The constitutive relations of initially stressed incompressible Mooney-Rivlin materials

46/2017

The constitutive relations of initially stressed incompressible Mooney-Rivlin materials

Tuesday 8th August 2017

Agosti, A.; Gower, A.L.; Ciarletta, P.

Initial stresses originate in soft materials by the occurrence of misfits in the undeformed microstruc- ture. Since the reference configuration is not stress-free, the effects of initial stresses on the hyperelastic behavior must be constitutively addressed. Notably, the free energy of an initially stressed material may not possess the same symmetry group as the one of the same material deforming from a naturally unstressed configuration. This work assumes an explicit dependence of the hyperelastic strain energy density on both the deformation gradient and the initial stress tensor, taking into account for their inde- pendent invariants. Using this theoretical framework, a constitutive equation is derived for an initially stressed body that naturally behaves as an incompressible Mooney-Rivlin material. The strain energy densities for initially stressed neo-Hookean and Mooney materials are derived as special sub–cases. By assuming the existence of a virtual state that is naturally stress-free, the resulting strain energy functions are proved to fulfill the required frame–independence constraints. In the case of plane strain condition, great simplifications arise in the expression of the constitutive relations. Finally, the resulting constitu- tive relations prove useful guidelines for designing non-destructive methods for the quantification of the underlying initial stresses in naturally isotropic materials.