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This work documents the formulation of an axial element that could handle implicit constitutive relations. This formulation uses nodal displacements, member forces, and reaction forces as primary unknowns. For some assumed values of these primary unknowns satisfying the nodal equilibrium equations, the computed stretch and stress in the members need not be in agreement with the constitutive relation. Hence, these primary unknowns are found such that the error in the constitutive relation for the members are minimized subjected to the constraint that the nodal equilibrium equations hold. Axial element developed using this concept is used to analyze a few trusses. The results of this formulation are in agreement with the results obtained from ABAQUS® when stress is an explicit function of the stretch ratio. Also, some statically determinate as well as indeterminate trusses are solved for an implicit constitutive relation and the reasonableness of the obtained results established.

We develop a methodology for solving truss problems involving viscoelastic materials where, of all the member forces that satisfy the nodal force equilibrium equation and nodal displacements that satisfy the displacement boundary conditions, those member forces and nodal displacements that satisfy the constitutive relation are sought. Since a rate type viscoelastic constitutive relation involves the rate of the stress or strain, this study explores the use of member forces, nodal displacements, and support reactions or their rates as independent variables. Assuming small deformations, the nodal force equilibrium and the displacement boundary condition results in a linear equality constraint between the independent variables. Then we find the unknown independent variables such that the root mean squared error in the constitutive relation of the members of the truss is minimized subject to the satisfaction of the linear constraint at selected times. The objective function is evaluated at selected times or integrated over subintervals of time. We explore six possible solution methods and benchmark them for their accuracy and efficiency. We study statically determinate and indeterminate truss whose members are modeled using rate and integral type viscoelastic constitutive relations for creep and oscillatory loading. For the standard linear solid model, we find that the proposed methods are more accurate than ABAQUS and, at times, require lesser computational wall time. We also demonstrate the applicability of the proposed methodology to fractional order and nonlinear viscoelastic constitutive relations.

It is customary to use a displacement-based formulation to seek solutions to boundary value problems for its computational efficacy. In displacement-based formulations, it is convenient to prescribe the constitutive relation for stress as an explicit function of displacement gradient. However, from a general theoretical point of view, the stress and displacement gradient could be related by an implicit function. This study develops techniques to solve boundary value problems when linearized strain and stress are related by an implicit function. Here both the stresses and the displacement are taken as unknowns. The stress field is constructed such that it satisfies the equilibrium equations identically within the element and the traction continuity requirements between the elements. A continuously differentiable displacement field is constructed, and the linearized strain is computed from this displacement field. Then, the unknown parameters in the stress and displacement field are estimated such that the constitutive relation holds in weak integral sense. Though in this procedure the number of unknowns has increased in comparison with the displacement formulation, both the strength and serviceability condition can be checked directly without any post-processing. Also, in this procedure, both the equilibrium equations and continuity of displacement are met exactly. The equation that is not satisfied exactly is the constitutive relation, which is an approximation anyway. The efficacy and accuracy of this method are benchmarked by studying some standard problems. Planar and three-dimensional truss elements have been developed and benchmarked. Then, a rectangular plane element is implemented and its performance recorded.

A methodology for obtaining implicit constitutive representations involving the Cauchy stress and the Hencky strain for isotropic materials undergoing a non-dissipative process is developed. Using this methodology, a general constitutive representation for a subclass of implicit models relating the Cauchy stress and the Hencky strain is obtained for an isotropic material with no internal constraints. It is shown that even for this subclass, unlike classical Green elasticity, one has to specify three potentials to relate the Cauchy stress and the Hencky strain. Then, a procedure to obtain implicit constitutive representations for isotropic materials with internal constraints is presented. As an illustration, it is shown that for incompressible materials the Cauchy stress and the Hencky strain could be related through a single potential. Finally, constitutive approximations are obtained when the displacement gradient is small.

All bodies are inhomogeneous at some scale but experience has shown that some of these bodies can be idealized as a homogeneous body. Here we examine which bodies can be idealized as a homogeneous body when they are subjected to a non-dissipative mechanical process. This is done by studying circumstances in which an inhomogeneous body admits pure stretch homogeneous deformations. Then, we devise experiments wherein these circumstances are prevented. If homogeneous deformation is observed in these devised experiments, the body could be modeled as a homogeneous body. We limit our analysis to a class of isotropic elastic bodies deforming from a stress free reference configuration whose Cauchy stress is explicitly related to left Cauchy-Green deformation tensor. It is further assumed that the constitutive relation is differentiable function of the position vector of material particles in the stress free reference configuration. Then, we find that a cuboid made of compressible and isotropic material could be modeled as a homogeneous body if it deforms homogeneously due to the application of the normal stresses on all of its six faces and the magnitude of the normal stresses on three orthogonal faces are different. A cuboid made of incompressible and isotropic material could be modeled as a homogeneous body, if it deforms homogeneously in two different biaxial experiments, such that the plane in which the forces are applied in the two biaxial experiments is mutually orthogonal. � 2014 Elsevier Ltd. All rights reserved.

