Dissimilar Elastic Half-planes Research Articles

Mathematical modeling of thermoelastic state in a tree-component piecewise-homogeneous plate containing a crack

Purpose. A two-dimensional mathematical model of the problem of thermoelasticity for three-component plate containing a crack has been built. The stress intensity coefficients in the vertices of the crack increase affecting strength of the body significantly. This leads to the growth of a crack and, as a result, to further local destruction of a material. Therefore, such a model reflects, to some extent, the destruction mechanism of the elements of engineering structures with cracks. Graphic dependences of stress intensity factors (SIFs) at the tops of the crack have been built. This would make it possible to obtain the critical values of constant temperature in the two joined dissimilar elastic half-planes containing an inclusion and a crack in order to prevent crack growth, which would not allow the local destruction of the body. Methodology. Based on the method of the function of a complex variable we have studied the two-dimensional thermoelastic state for body with crack as stress concentrators. As result, the problem of thermoelasticity was reduced to a system of two singular integral equations (SIE) of the first and second kind, a numerical solution of which was found by the method of mechanical quadratures. The two-dimensional mathematical model of the thermoelastic state has been built in order to determine the stress intensity coefficients at the top of the crack and inclusion. The systems of singular integral equations of the first and second kinds of the specified problem on closed (contour of inclusion) and open (crack) contours are constructed. Numerical solution of the integral equations in the case of constant temperature in the two joined dissimilar elastic half-planes containing the crack and an inclusion was obtained by the mechanical quadrature method. Influence of thermophysical and mechanical properties of an inclusion on the SIF sat the crack types was investigated. Graphic dependences of the stress intensity factors which characterize distribution of the intensity of stresses at the vertices of a crack have been built, as well as on its elastic and thermoelastic characteristics of inclusion. This would make it possible to analyze the intensity of stresses in the neighborhood of a crack vertices depending on the geometrical and mechanical factors. As a result, this allow to determine the critical values of temperature in the three-component plate containing a crack in order to prevent the growth of the crack, as well as to prevent the local destruction of the body. It was found that that the appropriate selection of mechanical and thermophysical characteristics of the components of a three-component plate containing a crack can be useful to achieve an improvement in body strength in terms of the mechanics of destruction by reducing stress intensity factors at the crack’s vertices. Originality. The solutions of the new two-dimensional problem of thermoelasticity for a specified region (a two joined dissimilar elastic half-planes containing inclusion and a crack) due to the action of constant temperature is obtained. The studied model is the generalization of the previous models to determine the two-dimensional thermoelastic state in a piecewise-homogeneous plate weakened by internal cracks. Practical value. The practical application of this model is a more complete description of the stress-strain state in piecewise homogeneous structural elements with cracks operating under temperature loads. The results of numerical calculations obtained from the solution of systems of equations and presented in the form of graphs can be used in the design of rational modes of operation of structural elements. This takes into account the possibility of preventing the growth of cracks by appropriate selection of composite components with appropriate mechanical characteristics.

Examining elastic interaction between a crack and the line of junction of dissimilar semi-infinite plates

We examined a two-dimensional mathematical model for the problem of elasticity theory on welded dissimilar elastic half-planes containing rectilinear cracks under the action of mechanical efforts on the shores of a crack. As a consequence, the intensity of stresses in the vicinity of tops of the cracks increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks when the water, contained in them, freezes to ice. It creates normal pressure on the shores of the cracks. Based on the application of the apparatus of singular integral equations (SIE), the problem is reduced to the system of SIE of the first kind on the contours of cracks. We obtained numerical solutions to the corresponding integral equation in particular cases of two welded dissimilar half-planes with one randomly-oriented crack, as well as a two-link irregular crack, which crosses the line of junction when the crack’s shores are exposed to uniformly distributed normal pressure. By employing these solutions, we determined stress intensity coefficients (SIC) at the tops of the crack, which are subsequently used to determine critical values of the normal pressure on the shores of the crack. We built graphic dependences of SIC, which characterize distribution of the intensity of stresses at the tops of a crack, on the angle of crack inclination and elastic characteristics of half-planes. This makes it possible to analyze the intensity of stresses in the vicinity of a crack’s tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of normal pressure on the shores of the crack at which the growth of the crack starts, as well as the local destruction of the body. It is shown that the proper selection of elastic characteristics of the components of welded dissimilar half-planes can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIC at the crack’s tops

