Implicit Constitutive Theories Research Articles

Solutions for the unsteady motion of porous elastic solids within the context of an implicit constitutive theory

In this work we investigate well-posedness for the partial differential equation stemming from the balance of linear momentum for an implicit constitutive relation that can describe the response of porous elastic solids whose material moduli depend on the density. We study heteroclinic travelling waves and obtain closed form analytic solutions for the resulting ordinary differential equation. Finally, we consider some special solutions for the partial differential equation and solve the resulting equations numerically.

Stress-rate-type strain-limiting models for solids resulting from implicit constitutive theory

The main objective of this work is two-fold. First, we investigate the stress-rate-type implicit constitutive relations for solids within the context of strain-limiting theory of material response. The relations we study are models for generalisations of elastic bodies whose strain depends on the stress and the stress rate. Secondly, we obtain travelling-wave solutions for some special cases that are nonlinear in the stress. These are the first notion of solutions available in the literature for this type of models describing stress-rate-type materials.

On lower-dimensional models of thin film flow, Part C: Derivation of a Reynolds type of equation for fluids with temperature and pressure dependent viscosity

This paper constitutes the third part of a series of works on lower-dimensional models in lubrication. In Part A, it was shown that implicit constitutive theory must be used in the modelling of incompressible fluids with pressure-dependent viscosity and that it is not possible to obtain a lower-dimensional model for the pressure just by letting the film thickness go to zero, as in the proof of the classical Reynolds equation. In Part B, a new method for deriving lower-dimensional models of thin-film flow of fluids with pressure-dependent viscosity was presented. Here, in Part C, we also incorporate the energy equation so as to include fluids with both temperature and pressure dependent viscosity. By asymptotic analysis of this system, as the film thickness goes to zero, we derive a simplified model of the flow. We also carry out an asymptotic analysis of the boundary condition, in the case where the normal stress is specified on one part of the boundary and the velocity on the remaining part.

An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279–319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.

A three dimensional constitutive model for plain cement concrete

The validity of some of the assumptions made during systematic development of constitutive relation – homogeneity, isotropy, and non-dissipative response – is examined experimentally by testing plain concrete cylinders in the circumferential displacement controlled uniaxial compression tests. The tested cylinders indicate that the surface strains are not uniform and that the principal direction of the strain varies with the magnitude of the applied load. Hence, these cylinders are either deforming from a stressed state or anisotropic or both. The percentage dissipation computed as the area between the loading and unloading curves normalized using the area under the unloading curve is less than 10 percent up to 90 percent of the peak load. Therefore the mechanical response of concrete can be considered as non-dissipative. Assuming that the magnitude of the residual stresses present in the concrete panels would be small compared to the stresses arising due to the applied load, these residual stresses are ignored. Consequently, using the implicit constitutive theory framework for isotropic and compressible materials undergoing a non-dissipative process from a stress-free reference configuration, a three-dimensional constitutive relation for plain concrete is proposed. The material parameters in the proposed model are estimated from the uniaxial compression test and equal biaxial compression test. These parameters are expressed using initial Young’s modulus, initial Poisson’s ratio, maximum uniaxial compressive stress, the axial and transverse strain corresponding to the maximum uniaxial compressive stress, maximum equal biaxial compressive stress and the equal biaxial strain corresponding to maximum equal biaxial compressive stress. The ability of the proposed model to capture the response of various grade and type of concrete in ten biaxial experiments reported in the literature is examined. It is found that despite the limiting assumptions, the compression-compression, compression-tension and tension-tension stress strain response of the concrete in these ten experiments are adequately captured by the proposed model with mean R2 value of 0.94.

Numerical and approximate analytical solutions for cylindrical and spherical annuli for a new class of elastic materials

Many materials that have been developed recently such as titanium alloys and polymeric composites exhibit nonlinear elasticity in the “small” strain regime, and the linearized theory cannot be used to describe the response. Recently, Rajagopal (Appl Math 48(4):279–319, 2003) introduced a new implicit constitutive theory which can be used to develop models to characterize the response of these newly fashioned materials. Here, we study the response of these new classes of elastic bodies within the context of two boundary value problems: the pressurization of a cylindrical annulus and a spherical shell. In the case of the cylindrical annulus, a stress function is introduced that automatically satisfies the equilibrium equation, and the compatibility equation for strain and the nonlinear constitutive equation is used to obtain the nonlinear compatibility equation in terms of the stress function. For the spherical shell, a displacement formulation is used to arrive at a nonlinear equation for the radial stress. The governing equations in both cases cannot be solved exactly, and we use an approximate technique, the variational iteration technique, to solve the problem. We show that this approximate solution agrees very well with the numerical solution of the governing equations, and the solution is different from that obtained in the classical linearized elasticity. In the case of spherical annulus with an internal pressure of 250 MPa, the hoop stress associated with the linearized solution overpredicts the numerical solution by about \(10\,\%\) at the inner radius and underpredicts by about \(7\,\%\) at the outer radius.

A note on some new classes of constitutive relations for elastic bodies

The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math., 48, 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A, 463, 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A, 463, 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.

