Computational-Micromechanics.md

 # Computation micro-mechanics
-In the computational micromechanics, we need to evaluate the stress tensor **P**(t) as a function of the strain path **F**(t). The nonLinearMechSolver allows obtaining this relation by different ways.
+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:
+```math
+ \bm{\nabla}\cdot\mathbf{P}_m= \mathbf{0}\,.
+```
+- Local material behavior for each constituent $`\alpha`$
+```math
+ \mathbf{P}_m(t)= \bm{\mathfrak{P}}^\alpha\left(\mathbf{F}(t),\mathbf{Z}\right)\,.
+```
+- Strain averaging theorem:
+```math
+\frac{1}{V_0}\int_{V_0} \mathbf{F}_m\,dV = \mathbf{F}_M \,.
+```
+- Stress averaging theorem:
+```math
+\frac{1}{V_0}\int_{V_0} \mathbf{P}_m\,dV = \mathbf{P}_M\,.
+```
+- Hill-Mandel macro-homogeneity condition:
+```math
+\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. 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.
 
 ## 1. Using a microscopic boundary condition
-### 1.1. Mesh
-1. Finite element mesh
-2. OK
-  * OKOK
-    - TT
-      Inline-style (hover to see title text):
-
-      ![alt text](images/markdown_logo.png "Title Text")
   
 ## 2. Using macroscopic BC
-OK $`a^2+b^2=c^2`$, one has
-```math
-a^2+b^2=c^2
-```
+