Showing papers on "Constitutive equation published in 2000"

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Abstract: In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

Constitutive models of rubber elasticity: A review

Abstract: A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory—Erman model for low stretch deformation and the Arruda—Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also addressed.

Continuum Mechanics and Theory of Materials

Abstract: 1 Kinematics.- 2 Balance Relations of Mechanics.- 3 Balance Relations of Thermodynamics.- 4 Objectivity.- 5 Classical Theories of Continuum Mechanics.- 6 Experimental Observation and Mathematical Modelling.- 7 General Theory of Mechanical Material Behaviour.- 8 Dual Variables.- 9 Elasticity.- 10 Viscoelasticity.- 11 Plasticity.- 12 Viscoplasticity.- 13 Constitutive Models in Thermomechanics.- References.

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

Abstract: We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, , of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G*(ω), in the Fourier frequency domain, and the stress relaxation modulus, Gr(t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing as a local power law. If the logarithmic slope of can be accurately determined, these estimates generally perform well at the frequency extremes.

On the plasticity of single crystals: free energy, microforces, plastic-strain gradients

Abstract: This study develops a general theory of crystalline plasticity based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. A viscoplastic regularization of the basic equations that obviates the need to determine the active slip systems is developed. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, the special case of single slip is discussed. Specific solutions are presented: one a single shear band connecting constant slip-states; one a periodic array of shear bands.

Biaxial Mechanical Evaluation of Planar Biological Materials

Abstract: A fundamental goal in constitutive modeling is to predict the mechanical behavior of a material under a generalized loading state. To achieve this goal, rigorous experimentation involving all relevant deformations is necessary to obtain both the form and material constants of a strain-energy density function. For both natural biological tissues and tissue-derived soft biomaterials, there exist many physiological, surgical, and medical device applications where rigorous constitutive models are required. Since biological tissues are generally considered incompressible, planar biaxial testing allows for a two-dimensional stress-state that can be used to characterize fully their mechanical properties. Application of biaxial testing to biological tissues initially developed as an extension of the techniques developed for the investigation of rubber elasticity [43, 57]. However, whereas for rubber-like materials the continuum scale is that of large polymer molecules, it is at the fiber-level (∼1 μm) for soft biological tissues. This is underscored by the fact that the fibers that comprise biological tissues exhibit finite nonlinear stress-strain responses and undergo large strains and rotations, which together induce complex mechanical behaviors not easily accounted for in classic constitutive models. Accounting for these behaviors by careful experimental evaluation and formulation of a constitutive model continues to be a challenging area in biomechanics. The focus of this paper is to describe a history of the application of biaxial testing techniques to soft planar tissues, their relation to relevant modern biomechanical constitutive theories, and important future trends.

A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals

Abstract: The energy balance equation for elastoplastic solids includes heat source terms that govern the conversion of some of the plastic work into heat. The remainder contributes to the stored energy of cold work due to the creation of crystal defects. This paper is concerned with the fraction β of the rate of plastic work converted into heating. We examine the status of the common assumption that β is a constant with regard to the thermodynamic foundations of thermoplasticity and experiments. A general internal-variable theory is introduced and restricted to abide by the second law of thermodynamics. Experimentally motivated assumptions reduce this theory to a special model of classical thermoplasticity. The only part of the internal energy not determined from the isothermal response is the stored energy of cold work, a function only of the internal variables. We show that this function can be inferred from stress and temperature data from a single adiabatic straining experiment. Experimental data from dynamic Kolsky-bar tests at various strain rates yield a unique stored energy function. Its knowledge is crucial for the determination of the thermomechanical response in non-isothermal processes. Such a prediction agrees well with results from dynamic tests at different rates. In these experiments, β is found to depend strongly on both strain and strain rate for various engineering materials. The model is successful in predicting this dependence. Requiring β to be constant is thus an approximation of dubious validity.

