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.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Growth of Pressure-Induced Fractures in the Vicinity of a Wellbore

Growth of pressure-induced fractures originating from a wellbore at an arbitrary angle to the direction of far field stresses is considered. A parametric study of hydraulic fracturing process from the point of view of fracture mechanics is presented in conditions of slow and fast pressurization rate. The study is based on the linear elastic fracture mechanics formulation that involves a complex hypersingular equation (CHSIE) for a plane with a circular opening and a system of arbitrary curvilinear cracks. Stepwise method is used to model fracture propagation. Dimensionless parameters influencing the fracture path are defined.     

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.     

A boundary integral method for multiple circular holes in an elastic half-plane

This paper presents a semi-analytical method for solving the problem of an isotropic elastic half-plane containing a large number of randomly distributed, non-overlapping, circular holes of arbitrary sizes. The boundary of the half-plane is assumed to be traction-free and a uniform far-field stress acts parallel to that boundary. The boundaries of the holes are assumed to be either traction-free or subjected to constant normal pressure. The analysis is based on solution of complex hypersingular integral equation with the unknown displacements 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. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-plane. Several examples available in the literature are re-examined and corrected, and new benchmark examples with multiple holes are included to demonstrate the effectiveness of the approach.     

A three-phase circular inhomogeneity with imperfect interface under thermomechanical loadings in plane elasticity

Summary The solution for a homogeneous circular inhomogeneity embedded in an infinite elastic matrix with a single interphase layer plays a fundamental role in many practical and theoretical applications. In particular, it serves as the basis for the solution of the generalized self-consistent method in the mechanics of composite materials. Thus, the study of three-phase problems is of great interest.A general method is presented for the rigorous solution of a three-phase circular inhomogeneity under thermomechanical loadings in plane elasticity. The bonding at the inhomogeneity-interphase interface is considered to be inperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Although the problem of a three-phase circular inhomogeneity with imperfect bonding has previously been studied, it seems that the explicit expressions for the complete solutions cannot be located in the literature. In this paper, it is found that stress field within the inhomogeneity is determined by three, in general, complex coefficients while the stress field in the matrix is controlled by three other, in general, complex coefficients. The role of the interphase layer as well as the influence of the imperfect bonding condition, on the stress fields, is manifested by their effect on the six, in general, complex coefficients.The exact closed-form solutions are applied to the design of a three-phase circular inhomogeneity. In particular, for specific thermomechanical loadings, it is shown that a uniform stress state within the inhomogeneity can be achieved with the imperfect interface model provided the imperfect interface parameters are suitably chosen.     

A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.     

波浪与弹性板作用的数值模拟

该文采用高阶有限元和边界元联合的方法求解波浪与弹性板的相互作用。其中流场采用边界元法求解,结构弹性响应方程采用基于Mindlin板理论的有限元方法求解,通过模态叠加技术实现了弹性板变形与流场相互作用的解耦。通过对一矩形板的计算,验证了该文方法与他人试验结果和数值模拟结果都吻合良好。利用这一模型进一步分析了波浪与弹性圆形板的作用问题,并对圆形板运动响应的收敛性进行了分析。     

Inclusion problem of a two-dimensional finite domain: The shape effect of matrix

With the advance in composite mechanics and micromechanics, there are increasing demands for analytical solutions of inclusion problems in a bounded domain. To echo this need, this study is focused on establishing explicit expressions of elastic fields for a 2D elastic domain containing a circular inclusion at center. Unlike the configuration in the classical Eshelby formulation, the elastic domain in this study is bounded and has shapes other than a circle. To circumvent the mathematical difficulty in solving Green’s function in a finite domain, an approach powered by complex potential method, which has been successfully employed to formulate the elastic fields for inclusion problems where matrix is unbounded or bounded by a circle, is extended to finite domains displaying complicated shapes, particularly, a Pascal’s limaçon and a curved square (an approximation of perfect square) in this study. In order to take advantage of the mathematical simplicity inherent in expressing a circular geometry, conformal mapping is used to transform the complex geometry of the finite domain of interest to a unit circle. The governing complex potentials, which capture the discontinuity on the inclusion–matrix interface due to the uniform eigenstrain within the inclusion, are formulated with the aid of Cauchy integral and then explicitly identified by satisfying the prescribed boundary conditions. In this study, the displacement fields for finite domains bounded by a Pascal’s limaçon and a curved square are obtained based on Dirichlet (displacement) boundary conditions imposed by the far field strain. In addition to asymptotical behaviors, firm agreement is also achieved when the analytical solutions based on complex potentials are compared with the FEM results. Furthermore, inverse of the conformal mapping is discussed here in order to get the explicit expression for elastic fields.     

Propagation of axisymmetric waves on the surface of a cylindrical cavity in elastic medium

A nondamped axisymmetric mode that propagates in an elastic cylindrical waveguide representing an extended cavity with a circular cross section in an infinite homogeneous medium is described. The wave dispersion in this system is analyzed and the similarity with and differences from other elastic media with one boundary are considered, including an infinite round rod and the surface of a half-space (Rayleigh wave). It is shown that, for axisymmetric waves in the cavity, a boundary frequency dependent on the curvature radius always exists, below which the waves are evanescent. A physical interpretation of results is given.     

Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

Summary An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.     

Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium

A planar crack of arbitrary shape in a 3D-anisotropic elastic medium subjected to an arbitrary external stress field is considered. An efficient numerical method of the solution of the problem is proposed. The problem is reduced to an integral equation for the crack opening vector on the crack surface. For discretization of this equation, Gaussian (radial) approximation functions centered at a system of nodes that covers the crack surface are used. For such functions, the elements of the matrix of the discretized problem are calculated in a quasi analytical form that involves standard non-singular integrals. If the node grid is regular, the matrix of the discretized system has Teoplitz’s structure, and the Fast Fourier Transform algorithm may be used for the calculation of matrix-vector products with such a matrix. It accelerate substantially the process of the iterative solution of the discretized system. Examples of the solutions for a circular crack in a transversally isotropic elastic medium are presented.     

Diffraction of a stress wave by a cylindrical cavity in an infinite elastic strip

The scattering of an incident plane shock wave by a cylindrical circular cavity in an infinite elastic strip is considered. In the Laplace transformed domain, boundary conditions at the plane surfaces and those at the circular hole are satisfied with the help of the Fourier transformation and the Schmidt method. A numerical Laplace inversion technique is taken to obtain the stresses in the physical space.     

The longitudinal shear problem for an array of cracks at the edge of a circular hole in an infinite elastic solid

Summary A Mellin-type transform technique reduces the longitudinal shear problem for a set of cracks at the edge of a circular hole in an infinite elastic solid to that of solving a system of integral equations. The stress intensity factors and crack formation energy are calculated. Three special cases are considered in detail and graphical results given.     

Feeble interfaces in bimaterials

Summary In the present paper, the thermal and thermo-elastic response of a bi-material to temperature changes is analyzed, when its interface exhibits a simultaneous weakness in traction transferring and heat flow conducting (feeble interface). Such a pathological behavior of an interface is described by two sets of constitutive relationships relating the heat flow passing through the interface to the temperature jump and the interfacial components of the traction to those of the displacement jump. The bimaterial model considered is that of a circular inhomogeneity in an elastic matrix with linear forms of the constitutive relationships. When the solutions of both heat conduction and thermoelastic problems with a perfect interface are known, the corresponding problems with a feeble interface are reduced to the solution of two dislocation problems: a heat conduction problem with an appropriate temperature dislocation applied across the interface, and an elasticity problem with an appropriate displacement dislocation of Somigliana type acting across the interface. For both dislocation problems, general representations of their solutions in terms of two-phase potential functions of complex variables are provided. Detailed analytical results are given for a circular inhomogeneity with a feeble interface disturbing a linear distribution of the temperature change in the matrix. In this case, the stress field within the inhomogeneity has a linear distribution and it vanishes for the limiting case of a sliding interface. For a specific value of the interface parameter H, which characterizes the thermal imperfection, there are no shear stresses within the inhomogeneity. Finally, since the constitutive laws describing the thermal and mechanical interface behavior correlate tensors of different order, the resulting fields in the system are drastically affected by the inhomogeneity size.     

