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A novel class of implicit constitutive relations is studied, wherein the stress and the linearized strain appear linearly, that describe material response in elastic porous bodies like rocks, ceramics, concrete, cement, bones and metals. The constitutive relation is applied to a body with a crack subjected to non-penetration conditions between the opposite crack faces. To treat well-posedness of a corresponding variational inequality, we rely on a new approximation by thresholding dilatation and apply the Lions existence theorem on pseudo-monotone variational inequalities. An analytical solution to a specific example (without crack) under uniform triaxial loading is constructed, wherein blow-up can take place at a finite load, and this difficulty is overcome within a thresholding approximation so that blow-up does not occur.

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method., Comment: [v2] minor modifications; to appear in ESAIM: M2AN

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.

In this note, we study the response of a viscoelastic body whose stress relaxation modulus and creep compliance depend on the density of the body in such a manner that the stress and strain appear linearly in the constitutive equation. Such models would be useful to study the response of porous viscoelastic bodies undergoing small deformations, as the moduli depend on the porosity, and hence the density. We study the problem of tension–torsion of cylinders of arbitrary cross-section within the context of this constitutive relation.

A new model is proposed to mimic the response of a class of seemingly viscoplastic materials. Using the proposed model, the steady, fully developed flow of the fluid is studied in a cylindrical pipe. The semi-inverse approach is applied to obtain an analytical solution for the velocity profile. The model is used to fit the shear-stress data of several supposedly viscoplastic materials reported in the literature. A numerical procedure is developed to solve the governing ODE and the procedure is validated by comparison with the analytical solution.

We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder–Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.

Wet granular materials develop both shear stresses and normal stresses during simple shear. Over a certain range of shear rate levels, both the shear stress and the normal stress that develop are l...

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.

A thermodynamic framework is proposed to capture the dissipative response of metals. In contrast to the conventional practice, a stressed reference configuration is assumed instead of a stress-free configuration. The second law of thermodynamics is converted into equality by prescribing a non-negative rate of dissipation function. Stress in the reference configuration evolves with time to satisfy the second law of thermodynamics. Appropriate constitutive prescriptions are made to model the experimentally observed cyclic response of metals such as mild steel, aluminum alloy, and titanium, to uniaxial loading. An appropriate choice of material parameters captures well the experimentally observed response for different loading protocols.

The aorta is a very complex organ comprising three layers, consisting of four kinds of tissues. It is an anisotropic, inhomogeneous, multiconstituent, and living organ that presents both a formidable challenge and a tremendous opportunity to a modeler to mathematically characterize its structure. Unfortunately, even the most sophisticated models in vogue do not faithfully take into consideration its various complexities, falling very short of putting into place a reasonable model, as they ignore many of the quintessential features that need to be taken into account. In this article, we address the various features that need to be taken into account to develop a meaningful model of the aorta.

Damage in concrete has been modelled using various approaches such as fracture mechanics, continuum damage mechanics and failure envelope theories. This study proposes a new approach to model the i...

In Part I, a density driven damage mechanics (D3-M) approach and its application to model mechanical damage in concrete are presented. In this study, chemical and chemo-mechanical damage in concret...

Aortic dissection occurs predominantly in the thoracic aorta and the mechanisms for the initiation and propagation of the tear in aortic dissection are not well understood. We study the tearing characteristics of the porcine thoracic aorta using a peeling test and we estimate the peeling energy per unit area in the ascending and the descending segments. The stretch and the peel force per unit width undergone by the peeled halves of a rectangular specimen are measured. We find that there can be significant variation in the stretch within the specimen and the stretch between the markers in the specimen varies with the dynamics of peeling. We found that in our experiment the stretch achieved in the peeled halves was such that it was in the range of the stretch at which the stress-stretch curve for the uniaxial experiment starts deviating from linearity. Higher peeling energy per unit area is required in the ascending aorta compared to the descending aorta. Longitudinal specimens required higher peeling energy per unit area when compared to the circumferential specimens.

The linear stability of channel flows driven by pressure drops is carried out for fluids that exhibit a yield stress, the mechanical response of which is prescribed by the Bingham constitutive relation or two of its regularizations: the model due to Allouche and co-workers that is usually referred to as the “simple model”, and Papanastasiou model. Despite the fact that these two regularized models provide a good approximation of the steady Poiseuille flow of a Bingham fluid, they fail to predict the stability characteristics of the exact Bingham model. The critical thresholds for the onset of turbulence predicted by using the simple and Papanastasiou models are essentially the same, but they differ significantly from that of the exact Bingham model. This discrepancy is shown to be due to the absence of energy dissipation in the rigid core of a Bingham fluid.

