Showing papers in "Mathematics and Mechanics of Solids in 2011"
TL;DR: In this article, two general models of fractional heat conduction for non-homogeneous anisotropic elastic solids are introduced and the constitutive equations for thermoelasticity theory are obtained.
Abstract: Two general models of fractional heat conduction for non-homogeneous anisotropic elastic solids are introduced and the constitutive equations for thermoelasticity theory are obtained, uniqueness an...
300 citations
TL;DR: In this article, the notions of Cauchy elastic and Green elastic bodies are discussed, and the authors find that such bodies need not be Green elastic, since the stress is not derivable from the stored energy.
Abstract: The term ‘elasticity’ seems to conjure different images in different minds. After a discussion of the various interpretations of elasticity espoused by the pioneers, we discuss the notions of Cauchy elastic and Green elastic bodies, and whether Cauchy elastic bodies that are not Green elastic are reasonable from a physical standpoint. We then discuss a class of models, more general than classical Cauchy elastic bodies, and we find that such bodies need not be Green elastic. While a stored energy can be associated with these materials, the stress is not derivable from the stored energy. One can delineate conditions under which these models are thermodynamically consistent in that they meet the second law of thermodynamics; more precisely, the general class of bodies that is being described is incapable of dissipation1 in any process whatsoever. These models not only add to the repertoire of the elasticians in modeling solids that are incapable of dissipation, but also they seem to provide an opportunity fo...
122 citations
TL;DR: In this article, a constitutive model for describing the elastic response of solids that does not stem from either classical Cauchy or Green elasticity was developed, where the linearized strain is related to the stress in a non-linear manner.
Abstract: In this paper we develop a constitutive model for describing the elastic response of solids that does not stem from either classical Cauchy or Green elasticity. In contrast to the classical theory, we show that it is possible to obtain a constitutive model wherein the linearized strain is related to the stress in a non-linear manner. The specific choice that we make allows for the stress to be arbitrarily large while the strain remains small (consistent with the assumption used in the linearization of the non-linear strain) or below some limiting value. Such models are worth investigating in detail as they have relevance to problems involving cracks as well as other problems wherein one finds strain singularities within the classical theory of linearized elasticity, and to models that exhibit limited stretch.
80 citations
TL;DR: In this article, the authors derived explicit expressions for the associated magneto-acoustic tensors in the case of an incompressible isotropic magnetoelastic material, and these were then used to study the propagation of incremental plane waves.
Abstract: In this paper, in the context of the quasi-magnetostatic approximation, we examine incremental motions superimposed on a static finite deformation of a magneto-elastic material in the presence of an applied magnetic field. Explicit expressions are obtained for the associated magneto-acoustic (or magneto-elastic moduli) tensors in the case of an incompressible isotropic magneto-elastic material, and these are then used to study the propagation of incremental plane waves. The propagation condition is derived in terms of a generalized acoustic tensor and the results are illustrated by obtaining explicit formulas in two special cases: first, when the material is undeformed but subject to a uniform bias field and, second for a prototype model of magneto-elastic interactions in the finite deformation regime. The results provide a basis for the experimental determination of the material parameters of a magneto-sensitive elastomer from measurements of the speed of incremental waves for different pre-strains, bias magnetic fields, and directions of propagation.
41 citations
TL;DR: In this article, a dynamical version of the three-dimensional translation gauge theory of dislocations is proposed, which uses the notions of dislocation density and dislocation current tensors as the translational field strengths and defines corresponding response quantities (pseudomoment stress and dislocations momentum flux).
Abstract: We propose a dynamical version of the three-dimensional translation gauge theory of dislocations. In our approach, we use the notions of dislocation density and dislocation current tensors as the translational field strengths and define corresponding response quantities (pseudomoment stress and dislocation momentum flux). For the equations of motion for dislocations, we derive a closed system of field equations in an elegant quasi-Maxwellian form. The dynamical Peach— Koehler force density is derived in this framework as well. Finally, we discuss the similarities and differences between Maxwell field theory and the dislocation gauge theory developed here.
39 citations
TL;DR: In this article, the spatial variation of material parameters for pressurized cylinders and spheres composed of either an incompressible Hookean, neo-Hookean or Mooney-Rivlin material was studied.
