Eshelby tensors for an ellipsoidal inclusion in a microstretch material

Eshelby tensors for an ellipsoidal inclusion in a microstretch material are derived in analytical form, involving only one-dimensional integral. As micropolar Eshelby tensor, the microstretch Eshelby tensors are not uniform inside of the ellipsoidal inclusion. However, different from micropolar Eshelby tensor, it is found that when the size of inclusion is large compared to the characteristic length of microstretch material, the microstretch Eshelby tensor cannot be reduced to the corresponding classical one. The reason for this is analyzed in details. It is found that under a pure hydrostatic loading, the bulk modulus of a microstretch material is not the same as the one in the corresponding classical material. A modified bulk modulus for the microstretch material is proposed, the microstretch Eshelby tensor is shown to be reduced to the modified classical Eshelby tensor at large size limit of inclusion. The fully analytical expressions of microstretch Eshelby tensors for a cylindrical inclusion are also derived.     

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

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.     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

On the generalized virial theorem and Eshelby tensors

The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule.     

考虑Steigmann-Ogden表面的微纳米圆柱解析模型

基于Steigmann-Ogden（S-O）表面理论,研究了圆柱形微纳米材料在轴向对压荷载作用下的力学性能。利用级数展开求解材料内部的弹性控制方程,获得了考虑表面效应时的域内解析表达式。当所得结果忽略表面弯曲参数时可退化为Gurtin-Murdoch（G-M）表面模型。用文献中有限元数值结果对本理论进行退化验证,结果得到良好一致性。在此基础上,讨论了表面弯曲参数和圆柱尺寸大小对材料特性的影响。结果显示：考虑了表面弯曲效应的S-O模型和G-M模型在应力分布中有很大的不同。另外,随着圆柱尺寸的减小,其表面效应对材料的力学特性的影响逐渐增大。     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

Strength certificate and the constraint equation relating stress and strain invariants for inhomogeneous rocks in bulk stress state

Mechanical models of rocks subjected to various bulk stress states are considered. The models contain the Nadai parameter, which takes into account various types of the bulk stress state, the potential energy of bulk variation, and the potential energy of variations in the shape of rock elements in the equilibrium stress-strain state. These variables permit determining the conditions under which rock massifs stay in equilibrium and the conditions under which rock can cease to be in equilibrium owing to the bulk stress state action; they also allow one to justify the physical parameters of the rock strength certificate under the bulk stress state action and determine the constraint equation relating the stress and strain invariants for inhomogeneous rocks.     

Scattered fracture of porous materials with brittle skeleton

A model of damage accumulation in a porous medium with a brittle skeleton saturated with a compressible fluid is formulated in the isothermal approximation. The model takes account of the skeleton elastic energy transformation into the surface energy of microcracks. In the case of arbitrary deformations of an anisotropic material, constitutive equations are obtained in a general form that is necessary and sufficient for the objectivity and thermodynamic consistency principles to be satisfied. We also formulate the kinetics equation ensuring that the scattered fracture dissipation is nonnegative for any loading history. For small deviations from the initial state, we propose an elastic potential which permits describing the principal characteristics of the behavior of a saturated porous medium with a brittle skeleton. We study the acoustic properties of the material under study and find their relationship with the strength criterion depending on the accumulated damage and the material current deformation. We consider the problem of scattered fracture of a saturated porous material in a neighborhood of a spherical cavity. We show that the cavity failure occurs if the Hadamard condition is violated.     

Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

The aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity.Kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models. Integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section shape change, are achieved. These integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives.     

Mechanical modeling of growth considering domain variation—Part II: Volumetric and surface growth involving Eshelby tensors

The growth of biological tissues is here described at the continuum scale of tissue elements. Relying on a previous work in Ganghoffer and Haussy (2005), the rephrasing of the balance laws for tissue elements under growth in terms of suitable Eshelby tensors is done in the present contribution, considering successively volumetric and surface growth. Balance laws for volumetric growth are written in both compatible and incompatible configurations, highlighting the material forces for growth associated to Eshelby tensors. Evolution laws for growth are written from the expression of the local dissipation in terms of a relation linking the growth velocity gradient to a growth-like Eshelby stress, in the spirit of configurational mechanics. Surface growth is next envisaged in terms of phenomena taking place in a varying reference configuration, relying on the setting up of a surface potential depending upon the surface transformation gradient and to the normal to the growing surface. The balance laws resulting from the stationnarity of the potential energy are expressed, involving surface Eshelby tensors associated to growth. Simulations of surface growth in both cases of fixed and moving generating surfaces evidence the interplay between diffusion of nutrients and the mechanical driving forces for growth.     