In this paper we study the deformation of a body with a notch subject to an anti-plane state of stress within the context of a new class of elastic models. These models stem as approximations of constitutive response functions for an elastic body that is defined within the context of an implicit constitutive relation between the stress and the deformation gradient. Gum metal and many metallic alloys are described well by such constitutive relations. We consider the state of anti-plane stress of a body with a smoothened V-notch within the context of constitutive relations for the linearized strain in terms of a power-law for the stretch. The problem is solved numerically and the convergence and the stability of the solution is studied., Comment: This version of the article has been accepted for publication, after peer review and is subject to Springer Nature's AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s10659-019-09724-0

We study the initiation of damage in a polymeric body in which there is a line defect due to the formation of a “weld line” that occurs when two polymer streams join together and then solidify. We show that damage initiates in the region of weakness, namely the “weld line” based on a criterion for damage that was developed earlier. We also show that if there are other stress concentrators also additionally present, such as a hole, then there is a competition between the stresses induced due to the weakness and the stress as a consequence of the stress concentrator (in this instance a hole). This study adds more credence to the criterion for the initiation of damage that is based completely on knowledge of information at the current configuration of the body, that is, the criterion for damage is not based on the value of quantities that also need information based on a reference configuration such as the stress or strain.

Knowledge of the mechanical properties of blood vessels and determining appropriate constitutive relations are essential in developing methodologies for accurate prognosis of vascular diseases. We examine the directional variation of the mechanical properties of the porcine thoracic aorta by performing uniaxial extension tests on dumbbell-shaped specimens cut at five different orientations with respect to the circumferential direction of the aorta. Specimens in all the orientations considered exhibit a nonlinear constitutive response that is typical of collagenous soft tissues. Shear strain under uniaxial extension demonstrates clearly discernible anisotropy of the mechanical response of the porcine aorta, and samples oriented at 45[Formula: see text] and 60[Formula: see text] with respect to the circumferential direction show a peculiar crescent-shaped shear strain-nominal stretch response not displayed by axial and circumferential specimens. Failure stress indicates decreasing tensile strength of the porcine aortic wall from the circumferential direction to the longitudinal direction. Furthermore, we determine the material parameters for the four-fiber-family and Gasser-Holzapfel-Ogden models from the mechanical response data of the circumferential and longitudinal specimens. It is shown how the material parameters derived from the uniaxial tests on circumferential and longitudinal specimens are insufficient to characterize the response of off-axis specimens.

Following a brief review of the theory of continuum mixtures, recent developments and applications emphasizing time-dependent behaviors of heterogeneous materials are described. Common approximations of mixture theory and related continuum homogenization schemes such as assigning material properties, boundary conditions and body forces are considered. Approaches to imposing restrictions due to the second law are discussed; traditional employment of the Clausius-Duhem inequality enforced for arbitrary processes is contrasted with maximization of the rate of entropy production applied to a functional space suitably restricted to conform with constitutive postulates. Remarks are made concerning related homogenization and thermodynamic developments, including poroelasticity, volume-averaging, local accompanying state, and linear irreversible thermodynamics. Applications to fibrous composite materials, and to additional classes of heterogeneous materials, are briefly discussed.

A new stabilized method is presented for coupled chemo-mechanical problems involving chemically reacting fluids flowing through deformable elastic solids. A mixture theory model is employed wherein kinematics is represented via an independent set of balance laws for each of the interacting constituents. A significant feature of the mixture model is the interactive force field in the momentum balance equations that couples the constituents implicitly at the level of the governing system of equations. The constitutive relations for the constituents in the mixture model are based on maximization of the rate of entropy production. Since each constituent is not discretely modeled and the interactive effects are mathematically coupled at the local continuum level, the resulting system serves as a physics-based reduced-order model for the complex microstructure of the material system. When constitutive equations are substituted into the balance laws, they give rise to a system of coupled nonlinear PDEs. Evolving nonlinearity and coupled chemo-mechanical effects give rise to spatially localized phenomena, namely boundary layers, shear bands, and steep gradients that appear at the reaction fronts. For large reaction rates, the balance of mass of the fluid becomes a singularly perturbed equation (reaction-dominated), which may exhibit boundary and/or internal layers. Likewise, for large reaction rates and/or low diffusivity, the balance of linear momentum for the fluid constituent also becomes a singularly perturbed PDE. Presence of these features in the solution requires stable numerical methods, and we present a variational multiscale (VMS)-based stabilized finite element method for the initial-boundary value problem. Mathematical attributes of the method are investigated via a range of numerical test cases that involve diffusion of chemically reacting fluids through nonlinear elastic solids. Enhanced stabilization features and higher spatial accuracy of the models and the methods are highlighted.

