Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-9, ExcInternalError()); + Assert((get_sigma(epsilon) - K_e*epsilon).norm() < 1e-6, ExcInternalError()); return K_e; } @@ -1153,7 +1153,7 @@ namespace LinearElastoplasticity update_values | update_gradients | update_quadrature_points | update_JxW_values); CopyData sample_copy_data (fe.dofs_per_cell); - + auto cell_worker = [this] (const CellIteratorType &cell, ScratchData &scratch_data, CopyData ©_data) diff --git a/Linear_Elastoplasticity/parameters.prm b/Linear_Elastoplasticity/parameters.prm index 3dc2c18a..540a3cc1 100644 --- a/Linear_Elastoplasticity/parameters.prm +++ b/Linear_Elastoplasticity/parameters.prm @@ -18,7 +18,7 @@ end subsection Geometry # Geometry to simulate # Options: Notched cylinder ; Tensile specimen - set Geometry type = Notched cylinder + set Geometry type = Tensile specimen # Global grid scaling factor set Grid scale = 1e-3 From a3eeabd7deeb14bdc659fbc6e5c041c76006794a Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Tue, 26 May 2020 22:57:15 +0200 Subject: [PATCH 18/20] Add theory PDF --- .../theory/theory-linear_elastoplasticity.pdf | Bin 0 -> 634322 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.pdf diff --git a/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.pdf b/Linear_Elastoplasticity/doc/theory/theory-linear_elastoplasticity.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2947b9d2af6ea9c651d5aa25b5ecec4a7d2b0040 GIT binary patch literal 634322 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readme --- Linear_Elastoplasticity/README.md | 68 ++++++++++++++++++++++++++++++- 1 file changed, 66 insertions(+), 2 deletions(-) diff --git a/Linear_Elastoplasticity/README.md b/Linear_Elastoplasticity/README.md index 679d241c..0fb5b4f7 100644 --- a/Linear_Elastoplasticity/README.md +++ b/Linear_Elastoplasticity/README.md @@ -1,4 +1,68 @@ ## Overview -Linear elasticity with linear isotropic hardening. -Solved on a notched cylindrical specimen or a tensile specimen. +There are some well-established experimental procedures that are commonly employed to investigate the plastic reponse of metals and similar materials. They are typically setup such that the stress and strain field in a region of the domain is predictable, which means that (i) the experiments are reliable and reproducable, and (ii) that can be more easily correlated to microstructural studies and numerical simulations, or vice versa. This is particularly useful when the materials exhibit complex constitutive responses (such as anisotropic plastic behaviour, or viscoplastic behaviour), as a well defined set of experiments can be used to understand the nature of the material response on both the macroscopic and microscopic levels. +In this code we have implemented basic linear elasticity with linear isotropic hardening. Although this is a very simple model of plasticity, it is useful as a stepping-stone to fortify one's understanding of the steps necessary to implement more elaborate models. The constiutive model(s) are used to examine the behaviour of two geometries, namely a notched cylindrical specimen and a tensile specimen that conforms to a certified standard. + +## Theory +The theory of linear elastoplasticity, starting from the absolute fundamentals moving all the way to the numerical model, involved several distinct topic that deserve thorough explanation. This includes: +* the governing equations of quasi-static elasticity, +* a generalised continuum framework for elastoplasticity, +* constiutive laws for elasticity and elastoplasticity with linear isotropic hardening (including the derivation of the elastoplastic tangent modulus in the continuous setting), +* time integration algorithms for elasto-plasticity (leading to the time-discrete setting and algorithmically correct elastoplastic tangent modulus), and finally +* the finite element discretisation itself. +Since its easier to do so, we have provided a [summary of the fundamental theory](./doc/theory/theory-linear_elastoplasticity.pdf) in the supplied PDF document. Unfortunately, though, the reader will have to satisfied with a marginally condensed (bullet point) form of the elastoplastic framework, constitutive laws and integration algorithms. + +## Recommended Literature +* J. C. Simo, and T. J. R. Hughes (2000), Computational Inelasticity. Springer Science & Business Media. ISBN: 978-0-387-97520-7. DOI: [10.1007/b98904](http://doi.org/10.1007/b98904) +* O. C. Zienkiewicz and R. L. Taylor (2005), The finite element method for solid and structural mechanics. Butterworth-Heinemann. ISBN: 978-1-85617-634-7. DOI: [10.1016/C2009-0-26332-X](https://doi.org/10.1016/C2009-0-26332-X) +* B. D. Reddy (2016) Theoretical and Numerical Elastoplasticity. In: J. Schroeder, P. Wriggers (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol 566, pp 177-194. Springer, Cham. ISBN: 978-3-319-31923-0. DOI: [10.1007/978-3-319-31925-4_7](https://doi.org/10.1007/978-3-319-31925-4_7) + +## Compiling and running +Similar to the example programs, run +``` +cmake -DDEAL_II_DIR=/path/to/deal.II . +``` +in this directory to configure the problem. +You can switch between debug and release mode by calling either +``` +make debug +``` +or +``` +make release +``` +The problem