Stress intensity factors for four interface-close cracks between a nonhomogeneous bonding layer and one of two dissimilar elastic half-planes

Abstract Stress intensity factors are estimated for four collinear interface cracks that are situated at the interface between a nonhomogeneous elastic bonding layer and one of two dissimilar elastic half-planes. Tensile stress is applied normal to the cracks. The material properties of the bonding layer vary continuously from those of the lower half-plane to those of the upper half-plane. To solve this problem, two solutions are combined: the solution for two inner collinear cracks and the solution for two outer collinear cracks. To satisfy the boundary conditions for the combined solution, the differences in the crack surface displacements are expanded as a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved using the Schmidt method so they satisfy the conditions inside the four cracks. The stress intensity factors are calculated numerically for the selected crack configurations.

Mode-I crack in a magnetoelectroelastic layer sandwiched by two elastic half-planes

In this paper we study the impermeable opening crack problem in a magnetoelectroelastic layer sandwiched between dissimilar elastic half-planes under in-plane mechanical, electric and magnetic loadings. Fourier transforms are applied to reduce the crack problem to dual integral equations, which can be further reduced to solving a system of simultaneous Fredholm integral equations. Numerical solutions of the integral equations can be obtained by using the collocation method, and the explicit form solution of the near crack-tip fields is obtained asymptotically. The field intensity factors are obtained and the effects of material properties, geometric size, and loadings are analyzed.

Thermal Stresses Around Two Parallel Interface Cracks Between a Nonhomogeneous Bonding Layer and Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

Composite materials consisting of two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer contain two interface cracks; one is situated at the lower interface between the layer and the lower half-plane, while the other is situated at the upper interface between the layer and the upper dissimilar half-plane. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique, and they are satisfied outside the cracks by expanding the differences in temperature and displacements at the crack surfaces using a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.

Random dynamic response and reliability of a crack in a functionally graded material layer between two dissimilar elastic half-planes

Abstract An analytical approach is presented for the random dynamic analysis of a functionally graded material (FGM) layer between two dissimilar elastic half-planes. This FGM layer contains a crack and its material properties vary randomly in the thickness direction, while their mean values are exponential functions of field position. The transient loadings applied on the crack faces are assumed to be stochastic processes of time. In order to obtain the solution, the FGM layer is divided into several sub-layers, and the material properties of each layer are reduced to random variables by an average method. A fundamental problem is constructed for the solution. Based on the use of Laplace and Fourier transforms, the boundary conditions are reduced to a set of singular integral equations, which can be solved by the Chebyshev polynomial expansions. Both stress intensity factor history with its statistics and dynamic reliability are analytically derived. Numerical calculations are provided to show the effects of related parameters.

Dynamic stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar elastic half-planes subject to an impact load

Some composite materials are constructed of two dissimilar half-planes bonded by a nonhomogeneous elastic layer. In the present study, a crack is situated at the interface between the upper half-plane and the bonding layer of such a material, and another crack is located at the interface between the lower half-plane and the bonding layer. The material properties of the bonding layer vary continuously from those of the lower half-plane to those of the upper half-plane. Incoming shock stress waves impinge upon the two interface cracks normal to their surfaces. Fourier transformations were used to reduce the boundary conditions for the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded in a series of functions that are zero-valued outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.

Thermal Stresses Around Two Collinear Interface Cracks between a Nonhomogeneous Bonding Layer and One of the Two Dissimilar Elastic Half-Planes Under Uniform Heat Flux

In composite materials, in which two dissimilar elastic half-planes are bonded by a nonhomogeneous elastic layer, two collinear cracks are situated at the interface between the nonhomogeneous elastic layer and one of the two dissimilar half-planes. The stress intensity factors are solved under uniform heat flux normal to the cracks. The material properties of the bonding layer vary continuously from the lower half-plane to the upper half-plane. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. In order to satisfy the boundary conditions outside the cracks, the differences in temperature and displacements at the crack surfaces are expanded in a series of functions that vanish outside the cracks. The unknown coefficients in each series are evaluated using the Schmidt method. The stress intensity factors were calculated numerically for selected crack configurations.

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.

