The expanding spherical inhomogeneity with transformation strain

Within the context of linear elastodynamics, the radiated fields (including inertia) for a spherical inhomogeneous (of different elastic constants) inclusion with dilatational transformation strain (or eigenstrain), expanding in general motion under applied loading, can been obtained on the basis of Eshelby’s equivalent inclusion method by using the strain field of the expanding homogeneous spherical inclusion (as a function of the eigenstrain) to determine the equivalent eigenstrain. With the equivalent dynamic eigenstrain (which is dependent on the boundary motion), the radiated fields for the inhomogeneous spherical expanding inclusion can be obtained. Based on them, the “driving force” (self-force) on the moving boundary can be computed, and this is the rate of mechanical work (with inertia) required to create an incremental region of inhomogeneity with eigenstrain, i.e with the elastic constants changing as the region of the eigenstrain expands. The self-force depends on the history of the motion, and, in the presence of external loading the driving force yields a Peach-Koehler type force, which exhibits coupling of the applied loading to the history of the motion of the boundary velocity.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Three-dimensional dynamic stress analysis on a penny-shaped crack interacting with a suddenly transformed spherical inclusion

In this paper, the interaction between a penny-shaped crack and a near-by suddenly transformed spherical inclusion in 3-dimensional solid is investigated to assess the dynamic effect of the transformation. To simplify the solution procedure, the current problem is divided into two sub-problems by using the superposition principle. A time domain boundary integral equation method (BIEM) is adopted to evaluate the stress and displacement fields. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. The impact effect of the spherical inclusion when it experiences a pure dilatational eigenstrain on the penny-shaped crack is studied. The relationship between the relative location of the inclusion and its impact effect on the time history of the Mode I crack intensity factor is discussed in detail.     

Dynamic potentials and Green’s functions of a quasi-plane magneto-electro-elastic medium with inclusion

The dynamic potentials of a quasi-plane magneto-electro-elastic medium of transversely isotropic symmetry with an inclusion of arbitrary shape are derived, and the dynamic potentials are finally governed by six scalar equations which can be regarded as inhomogeneous wave equations, Laplace and Helmholtz equations. The explicit expressions of the dynamic Green’s functions of this medium are also obtained both in the space–time domain and in the space–frequency domain. Closed-form expressions for the space–frequency representation of the dynamic potentials are given for the case when the inclusion is circular. The results are employed to obtain the generalized displacement fields of a circular inclusion undergoing uniform eigenstrain, eigenelectric field and eigenmagnetic field. In contrast to the corresponding static Eshelby inclusion problem the magneto-electro-elastic fields (i.e. strain, electric and magnetic field) inside the inclusion are non-uniform in the space–frequency domain.     

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.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

Binary inclusion model for the overall elasticity of imperfectly bonded composites

Summary. A micromechanical approach is developed to estimate the overall elastic moduli of composite materials with imperfectly bonded spherical fillers. The randomly dispersed particles are assumed to satisfy linear interfacial conditions where both tangential and normal interface displacement discontinuities are linearly related to the respective surface tractions. Using the generalized version of Eshelbys equivalent inclusion method proposed by Furuhashi et al. [6] the analysis of the heterogeneous medium reduces to the study of a corresponding homogeneous medium containing spherical inclusions with a proper distribution of eigenstrain and Somigliana dislocation fields. Based on the estimated pair-wise average of strain fields in two interacting imperfect fillers embedded in the homogeneous infinite matrix, the ensemble phase volume average of field quantities has been evaluated within a representative volume element containing a finite number of imperfect particles. For the case of a constant radial distribution function, results are in reasonable agreement with those based on the generalized self-consistent method and composite sphere assemblage proposed by Hashin [11].     

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.     

Analytical solutions for multilayered composite cylinders with harmonic quadratic eigenstrain in arbitrary layers

The eigenstrain problem of multilayered hollow and solid composite cylinders working in a constant magnetic field is investigated analytically in this paper. Each layer of the composite cylinder can undergo a harmonic and spatially varying eigenstrain. The eigenstrain is assumed to be a quadratic polynomial function of the radial coordinate. The closed-form elastic solutions are obtained by solving the inhomogeneous governing Bessel differential equations. Then, the effects of eigenstrain distribution, angular frequency and the intensity of the magnetic field on the radial displacement, radial stress, hoop stress, and axial stress are presented graphically.     

On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.     

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

Thermal stress-focusing effect in a spherical Zirconia inclusion with dynamically transforming strains

Some composite materials, such as Zirconia-toughened ceramics, are remarkable materials which have high strength, a high elastic modulus, and an improved toughness, etc. These good qualities are made possible through the stress-induced phase transformation of composite particles, which is accompanied by an impact cooling. When a spherical inclusion in an infinite elastic domain is suddenly subjected to an instantaneous phase transformation, stress waves occur at the surface of a spherical inclusion at the moment thermal impact is applied. The wave may accumulate at the center and show stress-focusing effects, even though the initial stress may be relatively small. This paper analyzes the thermal stress-focusing effect caused by the instantaneous anisotropic phase transformations in the spherical Zirconia inclusion. By use of ray theory, the numerical results give a clear indication of the mechanism of stress-focusing in an inclusion embedded in an infinite elastic medium.     