Abstract: We find the spatial variation of material parameters for pressurized cylinders and spheres composed of either an incompressible Hookean, neo-Hookean, or Mooney—Rivlin material so that during their axisymmetric deformations either the in-plane shear stress or the hoop stress has a desired spatial variation. It is shown that for a cylinder and a sphere made of an incompressible Hookean material, the shear modulus must be a linear function of the radius r for the hoop stress to be uniform through the thickness. For the in-plane shear stress to be constant through the cylinder (sphere) thickness, the shear modulus must be proportional to r2 (r3). For finite deformations of cylinders and spheres composed of either neo-Hookean or Mooney—Rivlin materials, the through-the-thickness variation of the material parameters is also determined, for either the in-plane shear stress or the hoop stress, to have a prespecified variation. We note that a constant hoop stress eliminates stress concentration near the innermost ...
30 citations
TL;DR: In this article, the small amplitude radial oscillation and infinitesimal stability about an equilibrium configuration of an arbitrary incompressible, isotropic and homogeneous elastic spherical shell under constant inflation pressure loading is studied for both thick and thin-walled shells and for a spherical cavity within an unbounded continuum.
Abstract: The small amplitude radial oscillation and infinitesimal stability about an equilibrium configuration of an arbitrary incompressible, isotropic and homogeneous elastic spherical shell under constant inflation pressure loading is studied for both thick- and thin-walled shells and for a spherical cavity within an unbounded continuum. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and the pressure response. It follows that points at which the pressure is stationary are unstable or neutrally stable. All results are expressed in terms of the shear response function for a general incompressible, isotropic elastic material, and specific results are illustrated for the Mooney—Rivlin and Gent material models, the latter having limited extensibility. The classical neo-Hookean material exhibits results that are lower bounds for both models. A criterion obtained by others to characterize the possible bifurcation from a spherical to an aspherical shape is c...
23 citations
TL;DR: An existence theory is provided for the full thermomechanical quasi-static evolution of a shape memory wire described by the Souza—Auricchio constitutive model and it is shown that, by imposing no such restriction, the original Souza-AurICchio model is ill-posed.
Abstract: We provide an existence theory for the full thermomechanical quasi-static evolution of a shape memory wire described by the Souza-Auricchio constitutive model. The analysis requires some mild restriction on the choice of the thermomechanical coupling term in the expression on the free energy of the material. This restriction slightly deviates from the original Souza-Auricchio modeling frame, while still being compatible with real situations. We additionally show that, by imposing no such restriction, the original Souza-Auricchio model is ill-posed.
19 citations
TL;DR: In this article, the authors gave new demonstrations of Reynolds' transport theorems for moving regions in Euclidean space, based on differential forms and Stokes' formula, and derived a corresponding surface transport theorem using the partition of unity and surface divergence theorem.
Abstract: This paper gives new demonstrations of Reynolds' transport theorems for moving regions in Euclidean space. For moving volume regions the proof is based on differential forms and Stokes' formula. Moving curves and surface regions are defined and the intrinsic normal time derivative is introduced. The corresponding surface transport theorem is derived using the partition of unity and the surface divergence theorem. A proof of the surface divergence theorem is also given.
18 citations
TL;DR: In this paper, the authors established a convolutional variational principle and a reciprocal principle in the context of the linear theory of thermoelasticity of type III, and formulated a characterization of the mixed boundary initial value problem in which the initial conditions are incorporated into the field equations.
Abstract: The aim of the present paper is to establish a convolutional type (Gurtin ME. Variational principles for linear elastodynamics. Arch Ration Mech Anal 1964; 16: 34—50) variational principle and a reciprocal principle in the context of the linear theory of thermoelasticity of type III. To establish the variational and reciprocal principles, we firstly formulate a characterization of the mixed boundary initial value problem in which the initial conditions are incorporated into the field equations.
18 citations
TL;DR: The problem of torsion superimposed on axial extension of a solid circular cylinder composed of an incompressible isotropic hyperelastic material has been extensively investigated in the literature as mentioned in this paper.
Abstract: The problem of torsion superimposed on axial extension of a solid circular cylinder composed of an incompressible isotropic hyperelastic material has been extensively investigated in the literature...
TL;DR: In this paper, a rigorous analysis of residual strains and stresses generated by growth in the axisymmetric Kirchhoff plate is given, where balance equations are derived via the global constraint principle, growth is incorporated via a multiplicative decomposition of the deformation gradient, and the system is closed by a response function.
