Finite Element Program for Elastic-Plastic Solids in MATLAB

This program implements finite element analysis for elastic-plastic solids, including stiffness matrix computation, deformation analysis, and node displacement solving. The implementation is based on established finite element analysis theories and methods.

MATLAB Implementation

1. Element Stiffness Matrix Calculation for Elastic-Plastic Solids

function stiffness = ComputeElementStiffness(youngs_modulus, poisson_ratio, thickness, node1_x, node1_y, node2_x, node2_y, node3_x, node3_y, analysis_type)
    % Calculate stiffness matrix for 3-node triangular element
    % Input parameters:
    % youngs_modulus - Young's modulus
    % poisson_ratio - Poisson's ratio
    % thickness - Element thickness
    % node1_x, node1_y, node2_x, node2_y, node3_x, node3_y - Coordinates of three nodes
    % analysis_type - Analysis type indicator (1 for plane stress, 2 for plane strain)
    % Output:
    % stiffness - Element stiffness matrix (6x6)

    % Calculate element area
    area = 0.5 * abs((node2_x - node1_x) * (node3_y - node1_y) - (node3_x - node1_x) * (node2_y - node1_y));
    
    % Calculate B matrix
    B_matrix = [node2_y - node3_y, 0; node3_x - node2_x, 0; 0, node3_y - node1_y; 0, node1_x - node3_x; node2_x - node1_x, 0; 0, node1_y - node2_y] / (2 * area);
    
    % Calculate D matrix
    if analysis_type == 1 % Plane stress
        D_matrix = (youngs_modulus / (1 - poisson_ratio^2)) * [1, poisson_ratio, 0; poisson_ratio, 1, 0; 0, 0, (1 - poisson_ratio) / 2];
    elseif analysis_type == 2 % Plane strain
        D_matrix = (youngs_modulus / ((1 + poisson_ratio) * (1 - 2 * poisson_ratio))) * [1 - poisson_ratio, poisson_ratio, 0; poisson_ratio, 1 - poisson_ratio, 0; 0, 0, (1 - 2 * poisson_ratio) / 2];
    else
        error('Invalid analysis type. Use 1 for plane stress or 2 for plane strain.');
    end
    
    % Calculate stiffness matrix
    stiffness = thickness * area * B_matrix' * D_matrix * B_matrix;
end

2. Global Matrix Assembly Function

function global_matrix = AssembleGlobalMatrix(global_matrix, element_matrix, node1, node2, node3)
    % Assemble element stiffness matrix into global stiffness matrix
    % Input parameters:
    % global_matrix - Global stiffness matrix
    % element_matrix - Element stiffness matrix
    % node1, node2, node3 - Node numbers of the element
    % Output:
    % global_matrix - Updated global stiffness matrix

    % Node degree of freedom numbers
    dof_indices = [2 * node1 - 1, 2 * node1, 2 * node2 - 1, 2 * node2, 2 * node3 - 1, 2 * node3];
    
    % Assemble stiffness matrix
    for row_idx = 1:6
        for col_idx = 1:6
            global_matrix(dof_indices(row_idx), dof_indices(col_idx)) = global_matrix(dof_indices(row_idx), dof_indices(col_idx)) + element_matrix(row_idx, col_idx);
        end
    end
end

3. Node Displacement Solver

function nodal_displacements = SolveNodeDisplacements(global_stiffness, force_vector, constraints)
    % Solve for node displacements
    % Input parameters:
    % global_stiffness - Global stiffness matrix
    % force_vector - Nodal force vector
    % constraints - Boundary conditions (node numbers and constrained directions)
    % Output:
    % nodal_displacements - Node displacement vector

    % Apply boundary conditions
    for constraint = constraints
        node_num = constraint(1);
        direction = constraint(2);
        dof = 2 * node_num - direction;
        global_stiffness(dof, :) = [];
        global_stiffness(:, dof) = [];
        force_vector(dof) = [];
    end
    
    % Solve linear system
    nodal_displacements = global_stiffness \ force_vector;
end

4. Main Program

% Main execution script
clear;
close all;

% Material properties
elastic_modulus = 210e9; % Young's modulus in Pascals
poissons_ratio = 0.3; % Poisson's ratio
element_thickness = 0.01; % Thickness in meters

% Node coordinates
node_coordinates = [0, 0; 1, 0; 1, 1]; % Coordinates of three nodes

% Element connectivity
element_connectivity = [1, 2, 3]; % Node numbers connected by the element

% Boundary conditions (node number and constrained direction)
boundary_constraints = [1, 1; 1, 2]; % Zero displacement in x and y directions for node 1

% Force vector
force_vector = zeros(6, 1); % Force vector for 6 degrees of freedom
force_vector(4) = -1000; % Apply -1000N force in y-direction at node 2

% Initialize global stiffness matrix
global_stiffness = zeros(6, 6); % Global stiffness matrix for 6 degrees of freedom

% Assemble global stiffness matrix
for elem_idx = 1:length(element_connectivity)
    n1_x = node_coordinates(element_connectivity(elem_idx), 1);
    n1_y = node_coordinates(element_connectivity(elem_idx), 2);
    n2_x = node_coordinates(element_connectivity(elem_idx) + 1, 1);
    n2_y = node_coordinates(element_connectivity(elem_idx) + 1, 2);
    n3_x = node_coordinates(element_connectivity(elem_idx) + 2, 1);
    n3_y = node_coordinates(element_connectivity(elem_idx) + 2, 2);
    
    elem_stiffness = ComputeElementStiffness(elastic_modulus, poissons_ratio, element_thickness, n1_x, n1_y, n2_x, n2_y, n3_x, n3_y, 1); % Plane stress analysis
    global_stiffness = AssembleGlobalMatrix(global_stiffness, elem_stiffness, element_connectivity(elem_idx), element_connectivity(elem_idx) + 1, element_connectivity(elem_idx) + 2);
end

% Solve for node displacements
nodal_displacements = SolveNodeDisplacements(global_stiffness, force_vector, boundary_constraints);

% Display results
disp('Nodal Displacements:');
disp(nodal_displacements);

Implementation Details

1. Element Stiffness Matrix Calculation: Computes the stiffness matrix for each element based on material properties and geometric information.
2. Global Matrix Assembly: Combines individual element stiffness matrices into the global stiffness matrix.
3. Node Displacement Solution: Applies boundary conditions and solves the resulting linear system to obtain node displacements.
4. Main Program: Sets up material properties, node coordinates, element connectivity, boundary conditions, and applied forces, then calls the appropriate functions to complete the finite element analysis.