Particularizing the constitutive equations of elastoplasticity of a transversely isotropic material

Abstract

A method for particularizing the constitutive equations describing the elastoplastic deformation of a transversely isotropic material is proposed. The method employs data of tension and internal-pressure base tests on tubular specimens. The calculated results and test data are compared Keywords: transversely isotropic material, plasticity, constitutive equations Introduction. Anisotropic materials are widely used in modern structures. There are various constitutive equations to describe the deformation of such materials beyond elasticity [1-7, 9-16, 18, 20-22] and, hence, various ways to particularize these equations. A theory of anisotropic elastoplasticity was proposed by Mises [20]. This theory was further developed by Hill [14] who suggested a plasticity criterion containing six constants in the principal axes of anisotropy that can be determined from uniaxial-tension and pure-shear tests. In [3,21], the Tresca yield criterion for isotropic materials is generalized to anisotropic materials. The anisotropy constants are determined, as in [14,20], in terms of the tensile and shear yield stresses. The relations between the components of the stress and strain deviators can be found in [11]. It is assumed that the expression for the bulk modulus for anisotropic materials is the same as for isotropic materials. The constitutive equations of a transversely isotropic material contain three coefficients determined from uniaxial-tension tests on specimens cut out along the principal axes of anisotropy and at an angle of 45°. Constitutive equations invariant under rotation of the principal axes of anisotropy were derived in [12]. To particularize the constitutive equations, it is necessary to determine six functions of six invariants for an orthotropic material and four functions of four invariants for a transversely isotropic material, which is very labor-intensive (from the experimental standpoint) task. Therefore, it was assumed in [13] that the plastic strains parallel and transverse to the axis of transverse isotropy are zero. The base tests in this theory are tension, torsion, and internal-pressure tests. Two functions depending on the first and second generalized invariants of the stress tensors and containing the anisotropy tensor are introduced in [1]. The constitutive equations were not particularized. The constitutive equations in [6] contain the compliance matrix and one nonlinear function.Base tests on flat specimens with different orientation of the principal axes of anisotropy are conducted to plot the initial sections of stress-strain curve for different tension directions and one complete stress-strain curve. The compliance coefficients depend on the elastic constants of the material and are assumed invariable beyond elasticity. In the case of transverse isotropy, the shear moduli appearing in the constitutive equations derived in [5,16] are determined in tests involving uniaxial tension along the axis of symmetry, torsion, and combined loading of tubular specimens in the elastic range. The constitutiv...