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Stress Strain Relationships

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

ProblemDisplacementsStrainStress
Plane stressu,vu,vεxx,εyy,εxy\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}σxx,σyy,σxy\sigma_{xx},\sigma_{yy},\sigma_{xy}
Plane strainu,vu,vεxx,εyy,εxy\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}σxx,σyy,σxy\sigma_{xx},\sigma_{yy},\sigma_{xy}
Axisymmetricu,vu,vεxx,εyy,εxy\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}σxx,σyy,σxy\sigma_{xx},\sigma_{yy},\sigma_{xy}
Plate Bendingwwκxx,κyy,κxy\kappa_{xx},\kappa_{yy},\kappa_{xy}Mxx,Myy,MxyM_{xx},M_{yy},M_{xy}
Generalu,v,wu,v,wεxx,εyy,εzz,εxy,εyz,εzx\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{zz},\varepsilon_{xy},\varepsilon_{yz},\varepsilon_{zx}σxx,σyy,σzz,σxy,σyz,σzx\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

κxx=−∂2w∂x2κxx=−∂2w∂y2κxx=−∂2w∂x∂y\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 stressE1−ν2[1ν0ν10001−ν2]\frac{E}{1 - \nu^{2}}\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \\ \end{bmatrix}
Plane strainE(1−ν)(1+ν)(1−2ν)[1ν1−ν0ν1−ν10001−2ν2(1−ν)]\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}
AxisymmetricE(1−ν)(1+ν)(1−2ν)[1ν1−νν1−ν0ν1−ν1ν1−ν0ν1−νν1−ν100001−2ν2(1−ν)]\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 bendingEt312(1−ν2)[1ν0ν10001−ν2]\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}
GeneralE(1−ν)(1+ν)(1−2ν)[1ν1−νν1−ν000ν1−ν1ν1−ν000ν1−νν1−ν10000001−2ν2(1−ν)0000001−2ν2(1−ν)0000001−2ν2(1−ν)]\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 stress11−νxyνyx[ExνxyEy0Ey0symmetricGxy]\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[Ex(1−νyzνzy)DEy(νxy−νzyνxz)D0Ex(1−νxzνzx)D0symmetricGxy]\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=vxyvyx−vzx(vxyvyz+vxz)−vzy(vxzvyz+vyz)D = v_{xy}v_{yx}-v_{zx}(v_{xy}v_{yz}+v_{xz})-v_{zy}(v_{xz}v_{yz}+v_{yz})
Axisymmetric[1Ex−νyxEy−νzxEz01Ey−νzyEz01Ez0symmetric1Gxy]−1\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 bendingt312[Ex1−νxyνyxνyxEy1−νxyνyx0Ey1−νxyνyx0symmetricGxy]\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[1Ex−νyxEy−νzxEz0001Ey−νzyEz0001Ez0001Gxy001Gyz0symmetric1Gxz]−1\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}