Stress fields and Eshelby forces on half-plane inhomogeneities with eigenstrains and strip inclusions meeting a free surface

The stress field due to a half-plane inhomogeneity with plane eigenstrain is obtained by a limiting procedure from the one of a circular Eshelby inhomogeneity/inclusion. This field, which requires tractions to be applied at infinity to be sustained, has minimum strain energy versus any other superposed homogeneous one, and is the Eshelby solution inside plus the Hill jump conditions. By superposition, the stresses due to an infinite strip (Eshelby property domain) inhomogeneity with eigenstrain are obtained, and, by superposition periodic strips or laminates can be obtained. By cancelling the stresses on a free-surface, strips of inclusions meeting a free surface are solved. They exhibit tensile stresses under the free surface, and logarithmic singularities in the tensile stress at the vertex, which may initiate cracking. The Eshelby self-forces on the boundary of circular and half-plane inhomogeneities are computed.     

Inhomogeneity effects on the crack driving force in elastic and elastic-plastic materials

This article evaluates the effect of material inhomogeneities on the crack-tip driving force in general inhomogeneous bodies and reports results for bimaterial composites. The theoretical model, based on Eshelby material forces, makes no assumptions about the distribution of the inhomogeneities or the constitutive properties of the materials. Inhomogeneities are modeled by making the stored energy have an explicit dependence on the reference coordinates. Then the material inhomogeneity effect on the crack-tip driving force is quantified by the term C_inh, which is the integral of the gradient of the stored energy in the direction of crack growth. The model is demonstrated by two model problems: (i) bimaterial elastic composite using asymptotic solutions and (ii) graded elastic and elastic-plastic compact tension specimen using numerical methods for stress analysis.     

Thermomechanics of volumetric growth in uniform bodies

A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of first- and second-order uniformity “transplants”. An essential role is played by the balance of canonical (material) momentum where the flux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only first-order in gradients, diffusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity “gradient” (first-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the “Mandel” stress, although the possible influence of “elastic” strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diffusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the first-order one, thus avoiding insuperable complications. Differential geometric aspects are developed where needed.     

Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems

At small length scales and/or in presence of large field gradients, the implicit long wavelength assumption of classical elasticity breaks down. Postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small-scale elastic phenomena, we herein extend Eshelby’s classical formulation for inclusions and inhomogeneities. While the modified size-dependent Eshelby’s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived, the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed-form solutions. To that end, Eshelby’s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities. The approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses. The present work is expected to find application towards nano-inclusions and certain types of composites in addition to being the basis for subsequent non-linear homogenization schemes.     

Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori-Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.     

Une transformation du problème d'élasticité linéaire en vue d'application au problème de l'inclusion et aux fonctions de Green

A simple transformation of the problem of the linear elastic structure is presented. The transformed problem corresponds to a new problem of linear elastic structure with different behaviour, geometry and prescribed forces and displacements. The transformed problem can be easier to study, or can correspond to cases with well-known solutions. By means of this transformation, the problem of ellipsoidal inclusion is transformed into a problem of spherical inclusion, the analytical results known for the Eshelby tensor for an isotropic or transversely isotropic matrix are extended to more general cases of matrix behaviour, and finally, close form expressions of the Green function for an infinite medium are derived for some cases of elastic behaviour without transversal isotropy or orthotropy.     

含圆形夹杂的无穷大平面电致伸缩材料的应力分析

利用电致伸缩基本方程,采用伪总应力和复变函数解法,并利用级数展开方法得出了含圆形夹杂的无限大电致伸缩材料应力场,在一般情况下,与Eshelby夹杂理论不同,在电致伸缩材料圆形夹杂内部应力场是非均匀的．     

Material force for dynamic crack propagation in multiphase micromorphic materials

Micromorphic theory, which considers material body as a continuous collection of deformable particles of finite size and inner structure; each has nine independent degrees of freedom describing the stretches and rotations of the particle in addition to the three classical translational degrees of freedom of its center, is briefly introduced in this work. The concept of material forces, which may also be referred as Eshelbian mechanics, is extended to micromorphic theory. The balance law of pseudo-momentum is formulated. The detailed expressions of Eshelby stress tensor, pseudo-momentum, and material forces are derived for thermoelastic micromorphic solid. It is found that the material forces are due to (1) body force and body moment, (2) temperature gradient and (3) material inhomogeneities in density, microinertia, and elastic coefficients. The general expression of material forces due to the presence of dynamically propagating crack front has also been derived. It is found that, at the crack front, material force is reduced to the J-integral in a very special and restrictive case.     

Elastic constants for granular materials modeled as first-order strain-gradient continua

Formulation of a stress–strain relationship is presented for a granular medium, which is modeled as a first-order strain-gradient continuum. The elastic constants used in the stress–strain relationship are derived as an explicit function of inter-particle stiffness, particle size, and packing density. It can be demonstrated that couple-stress continuum is a subclass of strain-gradient continua. The derived stress–strain relationship is simplified to obtain the expressions of elastic constants for a couple-stress continuum. The derived stress–strain relationship is compared with that of existing theories on strain- gradient models. The effects of inter-particle stiffness and particle size on material constants are discussed.