Abstract: Volumetric growth of an elastic body may give rise to residual stress. Here a rigorous analysis is given of the residual strains and stresses generated by growth in the axisymmetric Kirchhoff plate. Balance equations are derived via the Global Constraint Principle, growth is incorporated via a multiplicative decomposition of the deformation gradient, and the system is closed by a response function. The particular case of a compressible neo-Hookean material is analyzed, and the existence of residually stressed states is established.
TL;DR: In this paper, a finite strain model for the mechanical degradation of composite materials with multiple families of fine reinforcing fibers is developed and studied, and a rich variety of material and load-dependent transition possibilities are systematically uncovered using a combination of asymptotic and numerical techniques.
Abstract: A finite strain model for the mechanical degradation of composite materials with multiple families of fine reinforcing fibers is developed and studied. At any instant of time the matrix material may or may not be degrading with all, some, or none of the interpenetrating fibers also undergoing degradation. This multi-component description of damage is governed by coupled differential equations when more than one damage mechanism is active. These differential equations, and the threshold values of the strain invariants that activate the damage process, emerge naturally from a general framework that describes the response of dissipative systems under a maximum rate of dissipation postulate. In this context we then study uniaxial loadings when either a constant stretch or a constant force is suddenly applied to the composite. It is found that the initial type of degradation (e.g. degrading fibers in a non-degrading matrix) may transition to an alternative type of degradation (e.g. the degradation of all constituents) at some finite time into the process. A rich variety of material and load-dependent transition possibilities are systematically uncovered using a combination of asymptotic and numerical techniques. The resulting macroscopic behavior as the material weakens involves relaxation and creep phenomena that are formally similar to viscoelastic material behavior in solids even though the underlying processes are significantly different. Describing implants and tissue constructs containing biodegradable polymers is one possible area of application.
TL;DR: It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and is suitable for the formation of a polyconvex tissue strain energy function.
Abstract: Despite distinct mechanical functions, biological soft tissues have a common microstructure in which a ground matrix is reinforced by a collagen fibril network. The microstructural properties of the collagen network contribute to continuum mechanical tissue properties that are strongly anisotropic with tensile-compressive asymmetry. In this study, a novel approach based on a continuous distribution of collagen fibril volume fractions is developed to model fibril reinforced soft tissues as a nonlinearly elastic and anisotropic material. Compared with other approaches that use a normalized number of fibrils for the definition of the distribution function, this representation is based on a distribution parameter (i.e. volume fraction) that is commonly measured experimentally while also incorporating pre-stress of the collagen fibril network in a tissue natural configuration. After motivating the form of the collagen strain energy function, examples are provided for two volume fraction distribution functions. Consequently, collagen second-Piola Kirchhoff stress and elasticity tensors are derived, first in general form and then specifically for a model that may be used for immature bovine articular cartilage. It is shown that the proposed strain energy is a convex function of the deformation gradient tensor and, thus, is suitable for the formation of a polyconvex tissue strain energy function.
TL;DR: In this paper, the authors consider an antiplane model which describes the contact between a deformable cylinder and a rigid foundation, under the small deformation hypothesis, for quasistatic processes.
Abstract: We consider an antiplane model which describes the contact between a deformable cylinder and a rigid foundation, under the small deformation hypothesis, for quasistatic processes. The behaviour of the material is modelled using a viscoelastic constitutive law with long memory and the frictional contact is modelled using Tresca’s law. We focus on the weak solvability of the model, based on a weak formulation with dual Lagrange multipliers.
TL;DR: In this paper, the authors combined experiments at the grain level with the constitutive modeling of a single crystal to investigate deformation behaviors of a pure aluminum single crystal oriented for a single slip.
Abstract: This work combined experiments at the grain level with the constitutive modeling of a single crystal to investigate deformation behaviors of a pure aluminum single crystal oriented for a single slip. Simple compression of a [132] single crystal was conducted to obtain the macroscopic stress—strain response and observe the deformed shape of the sample. The evolution of the crystallographic orientation was measured by an electron backscattered diffraction system in a field emission scanning electron microscope. The heterogeneous in-plane surface deformation in a single crystal was calculated by using a digital image correlation technique. Finite element modeling based on a single crystal plasticity model agrees well with experimental results in an averaged sense, not at the individual dislocation level.
