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Showing papers in "Mathematics and Mechanics of Solids in 2003"


Journal ArticleDOI
TL;DR: In this article, a third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure using pantographic-type structures, where the deformation energies involve combinations of nodal displacements in the form of second-order or third-order finite differences.
Abstract: Until now, no third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure. In this paper, we prove that this is possible using pantographic-type structures. Their deformation energies involve combinations of nodal displacements havin the form of second-order or third-order finite differences. We establish the Gamma convergence of these energies to second and third gradient functionals. Some mechanical examples are provided so as to illustrate the special features of these homogenized models.

441 citations


Journal ArticleDOI
TL;DR: In this article, a set of simple sufficient conditions for the polyconvexity and coercivity of strain energy functions for transversely isotropic and orthotropic elastic solids is presented.
Abstract: We present a set of simple sufficient conditions for the polyconvexity and coercivity of strainenergy functions for transversely isotropic and orthotropic elastic solids. The formulation is based on appropriate function bases for the right stretch tensor in the polar decomposition of the deformation gradient and furnishes numerical analysts with a priori existence criteria for boundary-value problems.

66 citations


Journal ArticleDOI
TL;DR: An analysis of the air-damping effect on the frequency response of a micromachined beam resonator is presented in this paper, where the motion of the beam is analyzed based on the linear elastic beam theory.
Abstract: An analysis of the air-damping effect on the frequency response of a micromachined beam resonator is presented. The motion of the beam is analyzed based on the linear elastic beam theory. The air d...

59 citations


Journal ArticleDOI
TL;DR: A theory that can describe the mechanical aspects of cartilage growth is presented, based on a general thermomechanical theory for a mixture of an arbitrary number of growing elastic constituents and an inviscid fluid.
Abstract: The proteoglycan and collagen constituents of cartilage serve distinct mechanical roles. Changes to the mechanical loading conditions during cartilage growth lead to changes in the concentrations of these molecules and, consequently, the mechanical properties. The main aim of this paper is to present a theory that can describe the mechanical aspects of cartilage growth. The model for cartilage growth is based on a general thermomechanical theory for a mixture of an arbitrary number of growing elastic constituents and an inviscid fluid. Our development of a growth mixture theory is accomplished in two steps. First, the thermodynamics of growing elastic materials are considered. The resulting theory of growing thermoelastic materials is extended to continuum mixture theory. Using this general growth mixture theory, we then propose a cartilage growth model that includes two special types of internal constraints that are relevant to cartilage.

38 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate the helical shear problem for a circular cylindrical tube composed of an isotropic hyperelastic incompressible material and obtain explicit closed-form analytic expressions for two of the models considered.
Abstract: The purpose of this research is to investigate the helical shear problem for a circular cylindrical tube composed of an isotropic hyperelastic incompressible material. Three specific constitutive models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy-Green tensor. For limiting values of a material parameter, all of these models reduce to the classical neo-Hookean form. The stress fields and displacements are characterized for each of these models. For two of the models considered, explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.

26 citations


Journal ArticleDOI
L.J. Sudak1
TL;DR: In this paper, the authors derived analytical expressions for the elastic stress fields due to the presence of an exterior screw dislocation interacting with a three-phase circular inhomogeneity and derived the force on the dislocation.
Abstract: Analytical expressions are obtained for the elastic stress fields due to the presence an exterior screw dislocation interacting with a three-phase circular inhomogeneity. In addition, the force on the dislocation is also derived. The bonding at the inhomogeneity—interphase interface is considered to be imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase—matrix interface, the bonding is assumed to be perfect.The equilibrium position of the dislocation is discussed in detail for various imperfect interface conditions, interphase layer thicknesses and material property combinations. The results demonstrate that the relative thickness of the interphase layer, the influence of the imperfect bonding condition and the material property combinations are manifested by their effects on the equilibrium position and subsequent stability of the dislocation.

24 citations


Journal ArticleDOI
TL;DR: In this paper, the authors describe the underlining principle of ionic polymer-metal composite actuation/sensing phenomena using linear irreversible thermodynamics, and present a description of these phenomena in the linear regime, and in steady-state conditions using the standard Onsager formulation.
Abstract: Ion containing polymers display certain spectacular mechanoelectrical phenomena when suitably composited with a conductor phase such as a metal, a conductive polymer or graphite—sometimes called ionic polymer conductor composites. When subjected to a dynamic electric field they deform dynamically (actuation), and if dynamically deformed they generate a dynamic electric field (transduction or sensing). Here, we present a description of these phenomena in the linear regime, and in steady-state conditions using the standard Onsager formulation. We describe the underlining principle of ionic polymer-metal composite actuation/sensing phenomena using linear irreversible thermodynamics. When static conditions are imposed, a simple description of the mechanoelectric effect is possible based upon two forms of transport: ion transport (with a current density, J) and electro-osmotic solvent transport (with a flux, Q). The conjugate forces include the electric field, E, and the pressure gradient, -▿p. We also present...

