Stress-Strain Tensor Constitutive Equation Calculator (Isotropic Material)

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Stress-Strain Tensor Calculator

Strain Tensor Calculator

Stress Tensor [MPa], E (Young’s Modulus [GPa] ), and ν (Poisson’s Ratio):

σ Tensor:

E: ν:

Stress Tensor Calculator [MPa]

Strain Tensor, E (Young’s Modulus [GPa] ), and ν (Poisson’s Ratio):

ε Tensor:

E: ν:

Constitutive Equation for Isotropic Material (Hook’s Law)

Compute Strain Tensor

\[ \epsilon_{xx} = \frac{1}{E} \left( \sigma_{xx} - \nu \sigma_{yy} - \nu \sigma_{zz} \right) \]
\[ \epsilon_{yy} = \frac{1}{E} \left( \sigma_{yy} - \nu \sigma_{xx} - \nu \sigma_{zz} \right) \]
\[ \epsilon_{zz} = \frac{1}{E} \left( \sigma_{zz} - \nu \sigma_{xx} - \nu \sigma_{yy} \right) \]
\[ \epsilon_{xy} = \frac{1}{2G} \sigma_{xy} = \frac{1 + \nu}{E} \sigma_{xy} \]
\[ \epsilon_{xz} = \frac{1}{2G} \sigma_{xz} = \frac{1 + \nu}{E} \sigma_{xz} \]
\[ \epsilon_{yz} = \frac{1}{2G} \sigma_{yz} = \frac{1 + \nu}{E} \sigma_{yz} \]

Compute Stress Tensor

\[ \sigma_{xx} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{xx} + \nu (\epsilon_{yy} + \epsilon_{zz}) \right] \]
\[ \sigma_{yy} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{yy} + \nu (\epsilon_{xx} + \epsilon_{zz}) \right] \]
\[ \sigma_{zz} = \frac{E}{(1 + \nu)(1 - 2\nu)} \left[ (1 - \nu)\epsilon_{zz} + \nu (\epsilon_{xx} + \epsilon_{yy}) \right] \]
\[ \sigma_{xy} = \frac{E}{1 + \nu} \epsilon_{xy}, \quad \sigma_{xz} = \frac{E}{1 + \nu} \epsilon_{xz}, \quad \sigma_{yz} = \frac{E}{1 + \nu} \epsilon_{yz} \]

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