Min-Zhong Wang

Recent General Solutions in Linear Elasticity and Their Applications

A review is given on the progress in the study of general solutions of elasticity and their applications since 1972. Apart from summarizing and remarking the development of the general solution method in literature, this review aims to present the readers with a systematic and constructive scheme to develop general solutions from given governing differential equations and then to prove their completeness and investigate their nonuniqueness features. The effectiveness of the constructive scheme manifests itself in the fact that almost all the classic solutions, including not just classic displacement potentials but also classic stress functions, can be rederived by using this scheme. Furthermore, thanks to the systematic features of the scheme, it produces a constructive approach to study the completeness and nonuniqueness of general solutions and possesses more flexibility, which facilitates the extension of elastic general solution methods to more general systems governed by elliptic differential equations. Under the framework of this scheme, a comprehensive review is presented on wide application of general solutions in a variety of research areas, ranging from problems with different materials, isotropic or anisotropic, to various coupling problems, such as thermoelasticity, magnetoelasticity, piezoelectric elasticity, porous elasticity, and quasicrystal elasticity, and to problems of different engineering structures, for instance, the refined theories for beams and plates. There are 213 references cited in this review article. DOI: 10.1115/1.2909607

The quasi Eshelby property for rotational symmetrical inclusions of uniform eigencurvatures within an infinite plate

Although the Eshelby property does not hold for the non-ellipsoidal inclusions, some special properties have been found for the rotational symmetrical inclusions recently. In the present paper, an infinite plate containing a rotational symmetrical inclusion with uniform eigencurvatures is investigated. We mathematically prove that the rotational symmetrical inclusion in an infinite plate possesses the quasi Eshelby property. Namely, for an N-fold rotational symmetrical inclusion (N is an integer greater than 2 and unequal to 4), the arithmetic mean of curvatures at N rotational symmetrical points in the inclusion is identical with that of a circular inclusion and independent of the orientation of the inclusion. Meanwhile, in two corollaries, we point out that the curvature at the centre, the averaged curvature over the inclusion domain, and the line integral average of curvatures along any concentric circle of the inclusion are all the same as the arithmetic mean.

On the completeness of solutions of Boussinesq, Timpe, Love, and Michell in axisymmetric elasticity

The completeness of Loves solution was proved by Noll, Gurtin, and Carlson if the meridional half-section is z-convex. In this paper it is pointed out that the condition of z-convexity is unnecessary for the completeness of Loves solution.

A permeable interface crack between dissimilar thermopiezoelectric media

SummaryThis paper presents an explicit treatment of the generalized 2D thermopiezoelectric problem of an interfacial crack between two dissimilar thermopiezoelectric media by means of the extend Stroh formalism. In comparison with the other relevant studies, the present work has two features: one is that the crack is assumed to be a permeable slit across which the normal electric displacement and the tangential electric field are continuous. The other is that the heat loading is applied at infinity, rather than on the crack faces. As a result, the field intensity factors and the electric field inside the crack are obtained in explicit closed-forms, respectively. As examples, the solutions of several particular cases, including that of an impermeable crack and that of a homogeneous material with a crack are also presented. It is shown that the electric field inside a crack may be singular and oscillatory for the case of an interfacial crack, while for the case of a crack in a homogeneous medium it is linearly variable. Moreover, it is also found that for a homogeneous medium with a crack the stress intensity factors based on the impermeable model and permeable model are same, but the intensity factor of the electric displacement is not.

Constructivity and completeness of the general solutions in elastodynamics

SummaryThere are two general solutions in clastodynamics, which are the Cauchy-Kovalevski-Somigliana general solution (call it CKS for short) and the Boussinesq-Papkovich-Neuber general solution (BPN). In this paper, by constructing two new kinds of decomposition for any vector, the potential functions in CKS and BPN are constructed. Thus the completeness of the two general solutions gets proved in a pithy style. In addition, we also establish the evident relation between the two general solutions.