24 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigate the consequences of such an assumption and show that it is clearly inappropriate for many classes of inhomogeneous bodies, both qualitatively and quantitatively, with regard to local measures such as stresses and strains.
Abstract: It is quite common to approximate “mildly” inhomogeneous bodies as homogeneous bodies belonging to a certain constitutive class in view of the simplification that such an approximation accords. In this study, we investigate the consequences of such an assumption and we show that it is clearly inappropriate for many classes of inhomogeneous bodies. We choose specific boundary value problems to illustrate the fact that we could be grossly in error, both qualitatively and quantitatively, with regard to local measures such as stresses and strains. In the examples considered, we find that, for global quantities such as applied forces and moments, the error could be significant. Not only could the material parameters found from, say, an extension test and torsion test, which neglect the inhomogeneity of the body, be quite different from one that incorporates the inhomogeneity, but also the values for the material parameter in the homogenized approximation gleaned from these different experiments could be differ...

18 citations


Journal ArticleDOI
TL;DR: In this article, the authors extend this procedure to include crystals with uniform distribution of defects, where the behavior of defects is reckoned to amount for significant variations in the mechanical strength of crystalline materials.
Abstract: The continuum mechanics of crystals draws on the atomistic view of crystal structure by assuming that the symmetries of perfect (atomic) crystal lattices define the material symmetry groups of associated continua. Thus, for example, the discrete (point) group which gives the rotational symmetries of a perfect cubic lattice is employed, commonly, in defining the material symmetry group of the strain energy function of a linearly elastic continuum with cubic symmetry. In fact, the general procedure has also been used successfully in the context of nonlinear elasticity theory; it amounts to a constitutive assumption which transfers some aspect of a relevant discrete structure to the continuum model.Here I begin to extend this procedure so as to encompass crystals with uniform distribution of defects. The extension is of interest because the behavior of defects is reckoned to amount for significant variations in the mechanical strength of crystalline materials. So an understanding of the geometry of configura...

15 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give the overall picture relating to those special cases which give rise to analytical solutions for the two problems of granular flow through hoppers and the stress distributions at the base of stock piles.
Abstract: The flow of granular materials in the presence of gravity in hoppers and the storage of granular materials as a stock pile occur in many industrial situations. The governing ordinary differential equations for two-dimensional wedges and three-dimensional cones are highly nonlinear and there are no known general solutions, apart from that we have given for a special angle of internal friction. Here, we give the overall picture relating to those special cases which give rise to analytical solutions for the two problems of granular flow through hoppers and the stress distributions at the base of stock piles. These equations are fundamental to granular mechanics and previously only some special isolated exact solutions have been known. We list here a number of new exact analytical solutions applying for the two special cases of β= ±1, noting that β= sin φ where φ is the angle of internal friction. The case β= −1 corresponds to a non-physical material, but there are materials such as silica and alumina cake wh...

14 citations


Journal ArticleDOI
TL;DR: In this paper, the real boundary integral equation method is applied to solve interior and exterior mixed boundary value problems arising in a linear theory of antiplane elasticity which includes the effects of material microstructure.
Abstract: In the present paper we apply the real boundary integral equation method to solve interior and exterior mixed boundary value problems arising in a linear theory of anti-plane elasticity which includes the effects of material microstructure.

