On the response and symmetry of elastic materials with internal constraints
Elastic material is one where the Cauchy stress T is a function of the deformation gradient. The material is constrained if the accessible values of F are only those which belong to a submanifold \({\mathcal M}\) of the set of tensors with positive determinant. The constraint is internal if QF\(\in {\mathcal M}\) for all \(F\in {\mathcal M}\) and for all proper orthogonal tensors Q. Notice that the authors do not request that the identity tensor belong to \({\mathcal M}\); this fact renders certain points of the analysis subtler and certain assertions perhaps less palatable to some readers. A concept of group of symmetry k for the constraint is introduced and explored; examples are provided. A separate concept of group of material symmetry is successively introduced and studied as a subgroup of k; it includes transformations under which the constraint and the space of admissible active stresses are both invariant. Delicate questions arise due to the `interaction' between the two conditions of invariance and to the indeterminancy of active stress. Ample space is finally given to the response of some special types of materials and the corresponding representation formulae.
scientific article; zbMATH DE number 4024862