Elastic Solids

The system defining the behavior of a (3+1)D linear, isotropic, elastic solid is somewhat easier to handle numerically than the (2+1)D plate and (1+1)D beam systems which are derived from it; the physics is less obscured by modeling assumptions. Numerical simulation of the full (3+1)D system is, of course, much more computationally expensive.

Such a medium is characterized by its density, $\rho$, and two material parameters $\lambda$ and $\mu$, called the Lamé coefficients, which describe its resilience; there are two parameters because an solid will resist compressional and shear forces to different degrees. Other elastic parameters, which we have already made use of earlier in this chapter, can be defined in terms of these two constants. Young's modulus $E$ and Poisson's ratio $\nu$ can be written as

$$E = \frac{\mu(3\lambda+2\mu)}{\lambda+\mu}\hspace{0.5in}\nu = \frac{\lambda}{2(\lambda+\mu)}$$

We remark that $\mu$ is the same as $G$ that was used in the treatment of the Timoshenko beam (see §5.2), the Mindlin plate (see §5.4), and the Naghdi-Cooper shell model of §5.5.2. For the sake of generality, we allow all these parameters to be functions of $x$, $y$ and $z$.

The equations of motion of the solid can be written in terms of stress and displacement fields [77]. There are nine stresses: $\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{zz}$ are normal stresses in the direction indicated by the double subscript, and $\sigma_{xy}$, $\sigma_{xz}$, $\sigma_{yz}$, $\sigma_{yx}$, $\sigma_{zx}$, and $\sigma_{zy}$ are shear stresses. The displacements of a point in the medium from its equilibrium position are given by ${\bf d} = [w_{x},w_{y},w_{z}]^{T}$. If the material is assumed to be in rotational equilibrium, then we have

$$\sigma_{xy}=\sigma_{yx}\hspace{0.5in}\sigma_{xz}=\sigma_{zx}\hspace{0.5in}\sigma_{yz}=\sigma_{zy}$$

so that there are a total of six independent stresses acting at a given point in the solid [190].

Newton's Laws for a solid (neglecting body forces) are written as

$$\begin{eqnarray}\rho\frac{\partial^{2} w_{x}}{\partial t^{2}} &=&... ...\sigma_{yz}}{\partial y}+\frac{\partial \sigma_{zz}}{\partial z} \end{eqnarray}$$ (5.54a)

The stress-strain relation, or Hooke's Law [77] is expressed as a linear proportionality between the six stresses and spatial derivatives of the displacements (the strain):

$$\begin{eqnarray}\sigma_{xx} &=& 2\mu\frac{\partial w_{x}}{\partia... ...2\mu\frac{\partial w_{z}}{\partial z}+\lambda \nabla\cdot{\bf d} \end{eqnarray}$$ (5.55a)

$$\begin{eqnarray}\sigma_{xy} &=& \mu\left(\frac{\partial w_{x}}{\p... ...tial w_{y}}{\partial z}+\frac{\partial w_{z}}{\partial y}\right) \end{eqnarray}$$ (5.56a)

The systems (5.52), (5.53) and (5.54) taken together are sometimes called the Navier system [77,131]. By introducing velocities defined by

$$v_{x} \triangleq \frac{\partial w_{x}}{\partial t}\hspace{0.5in}v... ...y}}{\partial t}\hspace{0.5in}v_{z} \triangleq \frac{\partial w_{z}}{\partial t}$$

it is possible to manipulate these equations into the symmetric hyperbolic form of (3.1), with ${\bf w} = [v_{x}, v_{y}, v_{z}, \sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{xz},\sigma_{yz}]^{T}$, and

$${\bf P} = \begin{bmatrix}{\bf P}_{N}^{+}&\cdot\\ \cdot&{\bf P}_{N... ...bmatrix}\cdot&{\bf A}_{N3\times}\\ {\bf A}_{N3\times}^{T}&\cdot\\ \end{bmatrix}$$

with

$${\bf P}_{N}^{+} = {\rm diag}(\rho, \rho, \rho)\hspace{0.3in}{\bf ... ...1}{\mu}&0&0\\ 0&0&0&0&\frac{1}{\mu}&0\\ 0&0&0&0&0&\frac{1}{\mu}\\ \end{bmatrix}$$

and

$${\bf A}_{N1\times} = \begin{bmatrix}-1&0&0\\ 0&0&0\\ 0&0&0\\ 0&-1... ...in{bmatrix}0&0&0\\ 0&0&0\\ 0&0&-1\\ 0&0&0\\ -1&0&0\\ 0&-1&0\\ \end{bmatrix}^{T}$$