Matrix Issue Line 33, 35, 37

Hi. Line 33, 35, and 37 (v_a2, s1_a2, s2_a2) are supposed to create a matrix/array (not sure what the right term is) like Line 27 and Line 29 does.

Note that v_a2 uses K_a1 and Mu_a1 which are both matrix/array and then v_a2 is used for s1_a2 and s2_a2.

I am not sure how to accomplish this. Please help!

% Approach 1

E_m = 70000; % Elastic modulus of matrix, MPa
v_m = .33; % Poisson ratio of matrix
E_i = 250000; % Elastic modulus of inclusion, MPa
v_i = .2; % Poisson ratio of inclusion

% Fiber volume fraction
f_i = 0:0.01:1; 
    % 0 is 0% fiber and 1 is 100% fiber content
    % points to be plotted

K_m = (E_m)/(3*(1-2*v_m)); 

K_i = (E_i)/(3*(1-2*v_i));

Mu_m = (E_m)/(2*(1+v_m));

Mu_i = (E_i)/(2*(1+v_i));

% Iteration 1

s1_a1 = (1+v_m)/(3*(1-v_m));

s2_a1 = (2*(4-5*v_m))/(15*(1-v_m));

K_a1 = K_m*((1+f_i*(K_m/(K_m-K_i)-s1_a1)^-1)).^-1; 

Mu_a1 = Mu_m*((1+f_i*(Mu_m/(Mu_m-Mu_i)-s2_a1)^-1)).^-1;

% Iteration 2

v_a2 = (3*K_a1-2*Mu_a1)/(2*(3*K_a1+Mu_a1));

s1_a2 = (1+v_a2)/(3*(1-v_a2)); 

s2_a2 = (2*(4-5*v_a2))/(15*(1-v_a2));

K_a2 = K_m*(1+(f_i*((K_i/K_m)-1).*(1+((K_i./K_a1)-1)*s1_a2).^-1));

Mu_a2 = Mu_m*(1+(f_i*((Mu_i/Mu_m)-1).*(1+((Mu_i./Mu_a1)-1)*s2_a2).^-1));



% Plotting 
figure(1)
plot(f_i,K_a1, 'Color', 'red', 'LineWidth', 2);
hold on;
plot(f_i,K_a2, 'Color', 'black', 'LineWidth', 2);
hold off;
%plot(f_i,K_a3, 'Color', 'blue', 'LineWidth', 2);
%hold on;
%plot(f_i,K_a4, 'Color', 'cyan', 'LineWidth', 2);
%hold on;
%plot(f_i,K_a5, 'Color', 'yellow', 'LineWidth', 2);
%hold on;
%plot(f_i,K_Reuss, 'Color', 'magenta', 'LineWidth', 2);
%hold on;
%plot(f_i,K_Voigt, 'Color', 'green', 'LineWidth', 2);
%hold off;

grid off;
legend('K_a_1', 'K_a_2','K_a_3', 'K_a_4', 'K_a_5', 'K_R_e_u_s_s',...
    'K_V_o_i_g_t')
legend('Location','northwest','FontSize',12)
title('Approach 1: Bulk Modulus vs. Inclusion Volume Fraction')
xlabel('Inclusion Volume Fraction, f_i')
ylabel('Bulk Modulus, K [MPa]')

Answer 1

[Voss]

Use element-wise division (./) in the calculations on those lines, and use element-wise multiplication (.*) in the subsequent calculations of K_a2 and Mu_a2, since they are using vectors now.

% Approach 1

E_m = 70000; % Elastic modulus of matrix, MPa
v_m = .33; % Poisson ratio of matrix
E_i = 250000; % Elastic modulus of inclusion, MPa
v_i = .2; % Poisson ratio of inclusion

% Fiber volume fraction
f_i = 0:0.01:1; 
    % 0 is 0% fiber and 1 is 100% fiber content
    % points to be plotted

K_m = (E_m)/(3*(1-2*v_m)); 

K_i = (E_i)/(3*(1-2*v_i));

Mu_m = (E_m)/(2*(1+v_m));

Mu_i = (E_i)/(2*(1+v_i));

% Iteration 1

s1_a1 = (1+v_m)/(3*(1-v_m));

s2_a1 = (2*(4-5*v_m))/(15*(1-v_m));

K_a1 = K_m*((1+f_i*(K_m/(K_m-K_i)-s1_a1)^-1)).^-1; 

Mu_a1 = Mu_m*((1+f_i*(Mu_m/(Mu_m-Mu_i)-s2_a1)^-1)).^-1;

