539.3 Within the framework of the approach proposed in the first part of the work, the two-dimensional boundary-value problem for an isotropic body with noncanonical elastic inclusion is reduced to a finite system of linear algebraic equations. It is shown that the solution of this problem for elastic inclusions with small radius of curvature at the tip and/or cusps describes the intensity and concentration of stresses in the composition. For some special examples, we reveal the influence of elastic properties of the components of the composition and configuration of the inclusions on its local stress-strain state. It is also established that, unlike the method of perturbation of the shape of the boundary, this method is applicable to the determination of the concentration and intensity of stresses in the vicinity of the tips of elastic inclusions with small radii of curvature, including the inclusions whose elastic properties are close to the elastic properties of the matrix.In the first part of the present work [1], we suggested a method for the determination of the stress-strain state of isotropic bodies with curvilinear foreign elastic inclusions in the two-dimensional case. Here, we present some examples of application of this method. For inclusions of noncanonical shapes, the problem was reduced to the solution of a system of linear algebraic equations. It was shown that the investigated approach can be used for the determination of the concentration and intensity of stresses near the tips of inclusions with small radii of curvature. The influence of the configuration and elastic properties of inclusions on the local stress-strain state of the composition is analyzed for several special cases. Reduction of the Two-Dimensional Problem to a Finite System of Linear Algebraic EquationsLet us now return to the statement of the problem from [ 1 ]. We restrict ourselves to the case of a smooth contour L and present basic relationships of the suggested approach for the evaluation of a single complex stress potential ___l (Z). In what follows, the functions of z are denoted by underlined symbols.Let % (~) and ~l (~) be the limiting values on the contour ~/ (in a sense of transformation (8) The functions tp2(• ) and ~2 (0)
We consider thin sloping shells of a positive or zero Gaussian curvature satisfying the Kirchhoff-Love hypotheses with through cracklike holes. By using a method of perturbation of the shape of the boundary, we obtained numerical and analytic results in the case of"small" cracklike defects with a negligible but finite radius of curvature at the tips. By using new formulas of the Irwin type, we calculated the corresponding generalized stress intensity factors (factors of 1/~/p + 2z I in the stress distribution). As an example, we consider the elastic equilibrium of a spherical shell with a square rounded hole or an absolutely rigid inclusion.Presently, shells are a basic structure in different fields of industry because, not being heavy, they can carry considerable tensile and bending loadings.We consider only thin elastic isotropic homogeneous sloping shells of constant thickness with cracklike holes.I By calculating bending stresses, we use theories based on the Kirchhoff-Love hypotheses. By using the method of perturbation of the shape of the boundary, we obtained particular results in the case of "small" and, sometimes, "middle" cracklike holes and absolutely rigid inclusions with a small but nonzero radius of curvature at the tips. We calculate the stress intensity factors for cracks with the same accuracy as for the solved problem of an elliptic hole in sloping shells. We consider so-called convex shells with positive or zero Gaussian curvature RIR2 R12where R i are the radii of curvature of the shell.The problem is to determine the elastic equilibrium of thin sloping shells with lateral through nonreinforced holes, in particular, cracks or absolutely rigid inclusions, i.e., to calculate the corresponding stress intensity factors with regard for the radius of curvature at the tips of defect. The external admissible loading is assumed to be selfbalanced when the principal vector and the principal moment of forces applied to the edge of the shell L 0 or hole L l are equal to zero. In the case of a simply connected infinite region (an infinite shell with a through lateral hole), the uniqueness conditions of small complex elastic displacements are satisfied automatically.
eij = ~-~ k(so)(5o~3ij + ~-~ g(t2) (oO -(~o~3ij),Here, ~ij is the Kronecker symbol, aij and ~ij are the components of the strain and stress tensors, t o is the reduced intensity of tangential stresses, s o and (5 o are the mean strain and stress, respectively, K is the bulk Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv.
Within the framework of the classical theory of elasticity and the Leonov-Rusynko theory of macrostresses, we study the beginning of brittle fracture of a microinhomogeneous body with a slot-like defect of arbitrary size and sharpness. We determine the influence of the parameter p of material inhomogeneity, the linear size 2a of the defect, and the radius of curvature rt of the tip of the defect on the stress concentration and the characteristics of fracture of the material under tension. The effect of micro-and macrodefects whose configuration varies from cracks (ri --0) to cylindrical holes (r~ ---a), on the strength of materials is investigated from a common position.Microinhomogeneities randomly distributed in a body substantially disturb local stresses. Therefore, in some materials (e.g., gray cast iron and concrete), artificially made notches of certain sizes weakly or even do not affect strength. Therefore, in investigating the strength of bodies, it is necessary to take into account, along with the geometry of macrodefects, the actual structure of the material (microdefects and microinhomogeneities). It is known that fracture of specimens with stress concentrators, even ones made of rather brittle materials, occurs under loads much higher than those predicted by the formula %--r a where Op and Ou are the ultimate strengths of specimens with and without stress concentrators, respectively, and r a is the theoretical stress concentration factor [1]. For this reason, in engineering calculations, one usually employs the expressionwhere k a is the effective stress concentration factor, which is determined experimentally for various concentrators and materials and can be found in handbooks [2][3][4][5]. Deviation of the value of k, from the theoretical value is mainly caused by structural microinhomogeneities of the material and microplastic strains [3,4,[6][7][8]. , having developed a theory of macrostresses, substantially extended the framework of the theory of brittle fracture of microinhomogeneous materials. Theory of MacrostressesThis theory is based on a model of a microinhomogeneous but macrohomogeneous body. A body is called macrohomogeneous if the mechanical properties of all elementary volumes V 0 of certain size arbitrarily cut out from the body are identical. A sphere of radius p (which is taken as a structural parameter of the material) is chosen as the elementary volume.The stress-strain state of the body is characterized by a macroelongation em of diameter 2p and macrostresses calculated according to Hooke's law s m = 2G~ m + ~'ec, where G is the shear modulus, ~ is the Lam6 constant, Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv.
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.
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