Analysis and approximation of a strain-limiting nonlinear elastic model

Elastic solids with strain-limiting response to external loading represent an interesting class of material models, capable of describing stress concentration at strains with small magnitude. A theoretical justification of this class of models comes naturally from implicit constitutive theory. We investigate mathematical properties of static deformations for such strain-limiting nonlinear models. Focusing on the spatially periodic setting, we obtain results concerning existence, uniqueness and regularity of weak solutions, and existence of renormalized solutions for the full range of the positive scalar parameter featuring in the model. These solutions are constructed via a Fourier spectral method. We formulate a sufficient condition for ensuring that a renormalized solution is in fact a weak solution, and we comment on the extension of the analysis to nonperiodic boundary-value problems.

On strong ellipticity for implicit and strain-limiting theories of elasticity

There has been increasing interest in the archival literature devoted to the study of implicit constitutive theories for non-dissipative materials generalizing the classical Green and Cauchy notions of elasticity, and for the special case of strain-limiting models for which strains remain bounded, even infinitesimal, while stresses can become arbitrarily large. This paper addresses the question of strong ellipticity for several classes of these models. A general approach for studying strong ellipticity for implicit theories is introduced and it is noted that there is a close connection between the questions of strong ellipticity and the existence of an equivalent Cauchy elastic formulation. For most of the models studied to date, it is shown that strong ellipticity holds if the Green–St. Venant strain is small enough, whereas it fails to hold for large strain. The large strain failure of strong ellipticity is generally associated with extreme compression.

Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations

In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered.

A note on novel generalizations of the Maxwell fluid model

In this note we generalize the classical viscoelastic fluid model due to Maxwell to allow the relaxation time and the viscosity to depend on the stress. Such models are very useful in describing the response of geological and many other polymeric fluids. The procedure that is adopted to carry out the generalization allows one to have the springs and dashpots and their three dimensional generalizations, to be described by implicit constitutive theories thereby greatly enlarging the class of constitutive models that can be put into place to describe the response of complex viscoelastic fluids.

On compressible Korteweg fluid-like materials

We provide a thermodynamic basis for compressible fluids of a Korteweg type that are characterized by the presence of the dyadic product of the density gradients ∇ ϱ ⊗ ∇ ϱ in the constitutive equation for the Cauchy stress. Our approach does not need to introduce any new or non-standard concepts such as multipolarity or interstitial working and is based on prescribing the constitutive equations for two scalars: the entropy and the entropy production. In comparison with the Navier–Stokes–Fourier fluids we suppose that the entropy is not only a function of the internal energy and the density but also of the density gradient. The entropy production takes the same form as for a Navier–Stokes–Fourier fluid. For a Navier–Stokes–Fourier fluid one can express the entropy production equivalently in terms of either thermodynamic affinities or thermodynamic fluxes. Following the ideas of K.R. Rajagopal concerning the systematic development of implicit constitutive theory and primary role of thermodynamic fluxes (such as force) that are cause of effects in thermodynamic affinities (such as deformation) in considered processes, we further proceed with a constitutive equation for entropy production expressed in terms of thermodynamic fluxes. The constitutive equation for the Cauchy stress is then obtained by maximizing the form of the rate of entropy production with respect to thermodynamic fluxes keeping as the constraint the equation expressing the fact that the entropy production is the scalar product of thermodynamic fluxes and thermodynamic affinities. We also look at how the form of the constitutive equation changes if the material in question is incompressible or if the processes take place at constant temperature. In addition, we provide several specific examples for the form of the internal energy and make the link to models proposed earlier. Starting with fully implicit constitutive equation for the entropy production, we also outline how the methodology presented here can be extended to non-Newtonian fluid models containing the Korteweg tensor in a straightforward manner.

On the Oberbeck–Boussinesq approximation for fluids with pressure dependent viscosities

We derive, by using a rigorous perturbative approach, the Oberbeck–Boussinesq approximation within the framework of one of the simplest implicit constitutive theories: that of fluids with pressure dependent viscosity. To illustrate an application we use the equations derived here to study the flow of a fluid in a channel with heated walls.

The elasticity of elasticity

In this note we assert that the usual interpretation of what one means by “elasticity” is much too insular and illustrates our thesis by introducing implicit constitutive theories that can describe the non-dissipative response of solids. There is another important aspect to the introduction of such an implicit approach to the non-dissipative response of solids, the development of a hierarchy of approximations wherein, while the strains are infinitesimal the relationship between the stress and the linearized strain is non-linear. Such approximations would not be logically consistent within the context of explicit theories of Cauchy elasticity or Green elasticity that are currently popular.

On implicit constitutive theories for fluids

We consider generalizations of fluid models wherein the fluid is assumed to be incompressible, but with the viscosity depending on the pressure. We show that a natural setting for the development of such models is a class of implicit constitutive relations, which, in addition to the fluid model described here, provides a means for developing other complex models for viscoelastic fluids which cannot be set within the ambit of classical explicit constitutive relations for the stress in terms of the histories of appropriate kinematical variables.

On Implicit Constitutive Theories

In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).