11.3 Equations for plane strain and plane stress

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11.3 Equations for plane strain and plane stress

The fundamental physical quantities are the stress tensor $\sigma$ , the displacements u, and the strain tensor $\epsilon$ :

$\begin{displaymath} \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\end{displaymath}$

The suffix notation for tensors means that the suffices

and

are understood to vary over the range 1 to

, where

is the number of space dimensions, unless a suffix is repeated in a term, in which case it is understood to be summed from 1 to

(Einstein's summation convention).

The equations for equilibrium are:

$\begin{displaymath} {\rm div}~ \sigma + {\bf F} = 0\end{displaymath}$

where F is a body force per unit volume.

The stress-strain relation is:

$\begin{displaymath} \sigma = 2\mu\epsilon + \lambda I {\rm~trace}~\epsilon \end{displaymath}$

where $\mu = E/(1 + \nu)$ and $\lambda = \nu E /(1 - \nu^2)$ or

$\begin{displaymath} \sigma_{ij} = \mu \left( \frac{\partial u_i}{\partial x_j} ... ...ght) + \lambda \delta_{ij} \frac{\partial u_k}{\partial x_k}\end{displaymath}$

Here E is Young's modulus and $\nu$ is Poisson's ratio. I is the identity tensor, and $\delta_{ij}$ are its coefficients.

Combining this with the equilibrium relation gives:

$\begin{displaymath} \frac{\partial }{\partial x_j} \left( \mu \frac{\partial u_i... ...( \lambda \frac{\partial u_j}{\partial x_j} \right) + F_i = 0\end{displaymath}$ (1)

Many boundary conditions are possible, but the most straightforward are

prescribed displacement
stress	$\sigma_{ij}n_j = h_i$

In finite element terms, the first is an essential condition, and the second is a natural condition. As will appear, the equations have been formulated to facilitate the imposition of natural conditions.

You may notice that the second and third terms look as if they are similar, provided $\mu$ is constant, so the order of differentiation could be exchanged. This is not done in the Fasttalk code, because doing so would generate a different boundary integral, which would generally not represent the boundary stress.

So far, we have assumed 3D coefficients, with corresponding coefficients. There are two common circumstances in which a reduction to 2D is possible:

: plane strain, in which all terms of $\epsilon$ involving -coefficients are zero
: plane stress, in which all terms of $\sigma$ involving coefficients are zero

The first is relatively straightforward, in that equation (1) may be written for the restricted range of and ; $\sigma_{33}$ is not zero, but is easily evaluated. The case of plane stress is a little more difficult; (1) now involves $\epsilon_{33}$ , which is not zero, and must be eliminated before a valid equation over the 2D range of indices can be written. The elimination can be done using the equation from plane stress:

$\begin{displaymath} \sigma_{33} = 0 = 2\mu\epsilon_{33} + 3\lambda~\epsilon_{33} \end{displaymath}$

$\begin{displaymath} \epsilon_{33} = - \frac{3\lambda}{3\lambda + 2\mu} (\epsilon_{11} + \epsilon_{22})\end{displaymath}$

The nett result is that where the 2D range of indices is now understood:

$\begin{displaymath} \sigma_{ij} = 2\mu\epsilon_{ij} + \frac{2\lambda\mu}{3\lambda + 2\mu}~\delta_{ij} \epsilon_{kk} \end{displaymath}$

The problems described below will show the advantages of Fasttalk's use of tensor notation. Once the equations have been written this way, they can not only be directly transcribed into Fasttalk, but natural boundary conditions take the simplest possible form, provided the original format is adhered to. As a rule of thumb, the latter requirement will usually be met if you make no simplifications based on the assumption that the elastic constants are in fact constant in space.

Thermal heat stress can be imposed by equating it to a volume rate of displacement $\alpha \nabla T$ where $\alpha$ is the coefficient of expansion; the thermal body force is then

$\begin{displaymath} F_i = - \alpha E \nabla T\end{displaymath}$

The temperature

can be assumed to be independent of the stress-strain, and may satisfy its own diffusion equation

$\begin{displaymath} \rho c_p \frac{\partial T}{\partial t} = \nabla.(k \nabla T)\end{displaymath}$

where $\rho$ is the density,

is the specific heat, and

is the thermal conductivity. Common boundary conditions will be the prescription of

, or of the flux $k \nabla T$ .

Once the displacements and the strains have been calculated, two useful results can be calculated. These are

$\begin{displaymath} \pi=\epsilon_{11}+\epsilon_{22}= \frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}\end{displaymath}$

which is the divergence of the displacement and is proportional to the trace of the normal stress tensor (that is, to the pressure), and the maximum shearing stress $\tau$ given by (apart from normalising factors)

$\begin{displaymath} \tau^2=\left( \frac{\partial u_1}{\partial x_1}-\frac{\part... ..._1}{\partial x_2}+\frac{\partial u_2}{\partial x_1} \right)^2\end{displaymath}$

The structure will be most likely to fail at the point where the maximum shearing stress is greatest.