% Iteration 2

% v_a2 = (3*K_a1-2*Mu_a1)/(2*(3*K_a1+Mu_a1));
% s1_a2 = (1+v_a2)/(3*(1-v_a2));
% s2_a2 = (2*(4-5*v_a2))/(15*(1-v_a2));
v_a2 = (3*K_a1-2*Mu_a1)./(2*(3*K_a1+Mu_a1))

v_a2 = 1×101

0.3300 0.3290 0.3281 0.3270 0.3260 0.3250 0.3239 0.3228 0.3216 0.3204 0.3192 0.3180 0.3167 0.3154 0.3141 0.3127 0.3113 0.3098 0.3083 0.3068 0.3052 0.3035 0.3018 0.3001 0.2983 0.2964 0.2945 0.2925 0.2904 0.2883

s1_a2 = (1+v_a2)./(3*(1-v_a2))

s1_a2 = 1×101

0.6617 0.6603 0.6588 0.6573 0.6558 0.6542 0.6527 0.6510 0.6494 0.6477 0.6460 0.6442 0.6424 0.6405 0.6386 0.6366 0.6347 0.6326 0.6305 0.6283 0.6261 0.6239 0.6215 0.6192 0.6167 0.6142 0.6116 0.6089 0.6062 0.6034

s2_a2 = (2*(4-5*v_a2))./(15*(1-v_a2))

s2_a2 = 1×101

0.4677 0.4679 0.4682 0.4685 0.4688 0.4692 0.4695 0.4698 0.4701 0.4705 0.4708 0.4712 0.4715 0.4719 0.4723 0.4727 0.4731 0.4735 0.4739 0.4743 0.4748 0.4752 0.4757 0.4762 0.4767 0.4772 0.4777 0.4782 0.4788 0.4793

% K_a2 = K_m*(1+(f_i*((K_i/K_m)-1).*(1+((K_i./K_a1)-1)*s1_a2).^-1));
% Mu_a2 = Mu_m*(1+(f_i*((Mu_i/Mu_m)-1).*(1+((Mu_i./Mu_a1)-1)*s2_a2).^-1));
K_a2 = K_m*(1+(f_i.*((K_i/K_m)-1).*(1+((K_i./K_a1)-1).*s1_a2).^-1))

K_a2 = 1×101

1.0e+05 * 0.6863 0.6905 0.6947 0.6991 0.7034 0.7078 0.7123 0.7168 0.7214 0.7260 0.7307 0.7354 0.7402 0.7451 0.7500 0.7550 0.7600 0.7651 0.7703 0.7755 0.7808 0.7862 0.7916 0.7971 0.8027 0.8083 0.8140 0.8198 0.8257 0.8317

Mu_a2 = Mu_m*(1+(f_i.*((Mu_i/Mu_m)-1).*(1+((Mu_i./Mu_a1)-1).*s2_a2).^-1))

Mu_a2 = 1×101

1.0e+07 * 0.0026 0.0027 0.0027 0.0027 0.0028 0.0028 0.0028 0.0029 0.0029 0.0030 0.0030 0.0030 0.0031 0.0031 0.0032 0.0032 0.0032 0.0033 0.0033 0.0034 0.0034 0.0035 0.0035 0.0036 0.0036 0.0037 0.0038 0.0038 0.0039 0.0039

% Plotting 
figure(1)
plot(f_i,K_a1, 'Color', 'red', 'LineWidth', 2);
hold on;
plot(f_i,K_a2, 'Color', 'black', 'LineWidth', 2);
hold off;
%plot(f_i,K_a3, 'Color', 'blue', 'LineWidth', 2);
%hold on;
%plot(f_i,K_a4, 'Color', 'cyan', 'LineWidth', 2);
%hold on;
%plot(f_i,K_a5, 'Color', 'yellow', 'LineWidth', 2);
%hold on;
%plot(f_i,K_Reuss, 'Color', 'magenta', 'LineWidth', 2);
%hold on;
%plot(f_i,K_Voigt, 'Color', 'green', 'LineWidth', 2);
%hold off;

grid off;
legend('K_a_1', 'K_a_2','K_a_3', 'K_a_4', 'K_a_5', 'K_R_e_u_s_s',...
    'K_V_o_i_g_t')

Warning: Ignoring extra legend entries.

legend('Location','northwest','FontSize',12)
title('Approach 1: Bulk Modulus vs. Inclusion Volume Fraction')
xlabel('Inclusion Volume Fraction, f_i')
ylabel('Bulk Modulus, K [MPa]')