This paper studies the classical problem of extension, inflation, and circumferential shearing of an annular cylinder for a new class of compressible elastic bodies wherein the left Cauchy–Green deformation tensor is given as a function of the Cauchy stress tensor. We use a semi-inverse method to study the problem by assuming forms for both the deformation field and the stress field. Focusing our attention on to three specific constitutive relations and two geometries corresponding to thick and thin annular cylinders, we study the qualitative features of the governing differential equations. The models are chosen so that they exhibit qualitatively different response features, one of them displaying a limiting stretch. The classical assumption that the hoop and axial stresses are nearly constant through the thickness of the thin annular cylinder subjected to inflation holds for this class of elastic bodies too. However, for thick-walled annular cylinders subjected to inflation at constant length and for a class of models that exhibits limiting stretch we find that stress “boundary layers” form and that the radial stretch is not monotonic.

Using a special model that belongs to a new class of elastic bodies wherein the Cauchy–Green stretch is given in terms of the Cauchy stress and its invariants, within the context of the spherical inflation of a spherical annulus, we show that interesting phenomena like the development of “stress boundary layers” manifest themselves. We consider two cases of boundary value problems, one in which there is a cavity in a sphere and the other in which there is a rigid spherical inclusion in a sphere. We show that in the case of a rigid inclusion, it is possible for a pronounced “stress boundary” layer to develop, in that the values of the stresses within this boundary layer that is adjacent to a spherical inclusion are much larger than external to it. We also show that in the case of both the cavity and a rigid inclusion, the stress concentration is an order of magnitude higher than the increase in the deformation gradient, that is, the stress and the stretch do not scale in a similar manner. While the stress adjacent to a rigid inclusion can be 2500 times the applied radial stress, the maximum stretch, which occurs at the rigid inclusion is about 10. While the variation in the stresses are linear in thin walled annular regions, we find that in thick walled annular regions, the variation of the stresses is non-linear.

Here we investigate whether the deformation observed in an experiment in which the porcine circumflex coronary artery is subjected to inflation at constant length included in the class, $$r$$ $$=$$ $$r(R)$$ , $$\theta$$ $$=$$ $$\Theta$$ , $$z$$ $$=$$ $$\Lambda Z$$ . We find that this is not the case and discuss its implications in the study of the mechanics of this artery. Moreover, we identify and quantify the uncertainty in the value of the invariants of the left Cauchy–Green tensor inferred from the 2D motion of markers affixed to the surface of the test specimen, and suggest that 3D tracking of markers is needed due to inherent bending and twisting induced by pressurization in vitro.

We study the inflation, extension, torsion and shearing of an isotropic inhomogeneous compressible annular right circular cylinder. Current approaches to homogenization that appeal to an equivalence in the stored energies could lead to serious errors in the estimate for stresses in a inhomogeneous body as stresses depend on the derivatives of the stored energy with respect to the deformation gradient. This is a serious drawback as many a time failures are determined by the stresses. The study demonstrates that, in particular, great caution should be exercised in homogenization, especially if an inhomogeneous body is to be approximated by a homogeneous body belonging to the same class. Comparison of local measures, such as stresses, reveal that their values in the case of the inhomogeneous body and its homogeneous counterpart can be both qualitatively and quantitatively far apart. Even the differences in global measures like the axial load, torque, etc., are found to be significant between the inhomogeneous body and its homogeneous counterpart. It is also shown that the material parameters characterizing the homogenous approximation gleaned from correlations from different experiments, performed on the same inhomogeneous body, can be quite different.

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 different. In the process of elucidating our thesis, we investigate an important class of deformations, which in view of the paucity of boundary value problems that have been solved for non-linear inhomogeneous solids is worth documenting in its own right.

We show that even when the constitutive relations for the elastic response of compressible bodies satisfy the empirical, Baker–Ericksen and ordered forces inequalities and ellipticity restrictions, they could result in physically unrealistic response. We pick a constitutive relation that satisfies the above inequalities and is also globally elliptic but its response to both uniaxial extension and inflation at constant length is physically unrealistic, based on the available experimental observations. However, the Coleman–Noll, pressure–compression, tension–extension and invertibility of force-stretch conditions and that proposed in Carroll and McCarthy (1995) are violated by this constitutive relation.

Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error. Copyright � 2013 by ASME.

There are several problems wherein it would be convenient to have representations for the stress from a stressed configuration as reference. We assume that the body undergoes finite elastic deformations (i.e., non-dissipative processes) from a stressed reference configuration but it is possible that a dissipative (inelastic) process could be the cause for the origin of the stresses in the reference configuration. We restrict ourselves to determining the representation for the stress in a body, from a stressed reference configuration, whose symmetry group in the stress free configuration (that is accessible from the stressed reference configuration) coincides with the full orthogonal group. We also find restrictions on the constitutive relation so that we have a model that exhibits physically acceptable response characteristics. � 2008 Elsevier Ltd. All rights reserved.

All real bodies are inhomogeneous, though in many such bodies the inhomogeneity is “mild” in that the response of the bodies can be “approximated” well by the response of a homogeneous approximation. In this study we explore the status of such approximations when one is concerned with bodies whose response is nonlinear. We find that significant departures in response can occur between that of a “mildly” inhomogeneous body and its homogeneous approximation (if the approximate model is restricted to a certain class), both quantitatively and qualitatively. We illustrate this fact within the context of a specific boundary value problem, the inflation of an inhomogeneous spherical shell. We also discuss the inappropriateness of homogenization procedures that lead to a homogenized stored energy for the body when in fact what is required is a homogenized model that predicts the appropriate stresses as they invariably determine the failure or integrity of the body.