Mechanical Response of Polymers: An Introduction

Abstract: Preface 1. Discussion of response of a viscoelastic material 2. Constitutive equations for one-dimensional response of viscoelastic materials: mechanical analogs 3. Constitutive equations for one-dimensional linear response of a viscoelastic material 4. Some features of the linear response of viscoelastic materials 5. Histories with constant strain or stress rates 6. Sinusoidal oscillations 7. Constitutive equation for three dimensional linear isotropic viscoelastic materials 8. Axial load, bending and torsion 9. Dynamics of bodies with viscoelastic support 10. Boundary value problems for linear isotropic viscoelastic materials 11. Influence of temperature Appendices References Index.

Aging and rheology in soft materials

Abstract: We study theoretically the role of aging in the rheology of soft materials. We define several generalized rheological response functions suited to aging samples (in which time translation invariance is lost). These are then used to study aging effects within a simple scalar model (the "soft glassy rheology" or SGR model) whose constitutive equations relate shear stress to shear strain among a set of elastic elements, with distributed yield thresholds, undergoing activated dynamics governed by a "noise temperature," x. (Between yields, each element follows affinely the applied shear.) For 1 < x < 2 there is a power-law fluid regime in which transients occur, but no aging. For x < 1, the model has a macroscopic yield stress. So long as this yield stress is not exceeded, aging occurs, with a sample's apparent relaxation time being of order its own age. The (age-dependent) linear viscoelastic loss modulus G[double-prime](omega,t) rises as frequency is lowered, but falls with age t, so as to always remain less than G[prime](omega,t) (which is nearly constant). Significant aging is also predicted for the stress overshoot in nonlinear shear startup and for the creep compliance. Though obviously oversimplified, the SGR model may provide a valuable paradigm for the experimental and theoretical study of rheological aging phenomena in soft solids. ©2000 Society of Rheology.

Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation

Abstract: The paper presents a phenomenological material model for a superimposed elastic–viscoelastic–plastoelastic stress response with damage at large strains and considers details of its numerical implementation. The formulation is suitable for the simulation of carbon-black filled rubbers in monotonic and cyclic deformation processes under isothermal conditions. The underlying key approach is an experimentally motivated a priori decomposition of the local stress response into three constitutive branches which act in parallel: a rubber–elastic ground–stress response, a rate-dependent viscoelastic overstress response and a rate-independent plastoelastic overstress response. The damage is assumed to act isotropically on all three branches. These three branches are represented in a completely analogous format within separate eigenvalue spaces, where we apply a recently proposed compact setting of finite inelasticity based on developing reference metric tensors. On the numerical side, we propose a time integration scheme which exploits intrinsically the modular structure of the proposed constitutive model. This is achieved on the basis of a convenient operator split of the local evolution system, which we decouple into a stress evolution problem and a parameter evolution problem. The constitutive functions involved in the proposed model are specified for a particular filled rubber on the basis of a parameter identification process. The paper concludes with some numerical examples which demonstrate the overall response of the proposed model by means of a representative set of numerical examples.

A kinematic hardening constitutive model for natural clays with loss of structure

Abstract: A rate-independent constitutive model for natural clays is presented, formulated within the framework of kinematic hardening with elements of bounding surface plasticity. The modelling framework is intended to include effects of damage to structure caused by irrecoverable plastic strains caused by sampling, laboratory testing, or geotechnical loading. The incorporation of a structure measure allows the size of the bounding surface to decay with plastic deformations. This model can be seen as a logical extension from the Cam-clay model. The steady fall of stiffness with strain towards the Cam-clay value is controlled by a particular interpolation function. This ensures a smooth degree of approach between a kinematically hardening bubble (which is the boundary of the elastic region) and the bounding surface during their relative translation with stress history. The model describes the essential phenomena of pre-failure behaviour of natural clays: stiffness variation with strain, volumetric change accompanyi...

Microplane Model M4 for Concrete. I: Formulation with Work-Conjugate Deviatoric Stress

Abstract: The first part of this two-part study presents a new improved microplane constitutive model for concrete, representing the fourth version in the line of microplane models developed at Northwestern University. The constitutive law is characterized as a relation between the normal, volumetric, deviatoric, and shear stresses and strains on planes of various orientations, called the microplanes. The strain components on the microplanes are the projections of the continuum strain tensor, and the continuum stresses are obtained from the microplane stress components according to the principle of virtual work. The improvements include (1) a work-conjugate volumetric deviatoric split—the main improvement, facilitating physical interpretation of stress components; (2) additional horizontal boundaries (yield limits) for the normal and deviatoric microplane stress components, making it possible to control the curvature at the peaks of stress-strain curves; (3) an improved nonlinear frictional yield surface with plasticity asymptote; (4) a simpler and more effective fitting procedure with sequential identification of material parameters; (5) a method to control the steepness and tail length of postpeak softening; and (6) damage modeling with a reduction of unloading stiffness and crack-closing boundary. The second part of this study, by Caner and Bazant, will present an algorithm for implementing the model in structural analysis programs and provide experimental verification and calibration by test data.

Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Abstract: Previous studies by the authors and their co-workers show that the structure of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate-independent plasticity models. For example, the Boyd–Lagoudas polynomial hardening model has a stress-elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate-independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor-transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models, is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost. Copyright © 2000 John Wiley & Sons, Ltd.

Grain-size effect in viscoplastic polycrystals at moderate strains

Abstract: This work deals with the prediction of grain-size dependent hardening in FCC and BCC polycrystalline metals at moderately high strains (2–30%). The model considers 3–D, polycrystalline aggregates of purely viscoplastic crystals, and simulates quasi-static deformation histories with a hybrid finite element method implemented for parallel computation. The hardening response of the individual crystals is considered to be isotropic, but modified to include a physically motivated measure of lattice incompatibility which is supposed to model, in the continuum setting, the resistance to plastic flow provided by lattice defects. The length-scale in constitutive response that is required on dimensional grounds appears naturally from physical considerations. The grain-size effect in FCC polycrystals and development of Stage IV hardening in a BCC material are examined. Though the grain-size does not enter explicitly into the constitutive model, an inverse relationship between the macroscopic flow stress and grain-size is predicted, in agreement with experimental results for deformation of FCC polycrystals having grain-sizes below 100 microns and at strains beyond the initial yield (>2%). The development of lattice incompatibility is further shown to predict a transition to Stage IV (linear) hardening upon saturation of Stage III (parabolic) hardening.

Computational mechanics of the heart : From tissue structure to ventricular function

Abstract: Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a pole-zero constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure-volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media

Abstract: Higher order (so-called strain gradient) homogenised equations are rigorously derived for an infinitely extended periodic elastic medium with the periodicity cell of a small size e, in the presence of a fixed body force f, via a combination of variational and asymptotic techniques. The coefficients of these equations are explicitly related to solutions of higher order unit cell problems. The related higher order homogenised solutions are shown to be best possible in a certain variational sense, and it is shown that these solutions are close to the actual solutions up to higher orders in e. We derive a rigorous full asymptotic expansion for the energy I e, f and also show that its higher order terms are determined by the higher order homogenised solutions. The resulting variational construction generates higher order effective constitutive relations which are in agreement with those proposed by phenomenological strain gradient theories.

A three-phase model for unsaturated soils

Abstract: A comprehensive framework to define the constitutive behaviour of unsaturated soils is developed within the theory of mixtures applied to three-phase porous media. Each of the three phases is endowed with its own strain and stress. Elastic and elastic–plastic constitutive equations are developed. Particular emphasis is laid on the interactions between the phases both in the elastic and plastic regimes. Nevertheless, the clear structure of the constitutive equations requires a minimal number of material parameters. Their identification is provided: in particular, it incorporates directly the soil–water characteristic curve. Crucial to the formulation is an appropriate definition of the effective stress. The coupled influence of this effective stress and of suction makes it possible to describe qualitatively many of the characteristic features observed in experiments, e.g. for normally consolidated soils, a plastic behaviour up to air entry followed by an elastic behaviour at increasing suctions, and, on the way back, an elastic behaviour, unless compression is applied in which case plastic collapse occurs. Copyright © 2000 John Wiley & Sons, Ltd.