Journal ArticleDOI
TL;DR: This paper studies how the theory and constitutive restrictions depend upon the choice of the base curve, and it is shown how this choice has major qualitative consequences, which are illustrated with several concrete examples.
Abstract: In this paper, we treat several aspects of the induced geometrically exact theory of shearable rods, of central importance for contact problems, for which the regularity of solutions depends crucially on the presence of shearability. (An induced theory is one derived from three-dimensional theory by the imposition of constraints. Because the role of thickness enters into our theory in an essential way, it is an exact version of what has been called the theory of “moderately thick” rods.) In particular, we study how the theory and constitutive restrictions depend upon the choice of the base curve, and we show how this choice has major qualitative consequences, which are illustrated with several concrete examples.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the response of a pressurized spherical elastomeric membrane at a temperature high enough for the scission-crosslinking process to occur.
Abstract: When an elastomeric material is held at a fixed state of stretch at a sufficiently high temperature, macromolecular network junctions undergo time-dependent scission and the applied force relaxes with time. The affected molecules recoil and crosslink to form a new network that is stress-free in a new reference configuration. A constitutive equation that accounts for this process as the elastomer undergoes a time-dependent deformation has been presented elsewhere. This constitutive equation is used to study the response of a pressurized spherical elastomeric membrane at a temperature high enough for the scission-crosslinking process to occur. It is shown that the membrane diameter increases with time. Moreover, there can be a finite time when the membrane diameter increases at an infinite rate, i.e. there is “runaway inflation”. This event depends on the elastic properties of the molecular networks, the rate of scission, the rate of formation of new networks, and the history of the increase in the membrane...

Journal ArticleDOI
TL;DR: In this article, the consequences of Sih's fracture criterion in a material which behaves according to the Cosserat-type model with free or constrained rotations are analyzed.
Abstract: In this paper, we analyze the consequences of Sih’s fracture criterion in a material which behaves according to the Cosserat-type model with free or constrained rotations. Both models are considere...

Journal ArticleDOI
TL;DR: In this article, an energy-like measure E(θ) of the solution in the region between arbitrary θ and θ = α is defined, and proven to be positive definite provided that b/a < eπ.
Abstract: Solutions of the biharmonic equation are considered in the curvilinear rectangular region 0 ≤ θ ≤ α, a ≤ r ≤ b in the presence of boundary conditions φ = φr = 0 on the edges r = a, r = b, φ = φθ = 0 on the edge θ = α, (r, θ) denoting plane polar coordinates, a, b, α(< 2π) being constants; non-null boundary conditions are envisaged on the other edge θ = 0, involving the specification of φ, φθ thereon. An energy-like measure E(θ) of the solution in the region between arbitrary θ and θ = α is defined, and is proven to be positive definite provided that b/a < eπ. It is established that E(θ) / E(0) decays (at least) exponentially with respect to θ, under the aforementioned restriction on b/a. Additionally, a principle of the Dirichlet type is established (again provided b/a < eπ), which provides an upper bound for E(0) in terms of data (φ and φθ) prescribed on the edge θ = 0. When combined with the earlier result we obtain an explicit upper decay estimate for E(θ). The estimate can be regarded as a version of ...

Journal ArticleDOI
TL;DR: In this article, Chen and Haughton employed the nonlinear stability analysis to study the full non-homogeneous stability of the spherically symmetric deformation of an elastic thick-walled sphere.
Abstract: The nonlinear stability analysis introduced by Chen and Haughton [1] is employed to study the full nonlinear stability of the non-homogeneous spherically symmetric deformation of an elastic thick-walled sphere. The shell is composed of an arbitrary homogeneous, incompressible elastic material. The stability criterion ultimately requires the solution of a third-order nonlinear ordinary differential equation. Numerical calculations performed for a wide variety of well-known incompressible materials are then compared with existing bifurcation results and are found to be identical. Further analysis and comparison between stability and bifurcation are conducted for the case of thin shells and we prove by direct calculation that the two criteria are identical for all modes and all materials.

Journal ArticleDOI
TL;DR: In this paper, generalized durability diagrams and their properties are considered for a material under a multiaxial loading given by an arbitrary function of time, and material strength and durability under such loading is described in terms of durability, safety factor, and normalized equivalent stress.
Abstract: Generalized durability diagrams and their properties are considered for a material under a multiaxial loading given by an arbitrary function of time. Material strength and durability under such loading is described in terms of durability, safety factor, and normalized equivalent stress. Relations between these functionals are analysed. Some material properties including time and load stability, self-degradation (aging), monotonous damaging are discussed. Phenomenological strength conditions are presented in terms of the normalized equivalent stress. It is shown that the damage based durability analysis is reduced to a particular case of such strength conditions. Examples of the reduction are presented for some known durability models. The approach is applicable to the strength and durability description at creep and impact loading and their combination.

