The two-dimensional stressed state of a multiconnected anisotropic body with cavities and cracks

We present a method of determining the two-dimensional generalized stress-strain state and the stress intensity factors for an anisotropic body with cylindrical cavities and plane cracks. The method is based on the use of generalized complex potentials, conformal mappings, the method of least squares, and numerical passage to the limit to determine the stress intensity factors. We apply the method to study the stress-strain state and the change in stress intensity factors as functions of the geometric and elastic characteristics of an orthotropic cylinder with one or two cracks, an infinite anisotropic body with elliptic cavities and cracks, and an infinite body with a curvilinear cavity. Five figures. Six tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 45–56.     

The stress distribution in an anisotropic half-plane with two elliptic holes or cracks

By use of the method of complex potentials, conformal mappings and least squares this problem is reduced to solving a system of linear algebraic equations with respect to the unknown constants that occur in the required functions. We describe the results of numerical studies of the variation of the stress intensity factors for cracks in an anisotropic half-plane under tension of the half-plane and force on its boundary. Two figures, two tables. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 57–61.     

On the fundamental relations of the two-dimensional theory of cracks of an anisotropic body

Representations obtained by the author in an earlier paper for the complex potentials of the general two-dimensional problem of the theory of cracks of an anisotropic body are brought into more convenient form. A new form is introduced for the stress intensity factors. For certain types of load the value of the stress intensity factors are exhibited. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 18–26     

Electroelastic State of a Multiply Connected Piezoelectric Half Plane with Holes and Cracks

A method has been proposed [1] for solving two-dimensional electroelasticity problems using generalized complex potentials. General representations of complex potentials for a multiply coupled region have been studied [2] and a method for calculating the stress intensity factors and induction has been introduced. In this article, general expressions are obtained for the complex potentials for a multiply connected half plane with arbitrarily positioned holes and rectilinear cracks and the electroelastic state of a half plane with a single elliptical hole or rectilinear crack is studied.     

楔型向错偶极子和裂纹的干涉效应

研究了晶体材料中一个楔型向错偶极子与裂纹的弹性干涉效应．运用复变函数方法获得了复势函数和应力场的封闭形式解答,导出了裂纹尖端应力强度因子和作用在向错偶极子中心点像力的解析表达式．获得了向错偶极子的位置、方向和偶臂长度对裂纹尖端应力强度因子的影响规律,并讨论了裂纹附近向错偶极子的平衡位置．结果表明向错偶极子靠近裂纹尖端时,对应力强度因子有明显的屏蔽或反屏蔽作用．     

Antiplane strain of a multiconnected anisotropic body with longitudinal cavities and plane cracks

Applying the theory of generalized complex potentials and the method of least squares, we solve the problem of the stressed state of a multiconnected anisotropic body under an antiplane strain. The problem is reduced to a system of linear algebraic equations in the unknown constants that occur in the required functions. By numerical studies we exhibit the influence of the elastic and geometric characteristics on the stress distribution and the variation of the stress intensity factors in a cylinder with one or two cracks and in an infinite body with circular cavities and cracks. One figure. Six tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25. 1995, pp. 56–62.     

Stress concentration in an anisotropic half-plane with holes and cracks

On the basis of general representations of the generalized complex potentials for a multiconnected half-plane, which the authors have obtained, we solve problems for a multiconnected half-plane with holes and cracks when external forces or dies act on the boundary of the half-plane. Using conformal mapping for an ellipse and the method of least squares, we reduce these problems to solving a system of linear algebraic equations. For different anisotropic materials we give the results of studies of the stress distributions and the variation of the stress intensity factors for a half-plane with a crack in the case of tension at infinity, internal pressure on the edges of the crack, and the action of normal forces on the rectilinear boundary. Two figures, 2 tables. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 63–72.     

