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predict_ss_power.m
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% Example script of using MKS linkages for
% predicting stress--strain curves of composites
% with phases exhibiting power-law strain hardening
%
% Linkages are calibrated as described in Latypov et al., 2018
% https://arxiv.org/abs/1809.07484
%
% Marat I. Latypov ([email protected])
% November 2018
clear all; close all;
%% Load data
% make dependencies available
addpath('vendor')
% load microstructure examples
load('data\ms_examples');
% load calibrated linkages
load('data\linkages');
% pick a microstructure
ims(:,:,:) = ms(randi(size(ms,1),1),:,:,:);
%% Hardening parameters
% hard phase
m_tay = 2.4;
h_0_2 = m_tay*563e3;
s_0_2 = m_tay*406;
s_sat_2 = m_tay*971;
% soft phase
h_0_1 = 3*1.e3;
s_0_1 = 95*3;
s_sat_1 = 412*3;
h_a = 2.25;
%% model settings
% macroscopically imposed strain rate
rate = 1e-3;
% time settings
dt = 1;
tmin = 0;
tmax = 100.0;
time = tmin:dt:tmax;
ims = ims+1;
[pc,f2] = get_pc_scores(ims);
%% allocate
e = zeros(size(time));
e_1 = zeros(size(time));
e_2 = zeros(size(time));
x1 = zeros(size(time));
x2 = zeros(size(time));
s = zeros(size(time));
s_low = zeros(size(time));
s_upp = zeros(size(time));
s_1 = zeros(size(time));
s_2 = zeros(size(time));
%% initialize
eta = s_0_2/s_0_1;
s_1(1) = s_0_1;
s_2(1) = s_0_2;
tic
for ii = 2:numel(time)
t = time(ii-1);
tau = time(ii);
dt = tau-t;
x1_vars = [pc(ind_x1),(eta - 1)];
x2_vars = [pc(ind_x2),(eta - 1)];
x1_v = mono_eval(x1_vars,pv_x1);
x2_v = mono_eval(x2_vars,pv_x2);
x1(ii) = 1 + predict(link_x1,x1_v);
x2(ii) = 1 + predict(link_x2,x2_v);
rate_1 = rate*x1(ii);
rate_2 = rate*x2(ii);
s_dot_1 = rate_1*h_0_1*(1-s_1(ii-1)/s_sat_1)^h_a;
s_dot_2 = rate_2*h_0_2*(1-s_2(ii-1)/s_sat_2)^h_a;
s_1(ii) = s_1(ii-1) + s_dot_1*dt;
s_2(ii) = s_2(ii-1) + s_dot_2*dt;
eta = s_2(ii)/s_1(ii);
s_vars = [pc(ind_s),(eta - 1)];
s_v = mono_eval(s_vars,pv_s);
s(ii) = s_1(ii)*(1 + predict(link_s,s_v));
s_upp(ii) = s_1(ii)*(1-f2) + s_2(ii)*f2;
s_low(ii) = (1-f2)/s_1(ii) + f2/s_2(ii);
s_low(ii) = 1/s_low(ii);
e(ii) = e(ii-1) + rate*dt;
e_1(ii) = e_1(ii-1) + rate_1*dt;
e_2(ii) = e_2(ii-1) + rate_2*dt;
end
toc
figure; plot(e,s,'Color','k','LineWidth',3)
hold on; plot(e,s_low,'Color','g','LineWidth',2)
hold on; plot(e,s_upp,'Color','r','LineWidth',2)
xlabel('Strain')
ylabel('Stress')