On the theory of constitutive equations for composites with different resistance in compression and tension

Abstract

The paper proposes a new approach to the construction of nonlinear constitutive equations for the description of elastic deformation of fibrous composites with different resistance in compression and tension. The framework of the approach is the generalized rheological method of constructing constitutive equations of the uniaxial stress–strain state. Transition to the tensor equations uses the hypothesis of existence of a special cone of all possible compression states in the space of strains. The cone structure is governed by the reinforcement pattern of a composite. When imposed with strains in the cone, the composite behaves as a linearly elastic medium, while beyond the cone, the composite is nonlinear, with higher stiffness conditioned by tension of high-modulus reinforcement fibers. It is highly critical that the constitutive equations obtained using this approach belong to the theory of hyperelasticity. The elastic potentials of stresses and strains are obtained for these equations, which guarantees implementation of fundamental principles of equilibrium thermodynamics. Application of the developed general theory is exemplified by construction of the constitutive equations for the plane stress state of a thin unidirectional fiber reinforced composite plate and for a multilayer plate with different-direction reinforcement of layers. Using the finite element method in combination with the initial stress method, the stress and strain patterns are analyzed in rectangular one-, two- and three-layer carbon fiber reinforcement polymer plates with a circular cut under the action of external stresses at the boundary.