Linear elasticity¶
Editor: Weipeng Xu & Copilot
Last modified: 24/09/2025
Governing equations¶
Momentum equation¶
The Cauchy momentum equation describes the motion of a continuum via the Newton's second law and can be expressed as the following second-order elliptic PDE: $$ \begin{align*} \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{b} = \rho\ddot{\boldsymbol{u}} \quad &\text{in } \Omega \\ \boldsymbol{u} = \boldsymbol{g} \quad &\text{on } \Gamma_D \\ \boldsymbol{\sigma} \cdot \boldsymbol{n} = \boldsymbol{t} \quad &\text{on } \Gamma_N \end{align*} $$ where $\boldsymbol{\sigma}$ is the Cauchy stress, $\boldsymbol{b}$ is the body force per unit volume, $\boldsymbol{u}$ is the displacement field, $\rho$ is the density, $\ddot{\boldsymbol{u}}$ is the acceleration field, $\boldsymbol{g}$ is the prescribed displacement on the Dirichlet boundary $\Gamma_D$, and $\boldsymbol{t}$ is the prescribed traction on the Neumann boundary $\Gamma_N$. The index notation ($i=1,2,3$) of the monmentum equation is: $$ \begin{align*} \frac{\partial \sigma_{ij}}{\partial x_j} + b_i = \rho \frac{\partial^2 u_i}{\partial t^2} \quad &\text{in } \Omega\\ u_i = g_i \quad &\text{on } \Gamma_D \\ \sigma_{ij} n_j = t_i \quad &\text{on } \Gamma_N \end{align*} $$
Strain-displacement relation¶
The strain-displacement relation for small deformations is given by: $$ \boldsymbol{\varepsilon} = \frac{1}{2} \left[ \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right] $$ where $\boldsymbol{\varepsilon}$ is the Cauchy strain tensor. The index notation is: $$ \varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) $$
Constitutive relation¶
The constitutive relation for a linear elastic isotropic material is given by Hooke's law: $$ \boldsymbol{\sigma} = \boldsymbol{C} : \boldsymbol{\varepsilon} $$ where $\boldsymbol{C}$ is the fourth-order elasticity tensor. The index notation is: $$ \sigma_{ij} = C_{ijkl} \varepsilon_{kl} $$
Discussion on the constitutive relation¶
Isotropic materials¶
For isotropic materials, the fourth-order elasticity tensor can be expressed in terms of the Lamé parameters $\lambda$ and $\mu$ (see tensor.ipynb for more details): $$ C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{j}) $$
where $\delta_{ij}$ is the Kronecker delta.
Another commonly used form is interms of bulk modulus $K$ and shear modulus $G$: $$ C_{ijkl} = K \delta_{ij} \delta_{kl} + G (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} - \frac{2}{3} \delta_{ij} \delta_{kl}) $$
So we can conclude the relations between the two sets of parameters are: $$ K = \lambda + \frac{2}{3} \mu, \quad G = \mu $$ Then the constitutive relation can be expressed as: $$ \begin{aligned} \sigma_{ij} &= C_{ijkl} \varepsilon_{kl}\\ &= \left[ \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}) \right] \varepsilon_{kl} \\ &= \lambda \delta_{ij} \varepsilon_{kk} + \mu (\varepsilon_{ij} + \varepsilon_{ji})\\ &=\lambda \delta_{ij} \varepsilon_{kk} + 2 \mu \varepsilon_{ij} \end{aligned} $$ or $$ \sigma_{ij} = K \delta_{ij} \varepsilon_{kk} + 2 G \left( \varepsilon_{ij} - \frac{1}{3} \delta_{ij} \varepsilon_{kk} \right) $$