Journal ArticleDOI
TL;DR: In this article, Buryachenko et al. considered a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions.
Abstract: We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions and subjected to inhomogeneous boundary conditions. We use the multiparticle effective field method (MEFM) based on the theory of functions of random variables and Green's functions; for references, see Buryachenko, V.A. Applied Mechanics Reviews, 54, 1-47 (2001). Within this method, we derive a hierarchy of statistical moment equations for conditional averages of the stresses in the inclusions. The hierarchy is established by introducing the notion of an effective field. In this way the interaction of different inclusions is taken directly into account in the framework of the homogeneity hypothesis of the effective field. The non-local integral equation for the statistical average of stresses inside the inclusions is solved by three different methods: the quadrature method, the iteration method, and the Fourier transform method with subsequent ...

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of a single elastic inhomogeneity embedded within an infinite elastic matrix in anti-plane shear and examined the design of this inhomogeneous to achieve (stress) neutrality when a non-uniform stress field is prescribed in the surrounding matrix.
Abstract: In this paper, we consider the problem of a single elastic inhomogeneity embedded within an infinite elastic matrix in anti-plane shear. In particular, we examine the design of this inhomogeneity to achieve (stress) neutrality when a non-uniform stress field is prescribed in the surrounding matrix. Since it is known that neutral elastic inhomogeneities do not exist when the inhomogeneity is assumed to be perfectly bonded to the matrix, the design method presented here is based on the assumption of an imperfect interface and the appropriate choice of the (single) interface parameter (characterizing the imperfect interface) to achieve the desired neutrality. Specifically, in the case of a homogeneously imperfect interface, it is shown that the circular inhomogeneity is neutral if and only if the prescribed non-uniform stress field in the surrounding matrix belongs to a certain class of polynomial functions. In the case of an inhomogeneously imperfect interface, neutrality is established for circular and ell...

Journal ArticleDOI
TL;DR: In this article, the authors apply a novel perturbation technique developed by Elias-Zuniga, called the Elliptic Balance Method (EBM), to obtain the approximate solution of nonlinear two-degree-of-freedom systems.
Abstract: In many practical situations a pair of coupled, damped, homogeneous nonlinear ordinary differential equations model the dynamical behavior of mechanical systems. For example, these equations arise during the process of studying the mechanical response of systems such as strings, beams, absorbers, plates, and so on. In general, the exact solution of this sort of equation is unknown and hence, numerical integration, perturbation techniques or geometrical methods have been applied to obtain their approximate solution. A large number of studies of the nonlinear behavior of such systems have been made using perturbation techniques; however, the vast majority have dealt with weakly nonlinear systems, i.e. with small values of the nonlinear parameter e or the damping parameter ν. The objective of this work is to apply a novel perturbation technique developed by Elias-Zuniga, called the Elliptic Balance Method (EBM), to obtain the approximate solution of nonlinear two-degree-of-freedom systems. Two examples are p...

Journal ArticleDOI
TL;DR: In this paper, the authors give sufficient conditions for the existence and uniqueness of the classical and generalized solutions of a planar elastic, Reissner-type cantilever beam.
Abstract: We give the sufficient conditions for the existence and uniqueness of the classical and generalized solutions of a planar elastic, Reissner-type cantilever beam. The proof is rather general and assumes point loads from the outset. The existence and uniqueness of the finite element solution is also proven.

Journal ArticleDOI
TL;DR: In this article, the rate-type constitutive equations compatible with the dissipation postulate are derived in the initial and actual, as well as in the relaxed, configurations of anisotropic elasto-plastic materials.
Abstract: This paper deals with anisotropic elasto-plastic materials with relaxed configuration and internal variables, which undergo large deformations and obey the Il'yushin-type dissipation postulate. We emphasize new consequences of the dissipation postulate, when the definitions are built in the total strain space, and the irreversible behavior is described by the appropriate differential systems. The normality conditions prove that the appropriate stress release rates are collinear with the interior normal to the current yield surface and lead to the rate-type constitutive equations compatible with the dissipation postulate. The rate-type constitutive equations are derived in the initial and actual, as well as in the relaxed, configurations. For anisotropic Σ models, the rates of irreversible variables are linked through the flow rule in Σ space up to a term, generated by a gk-invariant, skew-symmetric tensor. Here Σ is Mandel's non-symmetric stress measure, and gk characterizes the pre-existing anisotropy. D...