Application of invariant integrals to the problems of defect identification

A problem of parameters identification for embedded defects in a linear elastic body using results of static tests is considered. A method, based on the use of invariant integrals is developed for solving this problem. A problem on identification the spherical inclusion parameters is considered as an example of the proposed approach application. It is shown that the radius, elastic moduli and coordinates of a spherical inclusion center are determined from one uniaxial tension (compression) test. The explicit formulae expressing the spherical inclusion parameters by means of the values of corresponding invariant integrals are obtained for the case when a spherical defect is located in an infinite elastic solid. If the defect is located in a bounded elastic body, the formulae can be considered as approximate ones. The values of the invariant integrals can be calculated from the experimental data if both applied loads and displacements are measured on the surface of the body in the static test. A numerical analysis of the obtained explicit formulae is fulfilled. It is shown that the formulae give a good approximation of the spherical inclusion parameters even in the case when the inclusion is located close enough to the surface of the body.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

Evolution equation of moving defects: dislocations and inclusions

Evolution equations, or equations of motion, of moving defects are the balance of the “driving forces”, in the presence of external loading. The “driving forces” are defined as the configurational forces on the basis of Noether’s theorem, which governs the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the ensuing dynamic J integral equals the change in the Lagrangean if and only if the linear momentum is preserved. Dislocations and inclusions are “defects” that possess self-stresses, and the total driving force for these defects consists only of two terms, one expressing the “ self-force” due to the self-stresses, and the other the effect of the external loading on the change of configuration (Peach–Koehler force). For a spherically expanding (including inertia effects) Eshelby (constrained) inclusion with dilatational eigenstrain (or transformation strain) in general subsonic motion, the dynamic J integral, which equals the energy-release rate, was calculated. By a limiting process as the radius tends to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which is the rate of the mechanical work required to create an incremental region of eigenstrain (or transformation strain) of a dynamic phase boundary. The total driving force (due to external loading and due to self-forces) must be equal to zero, in the absence of dissipation, and the evolution equation for a plane boundary with eigenstrain is presented. The equation applied to many strips of eigenstrain provides a system to solve for the position/ evolution of strips of eigenstrain.     

Steady-state response of a Cosserat medium with a spherical inclusion

Summary Two concepts of asymmetric eigenstrain and eigentorsion are employed to derive a general steady-state theory of inhomogeneous anisotropic micropolar media containing defects with the help of Green's function technique. In particular, a dynamic inclusion problem for homogeneous isotropic centrosymmetric micropolar elasticity is investigated. By means of Green's functions an exact closed-form solution is presented for the case of a spherical inclusion embedded in an infinitely extended Cosserat medium. With this result, the micropolar dynamic Eshelby tensors for the inside and outside elastic fields of the spherical inclusion are defined and determined. It is confirmed that the classical dynamic and static Eshelby tensors are obtained as two special cases of the micropolar dynamic Eshelby tensors, respectively.     

Overall behavior of composites with periodic multi-inhomogeneities

When applying the equivalent inclusion method (EIM) to a composite material with non-dilute distribution of reinforcement particles, due to the complex interaction between the particles, the homogenizing eigenstrain field will in general be highly nonlinear. The interaction becomes more complex, when the reinforcements are multi-phase particles, i.e., the core inhomogeneity is surrounded by many layers of coatings. In this paper, a treatment for an accurate determination of the distribution of homogenizing eigenstrain fields corresponding to composites with non-dilute periodic distribution of multi-phase reinforcement particles is given. The proposed method is applicable to problems, where the reinforcement particles have: very thick coatings; functionally graded (FG) coatings; or coatings with variable thicknesses. Strong dependence of the overall response of composites on the microstructure of their reinforcement particles is well recognized. The theory is extended to estimate the effective elastic moduli of such composites.     

Dynamics of a thick-walled dielectric elastomer spherical shell

The nonlinear vibration response of a thick-walled spherical shell subjected to the mechanical pressure and electric field is studied in this paper. When subject to an electric field through the thickness of the spherical shell, the material expands in plane and contracts in thickness. The dielectric elastomer is assumed to be isotropic and neo-Hookean. Based on simple geometrical and spherical capacitor assumptions, we deduce an explicit analytical equation of motion of the dielectric elastomer spherical shell. The dynamic behaviors of the spherical shell under a constant electric loading and periodic electric loading are analyzed. In addition, the critical voltage is calculated in terms of various loading.     

Elastic analysis of the half plane inhomogeneity problem

Summary The paper presents an analytical solution for the elastic field in the vicitity of a semi-circular inhomogeneity, embedded at the free surface of an elastic half-plane. This bi-material system is loaded by uniform remote tension or a constant eigenstrain sustained by the inhomogeneity.     

Effects of interfacial excess energy on the elastic field of a nano-inhomogeneity

A semi-analytical approach based on a variational framework is developed to obtain the three-dimensional solution for a nano-scale inhomogeneity with arbitrary eigenstrains embedded in a matrix of infinite extent. Both the inhomogeneity and the matrix can be elastically anisotropic. The Gurtin–Murdoch surface/interface model is used to describe the elastic behavior of the inhomogeneity/matrix interface. The displacement fields in the inhomogeneity and the matrix are represented, respectively, by two sets of polynomials. Coefficients of these polynomials are determined by solving a system of linear algebraic equations that are derived from minimizing the total potential energy of the system. In the case of an isotropic spherical inhomogeneity with dilatational eigenstrain in an isotropic matrix, our solution shows an excellent agreement with the corresponding analytical solution available in the literature. To demonstrate the capabilities of the method developed here and to investigate the effect of interfacial excess energy, numerical examples are also presented when the inhomogeneity and matrix are both elastically anisotropic. Both dilatational and pure shear eigenstrains are considered in these examples.