There has been considerable interest recently in the development of connected and autonomous vehicles (CAVs). Automatic vehicle following capability is central for CAVs; in this article, we provide a review of the critical issues in the longitudinal control design for automatic vehicle following systems (AVFS) employed by CAVs. This expository review differs from others in providing a review of underlying methodologies for design of AVFS and the impact of AVFS on traffic mobility and safety.

The goal of this study is to model the creep and recovery response of fine asphalt mixtures using a thermodynamically consistent nonlinear viscoelastic model. The model considers asphalt mixture to...

This study presents a thermodynamic framework for describing the mechanical response of viscoelastic polymers undergoing microstructural changes within the framework of a body possessing multiple natural configurations. In order to incorporate the effect of microstructural changes on the macroscopic response of polymers, we assume that the polymer under consideration consists of two networks. In its initial configuration, the polymer is comprised of an original microstructure (initial network) and as the polymer is deformed, a new network is formed. The newly formed network has a new reference configuration. When subjected to mechanical loading, the natural configuration of the polymer evolves between these initial and final configurations. We use the proposed constitutive model in order to describe the macroscopic response of a polyoxymethylene (POM) polymer under various loading histories. The predictions of the model developed are in good agreement with experimental data that is available.

The kinematics of an inelastic solid established in terms of Laplace stretch and its rate are decomposed into one stretch that describes an elastic response, another stretch that describes an inelastic response, and their rates. These kinematics are a direct consequence of Laplace stretch belonging to the group of all real, $$3 \times 3$$ , upper-triangular matrices with positive diagonal elements. The Laplace stretch follows from a Gram–Schmidt decomposition of the deformation gradient.

Various disparate restrictions are assumed with regard to bodies undergoing thermodynamic processes, in addition to the second law, to cull the class of constitutive relations for describing their response. In this paper we critically evaluate some of these additional restrictions. After a brief discussion of the various forms of the second law due to the pioneers in the field we turn our attention to an assessment of additional stringent restrictions due to Onsager, Prigogine, Ziegler, and Edelen. With the use of specific examples we show the pitfalls and inadequacies of these additional restrictions. The importance of identifying the “cause” (which we refer to as the “determinant” which is incorrectly referred to as “flux”) and its “effect” (which we refer to as the “resultant” which is incorrectly referred to as “affinity”) and not merely splitting the “determinant” on the basis of mathematical ease rather than understanding the underlying physics, is emphasized by considering specific examples.

In this article, we develop a new implicit constitutive relation, which is based on a thermodynamic foundation that relates the Hencky strain to the Cauchy stress, by assuming a structure for the Gibbs potential based on the Cauchy stress. We study the tension/compression of a cylinder, biaxial stretching of a thin plate and simple shear within the context of our constitutive relation. We then compare the predictions of the constitutive relation that we develop and that of Ogden’s constitutive relation with the experiments of Treloar concerning tension/compression of a cylinder, and we show that the predictions of our constitutive relation provide a better description than Ogden’s model, with fewer material moduli.

In this paper, we use an implicit generalization of the celebrated Kelvin–Voigt solid constitutive equation to study the problem of circumferential shearing of a cylindrical annulus of viscoelastic material. This generalization of the Kelvin–Voigt model allows one to take into consideration the possibility of the body undergoing shear thinning or shear thickening and allows the material moduli to depend on the mechanical pressure. Most importantly, the constitutive relation is a relation in the true mathematical sense in that neither the stress, nor the strain nor the symmetric part of the velocity gradient can be expressed explicitly in terms of the other quantities, thereby making the balance of linear momentum and the constitutive relation requiring solution simultaneously, unlike the case of the Kelvin–Voigt model where we can substitute the constitutive expression for the stress into the equilibrium equations to obtain an equation for the displacement field.

Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of the elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy–Green tensor is expressed as a function of stress, is of interest. We show how to use this new perspective in the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered.

In this study, we computationally corroborate the flow of rock glaciers against borehole measurements, within the context of a model previously developed (2020). The model is, here, tested against the simulation of the sliding motion of the Murtel-Corvatsch alpine glacier, which is characterized in detail in the literature with internal structure description and borehole deformations measurement. The capability of the model to take into account the composition of the rock glacier, as a mixture of ice and rock and sand grains with the local impact of pressure and heat transfer, results in the accurate detection of the internal sliding. With careful calibration of the model parameters, the computed numerical solution of the model reports a relative error of 1.8% and of 0.3% in the reproduction of the measured shear zone velocity and of the ratio of measured shear zone deformation over top surface deformation, respectively. Furthermore a deeper understanding of the role of the model parameters involved in the simulation of such a process is also gained and we discuss the same in detail.

In this note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.