The problem concerned with the limits of applicability of the deformation theory of plasticity

Abstract

The development of certain propositions of the investigation [6] is given in this paper, and the problem concerned with the limits of applicability of the deformation theory of plasticity [2] is considered for plane strain of a linear anisotropic medium. The results obtained are generalized for the three-dimensional case with certain additional assumptions.1. It is assumed [6] that the ratio of an increment of shear strength (dS m) to the value of the elemental shear strain (dTn/) depends only on the angle (co) between the directions m and/(n), so that the relation The choice of the hardening function in this form or another is not connected with the essence of the model thus proposed [6]. In each particular case the analytical representation of F(w) should be regarded as one of the possible forms of approximation, of the sensitivity of the material to the deformation anisotropy, that is convenient for the subsequent analytical calculations.In [6] the case of a logarithmic hardening functionis considered. As F(w), being a generalization of (1.2), we shall consider the expressionBy means of going to the limit, h ~ 0, we obtain from the formula (1.3) the logarithmic hardening function for P0 =l-hlnc. 2. For a monotonic plastic strain the problem of determining the connection between the stresses and strains for a given F(w) reduces to the determination of the three unknown functions (o, t), a, (t), ~ (t) (a~,2 (t) ~, o, "a = Here ~'s is the shear strength of the initially isotropic material; T(t), 0(t) are respectively the value of the maximum shear stress and the angle between /~ s, o Fig. 2 the directions of the principal stresses at an arbitrary instant of time t and at the instant to corresponding to the beginning of plastic strain: ~2 ~ (t) = cD l (t) --~, (to), T(t) = ~'V [e x (t) --e u