On reflected interactions in elastic solids containing inhomogeneities

Interactions in linear elastic solids containing inhomogeneities are examined using integral equations. Direct and reflected interactions are identified. Direct interactions occur simply because elastic fields emitted by inhomogeneities affect each other. Reflected interactions occur because elastic fields emitted by inhomogeneities are reflected by the specimen boundary back to the individual inhomogeneities. It is shown that the reflected interactions are of critical importance to analysis of representative volume elements. Further, the reflected interactions are expressed in simple terms, so that one can obtain explicit approximate expressions for the effective stiffness tensor for linear elastic solids containing ellipsoidal and non-ellipsoidal inhomogeneities. For ellipsoidal inhomogeneities, the new approximation is closely related to that of Mori and Tanaka. In general, the new approximation can be used to recover Ponte Castañeda–Willis׳ and Kanaun–Levin׳s approximations. Connections with Maxwell׳s approximation are established.     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Implementation of a high order lattice spring model for elasticity

In a lattice spring model (LSM), the material is discretised into particles linked by springs. However, LSMs always adopt linear springs, which results in a stiff approximation of the corresponding elastic solution. In this work, a high order LSM is proposed to overcome this limitation by introducing additional degrees of freedoms (DOFs) to the particles. Based on the energy minimisation principle and the local strain technique, equations for the stiffness matrices of high order LSM are derived. Relationships between micro spring parameters and macro material constants are derived from the Cauchy-born rules and the hyperelastic theory. Numerical examples show that the high order LSM can provide a better solution than that of the linear LSM and that the LSM is more suitable for modelling singularity and fracture problems.     

Thermodynamic nature and control of the elastic limit in solids

The elastic limit of a solid is implicit in its thermo-elastic properties and can be determined from the constitutive equations of internal energy and entropy in the elastic range. The second law of thermodynamics is responsible for this, as it sets an upper bound to the internal energy that a material can store during isothermal elastic deformation processes. A link between irreversibility and elasticity can thus be established, which allows for a better control of the properties of strength, ductility and elastic limit of the material. For elastic-plastic materials of practical interest it implies that the yield limit cannot be assigned independently of the elastic constitutive equations, although the current approaches do so. An application to elastic-plastic materials with linear thermo-elastic properties reveals that, in the one-dimensional case, all information on the entropy of the material can be drawn from standard uniaxial tests. An easy procedure can then be devised to design the preparation process of the material so that the desired combination of strength, ductility and elastic limit can be achieved within the admissible values for these quantities.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

On a theory of micropolar protoelastic material bodies and constitutive equations for nonlocal micropolar elastic continua

In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived.     

The fundamental equations in finite element method of coupled thermo-elastic plane problem

The fundamental equations in finite element method for unsteady temperature field elastic plane problem are derived on the bases of variational principle of coupled thermoelastic problems. In these derivations, elastic plane is divided into three nodes triangular elements, and time interval is divided into linear time elements, in which all the variables, including displacements and temperatures at various nodal points, are varied linearly with time. Two coupled sets of linear algebraic equations of all the unknown displacements and temperatures at every nodal point in every instant (i.e. the terminal values of time elements) are obtained. They are the fundamental equations of the said problem.The total energy in elastic body not only contains the potential energy and heat energy but also contains the kinetic energy, if the rate of change of temperature field with respect to the time in thermoelastic problem is large enough. And the change of displacement is included in the equations of heat conduction. For this reason the variational principle of coupled thermoelastic problems is employed. [1] In this paper, expressions of this principle for plane problems are given. The discretization is carried on then, and Hamilton's action and the potential action of heat flow of elements are derived. Finally they are assembled, so as to get the polar value of the action. And thus the groups of linear algebraic equations in matrix form are obtained.     

Plane strain and generalized plane stress problems for fibre-reinforced materials

In this paper the basic equations governing the plane strain or generalized plane stress deformations of a linear elastic material reinforced by a single family of parallel inextensible fibres are deduced. It is found that a single system of equations will cover all cases. The solutions for plane, half-plane and strip problems are evaluated and compared with those for an ideal fibre-reinforced material.     

