# Stress Strain Relationships

The relationship between stress and strain depends on the type of problem

ProblemDisplacementsStrainStress
Plane stress$u,v$$\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}$$\sigma_{xx},\sigma_{yy},\sigma_{xy}$
Plane strain$u,v$$\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}$$\sigma_{xx},\sigma_{yy},\sigma_{xy}$
Axisymmetric$u,v$$\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}$$\sigma_{xx},\sigma_{yy},\sigma_{xy}$
Plate Bending$w$$\kappa_{xx},\kappa_{yy},\kappa_{xy}$$M_{xx},M_{yy},M_{xy}$
General$u,v,w$$\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{zz},\varepsilon_{xy},\varepsilon_{yz},\varepsilon_{zx}$$\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{yz},\sigma_{zx}$

Where for beam and plate bending problems the relationship is between moment and curvature

\begin{aligned}\kappa_{xx} &= - \frac{\partial^{2}w}{\partial x^{2}}&\kappa_{xx} &= - \frac{\partial^{2}w}{\partial y^{2}}&\kappa_{xx} &= - \frac{\partial^{2}w}{\partial x\partial y}\end{aligned}

The stress-strain matrices for isotropic materials are

ProblemStress-strain matrix
Plane stress$\frac{E}{1 - \nu^{2}}\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \\ \end{bmatrix}$
Plane strain$\frac{E(1 - \nu)}{(1 + \nu)(1 - 2\nu)}\begin{bmatrix} 1 & \frac{\nu}{1 - \nu} & 0 \\ \frac{\nu}{1 - \nu} & 1 & 0 \\ 0 & 0 & \frac{1 - 2\nu}{2(1 - \nu)} \\ \end{bmatrix}$
Axisymmetric$\frac{E(1 - \nu)}{(1 + \nu)(1 - 2\nu)}\begin{bmatrix} 1 & \frac{\nu}{1 - \nu} & \frac{\nu}{1 - \nu} & 0 \\ \frac{\nu}{1 - \nu} & 1 & \frac{\nu}{1 - \nu} & 0 \\ \frac{\nu}{1 - \nu} & \frac{\nu}{1 - \nu} & 1 & 0 \\ 0 & 0 & 0 & \frac{1 - 2\nu}{2(1 - \nu)} \\ \end{bmatrix}$
Plate bending$\frac{Et^{3}}{12\left( 1 - \nu^{2} \right)}\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \\ \end{bmatrix}$
General$\frac{E(1 - \nu)}{(1 + \nu)(1 - 2\nu)}\begin{bmatrix} 1 & \frac{\nu}{1 - \nu} & \frac{\nu}{1 - \nu} & 0 & 0 & 0 \\ \frac{\nu}{1 - \nu} & 1 & \frac{\nu}{1 - \nu} & 0 & 0 & 0 \\ \frac{\nu}{1 - \nu} & \frac{\nu}{1 - \nu} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1 - 2\nu}{2(1 - \nu)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1 - 2\nu}{2(1 - \nu)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1 - 2\nu}{2(1 - \nu)} \\ \end{bmatrix}$

The stress-strain matrices for orthotropic materials:

ProblemStress-strain matrix
Plane stress$\frac{1}{1 - \nu_{xy}\nu_{yx}}\begin{bmatrix} E_{x} & \nu_{xy}E_{y} & 0 \\ & E_{y} & 0 \\ symmetric & & G_{xy} \\ \end{bmatrix}$
Plane strain$\begin{bmatrix}\frac{E_{x}\left( 1 - \nu_{yz}\nu_{zy} \right)}{D} & \frac{E_{y}\left( \nu_{xy} - \nu_{zy}\nu_{xz} \right)}{D} & 0 \\ & \frac{E_{x}\left( 1 - \nu_{xz}\nu_{zx} \right)}{D} & 0 \\ symmetric & & G_{xy} \\ \end{bmatrix}$

where

$D = v_{xy}v_{yx}-v_{zx}(v_{xy}v_{yz}+v_{xz})-v_{zy}(v_{xz}v_{yz}+v_{yz})$
Axisymmetric$\begin{bmatrix} \frac{1}{E_{x}} & \frac{- \nu_{yx}}{E_{y}} & \frac{- \nu_{zx}}{E_{z}} & 0 \\ & \frac{1}{E_{y}} & \frac{- \nu_{zy}}{E_{z}} & 0 \\ & & \frac{1}{E_{z}} & 0 \\ symmetric & & & \frac{1}{G_{xy}} \\ \end{bmatrix}^{- 1}$
Plate bending$\frac{t^{3}}{12}\begin{bmatrix} \frac{E_{x}}{1 - \nu_{xy}\nu_{yx}} & \frac{\nu_{yx}E_{y}}{1 - \nu_{xy}\nu_{yx}} & 0 \\ & \frac{E_{y}}{1 - \nu_{xy}\nu_{yx} } & 0 \\ symmetric & & G_{xy} \\ \end{bmatrix}$
General$\begin{bmatrix} \frac{1}{E_{x}} & \frac{- \nu_{yx}}{E_{y}} & \frac{- \nu_{zx}}{E_{z}} & 0 & 0 & 0 \\ & \frac{1}{E_{y}} & \frac{- \nu_{zy}}{E_{z}} & 0 & 0 & 0 \\ & & \frac{1}{E_{z}} & 0 & 0 & 0 \\ & & & \frac{1}{G_{xy}} & 0 & 0 \\ & & & & \frac{1}{G_{yz}} & 0 \\ symmetric & & & & & \frac{1}{G_{xz}} \\ \end{bmatrix}^{- 1}$