Fault rupture between dissimilar materials: Ill-posedness, regularization, and slip-pulse response

Abstract: Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of self-sustained slip pulses [Weertman, 1980] propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i.e., with a constant coefficient of friction) and a remote driving shear stress that is arbitrarily less than the corresponding frictional strength. Following Andrews and Ben-Zion [1997] (ABZ), we study numerically, with a two-dimensional (2-D) plane strain geometry, the propagation of ruptures along such a dissimilar material interface. However, this problem has been shown to be ill-posed for a wide range of elastic material contrasts [Renardy, 1992; Martins and Simoes, 1995; Adams, 1995]. Ranjith and Rice [2000] (RR) showed that when the generalized Rayleigh speed exists, as is the case for the material contrast studied by ABZ, the problem is ill-posed for all values of the coefficient of friction, f, whereas when it does not exist, the problem is ill-posed only for f greater than a critical value. We illustrate the ill-posedness by showing that in the unstable range the numerical solutions do not converge through grid size reduction. By contrast, convergence is achieved in the stable range but, not unexpectedly, only dying pulses are then observed. RR showed that among other regularization procedures, use of an experimentally based law [Prakash and Clifton, 1993; Prakash, 1998], in which the shear strength in response to an abrupt change in normal stress evolves continuously with time or slip toward the corresponding Coulomb strength, provides a regularization. (Classical slip weakening or rate- and state-dependent constitutive laws having the same kind of abrupt response as Coulomb friction also do not regularize the problem.) Convergence through grid size reduction is then achieved in the otherwise ill-posed range. For sufficiently rapid shear strength evolution, self-sustained pulses are observed. When the generalized Ray-leigh wave speed exists, they propagate essentially at that velocity and, in consistence with Weertman's [1980] analysis, the propagation occurs only in one direction, which is that of slip in the more compliant medium. When the generalized Rayleigh wave speed does not exist, similar self-sustained pulses propagate at about the slower S wave speed and in the same direction. RR also suggested that for sufficiently high coefficient of friction, another kind of (less unstable) self-sustained pulses, propagating at a velocity close to the slower P wave speed and in the opposite direction, could also exist. We numerically verify that prediction.

A theory of subgrain dislocation structures

Abstract: We develop a micromechanical theory of dislocation structures and finite deformation single crystal plasticity based on the direct generation of deformation microstructures and the computation of the attendant effective behavior. Specifically, we aim at describing the lamellar dislocation structures which develop at large strains under monotonic loading. These microstructures are regarded as instances of sequential lamination and treated accordingly. The present approach is based on the explicit construction of microstructures by recursive lamination and their subsequent equilibration in order to relax the incremental constitutive description of the material. The microstructures are permitted to evolve in complexity and fineness with increasing macroscopic deformation. The dislocation structures are deduced from the plastic deformation gradient field by recourse to Kroner's formula for the dislocation density tensor. The theory is rendered nonlocal by the consideration of the self-energy of the dislocations. Selected examples demonstrate the ability of the theory to generate complex microstructures, determine the softening effect which those microstructures have on the effective behavior of the crystal, and account for the dependence of the effective behavior on the size of the crystalline sample, or size effect. In this last regard, the theory predicts the effective behavior of the crystal to stiffen with decreasing sample size, in keeping with experiment. In contrast to strain-gradient theories of plasticity, the size effect occurs for nominally uniform macroscopic deformations. Also in contrast to strain-gradient theories, the dimensions of the microstructure depend sensitively on the loading geometry, the extent of macroscopic deformation and the size of the sample.

Mechanical properties and non-homogeneous deformation of open-cell nickel foams: application of the mechanics of cellular solids and of porous materials

Abstract: The mechanical properties of open-cell nickel foams are investigated for the range of densities used in industrial applications for energy storage. The obtained Young’s modulus, compression yield stress and tensile fracture stress are compared to the predictions of models based on periodic, Penrose and Voronoi beam networks. It is found that Gibson and Ashby’s model [L.J. Gibson, M.F. Ashby, Cellular Solids, Cambridge University Press, Cambridge, 1998] provides the proper scaling laws with respect to relative density for almost all investigated properties. The strong anisotropy of the observed overall responses can also be accounted for. The two-dimensional strain field during the tension of a nickel foam strip has been measured using a photomechanical technique. Non-homogeneous deformation patterns are shown to arise. The same technique is used to obtain the strain field around a circular hole in a nickel foam strip. The observed deformation fields are compared to the results of a finite element analysis using anisotropic compressible continuum plasticity.