Constitutive equations

In this section we present the constituve relations (i.e. relations between stress and strain tensors) for the case of hyperelasticity, which could be defined in terms of strain energy density $W($ $\varepsilon$$)$, which allows to evaluate stress components as partial derivatives:

$$\sigma_{ij}=$$$$\mbox{$\displaystyle\frac{\partial \,{W}}{\partial \,{\varepsilon _{ij}}}$}$$

For example, the Hooke's law is defined using following strain energy potential

$$W(\varepsilon ) =$$   $$\mbox{$\displaystyle\frac{1}{2}$}$$$$\varepsilon _{ij}$$$C$$$_{ijkl}\varepsilon _{kl}$$

where $C$ is forth order elasticity tensor. The equality of mixed derivatives $($$\mbox{$\displaystyle\frac{\partial ^{2}\,{W}}{\partial \,{\varepsilon _{ij}\varepsilon _{kl}}^{2}}$}$$=$   $\mbox{$\displaystyle\frac{\partial ^{2}\,{W}}{\partial \,{\varepsilon {kl}\varepsilon {ij}}^{2}}$}$$)$ and symmetry of stress and strain tensors $\mbox{$\displaystyle\frac{\partial \,{\sigma{ij}}}{\partial \,{\varepsilon _{kl}}}$}$$= C_{ijkl}=C_{jikl}=C_{ijlk}$ imply that there is in general maximum 21 independ components of the elasticity tensor. In the simplest case, the elasticity tensor for isotropic linear elastic material can be described by only two parameters: either Lameś parameters ( $\lambda, \mu$) or more usual parametrs being Young's modulus $E$ and Poisson's ratio $\nu$:

$$C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+2\mu {1\over 2}[\delta_{ik}\delta_{jl}+ \delta_{il}\delta_{jk}] \equiv$$   $C$$$=\lambda$$$1$$$\otimes$$$I$$$+2\mu$$$I$