TL;DR: In this paper, counterexamples to Korn's inequality were constructed for n = 2 and Γ = ∂Ω without any regularity assumptions on R. It is known that the above inequality holds in any dimension n and for any Γ ⊆ ∂ Ω, provided that R ∈ C(Ω).
Abstract: Let Ω be an open, bounded set in Rn (n ≥ 2) with Lipschitz boundary ∂Ω. Assume that Γ ⊆ ∂Ω has a positive Hn-1 measure and let R : Ω → SO(n) be given. We construct counterexamples to the following version of Korn’s inequality in the following two cases: (a) n = 2 and Γ ≠ ∂Ω; (b) n = 3 and Γ = ∂Ω. It is known that the above inequality holds in any dimension n and for any Γ ⊆ ∂Ω, provided that R ∈ C(Ω). However, the inequality is true for n = 2 and Γ = ∂Ω without any regularity assumptions on R.
TL;DR: In this article, a theorem and an example application are given to illustrate the restrictive nature of the virtual power principle, and to draw attention to its severe consequences: consequences that question the appropriateness of the conclusions that have been reported in the recent literature.
Abstract: A ‘principle of virtual power’ has been re-introduced in recent years into the continuum mechanics literature, and used in the modeling of material behavior that involves multiple length scales. In these works, the ‘principle’ is stated for arbitrary parts of a body and this flexibility is used, to a certain extent, to draw conclusions concerning the structure of the theory that results. However, this arbitrariness, when fully applied, carries many consequences that have been overlooked. Here, a theorem and an example application are given to illustrate the restrictive nature of the ‘principle’, as it has been stated, and to draw attention to its severe consequences: consequences that question the appropriateness of the conclusions that have been reported in the recent literature.
TL;DR: In this article, the propagation of torsional surface waves in an anisotropic porous half space in the presence of a gravity field was studied and it was shown that the velocity of the wave increases considerably as the porosity of the medium increases.
Abstract: In this paper we discuss the propagation of torsional surface waves in an anisotropic porous half space in the presence of a gravity field. Although the torsional wave does not propagate in an isotropic elastic half space in the absence of a gravity field, it is found that such a wave propagates in a gravitating isotropic half space. It also propagates in anisotropic half space in the presence or absence of a gravity field if the modulus of elasticity in the horizontal direction is more than that of the vertical direction. It is observed that poro-elasticity has a dominant effect on the velocity of the torsional surface wave. The velocity of the wave increases considerably as the porosity of the medium increases. The presence of a gravity field is also seen to increase the velocity of the torsional surface wave. The velocity depends to a large degree on the anisotropy factor. As a particular case, it is verified that the torsional surface wave does not propagate in perfect fluid under gravity.
TL;DR: In this paper, the Euler-Lagrange equations and the interface conditions are derived for homogeneous membranes with quadratic "Helfrich"-type energies with non-uniform spontaneous curvatures.
Abstract: Curvature elasticity is used to derive the equilibrium conditions that govern the mechanics of membrane–membrane adhesion. These include the Euler–Lagrange equations and the interface conditions which are derived here for the most general class of strain energies permissible for fluid surfaces. The theory is specialized for homogeneous membranes with quadratic ‘Helfrich’-type energies with non-uniform spontaneous curvatures. The results are employed to solve four-point boundary value problems that simulate the equilibrium shapes of lipid vesicles that adhere to each other. Numerical studies are conducted to investigate the effect of relative sizes, osmotic pressures, and adhesion-induced spontaneous curvature on the morphology of adhered vesicles.
TL;DR: In this paper, a damage detection method is formulated to estimate damage location and extent from non-kth perturbation terms of a specific set of eigenvectors and eigenvalues.
Abstract: A damage detection method is formulated to estimate damage location and extent from non-kth perturbation terms of a specific set of eigenvectors and eigenvalues. The perturbed eigenvalue problem is established from the perturbations of stiffness matrix, eigenvector, and eigenvalue. Then stiffness parameters are estimated from this equation using the Davidon—Fletcher—Powell quasi-Newton approach. The optimization algorithm is iterative, and its process is monitored by d-norm and t-norm indicators. A fixed—fixed beam with an odd number of elements is used as a test structure to investigate the applicability of the method. In a five elements beam, t-norm convergences of the second-order algorithm are more effective for small and large-percentage damages. In medium-percentage damages, convergences of the first-order algorithm are faster for both indicators. Convergences of the general-order perturbation method are more effective for small and medium-percentage damages. Meanwhile, convergences of this method a...