Bending of Circular Sandwich Plate on Elastic Foundation

Bending of elastic circular sandwich plate with light filler, which is on an elastic foundation, is considered. To describe the kinematics of plate bending that is nonsymmetric in sandwich thickness, broken normal hypotheses are accepted. The foundation reaction is described by the Winkler model. The load is local symmetrical. A set of equilibrium equations and their exact solution in displacements have been obtained. Numerical results for a metal-polymeric sandwich plate are given. __________ Translated from Problemy Prochnosti, No. 6, pp. 68 – 78, November – December, 2005.     

New higher-order bounds on effective transverse elastic moduli of three-phase fiber-reinforced composites with randomly located and interacting aligned circular fibers

A higher-order structure for three-phase composites containing randomly located yet unidirectionally aligned circular fibers is proposed to predict effective transverse elastic moduli based on the probabilistic spatial distribution of circular fibers, the pairwise fiber interactions, and the ensemble-area homogenization method. Specifically, the two inhomogeneity phases feature distinct elastic properties and sizes. In the special event, two-phase composites with same elastic properties and sizes of fibers are studied. Two non-equivalent formulations are considered in detail to derive effective transverse elastic moduli of two-phase composites leading to new higher-order bounds. Furthermore, the effective transverse elastic moduli for an incompressible matrix containing randomly located and identical circular rigid fibers and voids are derived. It is demonstrated that significant improvements in the singular problems and accuracy are achieved by the proposed methodology. Numerical examples and comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered to illustrate the potential of the present method.     

Elastic analysis for a radial crack in a circular ring

The plane elastic problem corresponding to a radial crack emanating from the internal boundary of a circular ring is considered. Uniform external tension on the outer boundary is chosen as the applied load. The stress intensity factors at the crack tip are found by using the recently derived ‘modified mappingcollocation’ technique. Accurate data are found for varying crack depths over a representative range of wall ratios for fracture mechanics applications to pressurized hollow circular cylinders.     

Fracture behaviour by two cracks around an elliptic rigid inclusion

An analysis is presented for the in-plane biaxial loading of an elastic system consisting of a partially bonded rigid elliptic inclusion embedded in an infinite matrix where the bond imperfections are two symmetric cracks. The boundary value problem is formulated through the complex variable technique in the case of linearly elastic matrix under general biaxial loading at infinity. The elastic solution is obtained for a simplified model by assuming incompressibility and plane strain conditions for the matrix. In particular, stress and displacement fields are represented and discussed.

Moreover, a stress criterion, which permits to take into account, either the crack extension at the interface or its deviation into the matrix, is considered to study the fracture response of the elastic system.     

Contact stresses in an infinite plate containing a smooth circular insert under thermal load

Summary. A solution is presented for the contact problem with a smooth circular elastic insert in an infinite, linearly elastic plate under thermal load. The thermal load considered here includes the temperature gradient applied at infinity and a uniform temperature change. One of the major difficult parts in solving the present problem is that separation may occur between an insert or inclusion and the surrounding matrix under a nonuniform expansion of the matrix and the insert due to a temperature change. Based on the complex potential theory and the method of analytical continuation, the problem is reduced to a singular integro-differential equation which is solved numerically. Results are presented for the contact angle and for the normal and circumferential stresses at the plate-insert interface. Exact solutions are found for the current problem for a special case when the material properties of the plate and the insert are identical.