A semi-analytical solution for multiple circular inhomogeneities in one of two joined isotropic elastic half-planes

The paper presents a semi-analytical method for solving the problem of two joined, dissimilar isotropic elastic half-planes, one of which contains a large number of arbitrary located, non-overlapping, perfectly bonded circular elastic inhomogeneities. In general, the inhomogeneities may have different elastic properties and sizes. The analysis is based on a solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. Apart from round-off, the only errors introduced into the solution are due to truncation of the Fourier series. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the inhomogeneities. Numerical examples are included to demonstrate the effectiveness of the approach.

204 異材界面近くに存在する3個のき裂周囲の応力拡大係数について(GS 材料力学(その1))

204 異材界面近くに存在する3個のき裂周囲の応力拡大係数について(GS 材料力学(その1))

Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape

A general method is presented for the rigorous solution of Eshelby’s problem concerned with an arbitrary shaped inclusion embedded within one of two dissimilar elastic half-planes in plane elasticity. The bonding between the half-planes is considered to be imperfect with the assumption that the interface imperfections are uniform.

面内力問題における等方性き裂と異材界面き裂に対して適用可能な応力関数

In this study, one set of stress function is shown on crack problem in an infinite isotropic plate. Features of obtained stress function are the plane elasticity problem that are common to both of an interface crack between two dissimilar elastic half planes and an infinite isotropic plate containing a central straight crack, and it is linearly independent at the crack tip. In some numerical examples, energy release rate for Young modulus ratio (E1/E2) has been obtained in an interface crack problem. Besides, the energy component of shearing stress has also been shown in these figures.

Investigation of the behavior of a griffith crack at the interface between two dissimilar orthotropic elastic half-planes for the opening crack mode

The behaviors of an interface crack between dissimilar orthotropic elastic half-planes subjected to uniform tension was reworked by use of the Schmidt method. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, of which the unknown variables are the jumps of the displacements across the crack surfaces. Numerical examples are provided for the stress intensity factors of the cracks. Contrary to the previous solution of the interface crack, it is found that the stress singularity of the present interface crack solution is of the same nature as that for the ordinary crack in homogeneous materials. When the materials from the two half planes are the same, an exact solution can be otained.

Stress intensity factors around a moving Griffith crack in a non-homogeneous layer between two dissimilar elastic half-planes

Stresses are determined in the vicinity of a propagating finite crack having a constant velocity in a non-homogeneous elastic layer sandwiched between two dissimilar elastic half-planes. The self-equilibrated system of pressure is applied to the crack surfaces. Application of the Fourier transform technique reduces the problem to that of solving dual integral equations. In order to solve the equations, the differences in the crack surface displacements are expanded in a series of functions that are equal to zero outside the crack. The unknown coefficients in the series are solved by the Schmidt method. The stress intensity factors are calculated numerically for a crack in a non-homogeneous layer between a half-plane made of epoxy resin and a half-plane made of aluminum.

Thermal stresses around a crack in the nonhomogeneous interfacial layer between two dissimilar elastic half-planes

Thermal stresses around a crack in the interfacial layer between two dissimilar elastic half-planes are solved. The surfaces of the crack are assumed to be insulated. The material constants of the interfacial layer are assumed to vary continuously from those of the upper half-plane to those of the lower half-plane. Uniform heat flows perpendicular the crack. Stress intensity factors are calculated numerically for several thicknesses of the interfacial layer.

Transient dynamic stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

Dynamic stresses around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes are obtained. The material constants vary continuously in the layer. An incoming shock stress wave impinges on the crack at right angles to the crack faces. In order to solve the problem, the interfacial layer is divided into several homogeneous layers that have different material properties. The boundary conditions are reduced to dual integral equations using the Fourier–Laplace transform technique. The equations are solved by expanding the differences of the crack faces in a series in the Laplace transform domain. The unknown coefficients in the series are determined using the Schmidt method. The stress intensity factors in the transform domain are inverted numerically in physical space. Using these numerical results, the stress intensity factors for a very thin layer can be estimated.

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.

K-0606 界面き裂を有するセラミックス被覆複合材の熱応力について(S05-2 複合材料システムの弾性数理モデリング(2))(S05 複合材料システムの弾性数理モデリング)

Thermal stresses around a crack in the interfacial layer between two dissimilar elastic half-planes are solved. The surfaces of the crack are assumed to be insulated. The material constants of the interfacial layer are assumed to vary continuously from those of the upper half-plane to those of the lower half-plane. Uniform heat flows perpendicular to the crack. Stress intensity factors are calculated numerically for several thicknesses of the interfacial layer. Using the numerical results, the stress intensity factors are obtained for an infinitesimally thin layer.