The distortion method in the theory of thermoelasticity of bodies with cracks

Кіт, Г. С.

doi:10.1007/bf00715589

Cited by 5 publications

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“…For numerical solution of Eqs. (4) and (6) we apply the technique proposed in [1]. We write the unknown function ot3(x ) as a product It follows from these results that K~ depends strongly on the relations between the shear moduli of the compound body.…”

Section: A"ij ~Ct'(~)dsrmentioning

confidence: 97%

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The effect of the interface between media on stress concentration in the vicinity of a crack

Khai¹,

Stepanyuk²

1998

J Math Sci

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The problem of determining the effect of the media interface in a piecewise-homogeneous body on the stress concentration in the vicinity of cracks located in one of the half-spaces is reduced to a system of two-dimensional singular equations of Newtonian potential type. We study the effect of the relations of the elastic constants of the materials of a body composed of two half-spaces on the stress intensity factors in the vicinity of a crack. We study different variants of the crack location relative to the interface.The relations between the elastic constants of the materials of a piecewise-homogeneous body have a strong effect on the stress concentration in the vicinity of cracks located in the body. These problems are mathematically quite complicated and cumbersome, as shown by the limited amount published on them. A piecewise-homogeneons body composed of two half-spaces is the simplest object for which it is possible to study the effect of the relations between the elastic constants of the materials on the stress concentration in the vicinity of cracks.Suppose an infinite piecewise-homogeneous body composed of two half-spaces contains in its lower halfspace a system of N planar cracks, arbitrarily situated. The surfaces of the cracks are subject to given selfbalancing external forces N/., j = 1, 3, n = 1, N. Here NI. and N2. are the shears, and N~. are the normal external forces prescribed on the n th crack. The upper half-space is characterized by its Poisson coefficient v t and its shear modulus G~. The lower half-space, weakened by the system of N cracks is characterized by the respective elastic constants v 2 and G 2 .To solve the problem of determining the stress concentrations in the vicinity of the cracks we choose a basic Cartesian coordinate system OXIoX2oX30 in such a way that the xtoox2o-plane coincides with the interface between the media. We also prescribe local Cartesian coordinate systems Oxl.x2.x3. with origin O. in the region S., n = 1, N, occupied by the nth crack so that the x~.O.x2.-plane coincides with the region of the crack S.. When this is done, the opposite faces of the cracks S. ~ correspond to the values x3. = _+0. The locations of the cracks in the lower half-space are defined by giving the distances between the origins of the coordinate systems, the direction cosines ej0 . , ej,,0, j = 1,---3, n = 1, N, of the vector d.0 joining the points O. and O (Fig. 1), and the direction cosines of the axes of the local coordinate systems O.xj. in the Ox~oX2oX3o coordinate system, which are given by Table 1.The external loads producing a strain on the piecewise-homogeneous body under consideration cause a mutual displacement of the opposite faces of the cracks. These displacements are characterized by functions ctj., j=l, 3, n=l,N. In the process of reducing the problem of determining the stress-strain state of this body with cracks to boundary integral equations it is necessary to have a solution that makes it possible to satisfy the boundary

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Section: A"ij ~Ct'(~)dsrmentioning

confidence: 97%

“…with the following relations: In relations (1) the functions r 9 u). (X.o) are the components of the displacement vector, determined by the displacement jumps on the opposite surfaces of the cracks using the following formulas [1 ]:…”

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confidence: 99%

The effect of the interface between media on stress concentration in the vicinity of a crack

Khai¹,

Stepanyuk²

1998

J Math Sci

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“…Displacements in the form (1) satisfy the Lam6 equation for steady vibrations [2] and the radiation condition for arbitrary values of the functions Auj. The stresses they cause satisfy the self-balancing condition on the surface AS.…”

Section: Y2 G/~n= Asys:--s-~mentioning

confidence: 99%

Boundary-integral formulation of three-dimensional problems of steady vibrations of an infinite body with a crack located on an open Lyapunov surface

Mikhas'kiv¹

1998

J Math Sci

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We construct integral representations of the solutions of three-dimensional problems in the form of a combination of Helmholtz potentials [1] whose coefficients are determined in terms of the geometric parameters of the crack. By satisfying the boundary conditions we obtain two-dimensional integral equations in the displacement jumps on the opposite surfaces of the crack when it is loaded by harmonic forces.The method of boundary integral equations has been used previously to solve three-dimensional dynamic problems of the theory of elasticity for an infinite body with one crack [3] or a system of planar cracks [5], as well as the axisymmetric problem [6,7] for a hemispherical crack.Consider an infinite elastic isotropic body with a crack located along an arbitrary open Lyapunov surface [1] (see figure). On the opposite surfaces of the crack S'there are self-balancing forces varying in time t according to a harmonic law. These forces are N § =-N-= iqexp (-iot), where lq is the amplitude of the vibrations and co is the frequency of vibrations. Partial contact of opposite surfaces of the crack is not taken into account, and we require that the Sommerfeld radiation conditions hold at infinity [I]. In this formulation the exponential time factor drops out of the solution and the problem reduces to finding the corresponding amplitude characteristics, which we denote by the superscript A over the functions.Imagining an arbitrary nonplanar surface of the crack S = S § partitioned into elementary regions AS, S = U AS, and assuming that these elementary regions are flat because of their smallness, to determine the displacements %, j = 1,---3, at a point y of the body from the jumps Au s of the displacements on the surface AS we apply the formulas of [5], which we write as follows: uj(y)=0hr22 0(.~3,-

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“…It should be pointed out that such a treatment has been proved appropriate, as evidenced by various scholars studying the crack problem without thermal effect [37][38][39].…”

Section: Intrinsic Links Between Fundamental Solutionsmentioning

confidence: 99%

Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional hexagonal quasi-crystal under thermal loading

2013

Proc. R. Soc. A.

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The distortion method in the theory of thermoelasticity of bodies with cracks

Cited by 5 publications

References 0 publications

The effect of the interface between media on stress concentration in the vicinity of a crack

The effect of the interface between media on stress concentration in the vicinity of a crack

Boundary-integral formulation of three-dimensional problems of steady vibrations of an infinite body with a crack located on an open Lyapunov surface

Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional hexagonal quasi-crystal under thermal loading

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