Fracture mechanics of a linearly strengthening body

Abstract

539,3 Supplementing and extending our results obtained earlier [ 1 ], here we consider the plane problem of small deformations of an elastic isotropic homogeneous body with a rectilinear crack of length 2/0 loaded at infinity. The approximation of the experimental curve "stress-deformation" by two straight lines has its advantages and disadvantages. Among the advantages, one should mention the simplicity of approximation of the diagram (3 -~, the possibility of dealing with nonlinear materials close to perfectly plastic and linearly elastic ones, the constant singularity in stresses near the crack tip rl s (r I << 10, s = -1/2), significant successes in the study of this problem [2,3], and a correlation with different approaches [1][2][3][4][5]. The disadvantages are the following: the loss of generality in the description of experimental curves (3 -e, the presence of a sudden change of slope at the diagram (3 -~, the dependence of the strengthening parameter of the material on the admissible deformation, and the small-scale yield or nonlinearity of the bodies. The value of the tangential modulus E t (modulus of strengthening) is 5-10 times less than the initial one E; however, there are nonlinear materials slightly differing from linear ones. We assume that conditions of their fracture by means of crack growth have been created.It is necessary to determine the effect of the physical nonlinearity and compressibility of the material (Poisson's ratio ta) on the stress intensity factors and to establish the difference between plane strain and a plane stressed state.We use the relation between small elastic deformations and stresses [2], which follows from formulas (1) and (16) where if C e > 1, and ~ = 0 if G e < 1. Here, e0 and (36 are the components of the deformation and stress tensors, respectively, sij are the components of the deviator stress tensor, (3kk = (311.4-(322 + (333, ~ij is the Kronecker symbol, and (3e is the stress intensity under uniaxial tension (e.g., (3e = (3,, under the Mises plasticity condition, (3s is the yield point under uniaxial tension).For a small-scale nonlinear loosening or plasticity, l under conditions of the Griffith problem and plane stressed state, the corresponding stress intensity factor for a normal tensile crack is [2] 1 The scale of plasticity or physical nonlinearity for cracked bodies is established in the literature by the ratio (K 1 / ~,~) 2 ~ Therefore, the elastic solution controls the smallness of the scale.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv.