A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem

Summary This is the first paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. A novel solution procedure has been developed to systematically solve a type of Fredholm integral equations based on symmetry, self-similarity, and invariant group arguments. In this paper, we consider a two-dimensional (2D) circular inclusion within a finite, circular representative volume element (RVE). The RVE is considered isotropic, linear elastic and is subjected to a displacement (Dirichlet) boundary condition. Starting from the 2D plane strain Navier equation and by using our new solution technique, we obtain the exact disturbance displacement and strain fields due to a prescribed constant eigenstrain field within the inclusion. The solution is characterized by the so-called Dirichlet-Eshelby tensor, which is provided in closed form for both the exterior and interior region of the inclusion. Some immediate applications of the Dirichlet-Eshelby tensor are discussed briefly.     

A circular inclusion with imperfect interface in finite plane elastostatics

We consider a circular elastic inclusion embedded in a particular class of harmonic materials subjected to remote uniform stresses. The imperfect interface can be rate dependent as well as rate independent. First, we study the situation in which both rate-depending slip and diffusional relaxation are present on the sharp inclusion-matrix imperfect interface. It is found that in general, the internal Piola stresses within the inclusion are spatially non-uniform and decay with two relaxation times. Interestingly, the average mean Piola stress within the circular inclusion is time independent. Some extreme cases for the imperfect interface are discussed in detail. Particularly, we find a simple condition leading to internal uniform Piola stresses that decay only with a single relaxation time. Second, we investigate a rate-independent spring-type imperfect interface on which normal and shear tractions are proportional to the corresponding displacement jumps. It is found that in general, the internal Piola stresses are intrinsically non-uniform. A special kind of the spring-type interface leading to internal uniform Piola stresses is also found.     

Antiplane eigenstrain problem of a circular inclusion in nonlocal elasticity

Summary The antiplane eigenstrain problem of a circular inclusion is investigated in nonlocal elasticity. The nonlocals tress fields are obtained analytically by generalizing the classical line force concept to the case of nonlocal elasticity. The nonlocal effects, stress concentration and nonlocal surface residual, are exhibited for the first time.     

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.     

Generalised plane strain problem for a circular inclusion in anisotropic elasticity

This paper is concerned with a generalised plane deformation problem in the linear theory of anisotropic elasticity. As is well known, the generalised plane deformation is the deformation of a body of infinite length bounded by a cylindrical surface, when all the stress and strain components exist but they are functions of two co-ordinates x₁, and x₂ only. It may be shown that if u₃ = 0, it is impossible to satisfy all the three equations of equilibrium of anisotropic elastic body. One has to choose u₃ as a non-zero function of x₁, x₂ for satisfying equations of equilibrium. In isotropic elasticity, u₃ = 0, makes the third equation of equilibrium identically equal to zero.The problem in this paper concerns an elastic circular cylindrical inclusion embedded in a matrix of different anisotropic material. The matrix and the inclusion are perfectly bonded at the interface. Each of the two materials possesses anisotropy of a general form with all the 21 elastic constants. The matrix is subjected to a uniform stress at infinity. The equations of elasticity theory demand that the rotation component ω₃ must also be prescribed at infinity. The complex variable technique is used and exact analytical expressions are derived for the elastic field in both the regions.     

A partially debonded circular inclusion interacting with a point singularity: A classical problem revisited

The classical problem for a partially debonded circular inhomogeneity is revisited. The interaction of the interfacial crack and a point singularity such as a point force and/or a dislocation is dealt with. Also the circular arc-shaped interfacial crack under remote stress is solved. This problem has been solved by many researchers for the cases of various loading types. However, lack of generality in the solution technique together with too complicated form of the solution makes it hard to grasp the structure of the solution. Based on the recently published technique for a perfectly bonded circular inhomogeneity, this problem is revisited. The resulting form of the solution is very simple, therefore, its structure is easily understood. Due to the merit of the present method, the image force on the edge dislocation near the tip of the interfacial crack is easily obtained.     

含界面层圆形夹杂的平面热弹性问题

研究了无穷远均匀拉伸条件下，含界面层圆形弹性夹杂的平面热弹性问题。运用Muskhelishvili复势理论的级数展开技术，将各区应力函数展开为合适的Taylor和Laurent级数，考虑边界上的力和位移连续性条件，将问题转化为线性方程组的求解，数值分析表明：总体上，软界面层可以有效的减小夹杂和基体的界面应力集中；硬界面层可以减小夹杂内的界面应力集中，但却增加了基体内的界面应力集中；此外，总体上，界面相热膨胀系数较基体相和夹杂相过高，不利于降低界面应力集中。     

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.     