Answer 2

[Mathieu NOE]

hello

I put the dots so that the multiplication and division are performed element by element between arrays , but the result looks strange to me (a modulus that drop until it gets negative seems wrong)

Dot product - MATLAB dot - MathWorks France

also , but this is my preference , instead of writing A .* (B.^-1) I prefer the simpler A./B

It's late now for me so I'll have a look again tomorrow

i'm surprised for example so see this equation that mix a modulus and poisson ratio as physically equivalent terms ... like adding tomatoes with oranges ??

v_a2 = (3*K_a1-2*Mu_a1)./(2*(3*K_a1+Mu_a1));

code

clc
clearvars
% Approach 1
E_m = 70000; % Elastic modulus of matrix, MPa
v_m = .33; % Poisson ratio of matrix
E_i = 250000; % Elastic modulus of inclusion, MPa
v_i = .2; % Poisson ratio of inclusion
% Fiber volume fraction
f_i = 0:0.01:1; 
    % 0 is 0% fiber and 1 is 100% fiber content
    % points to be plotted
K_m = (E_m)/(3*(1-2*v_m)); 
K_i = (E_i)/(3*(1-2*v_i));
Mu_m = (E_m)/(2*(1+v_m));
Mu_i = (E_i)/(2*(1+v_i));
% Iteration 1
s1_a1 = (1+v_m)/(3*(1-v_m));
s2_a1 = (2*(4-5*v_m))/(15*(1-v_m));
% K_a1 = K_m*((1+f_i*(K_m/(K_m-K_i)-s1_a1)^-1)).^-1; 
K_a1 = K_m./((1+f_i*(K_m/(K_m-K_i)-s1_a1)^-1)); 
% Mu_a1 = Mu_m*((1+f_i*(Mu_m/(Mu_m-Mu_i)-s2_a1)^-1)).^-1;
Mu_a1 = Mu_m./((1+f_i*(Mu_m/(Mu_m-Mu_i)-s2_a1)^-1));
% Iteration 2
v_a2 = (3*K_a1-2*Mu_a1)./(2*(3*K_a1+Mu_a1));
s1_a2 = (1+v_a2)./(3*(1-v_a2)); 
s2_a2 = (2*(4-5*v_a2))./(15*(1-v_a2));
% K_a2 = K_m*(1+(f_i*((K_i/K_m)-1).*(1+((K_i./K_a1)-1).*s1_a2).^-1));
K_a2 = K_m*(1+(f_i*((K_i/K_m)-1)./(1+((K_i./K_a1)-1).*s1_a2)));
% Mu_a2 = Mu_m*(1+(f_i*((Mu_i/Mu_m)-1).*(1+((Mu_i./Mu_a1)-1).*s2_a2).^-1));
Mu_a2 = Mu_m*(1+(f_i*((Mu_i/Mu_m)-1)./(1+((Mu_i./Mu_a1)-1).*s2_a2)));
% Plotting 
figure(1)
plot(f_i,K_a1, 'Color', 'red', 'LineWidth', 2);
hold on;
plot(f_i,K_a2, 'Color', 'black', 'LineWidth', 2);
% plot(f_i,K_a3, 'Color', 'blue', 'LineWidth', 2);
% plot(f_i,K_a4, 'Color', 'cyan', 'LineWidth', 2);
% plot(f_i,K_a5, 'Color', 'yellow', 'LineWidth', 2);
% plot(f_i,K_Reuss, 'Color', 'magenta', 'LineWidth', 2);
% plot(f_i,K_Voigt, 'Color', 'green', 'LineWidth', 2);
hold off;
grid off;
legend('K_a_1', 'K_a_2','K_a_3', 'K_a_4', 'K_a_5', 'K_R_e_u_s_s',...
    'K_V_o_i_g_t')
legend('Location','northwest','FontSize',12)
title('Approach 1: Bulk Modulus vs. Inclusion Volume Fraction')
xlabel('Inclusion Volume Fraction, f_i')
ylabel('Bulk Modulus, K [MPa]')

[Briana Canet]

Yes these results look strange to me as well. Please look into it! Appreciate it.

[Briana Canet]

The equations were provided by my instructor.

Answer 3

[Briana Canet]

K_a2 = K_m*(1+(f_i.*((K_i/K_m)-1).*(1+((K_i./K_a1)-1).*s1_a2).^-1))

Why is there a dot after :((K_i/K_m)-1)

K_i and K_m are constants non-arrays.

Why is there a dot immediately after K_i

Also constant

[Briana Canet]

Hi! There was something wrong my logic! I fixed it but thank you the dot operator tip helped.