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Showing papers on "Constitutive equation published in 1998"


Journal ArticleDOI
TL;DR: In this article, a dynamical theory of low-temperature shear deformation in amorphous solids is proposed based on molecular-dynamics simulations of a two-dimensional, two-component non-crystalline system.
Abstract: We propose a dynamical theory of low-temperature shear deformation in amorphous solids. Our analysis is based on molecular-dynamics simulations of a two-dimensional, two-component noncrystalline system. These numerical simulations reveal behavior typical of metallic glasses and other viscoplastic materials, specifically, reversible elastic deformation at small applied stresses, irreversible plastic deformation at larger stresses, a stress threshold above which unbounded plastic flow occurs, and a strong dependence of the state of the system on the history of past deformations. Microscopic observations suggest that a dynamically complete description of the macroscopic state of this deforming body requires specifying, in addition to stress and strain, certain average features of a population of two-state shear transformation zones. Our introduction of these state variables into the constitutive equations for this system is an extension of earlier models of creep in metallic glasses. In the treatment presented here, we specialize to temperatures far below the glass transition and postulate that irreversible motions are governed by local entropic fluctuations in the volumes of the transformation zones. In most respects, our theory is in good quantitative agreement with the rich variety of phenomena seen in the simulations. {copyright} {ital 1998} {ital The American Physical Society}

1,769 citations


Book
06 Nov 1998
TL;DR: In this article, the authors propose an approach for modeling multicomponent flows based on the classical theory of solutions of solutions, which they call well-posedness and well-posedness.
Abstract: Preliminaries.- Physical Reality, Corpuscular Models, Continuum Models.- Classical Continuum Theory.- Viscous and Inviscid Fluids and Elastic Solids.- Kinetic Theory.- Classical Theory of Solutions.- Continuum Theory.- Continuum Balance Equations for Multicomponent Fluids.- Mixture Equations.- Averaging Theory.- Ensemble Averaging.- Other Averages.- Averaged Equations.- Postulational and Averaging Approaches.- Modeling Multicomponent flows.- Closure Framework.- Relation of Microstructure to Constitutive Equations.- Maxwell-Boltzmann Dynamics.- Interfacial Area.- Equations of Motion for Dilute Flow.- Consequences.- Nature of the Equations.- Well-Posedness.- Solutions for Shearing Flows.- Wave Dynamics.

1,139 citations


Journal ArticleDOI
TL;DR: In this article, a detailed experimental investigation probing the material response of carbon black filled Chloroprene rubber subjected to different time-dependent strain histories is presented, based on the experimental data a new constitutive model has been developed.
Abstract: The mechanical behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Although these features of the rubbery constitutive response are well-recognized and important to its function, few models attempt to quantify these aspects of response perhaps due to the complex nature of the behavior and its apparent inconsistency with regard to current reasonably successful models of rubber elasticity. In this paper a detailed experimental investigation probing the material response of carbon black filled Chloroprene rubber subjected to different time-dependent strain histories is presented. Some of the key observations from the experiments are: (1) both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading; (2) the amount of carbon black particles does not strongly influence the normalized amount of hysteresis; (3) both filled and unfilled elastomers are strain-rate dependent and the rate dependence is higher during the uploading than during the unloading; (4) at fixed strain, the stress is observed to approach the same equilibrium level with relaxation time whether loading or unloading. Based on the experimental data a new constitutive model has been developed. The foundation of the model is that the mechanical behavior can be decomposed into two parts: an equilibrium network corresponding to the state that is approached in long time stress relaxation tests; and a second network capturing the non-linear rate-dependent deviation from the equilibrium state. The time-dependence of the second network is further assumed to be governed by the reptational motion of molecules having the ability to significantly change conformation and thereby relaxing the overall stress state. By comparing the predictions from the proposed three-dimensional constitutive model with experimental data for uniaxial compression and plane strain compression we conclude that the constitutive model predicts rate-dependence and relaxation behavior well.