Journal ArticleDOI
TL;DR: In this paper, the dynamics of a discretized model of an elastic bar in a hard device formed by a chain of point masses connected by nonlinear springs whose total length is a controlled parameter are studied.
Abstract: We study the dynamics of a discretized model of an elastic bar in a hard device formed by a chain of point masses connected by nonlinear springs whose total length is a controlled parameter. We compare the description of the system dynamics given by the first-order (gradient) dynamics, the second-order (Newtonian) dumped dynamics and the Relaxation Oscillation Theory. Using a technique based on Liapunov's second method, we prove a dynamic stability result concerning the above-mentioned ODEs.

Journal ArticleDOI
TL;DR: Using Shield's inverse theorem, new families of solutions describing spherical inflation for isotropic, homogeneous, and compressible elasticity were found in this paper, where it was shown that it is possible to compute spherical inflation in each case.
Abstract: Using Shield's inverse theorem, new families of solutions describing spherical inflation for isotropic, homogeneous, compressible elasticity are found. Cavitation is shown to be possible for each o...

Journal ArticleDOI
TL;DR: In this article, a model of Koiter's type for nonlinearly elastic shells with variable thickness was proposed, which generalizes a model recently proposed by P.G. Ciarlet for shells with constant thickness.
Abstract: In this paper, we propose a new model “of Koiter's type” for nonlinearly elastic shells with variable thickness, which generalizes a model recently proposed by P.G. Ciarlet for shells with constant thickness. We justify this model by means of an asymptotic analysis, by showing that its solution behaves either like that of a “membrane” or like that of a “flexural” shell as the thickness goes to zero.

Journal ArticleDOI
TL;DR: In this paper, the notion of energy-release rate is extended to the linearized incremental theory of pre-stressed and pre-polarized piezoelectric crystals under suitable restrictions on the pre-existing fields.
Abstract: Griffith's notion of energy-release rate is extended to the linearized incremental theory of pre-stressed and pre-polarized piezoelectric crystals under suitable restrictions on the pre-existing fields.

Journal ArticleDOI
TL;DR: There are six known families of deformations which are (isothermally) universal and controllable for incompressible, isotropic hyperelastic solids as mentioned in this paper.
Abstract: There are six known families of deformations which are (isothermally) universal and controllable for incompressible, isotropic hyperelastic solids. It is shown here that, independently of material, the first five of these families and a subclass of the sixth, no others, are compatible with non-uniform temperatures in the sense that a pressure exists to ensure static equilibrium, provided the temperature gradient is parallel to the divergence of the left Cauchy-Green strain tensor.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the plane stress unloading wave emanating from a suddenly punched circular hole in a thin elastic sheet, subjected to an initial finite deformation simple tension.
Abstract: We consider the plane stress unloading waves emanating from a suddenly punched circular hole in a thin elastic sheet, subjected to an initial finite deformation simple tension. The sheet is assumed to be hyperelastic, isotropic, incompressible and thin enough that the generalized plane stress approximation is valid. It is further assumed that the plugging of the hole is due to normal impact of a cylindrical projectile and that the impact velocity of the projectile is sufficiently high that the plugging can be assumed instantaneous. The governing equations of the problem are obtained in Lagrangian form with respect to a reference configuration that is the initial stretched configuration. A numerical scheme is proposed for the solution of the governing equations, and numerical results are obtained for the neo-Hookean strain energy function. However with some additional complication other isotropic strain energy functions can be considered.

Journal ArticleDOI
TL;DR: In this paper, a complete proof of the Graves and Weierstrass necessary conditions for minimizers for elastic materials is given for the case of an incompressible material, but only partial results have been established.
Abstract: The Graves and Weierstrass necessary conditions for minimizers are of central importance in the study of elastic materials that can undergo a change of phase. Proofs of these results applicable to the case of unconstrained materials abound. For the case of an incompressible material, however, only partial results have been established. Here, a complete proof is given.

Journal ArticleDOI
TL;DR: In this paper, the effect of stress-softening on the small amplitude transverse vibrational frequency of a stretched rubber membrane is investigated and the fundamental frequency is determined for a general incompressible and isotropic stresssoftening material.
Abstract: The effect of stress-softening on the small amplitude transverse vibrational frequency of a stretched rubber membrane is investigated. The fundamental frequency is determined for a general incompressible and isotropic stress-softening material. It is shown that for each fixed value of the stretch λ e (1,Λ), the fundamental frequency of the virgin material membrane is greater than the corresponding frequency of the same membrane preconditioned to a maximum previous stretch Λ. Moreover, the stress-softened membrane frequency for the same stretch decreases further with the degree of softening damage incurred upon increasing the maximum previous stretch. A specific damage function is introduced and explicit results are illustrated for three material models.