Changes on the linear-elasticity problem simulation using PSYDAC.

_toc.yml

@@ -58,7 +58,10 @@ parts:
           - file: chapter2/stokes-v2
       - file: chapter2/elliptic-curl
       - file: chapter2/elliptic-div
-      - file: chapter2/linear-elasticity
+      - file: chapter2/linear-elasticity-intro
+        sections:
+          - file: chapter2/linear-elasticity-v1
+          - file: chapter2/linear-elasticity-v2
   - caption: Nonlinear Problems 
     chapters:
       - file: chapter3/poisson

chapter2/linear-elasticity-intro.md

@@ -0,0 +1,137 @@
+# Linear Elasticity
+
+## The PDE problem
+The governing equations for small elastic deformations of a body $\Omega$ can be expressed as:
+
+$$
+\begin{aligned}
+    -\nabla \cdot \sigma(u) &= f & \text{in } \Omega \\
+    \sigma(u) &= C : \epsilon(u) & \text{in } \Omega \\
+\end{aligned}
+$$
+
+where : 
+ - $\Omega$ is the domain of interest,
+ - $u$ is the displacement vector field,
+ - $\sigma(u)$ is a second-order symmetric tensor, called stress tensor, 
+ - $f$ represents the load vector field (force per unit volume), 
+ <!-- $\kappa$ and $\mu$ are Lamé's elasticity parameters for the material, $I$ denotes the identity tensor,  -->
+ - $\epsilon(u)$ is a second-order symmetric tensor, called strain tensor and by definition $\epsilon(u) := \frac{1}{2}(\nabla u + (\nabla u)^T)$.
+ - $C$ is the elasticity tensor, which relates stress and strain. 
+
+Then, the strong formulation of linear elasticity is :
+Find $u \in V$ such that
+
+$$
+\begin{aligned}
+    -\nabla \cdot \sigma (u) &= f & \text{in } & \Omega \\
+    u &= 0 & \text{on } & \partial \Omega_D \\
+    \sigma(u) \cdot n &= g_T & \text{on } & \partial \Omega_T
+\end{aligned}
+$$
+
+where : 
+- $V = \{ v \in (H^1(\Omega))^3 : v = 0 \text{ on } \partial \Omega_D \}$,
+
+- $g_T$ is the traction vector on the part $\partial \Omega_T$ of the boundary,
+
+- $n$ is the outward normal vector on the boundary $\partial \Omega$,
+
+- $\partial \Omega_D \cap \partial \Omega_T = \emptyset$ and $\partial \Omega = \partial \Omega_D \cup \partial \Omega_T$. 
+- $\partial \Omega_D$ is the Dirichlet boundary condition where the displacement is fixed to zero, and $\partial \Omega_T$ is the Neumann boundary condition where the traction is prescribed.
+
+## The Variational Formulation
+The variational formulation of the linear elasticity equations involves forming the inner product of the PDE with a vector test function $ v \in V $ and integrating over the domain $ \Omega $. This yields:
+
+$$
+\int_{\Omega} - \nabla \cdot \sigma(u) \cdot v \, \mathrm{d} x = \int_{\Omega} f \cdot v \, \mathrm{d} x
+$$
+
+Integrating the term $ \nabla \cdot \sigma(u) \cdot v $ by parts, considering boundary conditions, we obtain:
+
+$$
+\int_{\Omega} \sigma(u) : \nabla v \, \mathrm{d} x = \int_{\Omega} f \cdot v \, \mathrm{d} x + \int_{\partial \Omega_T} g_T \cdot v \, \mathrm{d} s
+$$
+
+By using the symmetry of the stress tensor $ \sigma $ and its definition from $(2)$, we can notice that :
+
+$$
+\begin{aligned}
+\int_{\Omega} \sigma(u) : \nabla v \, \mathrm{d} x &= \int_{\Omega} \sigma(u) : \epsilon(v) \, \mathrm{d} x = \int_{\Omega} C : \epsilon(u) : \epsilon(v) \, \mathrm{d} x \\ &= \int_{\Omega} \epsilon(u) : C : \epsilon(v) \, \mathrm{d} x
+\end{aligned}
+$$
+
+This leads to the following variational formulation:
+
+$$
+\boxed{
+\begin{aligned}
+&\text{Find } u \in V \text{ such that:} \\
+&\qquad a(u, v) = L(v) \quad \forall v \in V
+\end{aligned}
+}
+$$
+
+with
+
+$$
+\begin{aligned}
+&a : 
+\begin{cases}
+V \times V \rightarrow \mathbb{R} \\
+(u, v) \longmapsto \int_{\Omega} \epsilon(u) : C : \epsilon(v) \, \mathrm{d} x
+\end{cases} \\[0.3cm]
+&L : 
+\begin{cases}
+V \rightarrow \mathbb{R} \\
+v \longmapsto \int_{\Omega} f \cdot v \, \mathrm{d} x + \int_{\partial \Omega_T} g_T \cdot v \, \mathrm{d} s
+\end{cases}
+\end{aligned}
+$$
+
+## Isotropic Materials
+For isotropic materials, the elasticity tensor $C$ can be expressed in terms of the Lamé parameters $\lambda$ and $\mu$ as follows:
+
+$$
+C := \lambda (\nabla \cdot u) I_3 + 2\mu \epsilon(u)
+$$
+Then, the stress tensor can be expressed as:
+$$\sigma(u) = \lambda (\nabla \cdot u) I_3 + 2\mu \epsilon(u)$$
+
+
+This leads to the variational formulation:
+$$
+\boxed{
+\begin{aligned}
+&\text{Find } u \in V \text{ such that:}
+\\
+&\qquad a(u, v) = L(v) \quad \forall v \in V
+\end{aligned}
+}
+$$
+
+with
+$$
+\begin{aligned}
+&a :\begin{cases}
+V \times V \rightarrow \mathbb{R} \\
+(u, v) \longmapsto \int_{\Omega} \sigma(u) : \epsilon(v) \, \mathrm{d} x
+\end{cases} \\[0.3cm]
+&L :\begin{cases}
+V \rightarrow \mathbb{R} \\
+v \longmapsto \int_{\Omega} f \cdot v \, \mathrm{d} x + \int_{\partial \Omega_T} g_T \cdot v \, \mathrm{d} s
+\end{cases}
+\end{aligned}
+$$
+
+With this formulation, the problem is well-posed under the assumption that the material is isotropic and the boundary conditions are properly defined. While $\frac{\lambda}{\mu}$ is not too large (typically $\frac{\lambda}{\mu} \leq 10^4$), the problem remains well-posed numerically. However, as $\frac{\lambda}{\mu}$ increases, the problem can become ill-posed, leading to numerical difficulties in finding a solution. The first notebook of this chapter illustrates the case of isotropic materials with $\frac{\lambda}{\mu} \leq 10^4$ and the second notebook is trying to illustrate the case of isotropic materials with $\frac{\lambda}{\mu} > 10^4$.
+
+<p align="center">
+
+| Material  | $\lambda$ (GPa) | $\mu$ (GPa) |
+|-----------|:--------------:|:-----------:|
+| Steel     | 120            | 80          |
+| Concrete  | 17             | 14          |
+| Rubber    | 0.16           | 0.00033     |
+
+</p>