ASL
0.1.7
Advanced Simulation Library
|
![]() |
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... | |
\[ \rho\ddot u_j =\nabla_i \sigma_{ij}, \]
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:
\[ F_{ij} = \nabla_j u_i + \delta_{ij} \]
\[ E_{ij}=\frac{1}{2}\left( \nabla_i u_j + \nabla_j u_i + \nabla_i u_k \nabla_j u_k \right) \]
\[ C_{ij}= F_{ki}F_{kj}= \left( \nabla_i u_j + \nabla_j u_i + \nabla_i u_k \nabla_j u_k + \delta_{ij} \right) \]
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:
\[ U=\frac{1}{2}\sigma_{ij}\varepsilon_{ij} \]
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:
\[ \lambda n_i\nabla_k u_k+\mu n_j \left( \nabla_i u_j + \nabla_j u_i \right) = F_i, \]
\[[\rho]=kg/m^3 \]
\[[\sigma_{ij}]=\frac{kg}{ms^2}=Pa\]
\[[\mu]=[K]=[\lambda]=Pa\]
\[\left[\frac{\mu}{\rho}\right]= \left[\frac{K}{\rho}\right]= \left[\frac{\lambda}{\rho}\right]=\frac{m^2}{s^2}\]