The Eshelby stress tensor,angular momentum tensor and dilatation flux in gradient elasticity

The Eshelby (static energy momentum) stress tensor, the angular momentum tensor and the dilatation flux are derived for anisotropic linear gradient elasticity in non-homogeneous materials. The divergence of these tensors gives the configurational forces, moments and work terms in gradient elasticity. There are several types of configurational forces, acting on the dislocation density and its gradient, on the inhomogeneities, proportional to the distortion, and linear and quadratic in the distortion gradient, and on the body force.     

Compatibility conditions and equations of motion in the linear micropolar theory of elasticity

An analog of Cesàro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.     

The energy momentum tensor in the presence of body forces and the Peach–Koehler force on a dislocation

A modified energy momentum tensor, in the presence of body forces, is introduced and used to construct the nonconserved J, M, and L integrals, and to derive the energetic forces associated with a defect motion within the material. The J integral is then applied to evaluate the Peach–Koehler force on an inclined edge dislocation within a large block due to its own weight. The equilibrium position of the dislocation is determined for different boundary conditions of interest in geomechanics.     

Material electromagnetic fields and material forces

Electromagnetic fields address configurational forces in a natural way through an energy–stress tensor, which reduces to the Maxwell tensor in the simplest case. This tensor is related to physical forces and to the Cauchy traction in a continuum. Material forces, as opposed to physical forces, are of a different nature as they act upon a site of a continuum where the possible material inhomogeneity is located. A material energy–stress tensor, which is reminiscent of the Maxwell stress, is associated with these forces. Through appropriate balance laws, a material momentum is also associated with material forces. The material momentum is of particular interest in electromagnetic materials as it is intimately related to the pseudomomentum of light [Peierls in Highlights of Condensed Matter Physics, pp. 237–255 (1985) and in Surprises in Theoretical Physics, pp. 91–99 (1979); Thellung in Ann. Phys. 127, 289–301 (1980)]. The balance law for the material momentum can be derived either from the classical physical laws or independently of them. This derivation, which is based on the material electromagnetic potentials and the related gauge transformations, is discussed and commented on for an electromagnetic body.     

Effective elasticity tensor of a periodic composite

The effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of Nemat-Nasseret al., also agree well with the numerical results for cubic lattices.     

The elastic energy-momentum tensor

The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Günther's and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.     

Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures.     

Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

Restudy of coupled field theories for micropolar continua (I)—Micropolar thermoelasticity

Problems of micropolar thermoelasticity have been presented and discussed by some authors in the traditional framework of micropolar continuum field theory. In this paper the theory of micropolar thermoelasticity is restudied. The reason why it was restricted to a linear one is analyzed. The rather general principle of virtual work and the new formulation for the virtual work of internal forces as well as the rather complete Hamilton principle in micropolar thermoelasticity are established. From this new Hamilton principle not only the equations of motion, the balance equation of entropy, the boundary conditions of stress, couple stress and heat, but also the boundary conditions of displacement, microrotation and temperature are simultaneously derived. Contributed by DAI Tian-min Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

On Symmetries and Anisotropies of Classical and Micropolar Linear Elasticities: A New Method Based upon a Complex Vector Basis and Some Systematic Results

A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity tensors, which correspond to the isotropy group and the seven crystal classes S ₂, C _2h , D _2h , D _3d , D _4h , D _6h and O _h , while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes just mentioned and the four crystal classes S ₆, C _4h , C_6h and T _h . From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group, the transverse isotropy groups C _{∞ h} and D _{∞ h} , and the nine centrosymmetric crystallographic point groups except C _6h and D _6h . This revised version was published online in August 2006 with corrections to the Cover Date.     

Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts

Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21]. It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25–27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29–31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11–13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34]. Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m, n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for themoments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

Micropolar hypo-elasticity

In this paper, the concept of hypo-elasticity is generalized to the micropolar continuum theory, and the general forms of the constitutive equations of the micropolar hypo-elastic materials are presented. A new co-rotational objective rate whose spin is the micropolar gyration tensor is introduced which describes the deformation of the material in view of an observer attached to the micro-structure. As special case, simplified versions of the proposed constitutive equations are given in which the same fourth-order elasticity tensors are used as in the micropolar linear elasticity. A 2-D finite element formulation for large elastic deformation of micropolar hypo-elastic media based on the simplified constitutive equations in conjunction with Jaumann and gyration rates is presented. As an example, buckling of a shallow arc is examined, and it is shown that an increase in the micropolar material parameters results in an increase in the buckling load of the arc. Also, it is shown that micropolar effects become important for deformations taking place at small scales.     

On smoothly growing bodies and the eshelby tensor

A continuum mechanical theory of growing bodies is presented. It is assumed that the various parts of the body are identifiable. The growth of a body is manifested by mapping the identifiable elements of the growing body into a material manifold. Kinematics and stress theory are formulated on the basis of an infinite dimensional differentiable bundle structure for the configuration space. Stresses representing the forces associated with the growth of the body are analogous to the Eshelby tensor.

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Sommario Si propone una teoria meccanica dei corpi di massa crescente. Si postula che le varie parti del corpo siano identificabili. La crescita del corpo si manifesta mediante l'applicazione degli elementi identificabili del corpo in una varietà materiale. La cinematica e la teoria degli sforzi vengono formulati sulla base di un fibrato differenziabile a infinite dimensioni per lo spazio delle configurazioni. Gli sforzi associati alla crescita del corpo sono analoghi al tensore di Eshelby.

     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

An integral representation of the surface-impedance tensor for incompressible elastic materials

A fundamental result in anisotropic elasticity and surface-wave theory is the integral representation for the surface-impedance tensor first derived by Barnett and Lothe in 1973. However, this representation is only valid for compressible materials but not valid for incompressible materials. In this paper the corresponding integral representation for the surface-impedance tensor valid for incompressible materials is derived and is used to establish the uniqueness of surface-wave speed and to obtain an expression for the tensor Green's function for the infinite space. Mathematics subject classifications (2000) 74B05, 74B15, 74B20, 74J15     

Evolution Model for Linearized Micropolar Plates by the Fourier Method

In this paper we justify a two-dimensional evolution and eigenvalue model for micropolar plates starting from three-dimensional linearly micropolar elasticity. A small parameter representing the thickness of the plate-like body is introduced in the problem. The asymptotics of the evolution and eigenvalue problems is then developed as this small parameter tends to zero. First the appropriate convergences of the eigenpairs of the three-dimensional problem to the eigenpairs of the two-dimensional eigenvalue problem for micropolar plates is shown. Then these convergences are used in the Fourier method to obtain the convergences of the solution of the three-dimensional evolution problem to the solution of the two-dimensional evolution plate model.      

Derivation and Justification of the Models of Rods and Plates From Linearized Three-Dimensional Micropolar Elasticity

In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross–section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner–Mindlin plate or the Timoshenko beam type.     

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.