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. Bibliography: 7 titles
Using the generalized complex potential and the method of least squares we solve the problem of the torsion of multiconnected anisotropic cylindrical rods. This problem is reduced to a system of linear algebraic equations in the unknowns that occur in the required function. By numerical studies we show the influence of the elastic and geometric characteristics of the rod on the stress distribution, the potential energy densities, and the variation of the stress intensity factor in an elliptic cylinder with one or two cracks and in an annular cylinder with a radial crack. Six figures. Three tables. Bibliography: ,~ titles.
539.3On the basis of numerical studies of the solution of the problem for an anisotropic plate with an arbitrary elliptic hole it is shown that a rectilinear slit can be modeled as an ellipse when the ratio of the semiaxes is less than 0.001, and as a triangle or other figure when the ratio of the sides is less than 0.001. It is shown that a slit cannot be modeled as a rectangular hole with any nonzero width. Two figures. Bibliography: 5 titles.In a previous paper [1] the authors have developed a method of determining the two-dimensional stressed state of a multiconnected anisotropic body with a finite number of elliptic cavities. In the present 0aper this method is used for numerical study of problems of the legitimacy of various models of cracks.Consider an infinite rectilinearly anisotropic multiconnected plate weakened by elliptic holes with boundaries Ll(l = ~ that may be arbitrarily located relative to one another, or may be tangent, intersect, form boundaries of curvilinear holes, or become rectilinear slits. External forces act on the boundaries of the holes, and also at infinity in the form of constant forces Crx, or, "Cx, ). There is no torque at infinity.Determining the stressed state of this plate reduces to finding the complex potentials r (j = 1, 2) from the boundary conditions [ 1 ] 2 2Re'~,(-I.t j, 1) By *~(zj)=(X,,,, Y,,) on L, (l=l-~), (I) j=l where 8 i =cosny-l.tjcosnx; cosruc dy cosny= dx Ixj are the roots of the known characteristic equation [2]; n is the exterior normal to the curve Lt, and X.l and Y.l are the components of the external forces on the curve L t . In this case the complex potentials *;(zj) defined in the regions Sj outside the ellipses Lit (/= ~ obtained from the given region by the affine transformations z2 = x + btjy have the form [ 1 ] L t=, n=, ;7[' Rjt(;~t-nil)" (2) Here I"j are known constants [1], ~jt are variables computed from the conformal mapping of the exterior of the unit disk I;" I > 1 onto the exterior of the ellipses tjl Zj -Zjl 0 = ejl(;j, Jr" mfl/;fl), and Bit and mjt are constants [1] that depend on the shape and size of the ellipses L t.By applying the method of least squares [1], from the boundary conditions on L t used to determine the unknown constants ajl p that occur in the function (2) we obtain a system of linear differential equations m=l Z Zs=I Zr=l Rsr(~2sr_msr)~sPr-I "I Rsr(~sr_msr)~s ='--g-I :
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.
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