936 citations


Journal ArticleDOI
TL;DR: In this paper, the deformation of a curved interface between solid phases was studied under the assumption of small strains in the bulk phases and neglecting accretion at the interfaces, and the authors showed that the free energy of the interface can depend on the normal and tangential components of the jump in displacement at the interface (stretch and slip), and the average of the projected strain in the tangent plane (average tangential strain).
Abstract: We discuss the deformation of a curved interface between solid phases, assuming small strains in the bulk phases and neglecting accretion at the interfaces. Such assumptions are relevant to the deformation of solid microstructures when atomic diffusion and the formation of defects such as dislocations are negligible. We base our theory on a constitutive equation giving the (excess) free energy ψ of the interface when the interfacial limits of the displacement fields in the abutting phases as well as the limits of the displacement gradients are known. Using general considerations of frame invariance, we show that ψ can depend on these quantities at most through: firstly the normal and tangential components of the jump in displacement at the interface (stretch and slip), secondly the average of the projected strain in the tangent plane (average tangential strain), thirdly the tangential component of the jump in the projected displacement gradient at the interface (relative tangential strain and rel...

824 citations


Journal ArticleDOI
TL;DR: In this article, a molecular constitutive equation for an idealized polymer architecture, called a pom-pom, has been proposed, which predicts rheology in both shear and extension.
Abstract: Polymer melts with long-chain side branches and more than one junction point, such as commercial low density polyethylene (LDPE), have extensional rheology characterized by extreme strain hardening, while the shear rheology is very shear thinning, much like that of unbranched polymers. Working with the tube model for entangled polymer melts, we propose a molecular constitutive equation for an idealized polymer architecture, which, like LDPE, has multiple branch points per molecule. The idealized molecule, called a “pom-pom,” has a single backbone with multiple branches emerging from each end. Because these branches are entangled with the surrounding molecules, the backbone can readily be stretched in an extensional flow, producing strain hardening. In start-up of shear, however, the backbone stretches only temporarily, and eventually collapses as the molecule is aligned, producing strain softening. Here we develop a differential/integral constitutive equation for this architecture, and show that it predicts rheology in both shear and extension that is qualitatively like that of LDPE, much more so than is possible with, for example, the K-BKZ integral constitutive equation.

724 citations


Journal ArticleDOI
TL;DR: In this article, a nonlinear evolution law for finite deformation viscoelasticity was proposed, which is not restricted to states close to the thermodynamic equilibrium, and upon appropriate linearization, it can recover several established models of finite linear viscoels and linear velocities.

629 citations


Journal ArticleDOI
TL;DR: In this article, a constitutive damage model for massive concrete is presented, mainly intended for the seismic analysis of gravity and arch dams, and an extension to account for the concrete strain-rate dependency, suitable for seismic analysis, is presented at the end.

434 citations


Journal ArticleDOI
TL;DR: In this article, strain gradient plasticity theory is used to model materials undergoing small-scale indentations, and a strong effect of including strain gradients in the constitutive description is found with hardness increasing by a factor of two or more over the relevant range of behavior.
Abstract: Indentation tests at scales on the order of one micron have shown that measured hardness increases significantly with decreasing indent size, a trend at odds with the size-independence implied by conventional plasticity theory. In this paper, strain gradient plasticity theory is used to model materials undergoing small-scale indentations. Finite element implementation of the theory as it pertains to indentation modeling is briefly reviewed. Results are presented for frictionless conical indentations. A strong effect of including strain gradients in the constitutive description is found with hardness increasing by a factor of two or more over the relevant range of behavior. The results are used to investigate the role of the two primary constitutive length parameters in the strain gradient theory. The study indicates that indentation may be the most effective test for measuring one of the length parameters.

399 citations


Journal ArticleDOI
TL;DR: In this paper, the authors analyzed nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials and found that some of them inevitably lead to residual stresses even at very late stages of the deformation process and, consequently, they are not capable of modeling complete separation in a widely open macroscopic crack.

392 citations


Journal ArticleDOI
TL;DR: In this article, the high-temperature deformation behavior of Ti6Al4V alloy has been investigated using split Hopkinson bar and fracture features and microstructures of the deformed specimens were studied by means of different microscopy techniques to understand the formation of adiabatic shear bands and the variations of dislocation features.

306 citations


Journal ArticleDOI
TL;DR: Fault slip data commonly are used to infer the orientations and relative magnitudes of either the principal stresses or the principal strain rates, which are not necessarily parallel or equal as discussed by the authors.
Abstract: Fault slip data commonly are used to infer the orientations and relative magnitudes of either the principal stresses or the principal strain rates, which are not necessarily parallel or equal. At the local scale of an individual fault, the shear plane and slip direction define the orientations of the local principal strain rate axes but not, in general, the local principal stress axes. At a large scale, the orientations of P and T axes maxima for sets of fault slip data do not provide accurate inversion solutions for either strain rate or stress. The quantitative inversion of such fault slip data, however, provides direct constraints on the orientations and relative magnitudes of the global principal strain rates. To interpret the inversion solution as constraining the global principal stresses requires that (1) the fault slip pattern must have a characteristic symmetry no lower than orthorhombic; (2) the material must be mechanically isotropic; and (3) there must be a linear constitutive relationship between the global stress and the global strain rate. Isotropic linear elastic constitutive equations are appropriate to describe the local deformation surrounding an individual slip discontinuity. Fault slip inversions, however, constrain the characteristics of a large-scale cataclastic flow, which is described by constitutive equations that are probably, but to an unknown degree, anisotropic and nonlinear. Such material behavior would not strictly satisfy the requirements for the stress interpretation. Thus, at the present state of knowledge, fault slip inversion solutions are most reliably interpreted as constraining the principal strain rates.

Journal ArticleDOI
TL;DR: In this paper, a framework based on elastic and viscous potentials is developed to fit the non-linear stress strain curves obtained at different strain rates with human cruciate ligaments and patellar tendons.

Journal ArticleDOI
TL;DR: In this paper, a hydrodynamic constitutive equation for low-density polymeric foam materials is presented for rigid polymeric foams with high rate impact loading, and the constitutive model has been implemented into finite-element program as a user-defined material subroutine.

Journal ArticleDOI
TL;DR: In this article, a unified elastic-viscoplastic constitutive model based on dislocation density considerations is proposed, which combines a combination of a kinetic equation, which describes the mechanical response of a material at a given microstructure in terms of dislocation glide, and evolution equations for internal variables characterising the micro-structure.

Journal ArticleDOI
TL;DR: In this article, a subloading surface model is proposed for predicting real soil deformation behavior, which is verified by predicting monotonic and cyclic loading behavior of sands under drained and undrained conditions and comparing them with test data.
Abstract: The subloading surface model fulfills the mechanical requirements for constitutive equations, i.e. the continuity condition, the smoothness condition and the work rate stiffness relaxation and describes pertinently the Masing effect. The constitutive equation of soils is formulated by introducing the subloading surface model and formulating the evolutional rule of rotational hardening for the description of anisotropy. The applicability of the constitutive equation to the prediction of real soil deformation behaviour is verified by predicting monotonic and cyclic loading behaviour of sands under drained and undrained conditions and comparing them with test data. © 1998 John Wiley & Sons, Ltd.


Journal ArticleDOI
TL;DR: In this article, a thermomechanical model of the primary shear zone is combined with a modelling of the contact problem at the tool-chip interface, and a friction law is introduced that accounts for temperature effects.
Abstract: In this paper the process of orthogonal cutting is studied by analytical means. A thermomechanical model of the primary shear zone is combined with a modelling of the contact problem at the tool-chip interface. A friction law is introduced that accounts for temperature effects. The effects of cutting conditions and material behaviour on the temperature distribution along the contact zone, on the mean friction and on the global cutting forces are evaluated. The experimental trends are shown to be well described by the proposed model. © 1998 Elsevier Science Ltd. All rights reserved.Keywords : A. cutting and forming, thermomechanical processes, B. viscoplastic material, friction.

Journal ArticleDOI
TL;DR: In this article, a mechanistic approach to uniaxial viscoelastic constitutive modeling of asphalt concrete is presented, which accounts for damage evolution under cyclic loading conditions.
Abstract: This paper presents a mechanistic approach to uniaxial viscoelastic constitutive modeling of asphalt concrete that accounts for damage evolution under cyclic loading conditions. Schapery's elastic-viscoelastic correspondence principle is applied as a means of separately evaluating viscoelasticity and time-dependent damage growth in asphalt concrete. The time-dependent damage growth in asphalt concrete is modeled by using a damage parameter based on a generalization of microcrack growth law. Internal state variable formulation was used in developing the analytical representation of the model. Tensile uniaxial fatigue tests were performed under the controlled-strain mode to determine model parameters. Then, the resulting constitutive equation was used to predict the stress-strain behavior of the same materials under controlled-stress mode. The constitutive equation proposed in this paper satisfactorily predicts the constitutive behavior of asphalt concrete all the way up to failure under various loading conditions including different stress-strain amplitudes, monotonic versus cyclic loadings, and different modes of loading.

Journal ArticleDOI
TL;DR: A simple three-coefficient exponential constitutive law provides an accurate prediction of stress-stretch behavior over a wide range of deformations and could provide substantial improvement in the evaluation and treatment of valvular disease, surgery, and replacement.
Abstract: Biaxial mechanical testing and theoretical continuum mechanics analysis are employed to formulate a constitutive law for cardiac mitral valve anterior and posterior leaflets. A strain energy description is formulated based on the fibrous architecture of the tissue, accurately describing the large deformation, highly nonlinear transversely isotropic material behavior. The results show that a simple three-coefficient exponential constitutive law provides an accurate prediction of stress-stretch behavior over a wide range of deformations. Regional heterogenity may be accommodated by spatially varying a single coefficient and incorporating collagen fiber angle. The application of this quantitative information to mechanical models and bioprosthetic development could provide substantial improvement in the evaluation and treatment of valvular disease, surgery, and replacement.

Journal ArticleDOI
TL;DR: In this paper, a constitutive framework for the elasto-viscoplastic response of metals that utilizes polycrystal plasticity is presented together with a corresponding numerical integration procedure, where single crystal equations are written in an intermediate configuration obtained by elastically unloading the deformed crystal without rotation from the current configuration to a stress-free state.

Journal ArticleDOI
TL;DR: In this article, the plastic spin concept in large deformation anisotropic elastoplasticity theories with tensorial internal variables, is proved to be a necessary constitutive ingredient, and its presence in the theory is demystified as something very simple and straightforward.

Journal ArticleDOI
TL;DR: In this article, a virtual internal bond (VIB) model with cohesive interactions between material particles is proposed as an alternative approach to modeling fracture, where a phenomenological "cohesive force law" is assumed to act between "material particles" which are not necessarily atoms.

Journal ArticleDOI
TL;DR: In this paper, the dependencies of granular soil behaviour on void ratio and stress are modelled within plasticity theory enriched by a modified stress-dilatancy law for easy future implementation in finite element code.

Journal ArticleDOI
TL;DR: In this paper, the authors used a finite element program to simulate the evolution of crystallographic texture in simple compression, plane-strain compression, and torsion under quasi-static conditions.
Abstract: Strain-rate and temperature-dependent constitutive equations for polycrystalline metals which are capable of modeling the initial and evolving anisotropy in ductile metallic materials owing to the evolution of crystallographic texture are reviewed and then specialized to reproduce the recently published stress-strain response of commercially pure b.c.c. tantalum for strains up to 60%, at strain rates from quasi-static to 30,000 s −1 , and temperatures from −200 to 525 °C (Hoge and Mukherjee, 1977; Vecchio, 1994; Nemat-Nasser and Isaacs, 1996). The constitutive equations have been implemented in a finite element program, and the computational capability is used to simulate the evolution of crystallographic texture in simple compression, plane-strain compression, and torsion under quasi-static conditions. A comparison of the predictions against corresponding experiments shows that the crystal plasticity-based model predicts the texture evolution and the macroscopic stress-strain curves satisfactorily. The computational capability is also used to simulate the dynamic Taylor rod-impact tests performed by Ting (1992) on pre-textured tantalum cylinders. The numerical simulations reasonably reproduce the final length and the ovalized macroscopic shape of the impact end of the cylinders observed in the experiments.

Journal ArticleDOI
TL;DR: In this article, numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model, the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbellmodel of Chilcott-Rallison (CR).
Abstract: Numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model; the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbell model of Chilcott–Rallison (CR). In the Giesekus model, intramolecular forces are described by a Hookean spring, whereas a finitely extensible spring whose modulus is given by the Warner approximation is used in both the FENE-P and CR models. Hydro dynamic drag on the beads is taken to be anisotropic for the Giesekus model and isotropic for the other two models. The CR and FENE-P models differ subtly in their approximate treatment of the nonlinear force law. The three models exhibit very similar rheological behavior in viscometric flow and steady elongational flow, with the notable exception that the viscosity for the CR model is shear-rate independent. Finite element simulations are performed by using two different formulations: the elastic-viscous split-stress gradient (EVSS-G) method and a new variant of this formulation, the discrete EVSS-G (DEVSS-G) formulation, in which the elliptic stabilization term is added only to the discrete version of the momentum equation, and the constitutive equation is solved directly in terms of the polymer contribution to the stress tensor. Calculations are performed for all models up to a Weissenberg number We, where the configuration tensor 〈QQ〉 loses positive definiteness. However, by locally refining the mesh in the gap region, the positive definiteness of 〈QQ〉 is recovered. The flow and stress fields predicted by the three constitutive equations are qualitatively similar. A `birefringent strand' of highly stretched polymer molecules, which appears to emanate from the rear stagnation point in the cylinder, strengthens as We is increased. Not surprisingly, the molecular extension computed for the Giesekus model is considerably larger than that of the two FENE spring models. The drag force on the cylinders differs for the FENE-P and CR models, because of the difference in the shear-thinning viscosity resulting from the different approximations used in these models.