The Arithmetic Mean Theorem of Eshelby Tensor for Exterior Points Outside the Rotational Symmetrical Inclusion

In 1957, Eshelby proved that the strain field within a homogeneous ellipsoidal inclusion embedded in an infinite isotropic media is uniform, when the eigenstrain prescribed in the inclusion is uniform. This property is usually referred to as the Eshelby property. Although the Eshelby property does not hold for the non-ellipsoidal inclusions, in recent studies we have successfully proved that the arithmetic mean of Eshelby tensors at N rotational symmetrical points inside an N-fold rotational symmetrical inclusion is constant and equals the Eshelby tensor for a circular inclusion, when N≥ 3 and N≠4. The property is named the quasi-Eshelby property or the arithmetic mean theorem of Eshelby tensors for interior points. In this paper, we investigate the elastic field outside the inclusion. By the Green formula and the knowledge of complex variable functions, we prove that the arithmetic mean of Eshelhy tensors at N rotational symmetrical points outside an N-fold rotational symmetrical inclusion is equal to zero, when N ≥3 and N i≠ 4. The property is referred to as the arithmetic mean theorem of Eshelby tensors for exterior points. Due to the quality of the Green function for plane strain problems, the fourfold rotational symmetrical inclusions are excluded from possessing the arithmetic mean theorem. At the same time, by the method proposed in this paper, we verify the quasi-Eshelby property which has been obtained in our previous work. As corollaries, two more special properties of Eshelby tensor for N-fold rotational symmetrical inclusions are presented which may be beneficial to the evaluation of effective material properties of composites. Finally, the circular inclusion is used to test the validity of the arithmetic mean theorem for exterior points by using the known solutions.

General Representations of Polynomial Elastic Fields

For hybrid finite elements based e.g. on complementary energy functional, polynomial trial functions of stress fields are required. Systematic schemes are given for the 2D and 3D elasticity, respectively. For 2D problems, the paper shows that there are maximal four independent polynomials for the n-th order homogeneous polynomial Airy stress functions: two for the first order, three for the second order, and four for the n-th order (n not less than 3). For 3D problems, there are 3(2n+1) independent polynomials for the n-th order homogeneous polynomial displacement. The corresponding stress field can be given directly. As examples, explicit expressions of the corresponding independent polynomials are listed: up to the 10-th order 1 for the 2D case and up to the third order for the 3D case.

The Fault in the Stress Analysis of Pseudo-Stress Function Method

Although Peng Yafei and his co-workers discovered some faults with the pseudo-stress function method suggested by Y. S. Lee in 1987, the authors did not provide convincing arguments. We investigate the crucial assumption in Lee’s method by rewriting it as the form of real part and imaginary part. Through a specific counterexample, we point out that the crucial assumption in Lee’s theory is untenable. Namely, for given Airy’s stress function, it cannot be guaranteed that the pseudo-stress function Λ(x,y) exists. The root cause of the fault with Lee’s method is found in this paper.

Equivalence between the Local Boundary Integral Equation and the Mean Value Theorem in the Theory of Elasticity

The boundary element method based on a boundary integral equation has been very successful in computational mechanics. Atluri et al. [4] recently developed a new meshless method using the local boundary integral equations. It eliminates the tedious step of mesh generation and thus greatly simplifies the numerical computation process. This paper shows the equivalence between the local boundary integral equation and the mean value theorem in the theory of elasticity. In addition, it gives new proofs for the mean value theorem of elasticity and its converse based on the concept of a companion solution.

The Proof of Strain Discontinuity in Eshelby Problems with Rotational Symmetrical Inclusions

One of the important approaches to study thermal stress in heterogeneous material/devices is the Eshelby equivalent inclusion method, which is based on the Eshelby uniform solution (the Eshelby property) for ellipsoid-like inclusions. Despite the non-uniformity in stresses fields, it has been proved \citep{WangXu2006} that rotational symmetrical inclusions satisfy the arithmetic mean theorem (the quasi-Eshelby property). That is, for N-fold rotational symmetric inclusions, the average (more accurately, the arithmetic mean) of the strains inside the inclusion equals the strain of the circular inclusion and vanishes for the points outside the inclusion. Similar to the Eshelby property, the quasi-Eshelby property can be used to study induced internal stress in micromechanics via the Eshelby equivalent inclusion method. Consequently, the Eshelby equivalent inclusion method can be extended to study inclusions of rotational symmetrical shape. In this paper, the discontinuity relation of the average strains in the Eshelby problem of rotational symmetrical inclusions obtained by [11] is proved by virtue of the stress continuity and the discontinuity of the displacement gradient.