Piecewise-homogeneous Plane Research Articles

Numerical modeling of 2-D smooth crack growth

Numerical algorithm to simulate 2-D smooth crack is presented. The stepwise method based on local criteria of propagation is used. Two crack propagation criteria are employed. At the first stage of propagation, the maximum tensile stress criterion is used to take into account the abrupt change in tangent direction. At subsequent stages, the assumption that the stress intensity factor (SIF) K_2= 0 at the current crack tip is exploited. The analytical formulae for calculating SIFs are given. The displacement discontinuities (DD) involved in these formulae are found from the numerical solution of a complex hypersingular integral equation (CHSIE) for a piecewise homogeneous plane with curvilinear cracks. The new mechanism of smooth approximation of the crack path by circular arcs at each propagation stage is suggested. Numerical results are given. They confirm the efficiency of the algorithm suggested.

An interface crack in an inhomogeneous stress field

The solutions for interface cracks of shear, opening and mixed modes (problems A, B, C) are obtained in elementary functions. In case C the crack surfaces partially overlap, slipping without friction. After removing the inhomogeneous stress field specified at the infinity the crack surfaces are loaded by normal and tangential stresses distributed according to a polynomial law. A detailed analysis is carried out for the solution corresponding to linear variation of these stresses. It develops that the strain-energy release rate varies slowly when going from problem C to problem A under compression-shear-bending of the piecewise-homogeneous plane, and from problem C to problem B when compression is replaced by tension. There is also a slow variation in the largest of moduli of the stress intensity factors with the elastic parameters. These results allow one to estimate the structure-toughness characteristics for inhomogeneous bodies by solving problems in simplified formulations.

Two approaches to the investigation of antiplane deformation of an isotropic solid with a thin elastic inclusion

An approach is proposed to the investigation of the state of stress and strain of a piecewise-homogeneous plane consisting of a matrix and a thin tunnel-like rectangular inclusion with rounded-off corners under the assumption that such a composite body is under antiplane deformation conditions. A numerical comparison is made of the results obtained in this paper and on the basis of an approximate model /1/. It is shown that they agree satisfactorily at sufficiently large distances from the vertices of the inclusion, when the inclusion is more pliable than the matrix.

Elastic equilibrium of circular piecewise-homogeneous media with a diametral crack

A method is proposed for solving boundary value problems of elasticity theory for circular piecewise-homogeneous media with a symmetric diametral crack, based on the application of vector relationships between the basis solutions of the equilibrium equations in polar and elliptic coordinates. The realization of this method results in infinite systems of linear algebraic equations of the second kind with exponentially decreasing matrix coefficients. The problem of the equilibrium of a two-component piecewise-homogeneous plane with a symmetric diametral crack is considered in an inner homogeneous domain. Asymptotic formulas for the stress intensity coefficients are obtained by expansion in a small parameter.

A piecewise-homogeneous plane with a separating boundary in the form of the sides of an angle and a symmetrical cut originating from the vertex

A plane problem of the equilibrium of an elastic plane formed by two different half-planes joined together along a boundary in the form of two sides of an angle, is considered. A symmetrical cut originates at the apex and a known, constant normal load is applied to its edges. A functional Wiener-Hopf equation of the problem is given and its exact closed solution is constructed. The stress intensity coefficient at the cut end is determined.

Elastic equilibrium of a piecewise-homogeneous plane with a thin hard inclusion intersecting the straight interface of the materials

Elastic equilibrium of a piecewise-homogeneous plane with a thin hard inclusion intersecting the straight interface of the materials

On the singularity of the stresses near the face of a thin elastic inclusion in a piecewise homogeneous plane

The asymptotic behaviour of normal stresses near the tip of a thin elastic inclusion situated near a line weld joining two dissimilar elastic half-planes, is studied. It is established that apart from the well-known root-type singularity /1/ two additional terms of the asymptotic expression exist which must not be neglected. One of them is of the order of unity, and the other contains an “imaginary singularity” and makes a singificant contribution to the state of stress when the distances between the face of the inclusion and the line separating the materials are small.

State of stress of a piecewise-homogeneous plane containing a band with four circular holes

State of stress of a piecewise-homogeneous plane containing a band with four circular holes

Elastic equilibrium of a composite plane containing an arbitrarily oriented thin elastic inclusion

The integral Mellin transform is used to study the stress-strain state of a piecewise homogeneous plane consisting of two bonded, isotropic half-planes, one of them containing an elastic, thin-walled inclusion of finite length. Solutions of the problems of stress concentration near an elastic inclusion in a homogeneous plane /1/ and the problem of a mathematical cut in a piecewise homogeneous plane /2/ are found to be particular cases of the results obtained in the present paper. However, unlike in /2/, the choice of the path around the pole in the process of estimating the integral equations obtained, is substantiated.

Influence of the shape of a thin inclusion on the temperature distribution in a piecewise-homogeneous plane

The plane stationary problem of heat conduction is considered without taking account of heat elimination through the side surfaces of a composite body with thinwalled interlayers on the interface of materials.

Stress distribution in a strip with a thin elastic inclusion: PMM, vol. 43, no. 3, 1979, pp. 542–549

The problem of the stress distribution in a strip with a rectilinear arbitrarily oriented thin-walled elastic inclusion of finite length is considered. The problem is reduced to finding the solution of a system of four singular integral equations by using a Fourier integral transform. Numerical values are presented for the stress intensity coefficients on the endfaces of the elastic inclusion. The state of stress of a piecewise-homogeneous plane with a thin-walled elastic inclusion of finite length was investigated in [1,2]. A solution of the corresponding periodic problem is obtained in [3].

Periodic problem for an elastic plane with thin-walled inclusions: PMM vol. 39, n≗ 3, 1975, pp. 520–529

The state of stress of a homogeneous and piecewise-homogeneous plane with a thin-walled elastic inclusion of finite length was investigated in [1–4]. The main attention in [1, 2] was paid to determining the difference in the shear stresses on opposite edges of the inclusion under the assumption that the displacements are equal. Consequently, the solution of the Prandtl integral equation, to which the problem reduced, is suitable for flexible inclusions with a sufficiently high elastic modulus. The approach in [3, 4] for the case of a homogeneous plane was somewhat different, where the relative displacement of the inclusion edges was taken into account while the difference in the originating stresses was neglected. Some results for a corresponding periodic problem were obtained in [5, 6]. The periodic problem of elastic equilibrium of a plane consisting of two bonded isotropic half-planes with finite thin-walled elastic inclusions on the straight line of separation of the materials is studied. A system of singular integral equa tions is obtained, whose solution is suitable for inclusions of any stiffness from absolutely stiff or flexible but inextensible to absolutely yielding (slits). Therefore, a close connection is established between crack theory and the theory of thin-walled elastic inclusions. In the limit case, the solution is obtained for the problem of one inclusion in a more accurate formulation than in [1–4].

A mixed boundary-value problem of the theory of elasticity for a piecewise-homogeneous plane with cuts

A mixed boundary-value problem of the theory of elasticity for a piecewise-homogeneous plane with cuts

First fundamental problem for a piecewise-homogeneous plane with a slit perpendicular to the line of separation: PMM vol.32, n≗4, 1968, pp.647–659

First fundamental problem for a piecewise-homogeneous plane with a slit perpendicular to the line of separation: PMM vol.32, n≗4, 1968, pp.647–659