Journal of Theoretical
and Applied Mechanics

30, 3, pp. 587-605, Warsaw 1992

### Constitutive relationships for elastic and plastic behaviour of isotropic matrix reinforced with three families of fibres

Stanisław Jemioło, Marek Kwieciński, Wiesław Wojewódzki
The nonpolynomial anisotropic tensor function theory is employed together with the theorems of their representations. The tensor function approach combines clarity with the necessary generality of the constitutive equations to be formulated with the help of this type of description. As known, constitutive relationships can be presented for virtually any deformable continuum provided the necessary tensor generators are established together with the minimal set of fundamental invariants. In tins paper the constitutive equations are formulated for nonlinear elasticity and perfect plasticity of innately anisotropic media. The isotropic matrix is reinforced with three families of straight fibres. Within each family the fibres are evenly distributed and made of the same material. Three cases of reinforcement are considered: a) nonorthogonal fibres, b) orthogonal fibres, c) orthogonal fibres, each family being made of the same material. The introduction of reinforcement causes the material symmetry group to be a finite one, which can be suitably characterized by parametric tensors. General constitutive relations are formulated to describe nonlinearly elastic behaviour. The flow rules and yield criteria are derived from the condition that the constitutive equations must be homogeneous zero order functions of strain rates. Derivation of those equations is based on the concept developed by Sawczuk, Stutz and Boehler. Their linearization leads to the relatively simple expressions. The simplified perfect plasticity models are proposed. The simplification consists in application of an equivalent stress tensor which is a tensor transformed by using a fourth-order symmetric tensor dependent of parametric tensors describing the material symmetry group.