may then be run with +``` +make run +``` + +## Problem setup and results +In this code we have two toy test cases set up to demonstrate either purely elastic (fully recoverable) material behaviour or elastoplasticity (dissipative) with linear isotropic hardening. +The constitutive parameters are entirely fictitious, and were chosen such that plastic flow would be easily observed in both cases. + +### Tensile specimen +In the first example, we investigate plastic behaviour of a dumbbell-shaped tensile specimen, which is dimensioned according the DIN 53504 type S3 specification. The supplied geometry is shown in the figure below. +![Problem geometry](./doc/results/tensile_specimen-geometry.png) +Due to the symmetry of the geometry, only one-eigth of the geometry need be modelled. +Naturally, the cut planes around which the model is reflected must be treated as symmetry boundary conditions. Note that we also attach manifolds to the curved surface in order to maintain the accuracy of the geometry upon refinement. + +These specimens are typically clamped in a tensile test machine, and stretched according to some experimental protocol. Accordingly, a portion of the wider surface of the specimen has a prescribed displacement in the axial direction (aligned with the X-Cartesian basis vector), while the out of plane displacement over the same subregion of the boundary is not permitted. The second condition is a rudimentary approximation to the real physical conditions, as the thickness of the specimen would change during clamping and during the course of the experiment itself. + +The next figure shows the geometry after it is lengthened by 20%. The plotted contours depict the local values of the isotropic hardening variable, which is an indicator of how much (non-recoverable) plastic flow has occured. +![Displaced solution](./doc/results/tensile_specimen-isotropic_hardening.png) +There are three regions where distinct and interesting behaviour may be noted. The first is the thin part of the specimen, where necking occurs. The strain in this region is nearly constant (which is precisely the intention of the design of the specimen); this leads to near uniform plastic flow in this region, which is ideal for experimental observations. The second interesting region is where the specimen starts to thicken. The change in curvature of the geometry causes a slight stress concentration in this location, leading to increased plastic flow. Again, by design this occurs out of the region of experimental observation. The third place where plastic flow is observed is near the clamped surface. This is a numerical artefact caused by the simplified boundary conditions. In reality, the thickness of the sample would reduce here and the amount of out-of-plane deformation would be less than that predicted here. + +### Notched cylinder +In the second example we model cylindrical specimen that is also placed in tension. However, a defect in the form of a notch is introduced into the specimen, with the idea that this will induce a localised stress concentration that will, in turn, lead to plastic flow. +The geometry is shown below; we may again apply symmetry arguments to reduce the size of the problem to be modelled. In this case (for which the geometry is created inside `deal.II`), we model only a half of the specimen. With a little effort, this could also be reduced to a one-eigth model. +![Problem geometry](./doc/results/notched_cylinder-geometry.png) + +Like before, the specimen is stretched along its axis such that its overall length increases by 20%. +As anticipated, there is indeed a significant amount of plastic flow at the reentrant corner of the notch where the stress concentration is predicted to be. +![Displaced solution](./doc/results/notched_cylinder-isotropic_hardening.png) +We also observe that the majority of the specimen is predicted to undergo some plastic flow, although the larger-radius section adjacent to the notch undergoes no plastic deformation. This could be a real phenomenon, but could also be an experimental artefact. Due to the refinement strategy chosen for this case, the mesh around the notch is more refined than in the bulk of the specimen. This means that the boundary of the bulk geometry is captured with differing accuracy as one looks towards the notch, and may artifically be inducing some non-uniform stress response.Further investigation and mesh refinement studies could render the answer to this outstanding question. + +## Future work / possibilities for extension +* Implement and investigate other hardening rules, such as kinematic hardening and anisotropic hardening. +* Many materials also exhibit rate-dependent responses, so using an elasto-viscoplastic constitutive law might be more appropriate in those cases. +* Due to the large deformation, it may be necessary to extend this model from the linear response regimen to one that incorporates the effects of geometric stiffening, i.e. finite strain elasto-plasticity. \ No newline at end of file From 4afafcc96293641b5a2f71f45efba42c640632a5 Mon Sep 17 00:00:00 2001 From: Jean-Paul Pelteret Date: Wed, 27 May 2020 12:27:19 +0200 Subject: [PATCH 20/20] Add result images 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