Elasticity
Physics
Collaboration diagram for Elasticity:

## Modules

| Elasticity Boundary Conditions

## Classes

class | asl::FDElasticityIncompressibleStatic
| Numerical method which computes homogenious isotropic elasticity equation. More...
class | asl::FDElasticityRelaxation
| Numerical method which computes homogenious isotropic elasticity equation. More...
class | asl::FDElasticity2
| Numerical method which computes homogenious isotropic elasticity equation. More...
class | asl::FDPoroElasticity
| Numerical method which computes homogenious isotropic poro-elasticity equation. More...

## Detailed Description

# Theory

Equation of motion:

\[ \rho\ddot u_j =\nabla_i \sigma_{ij}, \]

where \( \vec u \) is the displacement vector, \( \sigma_{i,j} \) is the Cauchy stress tensor. The strain-displacement equation is:

\[ \varepsilon_{ij}=\frac{1}{2}\left( \nabla_i u_j + \nabla_j u_i \right) \]

strain tensor is related to the stress tensor by the following equation:

\[ \sigma_{ij}=C_{ijkl}\varepsilon_{kl}, \]

where \( C_{ijkl} \) is the stiffness tensor which has the following properties:

\[ C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk} \]

There are several different tensors used in the elasticity theory:

- Deformation gradient tensor

  \[ F_{ij} = \nabla_j u_i + \delta_{ij} \]

- Finite strain tensor

  \[ E_{ij}=\frac{1}{2}\left( \nabla_i u_j + \nabla_j u_i + \nabla_i u_k \nabla_j u_k \right) \]

- Right Cauchy-Green deformation tensor

  \[ C_{ij}= F_{ki}F_{kj}= \left( \nabla_i u_j + \nabla_j u_i + \nabla_i u_k \nabla_j u_k + \delta_{ij} \right) \]

## Isotropic and homogeneous media

In the case of isotropic homogeneous media the stiffness tensor is defined by two modules: \( K \) is the bulk modulus (or incompressibility) and \( \mu \) is the shear modulus (or rigidity). The corresponding expression is:

\[ C_{ijkl}=K \delta_{ij}\delta_{kl}+\mu\left(\delta_{ik}\delta_{jl}+ \delta_{il}\delta_{jk}- \frac{2}{3}\delta_{ij}\delta_{kl}\right) \]

Stress-strain relation can be rewritten as follows:

\[ \sigma_{ij}=\lambda\delta_{ij}\varepsilon_{kk}+2\mu\varepsilon_{ij}, \]

where \( \lambda\equiv K-2/3\mu \) is Lam\'e's first parameter.

The media can be parametrized by \( v_p \) and \( v_s \) velocities:

\[ v_p=\sqrt{(\lambda+2\mu)/\rho},\;\; v_s=\sqrt{\mu/\rho}. \]

Equation of motion can be simplified for an isotropic case:

\[ \rho\ddot u_j =\nabla_j \lambda \nabla_k u_k+ \nabla_k \mu \nabla_k u_j+ \nabla_k \mu \nabla_j u_k, \]

In a homogeneous case it takes:

\[ \rho\ddot u_j =(\lambda+\mu)\nabla_j \nabla_k u_k+ \mu \Delta u_j \]

This equation is solved by following classes: asl::FDElasticity, asl::FDElasticity2

The potential energy density is:

\[ U=\frac{1}{2}\sigma_{ij}\varepsilon_{ij} \]

## Boundary Conditions

Free surface BC with surface orientations \( \vec n \) is:

\[ \sigma_{ij} n_j = F_i \]

For the isotropic and inhomogeneous case substitution of the strain-stress relation results:

\[ \lambda n_i\nabla_k u_k+\mu n_j \left( \nabla_i u_j + \nabla_j u_i \right) = F_i, \]

## Units

\[[\rho]=kg/m^3 \]

\[[\sigma_{ij}]=\frac{kg}{ms^2}=Pa\]

\[[\mu]=[K]=[\lambda]=Pa\]

\[\left[\frac{\mu}{\rho}\right]= \left[\frac{K}{\rho}\right]= \left[\frac{\lambda}{\rho}\right]=\frac{m^2}{s^2}\]