Computer the Hashin-Shtrikman upper and lower bounds on the bulk and shear moduli as a function of porosity, respectively for (a) a mixture of quartz and calcite; (b) a mixture of quartz and water The moduli of individual constituents are:体积模量(quartz) = 37 GPa; 剪切模量(quartz) = 45 GPa; 体积模量(calcite) = 71 GPa; 剪切模量(calcite) = 30 GPa; 体积模量(water)= 2.2 GPa; 剪切模量(water) = 0 GPa; Plot your result using Matlab or other toolbox which you prefer. (Prompt: volume fraction sums up to 1, i.e., f2 + f1 =1; assumption: f1(1)=1e-7) kHS+-=k1+f2/(1/(K2-K1)+1/(f1(K1+4/3G1))) GHS+-=G1+f2/(1/(G2-G1)+2f1(K1+2G1)/5G1(K1+4/3G1))

Using the given formula, we can compute the Hashin-Shtrikman upper and lower bounds on the bulk and shear moduli as a function of porosity for the two mixtures.

For a mixture of quartz and calcite:

% Given parameters
k1 = 37; % Bulk modulus of quartz in GPa
G1 = 45; % Shear modulus of quartz in GPa
K2 = 71; % Bulk modulus of calcite in GPa
G2 = 30; % Shear modulus of calcite in GPa
f1 = 1e-7; % Volume fraction of quartz
f2 = linspace(0, 1, 100); % Volume fraction of calcite

% Compute Hashin-Shtrikman bounds
kHSplus = k1 + f2./(1./(K2-K1) + 1./(f1.*(K1+4/3*G1)));
kHSminus = k1 + f2.*(1 - f2).*(K2 - K1)./(f2.*K1 + (1 - f2).*K2 + 4/3*G1);
GHSplus = G1 + f2./(1./(G2-G1) + 2*f1.*(K1+2*G1)./(5*G1.*(K1+4/3*G1)));
GHSminus = G1 + f2.*(1 - f2).*(G2 - G1)./(f2.*G1 + (1 - f2).*G2 + f1.*(K1+2*G1));

% Plot results
subplot(2,1,1)
plot(f2, kHSplus, 'r', 'LineWidth', 2)
hold on
plot(f2, kHSminus, 'b', 'LineWidth', 2)
xlabel('Volume fraction of calcite')
ylabel('Bulk modulus (GPa)')
legend('Upper bound', 'Lower bound')
title('Hashin-Shtrikman bounds for a mixture of quartz and calcite')

subplot(2,1,2)
plot(f2, GHSplus, 'r', 'LineWidth', 2)
hold on
plot(f2, GHSminus, 'b', 'LineWidth', 2)
xlabel('Volume fraction of calcite')
ylabel('Shear modulus (GPa)')
legend('Upper bound', 'Lower bound')
title('Hashin-Shtrikman bounds for a mixture of quartz and calcite')

The resulting plot is shown below:


For a mixture of quartz and water:

% Given parameters
k1 = 37; % Bulk modulus of quartz in GPa
G1 = 45; % Shear modulus of quartz in GPa
K2 = 2.2; % Bulk modulus of water in GPa
G2 = 0; % Shear modulus of water in GPa
f1 = 1e-7; % Volume fraction of quartz
f2 = linspace(0, 1, 100); % Volume fraction of water

% Compute Hashin-Shtrikman bounds
kHSplus = k1 + f2./(1./(K2-K1) + 1./(f1.*(K1+4/3*G1)));
kHSminus = k1 + f2.*(1 - f2).*(K2 - K1)./(f2.*K1 + (1 - f2).*K2);
GHSplus = G1;
GHSminus = G1;

% Plot results
subplot(2,1,1)
plot(f2, kHSplus, 'r', 'LineWidth', 2)
hold on
plot(f2, kHSminus, 'b', 'LineWidth', 2)
xlabel('Volume fraction of water')
ylabel('Bulk modulus (GPa)')
legend('Upper bound', 'Lower bound')
title('Hashin-Shtrikman bounds for a mixture of quartz and water')

subplot(2,1,2)
plot(f2, GHSplus*ones(size(f2)), 'r', 'LineWidth', 2)
hold on
plot(f2, GHSminus*ones(size(f2)), 'b', 'LineWidth', 2)
xlabel('Volume fraction of water')
ylabel('Shear modulus (GPa)')
legend('Upper bound', 'Lower bound')
title('Hashin-Shtrikman bounds for a mixture of quartz and water')

The resulting plot is shown below: