Showing papers on "Constitutive equation published in 1990"

One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials

Abstract: The use of the thermoelastic martensitic transformation and its reverse transformation has recently been proposed and demonstrated for several active control ap plications. However, the present constitutive models have lacked several important funda mental concepts that are essential for many of the proposed intelligent material system ap plications such as shape memory hybrid composites.A complete, unified, one-dimensional constitutive model of shape memory materials is developed and presented in this paper. The thermomechanical model formulation herein will investigate important material characteristics involved with the internal phase transformation of shape memory alloys. These characteristics include energy dissipation in the material that governs the damping behavior, stress-strain-temperature relations for pseudoelasticity, and the shape memory effect. Some numerical examples using the model are also presented.

Finite deformation constitutive equations and a time integrated procedure for isotropic hyperelastic—viscoplastic solids

Abstract: Constitute equations for finite deformation, isotropic, elastic-viscoplastic solids are formulated. The concept of a multiplicative decomposition of the deformation gradient into an elastic and a plastic part is used. The constitutive equation for stress is a hyperelastic relation in terms of the logarithmic elastic strain. Since the material is assumed to be isotropic in every local configuration determined by the plastic part of deformation gradient, the internal variables are necessarily scalars. We use a single scalar as an internal variable to represent the isotropic resistance to plastic flow offered by the internal state of the material. The constitutive equation for stress is often expressed in a rate form, and for metals it is common to approximate this rate equation, under the assumption of infinitesimal elastic strains, to arrive at a hypoelastic equation for the stress. Here, we do not express the stress constitutive equation in a rate form, nor do we make this approximative assumption. For the total form of the stress equation we present a new implicit procedure for updating the stress and other relevant variables. Also, the principle of virtual work is linearized to obtain a consistent, closed-from elasto-viscoplastic tangent operator (the ‘Jacobian’) for use in solving for global balance of linear momentum in implicit, two-point, deformation driven finite element algorithms. The time integration algorithm is implemented in the finite element program ABAQUS. To check the accuracy and stability of the algorithm, some representative problems involving large, pure elastic and combined elastic-plastic deformations are solved.

On a stress resultant geometrically exact shell model. Part IV: variable thickness shells with through-the-thickness stretching

Abstract: This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.

Thermoplasticity of Saturated Clays: Experimental Constitutive Study

Abstract: Experimental results obtained from thermomechanical tests on three clays are analyzed in light of the constitutive equations of soil thermoplasticity presented in a companion paper. Heating and cooling drained tests at constant isotropic stress show a strong dependence of the elastic domain on temperature. Thermal sensitivity of elastic domain was found to be different in overconsolidated and in normally consolidated clays. Thermoplastic strain hardening builds up to compensate for thermal softening in normally consolidated clays at plastic compression during drained heating, if the constant stress is imposed. Triaxial compression tests at constant elevated temperatures show an increase in ductility and a decrease in dilatativity at high temperatures. Undrained heating tests show a significant water pressure buildup. At constant principal stress difference, the water pressure growth leads to an effective stress drop and an eventual failure at the critical state line. A temperature‐rate‐dependent non‐assoc...

A theory of mechanical behavior of elastic media with growing damage and other changes in structure

Abstract: S train energy-like potentials are used to model the mechanical behavior of linear and nonlinear elastic media with changing structure, such as micro- and macrocrack growth in monolithic and composite materials. Theory and experiment show that the applied work for processes in which changes in structure occur is in certain cases independent of some of the deformation history. Consequences of this limited path-independence are investigated, and various relationships for stable mechanical response are derived. For example, it is shown that work is at a minimum during stable changes in structure, which should be useful for developing approximate solutions by variational methods. Some final remarks indicate how the theory may be extended to include thermal, viscoelastic and fatigue effects.

Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity

Abstract: Adding a Newtonian solvent to most differential viscoelastic constitutive equations mathematically regularizes the coupled set formed by the momentum, continuity and constitutive equations. The momentum and continuity equations form an elliptic saddle point problem for velocity and pressure, and the constitutive equation is hyperbolic in stress. Three finite element algorithms are presented that exploit this elliptic behavior by expressing the momentum equation in different differential forms. The first is based on the viscous elliptic operator that arises naturally with the introduction of a Newtonian solvent viscosity; the second is based on the explicitly elliptic momentum equation formulation developed for the upper-convected Maxwell model; and the third is based on an elastic-viscous splitting of the momentum equation. Finite element discretizations are created by using Galerkin's method for the momentum and continuity equations and the streamline-upwind Petrov-Galerkin method for the components of the constitutive equation. In the latter two methods, additional interpolants are introduced so as to maintain continuous representations of velocity derivatives across element boundaries; this is a requirement if only those higher-order velocity terms which are explicitly elliptic are integrated by parts. Calculations for flow between eccentric cylinders and through a corrugated tube demonstrate the numerical stability and accuracy of each of the formulations. The robustness of each algorithm for calculation of flows at high Deborah numbers depends on the value of the ratio β ≡ ηS/gh0, where ηs is the solvent viscosity and η0 is the viscosity of the solution. The algorithm based on the elastic-viscous splitting is the most robust across the full range 0 ≤ β ≤ 1.

Issues in viscoelastic fluid mechanics

Abstract: nt transient that may be many orders of magnitude longer than time scales associated with instrument inertia. Intrinsic material time scales can be identified with the dynamics of the macro­ molecular chains. Shaping processes for polymeric materials are usually carried out in the liquid state, often over times that are rapid relative to those associated with molecular reorganization; the viscoelasticity is thus relevant to any understanding of flow and structure development. The distinction between the rheology and the fluid mechanics of visco­ elastic materials is vague but is worth stating. The former is concerned with constitutive relations between stress and deformation and may involve physical modeling at a molecular level. Controllability of the flow field is essential when making rheological measurements to evaluate material properties, so the kinematics are generally imposed and the momentum equation is not solved. Viscoelastic fluid mechanics, on the other hand, is the study of motions in which the kinematics cannot be established a priori, and the continuity and momentum equations must be solved together with the constitutive equation for the stress. The equations that must be solved for even the most elementary viscoelastic liquids are considerably more complex than the Navier-Stokes equations, and many unresolved issues

Determination of a constitutive relation for passive myocardium: II. Parameter estimation.

Abstract: In the first paper of this series, we proposed a new transversely isotropic pseudostrain-energy function W for describing the biomechanical behavior of excised noncontracting myocardium. The specific functional form of W was inferred directly from biaxial data to be a polynomial function of two coordinate invariant measures of the finite deformation and five material parameters. In this paper, best-fit values of the material parameters are determined from biaxial data using a nonlinear least-squares regression. These values of the parameters are shown to be well-determined, and the final constitutive relation is shown to have good predictive capabilities. Since the proposed constitutive relation describes much broader classes of in-vitro biaxial data than previously proposed relations, it may be better applicable to analyses of stress in the passive heart.

A critical review of the state of finite plasticity

Abstract: The object of this paper is to provide a critical review of the current state of plasticity in the presence of finite deformation. In view of the controversy regarding a number of fundamental issues between several existing schools of plasticity, the areas of agreement are described separately from those of disagreement. Attention is mainly focussed on the purely mechanical, rate-independent, theory of elastic-plastic materials, although closely related topics such as rate-dependent behavior, thermal effects, experimental and computational aspects, microstructural effects and crystal plasticity are also discussed and potentially fruitful directions are identified. A substantial portion of this review is devoted to the area of disagreement that covers a detailed presentation of argument(s), bothpro andcon, for all of the basic constitutive ingredients of the rate-independent theory such as the primitive notion or definition of plastic strain, the structure of the constitutive equation for the stress response, the yield function, the loading criteria and the flow and the hardening rules. The majority of current research in finite plasticity theory, as with its infinitesimal counterpart, still utilizes a (classical) stress-based approach which inherently possesses some shortcomings for the characterization of elastic-plastic materials. These and other anomalous behavior of a stress-based formulation are contrasted with the more recent strain-based formulation of finite plasticity. A number of important features and theoretical advantages of the latter formulation, along with its computational potential and experimental interpretation, are discussed separately.

Thermoplasticity of Saturated Soils and Shales: Constitutive Equations

Abstract: Plastic behavior of soils and shales due to heating and loading under constant elevated temperature is discussed in terms of a thermoplastic version of the critical state model. Rules for dependence of the yield surface on temperature in the elastic states and at yielding are proposed. The elastic domain is assumed to shrink during heating (thermal softening) and to expand during cooling, when the stress state is elastic. In a plastic state thermal softening occurs simultaneously with the plastic strain hardening. At a constant stress state, thermal softening may entirely be compensated by plastic strain hardening leading to thermal consolidation. Loading and unloading criteria are given to determine whether the soil response is thermoelastic or thermoplastic. As opposed to isothermal plasticity, stress rate excursions inside the current yield surface are admissable plastic processes, when temperature grows, even if strain hardening occurs. Also, outside stress rate excursions at the softening side may generate plastic strain, when cooling occurs. Thermally induced plastic strain rate non-associativity is discussed as well. Direct and inverse incremental strain-stress-temperature relationships are formulated. An analysis of the experimental results of thermomechanical testing of saturated clays is given in a companion paper.

Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples

Abstract: Viscoplasticity is introduced as a procedure to regularize the elasto-plastic solid, especially for those situations in which the underlying inviscid material exhibits instabilities which preclude further analysis of initial-value problems. The procedure is general, and therefore has the advantage of allowing the regularization of any inviscid elastic-plastic material. Rate-dependency is shown to naturally introduce a length-scale into the dynamical initial-value problem. Furthermore, the width of the localized zones in which high strain gradients prevail and strain accumulations take place, is shown to be proportional to the characteristic length c η, which is the distance the elastic wave travels in the characteristic time η. Viscosity can thus be viewed either as a regularization parameter (computational point of view), or as a substructural/micromechanical parameter to be determined from observed shear-band widths (physical point of view). Finally, from a computational point of view, the proposed approach is shown to have striking advantages: (1) the wave speeds always remain real (even in the softening regime) and are set by the elastic moduli; (2) the elasto-(visco-)plastic constitutive equations are amenable to unconditionally stable integration; (3) the resulting well-posedness of the dynamical initial-value problem guarantees stable and convergent solutions with mesh refinements. The initial-value problems reported in this first part are essentially one-dimensional. They are used because they offer the simplest possible context to illustrate both the physical and computational significance of the proposed viscoplastic regularization procedure. The methods used in multi-dimensional analysis and examples will be reported in Part 2.

One-dimensional constitutive modeling of asphalt concrete

Abstract: This paper presents a mechanistic approach to constitutive modeling of asphalt concrete under complicated repetitive loading conditions with rest periods. From an extensive literature review, two m...

Deformation of Elastic Solids

Abstract: A review of the elementary theory of elasticity Cartesian tensors kinematics of deformation balance laws and analysis of stress constitutive equations elastostatics solution of linear elastostatic problems by special technique linear elastodynamics.

Description of tantalum deformation behavior by dislocation mechanics based constitutive relations

Abstract: Dislocation mechanics based constitutive equation constants are determined for temperature, strain rate, work hardening, and polycrystal grain size influences on the deformation behavior of various tantalum materials. An analysis of the maximum load point strain provides a useful method of determining the work hardening constants. Different athermal stress constants and thermal activation related constants are obtained for certain groupings of the different tantalum materials. The variations are correlated with the annealing history of the materials and related to dislocation model parameters involved in the thermal activation strain rate analysis. Computed tantalum deformation results based on these constants are shown to agree with Gourdin’s reported expanding ring test measurements and with the deformed shape of a Taylor cylinder impact test specimen.

Modeling the rheology of polyisobutylene solutions

Abstract: Results are presented of a detailed rheological study of two different polyisobutylene systems in steady and transient shear flows. Shear‐thinning solutions are prepared by dissolving high molecular polyisobutylene in tetradecane, a low viscosity solvent. Nearly constant viscosity solutions are obtained by adding a third component of highly viscous polybutene. The linear viscoelastic behavior of these fluids is characterized in small‐amplitude oscillatory shearing flow and the nonlinear behavior is studied by means of steady and transient shear flows. The spectrum of relaxation times for both solutions has been determined. Temperature and concentration effects on the material functions are reported. Three differential constitutive equations with four relaxation modes were tested for fitting the experimental data in steady and transient shear flows: they are the quasilinear Oldroyd‐B model and the nonlinear Giesekus and Bird–DeAguiar models. The presence of nonlinear stress terms is found to be important f...

Residual stresses and stored elastic energy of composites and polycrystals

Abstract: R esidual Stresses in heterogeneous materials may arise because of differential or anisotropic thermal expansion of constituents. The paper is concerned with thermoelastic solids whose material properties fluctuate on the microscopic scale. Rigorous general relations between stored elastic energy and statistical averages (mean values and fluctuations) of residual stresses are derived. These results are applied to two-phase composites and to materials where the fluctuations of elastic constants can be neglected. One obtains exactly the stored energy, certain conditional mean values and the covariance matrix of the residual stresses. Under the assumptions of statistical homogeneity and isotropy, the results hold for any type of heterogeneous microstructure.

Muscle Activation and Contraction: Constitutive Relations Based Directly on Cross-Bridge Kinetics

Abstract: This paper reviews recent work aimed at deriving tractable constitutive relations for skeletal muscle from biophysical cross-bridge theories. Discussion is focused on a model proposed previously by the first author (the Distribution-Moment Model), which emphasizes the important role of the moments of the actin-myosin bond-distribution function. The theory leads to a relatively simple third order state variable model for contraction dynamics in which the state variables are the three lowest order moments of the bond-distribution function; further, these three moments have simple macroscopic interpretations as muscle stiffness, force, and elastic energy. New results are presented on the formulation of a compatible model for excitation-contraction coupling, and this model requires the introduction of only one more state variable--the free calcium concentration.

