The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials

Residual stress is the stress present in the unloaded equilibrium configuration of a body. Because residual stresses can significantly affect the mechanical behavior of a component, the measurement of these stresses and the prediction of their effect on mechanical behavior are important objectives in many engineering problems. Common methods for the measurement of residual stresses include various destructive experiments in which the body is cut to relieve the residual stress. The resulting strain is measured and used to approximate the original residual stress in the intact body. In order to predict the mechanical behavior of a residually stressed body, a constitutive model is required that includes the influence of the residual stress.In this paper we present a method by which the data obtained from standard destructive experiments can be used to derive constitutive equations that describe the mechanical behavior of elastic residually stressed bodies. The derivation is based on the idea that for each infinitesimal neighborhood in a residually stressed body, there exists a corresponding stress free configuration. We refer to this stress free configuration as the virtual configuration of the infinitesimal neighborhood. The derivation requires that the constitutive equation for the stress free material be known and invertible; it is used to relate the residual stress to the deformation of the virtual configuration into the residually stressed configuration. Although the concept of the virtual configuration is central to the derivation, the geometry of this configuration need not be determined explicitly, and it need not be achievable experimentally, in order to construct the constitutive equation for the residually stressed body.The general mathematical forms of constitutive equations valid for residually stressed elastic materials have been derived previously for a number of cases. These general forms contain numerous unknown material-response functions or material constants that must be determined experimentally. In contrast, the method presented here results in a constitutive equation that is an explicit function of residual stress and includes only the material parameters required to describe the stress free material.After presenting the method for the derivation of constitutive equations, we explore the relationship between destructive experiments and the theory used in the derivation. Specifically, we discuss the use of the theory to improve the design of destructive experiments, and the use of destructive experiments to obtain the data required to construct the constitutive equation for a particular material.     

Initial stresses in elastic solids: Constitutive laws and acoustoelasticity

On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second- and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress.     

The Equations of Motion of a Plate with Residual Stress

We consider a three-dimensional body which is at rest in a cylindrical configuration of height 2ε. We assume that the material is residually stressed and that it responds elastically to deformations from the reference configuration. Under appropriate assumptions on the data, and using weak-convergence methods, we determine the limit, as ε goes to zero, of the elasto-dynamic problem. The plate problem obtained, as in the case without residual stress, splits into two problems: one governing the in-plane displacement and the other determining the out-of-plane motion.     

Three-fold symmetry restrictions on two-dimensional micropolar materials

Analysis of the mechanical properties of engineering materials with microstructure generally requires modification of the concept of a simple material. One approach is the theory of micropolar materials which introduces an independent rotation of a material element and the resulting stress and strain tensors are generally non-symmetric. In two-dimensional material models these microstructures are often represented by geometries which exhibit three-fold symmetry in the plane. In this work we investigate the form of the constitutive relations which this three-fold symmetry imposes. We show that three-fold symmetry requires both the stress and couple stress tensors to be isotropic in the plane. We obtain specific constitutive relations for an equilateral triangle structure and for a hexagonal or honeycomb structure and compare these results with the results of previous investigations of these two-dimensional material models.     

A second order constitutive theory for hyperelastic materials

The second order constitutive equation for a hyperelastic material with arbitrary symmetry is derived. In developing a second order theory, it is necessary to be discriminating in the choice of measures of deformation. Here the derivation is done in terms of the Biot strain, which has a direct physical interpretation in that its eigenvalues are the principal extensions of the deformation. The constitutive equation is specialized for the cases of isotropy and transverse isotropy. The isotropic equation derived here is compared with equations obtained by other authors in terms of the displacement gradient and the Green strain.     

Extension,inflation and torsion of a residually stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

Restrictions on nonlinear constitutive equations for elastic shells

Constitutive equations for the resultant forces and moments applied to a shell-like body necessarily couple the influences of the shell geometry and the constitutive nature of the three-dimensional material from which the shell is constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the complicated influence of the shell geometry on the constitutive response of the shell is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic shells which ensure that exact solutions of the shell equations are consistent with exact nonlinear solutions of the three-dimensional equations for homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of shells. Examples of the linear theories of a plate and a spherical shell are considered.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

Modelling of stress softening in elastomeric materials: foundations of simple theories

This work is concerned with formulation of constitutive relations for materials exhibiting the stress softening phenomenon (known as the Mullins effect) typical observed in elastomeric and other amorphous materials during loading–reloading cycles. It is assumed that microstructural changes in such materials during the deformation process can be represented by a single scalar-valued softening variable whose evolution is accompanied by microforces satisfying their own law of balance, besides the classical laws of mechanics underlying macroscopic deformation of a material. The constitutive equations are then derived in consistency with thermodynamics of irreversible processes with the restriction to purely mechanical theory. The general form of the derived constitutive equations is subsequently simplified through introduction of additional assumptions leading to various models of the stress softening phenomenon. As an illustration of the general theory, it is shown that the so-called pseudo-elastic model proposed in the literature may be derived without an ad hoc postulate of the variational principle.     

A Note on Residual Stress, Lattice Orientation and Dislocation Density in Crystalline Solids

The theory of residually stressed crystals is discussed in the context of the finite deformation theory of elastic?Cplastic solids. Here residual stress is assumed to be due to the prior plastic flow of a single crystal. On the basis of recent experiments, as well as the structure of the underlying theory, we advance the view that the lattice orientation field should be regarded as data. This stands in contrast to the classical theory in which dislocation density is assigned.     

Restrictions on nonlinear constitutive equations for elastic rods

Constitutive equations for the resultant forces and moments applied to a rod-like body necessarily couple the influences of the rod geometry and the constitutive nature of the three-dimensional material from which the rod was constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the influence of the rod geometry on the constitutive response of the rod is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic rods which ensure that exact solutions of the rod equations are consistent with exact nonlinear solutions of the three-dimensional equations for all homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of rods. Also, an example of a straight beam clamped at one end and subjected to a shear force at the other end is used to examine the validity of the proposed value for the transverse shear deformation coefficient.     

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.     

NEW POINTSOF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS( Ⅲ) ——RE_DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

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On some connections between equivalent single material and mixture theory models for fiber reinforced hyperelastic materials

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration.In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations.Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.     

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.