Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots

This paper deals with an elastic orthotropic inhomogeneity problem due to non-uniform eigenstrains. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Based on the polynomial conservation theorem, the induced stress field inside the inhomogeneity which is also linear, is determined by the evaluation of 10 unknown real coefficients. These coefficients are derived analytically based on the principle of minimum potential energy of the elastic inhomogeneity/matrix system together with the complex function method and conformal transformation. The resulting stress field in the inhomogeneity is verified using the continuity conditions for the normal and shear stresses on the boundary. In addition, the present analytic solution can be reduced to known results for the case of uniform eigenstrain.     

Elastic Energies in Circular Inhomogeneities: Imperfect Versus Perfect Interfaces

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity.     

Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains

In this paper a variational formulation of the equivalent eigenstrain method is established. A functional of the Hashin–Shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional. Moreover, it is also shown that the equivalent eigenstrain equation is the Euler–Lagrange equation of the potential energy of the inclusions. An approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains. Special attention is paid to possible symmetries of the problem. The variational formulation is illustrated by determination of effective linear elastic properties. In particular, material with a simple cubic microstructure is considered in detail. A solution for the polynomial radial basic eigenstrains approximation is found. In particular, for the homogeneous eigenstrain approximation, the effective moduli are derived in an exact closed form.     

Geometrical shape of in-plane inclusion characterized by polynomial internal stress field under uniform eigenstrains

The internal stress field of an inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains is studied. The study is restricted to the inclusion shapes defined by the polynomial mapping functions mapping the exterior of the inclusion onto the exterior of a unit circle. The inclusion shapes, giving a polynomial internal stress field, are determined for three types of inclusions, i.e., an inhomogeneous inclusion with an elastic modulus different from the surrounding matrix, an inhomogeneous inclusion with the same shear modulus but a different Poisson’s ratio from the surrounding matrix, and a homogeneous inclusion with the same elastic modulus as the surrounding matrix. Examples are presented, and several specific conclusions are achieved for the relation between the degree of the polynomial internal stress field and the degree of the mapping function defining the inclusion shape.     

On consistent micromechanical estimation of macroscopic elastic energy,coherence energy and phase transformation strains for SMA materials

An apparatus of micromechanics is used to isolate the key ingredients entering macroscopic Gibbs free energy function of a shape memory alloy (SMA) material. A new self-equilibrated eigenstrains influence moduli (SEIM) method is developed for consistent estimation of effective (macroscopic) thermostatic properties of solid materials, which in microscale can be regarded as amalgams of n-phase linear thermoelastic component materials with eigenstrains. The SEIM satisfy the self-consistency conditions, following from elastic reciprocity (Betti) theorem. The method allowed expressing macroscopic coherency energy and elastic complementary energy terms present in the general form of macroscopic Gibbs free energy of SMA materials in the form of semilinear and semiquadratic functions of the phase composition. Consistent SEIM estimates of elastic complementary energy, coherency energy and phase transformation strains corresponding to classical Reuss and Voigt conjectures are explicitly specified. The Voigt explicit relations served as inspiration for working out an original engineering practice-oriented semiexperimental SEIM estimates. They are especially conveniently applicable for an isotropic aggregate (composite) composed of a mixture of n isotropic phases. Using experimental data for NiTi alloy and adopting conjecture that it can be treated as an isotropic aggregate of two isotropic phases, it is shown that the NiTi coherency energy and macroscopic phase strain are practically not influenced by the difference in values of austenite and martensite elastic constants. It is shown that existence of nonzero fluctuating part of phase microeigenstrains field is responsible for building up of so-called stored energy of coherency, which is accumulated in pure martensitic phase after full completion of phase transition. Experimental data for NiTi alloy show that the stored coherency energy cannot be neglected as it considerably influences the characteristic phase transition temperatures of SMA material.     

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation 2

The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.     

Stress and electric fields in an infinite electrostrictive plate with an elliptic inhomogeneity

The stress and electric fields in electrostrictive materials under general electric loading at infinity are obtained in this paper. It is shown that the pseudo total stresses are continuous in the whole body. The elliptic inhomogeneity problem is first discussed in this paper and its solution is also given. The results show that the stress in the inhomogeneity is not uniform which is different from the solution of Eshelby theory for elastic materials. When the inhomogeneity and matrix have the same dielectric permittivity or the matrix is a non-electrostrictive material, the stress field is uniform in the inhomogeneity. The form of stress function is simple when the inhomogeneity degenerates to a circle.     

Inplane deformation of a circular inhomogeneity with imperfect interface

The plane elastic problem of a circular inhomogeneity with an imperfect interface of spring-constant-type is reduced to the solution of a Somigliana dislocation problem, when the solution for the corresponding problem with a perfect interface is known. The Burger's vector of the Somigliana dislocation is determined so that its components satisfy two interfacial conditions involving the traction components of the corresponding problem with a perfect interface. Employing complex variables, a two-phase potential solution to the Somigliana dislocation inhomogeneity problem is developed for a general form of the Burger's vector. Detailed results are reported for a uniform eigenstrain in the inhomogeneity, and for a remote uniform heat flow in the matrix. In the latter case, the inhomogeneity behaves as a void, when it begins to slide.     

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

In traditional continuum mechanics, the effect of surface energy is ignored as it is small compared to the bulk energy. For nanoscale materials and structures, however, the surface effects become significant due to the high surface/volume ratio. In this paper, two-dimensional elastic field of a nanoscale elliptical inhomogeneity embedded in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. By using the complex variable technique of Muskhelishvili, the analytic potential functions are obtained in the form of an infinite series. Selected numerical results are presented to study the size-dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. It is found that the elastic field of an elliptic inhomogeneity under uniform eigenstrain is no longer uniform when the interfacial stress effects are taken into account.     

Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix

This paper presents two different analytical methods to investigate the magneto-mechanical coupling effect for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. First, the magnetoelastic solution is expressed in terms of magnetoelastic Green's function that can be decoupled into elastic Green's function and magnetic Green's function. Second, the problem is analyzed by the equivalent inclusion method, and then, the formulation of the inhomogeneity problem can be decoupled into an elastic problem and a magnetic inhomogeneity problem connected by some eigenstrain and eigenmagnetic fields. For the piezomagnetic composites with a non-piezomagnetic matrix, these two solutions are completely equivalent each other though they are obtained by means of two different methods. Moreover, based upon the unified energy method, the effective magnetoelastic moduli of the composites are expressed explicitly in terms of phase properties and volume fractions. Then the dilute and Mori–Tanaka schemes are discussed, respectively. Finally, the calculations are made to predict the effective magnetoelastic moduli and illustrate the performance of each model.     

A unified treatment of the elastic elliptical inclusion under antiplane shear

Summary A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.     

Magnetostrictive composites in the dilute limit

We calculate the effective properties of a magnetostrictive composite in the dilute limit. The composite consists of well separated identical ellipsoidal particles of magnetostrictive material, surrounded by an elastic matrix. The free energy of the magnetostrictive particles is computed using the constrained theory of DeSimone and James [2002. A constrained theory of magnetoelasticity with applications to magnetic shape memory materials. J. Mech. Phys. Solids 50, 283-320], where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and demagnetization terms. By using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. The solution of the quadratic programming problem yields the effective properties of Ni₂MnGa and Terfenol-D composite systems. Numerical results show that the average strain of the composite depends strongly on the particle shape, the applied stress, and the elastic modulus of the matrix.     

New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

On the equivalent inclusion method and impotent eigenstrains

The equivalency equations and the nature of the solution are investigated when an inhomogeneity under applied stresses is simulated by an inclusion with eigenstrains. The equivalency equations by which the equivalent eigenstrain is obtained becomes singular when the inhomogeneity is void and the applied stress has a form of polynomials of coordinates of degree one. The solutions of the system of the equivalency equations are not uniquely determined. Explicit expressions are given for the impotent eigenstrains which do not generate any stress field throughout a material.This research was supported by the U.S. Army Research Office under Grant No. DAAG 29-77-G-0042 to Northwestern University.On leave of absence of Meiji University, Tokyo.     

On the Elastic Field of a Shpherical Inhomogeneity with an Imperfectly Bonded Interface

This study is devoted to the development of a unified and explicit elastic solution to the problem of a spherical inhomogeneity with an imperfectly bonded interface. Both tangential and normal displacement discontinuities at the interface are considered and a linear interfacial condition, which assumes that the tangential and the normal displacement jumps are proportional to the associated tractions, is adopted. The elastic disturbance due to the presence of an imperfectly bonded inhomogeneity is decomposed into two parts: the first is formulated in terms of an equivalent nonuniform eigenstrain distributed over a perfectly bonded spherical inclusion, while the second is formulated in terms of an imaginary Somigliana dislocation field which models the interfacial sliding and normal separation. The exact form of the equivalent nonuniform eigenstrain and the imaginary Somigliana dislocation are fully determined in this paper.     

Piezoelectric Study on Elliptic Inhomogeneity Problem Using Alternating Technique

This paper presents a novel efficient procedure to analyze the elliptical inhomogeneity problem in piezoelectric materials under electromechanical loadings. The electromechanical loadings considered in this paper include a point force and a point charge or a far-field anti-plane shear and in-plane electric field. The analytical continuation method together with alternating technique is used to derive the electroelastic fields in terms of the corresponding homogeneous solution. Compared to existing related papers, this approach could lead to some interesting simplifications in solution procedure and the derived analytical solution for singularity problems can be employed as a Green's function to investigate matrix cracking in the inclusion/matrix system. Numerical results are provided to show the effect of the material mismatch, the aspect ratio and the loading condition on the electroelastic field due to the presence of the inhomogeneity.     

Analysis of a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains

This paper studies a transversely isotropic rod containing a single cylindrical inclusion with axisymmetric eigenstrains. The analytical elastic solution is obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructural parameters and its evolution on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated through examples.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Analysis of a screw dislocation inside an elliptical inhomogeneity in piezoelectric solids

This paper formulates and examines the electro-elastic coupling effects resulting from the presence of a screw dislocation inside an elliptical piezoelectric inhomogeneity embedded in an infinite piezoelectric matrix. The general solution to this problem is obtained by conformal mapping and Laurent series expansion of the corresponding complex potentials. The appropriate expressions of the field potentials and the field components are given explicitly in both the inhomogeneity and the surrounding matrix using a perturbation technique. The internal energy and the force on the dislocation are computed and several specific examples are provided to illustrate the validity and versatility of the developed formulations.     

二次规划的混合能方法及杆系结构弹塑性分析

基于混合能理论构造了二次规划问题求解的新算法，并在杆系弹塑性增量分析中进行了成功的应用。所提出的算法一改传统的对单一规划变量（如位移或应力等）的求解策略，将问题演变成混合变量的形式，在混合空间下对问题进行算法构造，使问题的下一个增量步的求解可以充分利用增量步前的信息，由此极大地提高了算法的求解效率，对杆系弹塑性问题的数值分析充分地证实了这一点。