Geometric Nonlinearity

In an elastic region, if the deformation is less than 10^-3, nonlinearity is assumed to be due to geometry and not due to a nonlinear stress-strain relation. However the boundary between geometric and material nonlinearity cannot be drawn very clearly. Following Biot 1965, it is possible to compute the material deformation to second order as:  

$$\epsilon_{kl} = e_{kl} + 1/2 ( e_{k\mu} \omega_{\mu l} + e_{l\mu} \omega_{\mu k} ) + 1/2 ( \omega_{k\mu} \omega_{l\mu} )$$ (1)

$$e_{kl} = 1/2 ( {{\partial u_k} \over{\partial l}} + {{\par... ... u_k} \over{\partial l}} - {{\partial u_l} \over{\partial k}})$$ (2)

Neglecting the rotational tensor components $\omega$ in (1) results in the usual definition of the elastic symmetric strain tensor e. (All doubly appearing indices are summed over and the ranges are always i=1,2,3.) Following Fung 1965 we have a similar formulation for the Eulerian strain tensor:

 

$$\epsilon^{E}_{kl} = 1/2 ( {{\partial u_k} \over{\partial l}}... ...al u_m} \over{\partial k}} {{\partial u_m} \over{\partial l}}$$ (3)

Both equation (1) and (3) describe the deformation of a medium to a higher order than the usual elastic strain tensor e. The deformation is a pure geometrical property and not a material property. Consequently the linear Hooke's law:

$$\sigma_{ij}~=~c_{ijkl}~~\epsilon_{kl}$$ (4)

remains unmodified. The relation between stress and strain components is still linear. It is the computation of $\epsilon_{kl}$ that changes in each case. When considering the displacement gradients, we can see that higher order gradient components are related to stress components nonlinearly. However the parameters which link them, the stiffness coefficients c_ijkl, are still linear elastic parameters. All previous equations lead to an elastic wave equation of the form

$$\nabla~C~\nabla^t~u = \rho {{\partial^2 u}\over{\partial t^2}},$$ (5)

or in the geometrically nonlinear case to a slight modification of the previous expression:

$$\nabla~C~\nabla_g^t~u = \rho {{\partial^2 u}\over{\partial t^2}}$$ (6)

where $\nabla_g$ is the geometrically nonlinear derivative operator.