Stress
- Stress \(\;\sigma\in\mathbb{R}^{3 \times 3}\) describes internal forces (per area unit) induced by the local deformation (strain) in
any direction
- Constitutive Relation: Relation between stress and strain, characterize a type of material.
- For linear constitutive relation:
-
- \(\displaystyle \sigma_{ij}=\sum_{k,l} C_{ijkl}\,\epsilon_{kl}\;\;\), \(\;C\): stiffness tensor (81 coefficients)
- Strain energy/elastic potential energy: \(\displaystyle U= \frac{1}{2} \sum_{i,j,k,l} \sigma_{ij}(\epsilon)
\epsilon_{kl} = \frac{1}{2}\sum_{i,j,k,l}C_{ijkl}\,\epsilon_{ij}\,\epsilon_{kl}\)
- For homogeneous isotropic elastic material, constitutive relation simplifies to
-
- \(\sigma = 2\mu\,\epsilon + \lambda \mathrm{tr}(\epsilon)\,\mathrm{Id}\), \((\mu,\lambda)\): Lamé parameters
- Related to common mechanical modulus : Young' modulus \(Y\) and Poisson's ratio \(\nu\)
-
- \(\mu=\frac{Y}{2(1+\nu)}\), \(\lambda=\frac{Y\nu}{(1+\nu)(1-2\nu)}\)
