## 3 Finite Element Calculations
### Boundary Conditions
#### 3.2.1 Linear Elasticity  
In this study, we assume that macroscopic properties are invariant with respect to various boundary conditions. To investigate this, we specifically focus on testing two distinct boundary conditions: the kinematic uniform boundary conditions (KUBC) and the periodic boundary conditions:  
* Kinematic uniform boundary conditions (KUBC): Here, the displacement \(\underline{u}\) is imposed at a point \(\underline{x}\) on the boundary \(\partial V\). The conditions are defined such that: \[\underline{u}=\underset{\sim}{E}\cdot\underline{x}\quad\forall\underline{x} \in\partial V\] (5) Macroscopically, the material is considered isotropic. [PERSON]'s law is written as follows: \[\underset{\sim}{\Sigma}=[\underset{\sim}{C}]\cdot\underset{\sim}{E}\] (6) where: \[\Sigma\] : the second-rank macroscopic stress tensor; \[E\] : the second-rank macroscopic strain tensor; \[[C]\] : the fourth-rank macroscopic elastic property tensor of the composite; \[\underset{\sim}{E}\] is a symmetrical second-rank tensor that does not depend on \[\underline{x}\]. This implies that: \[\underset{\sim}{\langle\varepsilon\rangle}=\frac{1}{V}\int_{V}\underset{\sim} {\varepsilon}\mathrm{d}V=\underset{\sim}{E}\] (7) where \(<>\) represents the average function and \(\varepsilon\) is the local strain tensor. The macroscopic stress tensor is then defined by the spatial average: \[\underset{\sim}{\Sigma}=\langle\sigma\rangle=\frac{1}{V}\int_{V}\underset{ \sim}{\sigma}\mathrm{d}V\] (8) where \(\sigma\) represents the local stress tensor. * Static uniform boundary conditions (SUBC): The traction vector is prescribed at the boundary: \[\underset{\sim}{\sigma}\cdot\underline{u}=\underset{\sim}{\Sigma}\cdot \underline{u}\quad\forall\underline{x}\in\partial V\] (9) where \(n\) represents the normal surface vector. \(\underset{\sim}{\Sigma}\) is a symmetrical second-rank tensor independent of \(\underline{x}\). The vector normal to \(\partial V\) at \(\underline{x}\) is denoted by \(\underline{u}\). This implies that: \[\underset{\sim}{\Sigma}=[\underset{\sim}{C}]\cdot\underset{\sim}{E}\]0 (10) The macroscopic strain tensor is then defined as the spatial average: \[\underset{\sim}{\Sigma}=[\underset{\sim}{C}]\cdot\underset{\sim}{E}\]1 (11)
* Periodicity conditions (PERIODIC): In the context of a periodic structured medium, i.e., one with a repetitive pattern, the elementary volume \(V\) is meticulously characterized in terms of its geometric morphologies. The specific shape of this volume is designed in a manner that enables the replication of the entire spatial domain through the simple translational repetition of \(V\). In this study, the objective is to identify a solution field that conforms to a predetermined structure. This sought-after field is expected to capture and reflect the underlying periodicity of the medium. This periodicity is expressed by an equation that can be formulated as follows: \[\underset{\sim}{\Sigma}=[\underset{\sim}{C}]\cdot\underset{\sim}{E}\]2 (12)
where the fluctuation \(\underline{v}\) is periodic. It takes the same values at two homologous points on opposite faces of \(V\). The traction vector \(\sigma\cdot\underline{u}\) takes opposite values at two homologous points on opposite faces of \(V\). The proof of existence and uniqueness of solutions for the three specified boundary value problems advances with a heightened level of scientific rigor, particularly within the linear framework and accounting for potential rigid-body motions or translations. Note that the intricacies of these demonstrations are informed and fortified by results derived from the meticulous averaging computations detailed in the preceding section, as encapsulated by the expressions denoted in Equations (7) and (10). For homogeneous deformation conditions at the edges and periodicity, coupled with dual-stress conditions, a nuanced exploration is necessary.