Stress Concentration in an Anisotropic Half Space with Cylindrical Cavities and Plane Cracks

A method is proposed for determining the two-dimensional stressed state of a half space with a general rectilinear anisotropy. General representations of the complex potentials are obtained and studied, as well as expressions for the stresses and displacements, along with the boundary conditions for determining these functions. As an example, we solve for the stressed state of and calculate the stress intensity factors for a half plane (in the presence of a single elastic symmetry plane) with a circular (elliptical) hole and edge cracks. It is shown how the crack length, the closeness of a hole with a crack to the boundary, and the anisotropy of the material affect the stress concentration and stress intensity factors.     

Stressed state of an anisotropic body with elliptical holes, elastic inclusions, and cracks

A method is proposed for studying the two-dimensional stressed state of a multiply connected anisotropic body with cavities and elastic and rigid inclusions, as well as planar cracks and rigid laminar inclusions. Generalized complex potentials, conformal mapping, and the method of least squares are used. The problem is reduced to solving a system of linear algebraic equations. Formulas are given for finding the stress intensity factors in the case of cracks and laminar inclusions. For an anisotropic plate with a single elliptical hole or a crack and an elastic (rigid) inclusion, some numerical results are presented from a study of the effect of the rigidity of the inclusion and the closeness of the contours to one another on the distribution of stresses and the stress intensity factor. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 175–187, 1999.     

On a method of studying the stress state of orthotropic shells and plates weakened by a hole with cracks extending to its edge

On the basis of the Timoshenko kinematic hypothesis for shells we give a formulation of the problem of studying the stress-strain state of orthotropic plates and shells weakened by a combined stress concentrator (a hole with two symmetric cracks extending to its edge). We propose a method of solving such problems on the basis of the finite-element method. To simulate the singularity of the stresses and displacements in a neighborhood of the tip of a crack we apply special finite elements with degenerate faces and nodes displaced by 1/4 the length of an edge. The stress intensity factors are found in terms of the displacements of the nodes of such elements. We give the results of computation of the concentration coefficients and the stress intensity factors for spherical and cylindrical shells loaded by internal pressure and for a cylindrical shell and a plate under the action of a distending load with various concentrators: a circular hole, an isolated crack, and a combined concentrator. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 48–54.     

Numerical computation of the stress-strain state at the end of a crack under dynamic load

We obtain the values of the stress intensity factor at the end of a stationary crack under dynamic load using a numerical method. We compare the results with the existing experimental data. We establish that it is possible to apply the proposed method to compute stationary cracks under dynamic load. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 61–65.     

A hybrid method of studying the stress intensity factors at the tips of cracks in anisotropic plates

We propose an efficient new method of numerical analysis of the residual strength of finite plates weakened by cracks. The basis of the method is an alternating Schwartz procedure that makes it possible to achieve a successful combination of the algorithmicity and indifference to boundary conditions of the finite-element method and the application of the method of integral equations for studying singular stress fields. The numerical implementation of the scheme and the algorithm for determining the stress intensity factors at the tips of cracks is given as a software package written in FORTRAN. The efficiency of application of the computational methodology is illustrated by examples. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 34–40.     

The periodic problem for a system of coplanar thin ribbons under longitudinal shear

We give the general outline of the construction of systems of singular integral equations for linearly and circularly periodic two-dimensional problems of the theory of cracks and thinwalled inclusions and the solution of these problems using the method of collocations taking account of the limited number of intervals of integration. As an example we study the dependence of the generalized stress intensity factors on identical systems of thin elastic ribbons forming one, two, and three columnsin an infinite isotropic medium using the conditions of longitudinal shear under the influence of a uniform stress field at infinity. In a particular case we obtain results for the corresponding systems of cracks or absolutely rigid inclusions and films. The method makes it possible to study the interaction of rows of closely placed defects. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 91–99.     

A two-dimensional problem for an anisotropic body with a finite number of “tunnel” cracks

We give the results of studies of the stress state for an infinite anisotropic body with a number of planar cracks along a single plane. For the simplest types of load we prove that the stress intensity factors is independent of the type of anisotropy. We describe the results of numerical studies as functions of the geometric characteristics of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 27–34     

Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.     

Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal

For a one-dimensional (1D) hexagonal quasicrystal (QC), there is the periodic (x₁,x₂)-plane of atomic structures with the quasiperiodic direction x₃-axis along which there exists a phason displacement. The macroscopically collinear periodic cracks and/or rigid line inclusions are placed on the periodic (x₁,x₂)-plane for finding out the influence of phason displacement on the related physical quantities. These two models are reduced to the Riemann–Hilbert problem of periodic analytic functions to obtain the closed-form solutions for the antiplane sliding mode. It is found that the phonon and phason stress intensity factors of cracks as well as the phonon and phason stress field intensity factors of rigid line inclusions are not related to the coupling of phonon and phason fields. These mean that there is not the influence of phason displacement on both the phonon stress intensity factor (usual stress intensity factor) of cracks and the phonon stress field intensity factor of rigid line inclusions. However, the energy release rates of periodic cracks and/or rigid line inclusions are obtained and affected not only by the periodicity of cracks and/or rigid line inclusions but also by the phason displacement.     

The stress state of rock with an excavation and load-removing slits

We propose a method of determining the stress state of an anisotropic rock with an excavation, including the case in which there are load-removing slits extending to the surface. In cross section the excavation can have an arbitrary curvilinear outline. The rocks have any rectilinear anisotropy, and may have planes of elastic symmetry with an arbitrary inclination to the horizontal plane. The method is based on solving the two-dimensional problem of the theory of elasticity of an anisotropic body using generalized complex potentials, conformal mappings, and the method of least squares. The problem is reduced to solving a system of linear algebraic equations in the unknowns that occur in the complex potentials. We carry out detailed numerical studies on the distribution of stresses and lowering their concentration around the excavation by setting up a system of load-removing slits. The proposed method of studying the stress state of rocks around an excavation both with and without slits and makes it possible not only to establish the zone of high stress concentration, but also to choose the optimal combination and location of load-removing slits. Its application by specialists in the coal industry will make it possible to establish the stability of excavations taking account of their geometric parameters and the depth, length, and location of the load-removing slits, and the physico-mechanical characteristics of the rock. Five figures. One table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 28–35.     

The stress state of an anisotropic plate with two arbitrarily located elliptic holes or cracks

Using Lekhnitskii's method of complex potentials we study the stress state of an anisotropic plate with two arbitrary elliptic holes. We consider the cases when one or both of the holes become narrow slit-cracks, when the cracks extend to the edge of the hole, and when two cracks intersect or form a broken two-link crack. We give the results of numerical studies. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 33–43.     

The interaction between a screw dislocation and a rigid line in a confocal elliptical inhomogeneity

The interaction between a screw dislocation and an elastic elliptical inhomogeneity which contains a confocal rigid line is investigated. The screw dislocation is located inside either the elliptical inhomogeneity or the infinite matrix. By using the complex potential method, explicit series solutions of complex potentials are obtained. The image force acting on the screw dislocation and the stress intensity factor at the tip of the rigid line are derived. As a result, the analysis and discussion show that the influence of the rigid line on the interaction effects between a screw dislocation and an elliptical inhomogeneity is significant. The rigid line enhances the repulsive force exerted on the dislocation produced by the stiff inhomogeneity and abates the attractive force produced by the soft inhomogeneity. For the soft inhomogeneity, there is an unstable equilibrium position when the dislocation is inside the matrix and there is a stable equilibrium position when the dislocation is inside the inhomogeneity. The stress intensity factor contour around the rigid line tip shows that when a dislocation with positive burgers vector is in the upper half-plane, stress intensity factor will be positive; while in the lower half-plane, stress intensity factor will be negative; and in the x-axis, it will be zero. The absolute value of the stress intensity factor will increase when the dislocation approaches the tip of the rigid line. The stress intensity factor at the rigid line tip is enhanced by a harder matrix and abated by a softer matrix.