Elastic field equations

The derivation of the coupled elastic wave equations is similar to that of the uncoupled equations. In an elastic medium forces have to be in balance. This implies that the divergence of the elastic stress tensor $$\mathop{\mbox{${\bf \nabla}$}}_{\mbox{$\sim$}} \cdot {\bf \sigma_1}$$has to equal the product of mass density $\rho$ and the second temporal derivative of the particle displacement ${\bf u}$. We can also include a forcing term ${\bf F}$ and get

 

$$\displaystyle \mathop{\mbox{${\bf \nabla}$}}_{\mbox{$\sim$}}... ...1}~=~\rho~{\partial^2 \over{\partial t^2}} {\bf u_1}~-~{\bf F}.$$ (5)

The elastic strain ${\bf \epsilon_1}$ can be calculated by taking spatial derivatives $$\mathop{\mbox{${\bf \nabla}$}}_{\mbox{$\sim$}}$$ of the displacement ${\bf u}$, as follows:

 

$${\bf\epsilon_1}~=~ \displaystyle \mathop{\mbox{${\bf \nabla^T}$}}_{\mbox{$\sim$}} ~{\bf u_1}$$ (6)

Next we relate the components of the elastic stress ${\bf \sigma_1}$ to the components of the elastic strain ${\bf \epsilon_1}$, the electric displacement ${\bf \epsilon_2}$ and the first temporal derivative of the thermal displacement ${\partial \over{\partial t}}{\bf\epsilon_3}$ in the following equation:

 

$${\bf\sigma_1} ~=~ \displaystyle \mathop{\mbox{${\bf c_{11}}$... ...}_{\mbox{$\sim$}} ~{\partial \over{\partial t}} {\bf\epsilon_3}$$ (7)

Substituting Equation 7 into 5 gives a wave equation in terms of elastic displacements (omitting the source term):

 

$$\displaystyle \mathop{\mbox{${\bf \nabla}$}}_{\mbox{$\sim$}}... ...\epsilon_3})~= ~\rho~{\partial^2 \over{\partial t^2}} {\bf u_1}$$ (8)

This is a coupled wave equation for elastic displacements, since it includes effects produced by electric and thermal displacements as well.