Wave propagation in fractured porous media

A theory of wave propagation in fractured porous media is presented based on the double-porosity concept. The macroscopic constitutive relations and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. In microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law in fractured porous media. The macroscopic constitutive relations of elastic porous media saturated by one or two fluids and saturated fractured porous media can be obtained from the constitutive relations developed in the paper. In the simplest case, the final set of governing equations reduce to Biot's equations containing the same parameters as of Biot and Willis.Now at Izmir Institute of Technology, Anafartalar Cad. 904, Basmane 35230, Izmir, Turkey.     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

Numerical simulation on coupling behavior of Terfenol-D rods

Terfenol-D rods, as a kind of giant magnetostrictive materials, are often used as active elements of device for anti-vibration application due to its superior material properties. Their magneto-mechanical responses exhibited in many experiments are nonlinear and coupled. In order to have a good understanding on their coupling characters for accurate control, the numerical simulation on dynamic behavior of a Terfenol-D rod is conducted based on a nonlinear and coupling constitutive model proposed in this paper. The results show that the constitutive model can effectively describe some intrinsic coupling phenomena observed by experiments involving the maximum magnetostrictive strain of a Terfenol-D rod changing with pre-stresses and the corresponding dynamic responses show that the frequency and the amplification of the Terfenol-D rod change with magnetic bias field and pre-stresses, which are also consistent with experimental data and cannot be captured by previous constitutive model.     

Load transfer in fibre-reinforced composites with viscoelastic matrix: an analytical study

In A fibre-reinforced 2D composite material with elastic fibres and viscoelastic, isotropic matrix is studied. Starting from the solution of a reference-problem with elastic matrix material the elastic matrix parameters are substituted by their viscoelastic correspondents in the Laplace domain. For simplification the time-dependent solution is approximated by using limiting value theorems that give information about the time-dependent solution for t → 0 and t → ∞. Then the method of asymptotically equivalent functions is used and illustrated with examples of a steel fibre in a PMMA matrix. The analytical solutions are compared with their numerical counterparts. In summary it can be stated that this paper is a further contribution to the vast literature about the application of the correspondence principle to the solution of special problems of the linear viscoelasticity.     

A procedure for the static analysis of cable structures following elastic catenary theory

The paper proposes an unitary strategy for the static analysis of general cable nets under conservative loads. A form-finding is first performed in order to initialize the successive non linear analysis. The numerical procedures carried on in both steps, form finding and structural analysis of the net, employ a three dimensional elastic catenary element. Equilibrium conditions at internal nodes and kinematic compatibility at the end nodes of each cable are used to derive the global equations of the net. When the pre-stresses are high and the topology of the net is involved, an accurate initializing solution is essential for the convergence of the successive numeric non linear structural analysis (performed by Newton method). The numerical applications highlight the capability of the proposed procedure to solve three dimensional problems with taut and slack cables, out of plane distributed forces (modeling wind loads), point loads along the cables. The contemporary presence of cables and compression truss elements is also considered testing the effectiveness of the method in the analysis of tensegrity structures.     

On the Canonical Elastic Moduli of Linear Plane Anisotropic Elasticity

The linear, planar, anisotropic elastic equilibrium equations are transformed to canonical form, through linear transformations of both coordinates and unknown displacement functions, together with a linear combination of equations. Correspondingly, the six original material moduli are replaced by two canonical elastic moduli. Similar results have been reached by Olver in 1988. However, the method demonstrated in this paper is more concise and direct. As an example, the general solution to the canonical equations is obtained in the case of a pair of double roots.     

Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations

The aim of this article is to propose a simple way of describing a tumour as a linear elastic material from a reference configuration that is continuously evolving in time due to growth and remodelling. The main assumption allowing this simplification is that the tumour mass is a very ductile material, so that it can only sustain moderate stresses while the deformation induced by growth, that can actually be quite big, mainly induces a plastic reorganisation of malignant cells. In mathematical terms this means that the deformation gradient can be split into a volumetric growth term, a term describing the reorganisation of cells, and a term that can be approximated by means of the linear strain tensor. A dimensional analysis of the importance of the different terms also allows to introduce a second simplification consisting of decoupling the equations describing the growth of the tumour mass from those describing the flow of the interstitial fluid.     

有限尺寸多钉连接件钉传载荷计算的解析方法

提供一种确定多钉连接件中钉传载荷的解析方法，这个方法将被连接件看作弹性体，以经典结构力学以及弹性理论平面问题复变函数解法为基础，建立了求解钉传载荷的线性代数方程组并给出了若干算例。这个方法不仅具有合理的力学模型，而且具有计算的简捷性与适用的广泛性。     

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.     

Mechanics of Materials with Periodic Truss or Frame Micro-Structures

This paper describes the mechanics of materials with periodic skeletal micro-structures in infinite domains. The principal technical results consist of certain Korn-type inequalities that provide upper and lower bounds for the linear elastic strain energy in the material. Using these inequalities, existence and uniqueness results for the equations of linear elastic equilibrium are derived, and some asymptotic properties of the solutions are described. Particular attention is paid to the question of when a lattice structure can accurately be modeled as a pin-jointed truss, and when a rigid-node frame model must be employed. A practical technique for how to distinguish between the two types of material is given, and the distinct differences in their mechanical behavior are described.