Journal ArticleDOI
TL;DR: In this paper, a "compressible-Leonov model" is introduced, in which the elastic volume response is rigorously separated from the elasto-viscoplastic isochoric deformation.
Abstract: Constitutive equations for finite elastic-plastic deformation of polymers and metals are usually formulated by assuming an isotropic relation between the Jaumann rate of the Cauchy-stress tensor and the strain-rate tensor. However, the Jaumann-stress rate is known to display spurious non- physical behaviour in the elastic region. Replacing the Jaumann-stress rate by a Truesdell-stress rate results in an adequate description in the elastic region, but gives rise to a volume decrease during plastic flow in tensile deformation. In this paper a "compressible-Leonov model" is introduced, in which the elastic volume response is rigorously separated from the elasto-viscoplastic isochoric de- formation. This has the advantage that the model can be extended in a straightforward way to include a spectrum of relaxation times. It is shown that in the limit of small elastic strains, the compressible Leonov model reduces to the Jaumann-stress rate model, but diverges from the Truesdell-stress rate model. Finally, a comparison is made of the above mentioned models in a homogeneous uniaxial ten- sile test and a homogeneous plane-stress shear test, using polycarbonate (PC) as a model system. All models considered in this paper are "single mode" models (i.e. one relaxation time), and, therefore, cannot describe the full (non)linear viscoelastic region, nor the strain-hardening or strain-softening response.

Journal ArticleDOI
TL;DR: In this article, a semi-empirical constitutive law for the brittle deformation of intact Westerly granite is presented, which can be extended to larger displacements, dominated by localized deformation, by including a displacement-weakening break-down region terminating in a frictional sliding regime.
Abstract: A semiempirical constitutive law is presented for the brittle deformation of intact Westerly granite. The law can be extended to larger displacements, dominated by localized deformation, by including a displacement-weakening break-down region terminating in a frictional sliding regime often described by a rate- and state-dependent constitutive law. The intact deformation law, based on an Arrhenius type rate equation, relates inelastic strain rate to confining pressure Pc, differential stress σΔ, inelastic strain ei and temperature T. The basic form of the law for deformation prior to fault nucleation is lne˙i=c-(E*/RT)+(σΔ/aσo)sin-a(πei/2eo) where σo and eo are normalization constants (dependent on confining pressure), a is rate sensitivity of stress, and α is a shape parameter. At room temperature, eight experimentally determined coefficients are needed to fully describe the stress-strain-strain rate response for Westerly granite from initial loading to failure. Temperature dependence requires apparent activation energy (E* ∼ 90 kJ/mol) and one additional experimentally determined coefficient. The similarity between the prefailure constitutive law for intact rock and the rate- and state-dependent friction laws for frictional sliding on fracture surfaces suggests a close connection between these brittle phenomena.

Journal ArticleDOI
TL;DR: In this article, a modified version of the classic linear viscoelastic Jeffreys model is proposed and the corresponding five-parameter equation with fractional derivatives of different orders of the stress and rate of strain is stated.
Abstract: Based on the classic linear viscoelastic Jeffreys model, a modified Jeffreys model is suggested. The corresponding five-parameter equation with fractional derivatives of different orders of the stress and rate of strain is stated and the characteristic material functions of the linear viscoelasticity theory, such as the dynamic moduli, are derived. The comparison between the measured dynamic moduli of Sesbania gel and xanthan gum and the theoretical predictions of the proposed phenomenological model shows an excellent agreement.

Journal ArticleDOI
TL;DR: In this paper, it is shown that the macroscale field equations derived from mixture theories can be reformulated in terms of the measurable quantities involved in the macro-scale theories, including the fundamental inequality obtained from the second law, entail the existence of a macroscale C-potential upon which a thermo- dynamically consistent formulation of the constitutive equations can be firmly founded.

Journal ArticleDOI
TL;DR: In this paper, a modified version of the Cam-Clay model of critical state soil mechanics is reformulated to include finite deformation effects, which is appropriate for cases involving large plastic volumetric strains, and the use of a class of two-invariant stored energy functions.