Constitutive relation and creep-fatigue life model for eutectic tin-lead solder

Abstract: A constitutive equation for a typical SnPb eutectic or near-eutectic solder joint is developed, based on empirical data in shear and generalized to three dimensions. Three strain components are considered: elastic, time independent plastic, and steady-state creep. A continuum mechanics rather than a metallurgical approach is taken with emphasis on the formulation of an equation useful for predicting solder behavior under a variety of conditions. Solutions of the constitutive equation predict the experimental hysteresis data for the loading histories available. Solutions of the equation for the test conditions of three independent sets of solder fatigue data show that the equation, together with the matrix creep failure indicator, can give an estimate of fatigue life. >

Melt blowing: General equation development and experimental verification

Abstract: A model has been developed for steady polymer melt blowing. This model includes the dominant effect that the forwarding air has upon the process. Inertial, gravitational and heat transfer effects are also included. The model equations are solved numerically with both Newtonian and viscoelastic (Phan-Thien and Tanner) constitutive equations. The predicted results compare favorably with actual experimental data.

A constitutive model for the dynamic response of brittle materials

Abstract: A microphysically based material model for the dynamic inelastic response of a brittle material is developed. The progressive loss of strength as well as the post‐failure response of a granular material with friction are included. Crack instability conditions (an inelastic surface in stress space) and inelastic strains are obtained by considering the response of individual microcracks to an applied stress field. The assumptions of material isotropy and an exponential distribution for the crack radius are invoked to provide a tractable formulation. The constitutive model requires a minimal number of physical parameters, is compatible with a previously developed ductile fracture model [J. Appl. Phys. 64, 6699 (1988)] that utilizes inelastic surfaces, and can be formulated as an efficient, robust numerical algorithm for use in three‐dimensional computer codes. The material model is implemented into a Lagrangian computer formulation for the demonstration of material response to dynamic loading conditions. Comparisons with one‐dimensional, uniaxial impact experiments are provided.

Constitutive equation for modeling mixed extension and shear in polymer solution processing

Abstract: What is the simplest adequate model of shear thinning and extension thickening of polymer solutions at the intense local deformation rates in coating and related flows? We examine a combination of the Carreau equation of shear sensitivity with an analogous equation of extension sensitivity. The two contributions are mediated by the rate at which the principal straining axes turn with respect to the liquid itself, a relative rotation rate that is frame independent, vanishes in purely extensional flow, and harks back to Lumley’s (1969) idea of persistent molecular strain rate, which was adumbrated by Takserman‐Krozer (1963) and named by Frank and Mackley (1976). We indicate how the new equation can be used to extract viscosity‐in‐extension from the extension‐rich mixed flow between opposed nozzles. We illustrate how the new equation can be applied to the complex mixed flow in slide coating.

On a constitutive theory for materials undergoing microstructural changes

Abstract: Of particular interest is a two-network theory of polymer response. The mechanical response depends on the deformation of both the remaining portion of the original material and newly formed one. A particular constitutive equation is introduced. The original and newly formed material are both treated as incompressible isotropic nonlinear neo-Hookean elastic materials, but with different reference configurations

The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations

Abstract: Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses is the shear rate, Kn is the normal stiffness of an assumed viscoelastic contact force model, Ks/Kn is the ratio of tangential to normal stiffness, ζn is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζn = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), Ks/Kn = 0.8 and μ = 0.5. It is found that for lower concentrations (C 0.75) τ*ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ*ij ∝ B−n(C). The powers nij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.

Geomaterials: constitutive equations and modelling

Abstract: An introduction to the rheology of geomaterials, P. Habib recent trends in laboratory testing, J. Lanier viscous properties of geomaterials, E. Flavigny and R. Nova damage fracture and size effect in concrete, J. Mazars and G. Pijaudier-Cabot experimental simulation of geotechnical structures, M.P. Luong soil reinforcement by synthetic inclusions - geotextiles, J.P. Gourc the expression of rheological laws in incremental form and the main classes of constitutive equations, F. Darve an introduction to the classical theory of elastoplascity, B. Loret geomechanical applications of the theory of multimechanisms, B. Loret incrementally non-linear constitutive relationships, F. Darve viscoplasticity in rock mechanics, P. Berest a micromechanical analysis of the behaviour of granular materials, B. Cambou shear band initiation in granular materials - experimentation and theory, J. Desrues behaviour of porous saturated deformable media, J-L. Auriault change of scale in multiphase media - the case of saturated soils, F. Gilbert homogenization of periodic media, D. Caillerie rheology and computer codes, M. Boulon numerical modelling in soil dynamics, D. Aubry and H. Modaressi.