Computational-Micromechanics.md

@@ -2,19 +2,19 @@
 In the computational micromechanics, we need to evaluate the stress tensor $`\mathbf{P}_M(t)`$ as a function of the strain path $`\mathbf{F}_M(t)`$. With a representative volume elmeent (RVE) $`V_0`$, a microscopic boundary value problem (mBVP) is defined as
 
 - Local balance:
-$$\bm{\nabla}\cdot\mathbf{P}_m= \mathbf{0}\,.$$
+$$`\bm{\nabla}\cdot\mathbf{P}_m= \mathbf{0}`\,.$$
 
 - Local material behavior for each constituent $`\alpha`$
-$$\mathbf{P}_m(t)= \bm{\mathfrak{P}}^\alpha\left(\mathbf{F}(t),\mathbf{Z}\right)\,.$$
+$$`\mathbf{P}_m(t)= \bm{\mathfrak{P}}^\alpha\left(\mathbf{F}(t),\mathbf{Z}\right)`\,.$$
 
 - Strain averaging theorem:
-$$\frac{1}{V_0}\int_{V_0} \mathbf{F}_m\,dV = \mathbf{F}_M \,.$$
+$$`\frac{1}{V_0}\int_{V_0} \mathbf{F}_m\,dV = \mathbf{F}_M` \,.$$
 
 - Stress averaging theorem:
-$$\frac{1}{V_0}\int_{V_0} \mathbf{P}_m\,dV = \mathbf{P}_M\,.$$
+$$`\frac{1}{V_0}\int_{V_0} \mathbf{P}_m\,dV = \mathbf{P}_M`\,.$$
 
 - Hill-Mandel macro-homogeneity condition:
-$$\frac{1}{V_0}\int_{V_0} \mathbf{P}_m : \dot{\mathbf{F}}_m\,dV = \mathbf{P}_M: \dot{\mathbf{F}}_M\,.$$
+$$`\frac{1}{V_0}\int_{V_0} \mathbf{P}_m : \dot{\mathbf{F}}_m\,dV = \mathbf{P}_M: \dot{\mathbf{F}}_M`\,.$$
 
 In general, the strain averaging theorem and the Hill-Mandel condition are sastified a priori using a microscopic boundary condition (micro-BC). Four caterogies can be found: (i) Periodic BC (PBC), (ii) Displacement BC (DBC), (iii) Minimal Kinematic BC (MKBC), and (iv) Mixed BC (MBC). The solution of this mBVP is carried out using nonLinearMechSolver, in wihch the relation between $`\mathbf{P}_M(t)`$ and $`\mathbf{F}_M(t)`$ can be obtained by different ways.