The interaction of a curved crack with a circular elastic inclusion

The solution to a curved matrix crack interacting with a circular elastic inclusion is presented. The problem is formulated using the Kolosov–Muskhelishvili complex stress potential technique where the crack is represented by an unknown distribution of dislocations. After an appropriate parameterization, the resulting singular integral equations are solved with the Lobatto-Chebyshev quadrature technique. The accuracy of the current solution is shown by comparing these results to previously published results. A preliminary investigation is conducted to study the effects of crack curvature and inclusion stiffness on the stress intensity factors and it is shown that in certain instances, the effect of the crack curvature and the inclusion stiffness are competing influences.     

Approximate analytical solution for the problem of an inclusion in a viscoelastic solid under finite strains

The approximate analytical solution of a quasi-static plane problem of the theory of viscoelasticity is obtained under finite strains. This is the problem of the stress–strain state in an infinite body with circular viscoelastic inclusion. The perturbation technique, Laplace transform, and complex Kolosov–Muskhelishvili’s potentials are used for the solution. The numerical results are presented. The nonlinear effects and the effects of viscosity are estimated.     

A piezoelectric screw dislocation interacting with a nonuniformly coated circular inclusion

This paper investigated the interaction of a piezoelectric screw dislocation with a nonuniformly coated circular inclusion in an unbounded piezoelectric matrix subjected to remote antiplane shear and electric fields. In addition to having a discontinuous displacement and a discontinuous electric potential across the slip plane, the dislocation is subjected to a line force and a line charge at the core. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general solutions in an explicit series form. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution for the corresponding homogeneous problem. The presented series solutions have rapid convergence which is guaranteed numerically. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. The numerical results show that the varying thickness of the interphase layer will exert a significant influence on the shear stress and electric field within the circular inclusion, and on the direction and magnitude of the image force. This solution can be used as Green’s function for the analysis of the corresponding piezoelectric matrix cracking problem.     

The inclusion problem with a crack crossing the boundary

The problem of an elastic plane containing an elastic inclusion is considered. It is assumed that both the plane and the inclusion contain a radial crack and the two cracks are collinear. The problem is formulated in terms of a system of singular integral equations. In the interesting limiting cases in which the crack tips approach the interface from either one or both sides, the dominant parts of the kernels become generalized Cauchy kernels giving rise to stress singularities of other than 13-1 power. For these unusual cases of a crack terminating at or crossing the interface stress intensity factors are defined and some detailed results are given for various crack-inclusion geometries and material combinations.     

A finite element model of the stress field in a star-shaped inclusion

The elastic stress field in the neighborhood of an inclusion in the shape of a five-pointed star embedded in a homogeneous matrix subject to a remote uniaxial strain is determined using the finite element method in both plane stress and plane strain. The results do not support a recent preliminary analytical result that both the individual stress components and the effective stress distribution inside the inclusion should be uniform.     

Separation of a smooth circular inclusion from a matrix

The paper treats the separation of a smooth circular inclusion from a matrix, as the latter is deformed uniformly. Using finite integral transforms, the problem of finding the extent of separation and the contact pressure is reduced to the solution of a Fredholm integral equation with a weakly singular kernel. The singular part of the kernel can be removed and the equation made suitable for an effective numerical solution. Explicit results are given for general combinations of materials and several cases of loading.     

A debonding and a crack on a circular rigid inclusion subjected to rotation

A model of debonding and crack occurring from a circular rigid inclusion in an infinite plate is analyzed under the loading condition of the inclusion rotation. A mapping function of a sum of fractional expressions and complex stress functions are used for the analysis of the mixed boundary value problem of the plane elasticity. The following values are obtained: values of stress singularity at the debonded tip where only a debonding exists i.e. without a crack; stress intensity factors just after the occurrence of a crack; and the energy release rates for the debonding propagation and the crack occurrence. Furthermore we discuss which phenomenon occurs, the debonding propagation or the crack occurrence at the debonded tip. We also discuss from which tip the debonding propagation occurs.

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Résumé On étudie un modèle de décollement et de fissuration se produisant au départ d'une inclusion circulaire rigide dans une plaque infinie sujette à rotation. Pour l'analyse du problème d'évaluation du contour mixte en élasticité plane, on recourt à une fonction de représentation constituée d'une somme d'expression fractionnaires et de fonctions complexes de contraintes. On obtient les valuers singulières de la contrainte à l'extrémité du décollement, lorsque seul apparaît un decollement, sans fissuration, et les facteurs d'intensité de contraintes immédiatement après l'apparition de la fissure. On obtient également la vitesse de rotation de l'énergie associée à la propagation du décollement et à l'apparition de la fissure. On discute en outre de l'occurence de l'un ou de l'autre de ces deux phénomènes, ainsi que de quel côté se produit la propagation du décollement.

     

Pile-up of screw dislocations at a circular inclusion

The behaviour of a pile-up of screw dislocations at a circular inclusion, with its tip away from the interface is analyzed using the method of continuously distributed dislocations. This leads for the first time, to a distribution function representing a shear crack at the inclusion. The stress required to extend the crack is derived and some new conclusions drawn on the deformation and fracture behaviour of the two-phase systems.     

A variational approach to a circular hyperelastic membrane problem

Summary The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight.It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.