TL;DR: In this article, the authors investigated the asymptotic structure of a boundary-value problem proposed recently in connection with in-plane instabilities of spinning disks, assuming an orthotropic elastic material with cylindrical symmetry.
Abstract: This work investigates the asymptotic structure of a boundary-value problem proposed recently in connection with in-plane instabilities of spinning disks. Assuming an orthotropic elastic material with cylindrical symmetry we consider a perturbation with respect to the constitutive behavior. The material is assumed to be very stiff in the azimuthal direction, a situation which is commonly encountered in the case of composite flywheels based on hoop-wound carbon fibers in a flexible polyurethane resin. The accuracy of the asymptotic strategy is confirmed by a number of direct computer simulations of the original problem.
TL;DR: In this article, a general theory of growing nonlinear elastic Kirchhoff plate is described, and a complete geometric description of incompatibility with simple examples is given, as well as the stability of growing kirchhoff plates.
Abstract: Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed.
TL;DR: In this paper, a simplified theory based on the thickness-wise expansion of the potential energy truncated at third order in thickness is proposed, which satisfies the stability condition of Legendre-Hadamard which is necessary for the existence of a minimizer.
Abstract: For laminated plates, the displacement field can be approximated in each layer by a third-order Taylor—Young expansion in thickness. These involve (sections 3.1 and 3.2) that the highest order term of transverse shear is of first-order in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. These lead to an analytical expression for two-dimensional potential energy in terms of the zero-order displacement field and its derivatives that includes non-standard transverse shearing energy and coupled bending—stretching energy. As a consequence this potential energy satisfies the stability condition of Legendre—Hadamard which is necessary for the existence of a minimizer.
TL;DR: In this article, the bipotential convex covers are used to capture the blurred constitutive laws of a family of graphs instead of a single graph, which is a special case of our problem.
Abstract: In many practical situations, uncertainties affect the mechanical behavior that is given by a family of graphs instead of a single graph. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws, using bipotential convex covers.
TL;DR: In this article, simple analytical expressions for the contact pressure applied to a spherical cavity that is expanding with constant velocity in an infinite compressible elastic perfectly plastic material with large deformations were obtained by using the kinematics of isochoric flow with a modified value of the shear modulus.
Abstract: Simple analytical expressions are obtained for the contact pressure applied to a spherical cavity that is expanding with constant velocity in an infinite compressible elastic perfectly plastic material with large deformations. The solution is obtained by using the kinematics of isochoric flow with a modified value of the shear modulus and with a physical boundary condition on the radial stress at the location of the precursor wave associated with compressible response. Comparison of the predictions with those of full numerical simulations and non-linear similarity solutions indicates that this modified incompressible solution predicts reasonably accurate results for the transient decay of the contact pressure from its shocked value to its asymptotic value.
TL;DR: In this article, a Cosserat-based 3D-1D dimensional reduction for a viscoelastic finite strain model is presented, where the numerical resolution of the reduced coupled minimization/evolution problem is based on a splittin...
Abstract: We present a Cosserat-based 3D–1D dimensional reduction for a viscoelastic finite strain model. The numerical resolution of the reduced coupled minimization/evolution problem is based on a splittin...
TL;DR: In this paper, the dynamic azimuthal shearing of a mixture of a transversely isotropic viscoelastic material which is surrounded by a voxelized mesh is studied.
Abstract: Motivated by studies on a type of brain injury known as diffuse axonal injury, the dynamic azimuthal shearing of a mixture of a transversely isotropic viscoelastic material which is surrounded by a...
TL;DR: In this article, a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney was established, and the connection provided a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney-Rivlin solids, and for the general third-order elasticity models of incompressible solids.
Abstract: We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc Amer Math Soc1950; 1: 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica1967; 3: 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney‐Rivlin solids, and for the general third- and fourth-order elasticity models of incompressible solids.
TL;DR: In this article, the authors considered a plate made of Ciarlet-Geymonat type materials and obtained a nonlinear membrane model and a non-linear membrane inextensional bending model as announced by Pruchnicki.
Abstract: This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet—Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.