1.2. Elastic waves

Similar to Section 1.1 the geometry is a given open domain \(\Omega\subset\mathbb R^3\), and time interval \([0,T], T>0\)

The constituents are the displacement \(u:[0,T]\times\Omega\to\mathbb R^3\) (unit \(m\)), the particle velocity \(v=\partial_t u\) (unit \(m/s\)), the strain tensor

\[\varepsilon(u) = \frac{1}{2}\left(\nabla u+ (\nabla u)^\top\right)\]

(unit \(1\)) the stress tensor \(\sigma:[0,T]\times\Omega\to \mathbb R^{3\times 3}_{sym},\) (unit \(kg/(ms^2)\)) where \(\mathbb R^{3\times 3}_{sym}\) denotes the space of symmetric, real three-by-three matrices.

We either choose \(u\) or \(v,\sigma\) as our primal unknowns.

Material parameters here are the mass density \(\rho:\Omega\to (0,\infty)\) (unit \(kg/m^3\)) and the material stiffness given by Hooke’s tensor (also known as elastic tensor) \(C:\Omega\to \mathcal L(\mathbb R^{3\times 3}_{sym},\mathbb R^{3\times 3}_{sym})\) (unit \(kg/(ms^2)\)). Note that here, differing from linear acoustics, we already assumed that the density \(\rho\) is uniform in time.

As balancing law we impose linear conservation of momentum

(1.12)\[\rho \partial_t v = \nabla\cdot \sigma\]

where the divergence \(\nabla\cdot\sigma\) has to be understood column wise, i.e. \((\nabla\cdot\sigma)_i=\sum_j \partial_{x_j}\sigma_{j,i}\). The material law to link stress and strain is Hooke’s law given by

(1.13)\[\sigma = C(\varepsilon(u))\]

Provided boundary and initial data this leads to the following first and second order formulation.

\[\begin{split}\begin{aligned} \rho\partial_tv-\nabla\cdot \sigma &= f,&\text{on }& (0,T)\times\Omega,\\ \partial_t \sigma-C(\varepsilon(v))&=0,&\text{on }& (0,T)\times\Omega,\\ v(0)&=v_0,&\text{on }& \Omega,\\ \sigma(0)&=C\varepsilon(u_0),&\text{on }& \Omega,\\ v(t) &= \partial_t u_V(t),&\text{on }& \partial\Omega, \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \rho\partial_t^2 u-\nabla\cdot C(\varepsilon(u))&=f,&\text{on }& (0,T)\times \Omega, \\ u(0,\cdot)&=u_0,&\text{on }& (0,T)\times\Omega, \\ \partial_t u(0,\cdot)&=v_0,&\text{on }& \Omega, \\ u &= u_V,&\text{on }&\partial \Omega, \end{aligned}\end{split}\]

where we assumed dynamic boundary conditions. Alternatively one may assume static boundary conditions, namely conditions on the normal component of \(\sigma\).

Hooke’s tensor \(C\), also called the 4-index elastic tensor has 81 components. In isotropic elasitcity it is given by only two parameters, namely the so-called Lamé parameters \(\mu,\lambda\)

\[C(\varepsilon) = 2\mu\varepsilon+\lambda\mathrm{tr}(\varepsilon)I=2\mu(\varepsilon-\frac{1}{3}\mathrm{tr}(\varepsilon I))+\kappa\mathrm{tr}(\varepsilon)I.\]

Here the two parameters \(\mu,\lambda\) correspond to the decomposition of waves into shear waves, depending on the shear modulus \(\mu\) and compressional waves (pressure waves) depending on the compression modulus \(\kappa=\frac{2}{3}+\lambda\). Then the linear second order elastic wave equation in isotropic and homogeneous media takes the form

\[\rho\partial_t^2 u+\mu \nabla\times\nabla\times u-3\kappa\nabla(\nabla\cdot u)=f\]

Assuming a vanishing shear modulus \(\mu\to 0\) leads to compressional waves only, and thus to the linear acoustic wave equation for the hydrostatic pressure \(p=\frac{1}{3}\mathrm{tr}\sigma\) already introduced in Section 1.1. Note however, that historically the sign conventions for pressure and stress are flipped in fluid and solid mechanics.