Showing papers on "Constitutive equation published in 1999"

Mechanism-based strain gradient plasticity— I. Theory

Abstract: A mechanism-based theory of strain gradient plasticity (MSG) is proposed based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient. This theory is motivated by our recent analysis of indentation experiments which strongly suggest a linear dependence of the square of plastic flow stress on strain gradient. While such linear dependence is predicted by the Taylor hardening model relating the flow stress to dislocation density, existing theories of strain gradient plasticity have failed to explain such behavior. We believe that a mesoscale theory of plasticity should not only be based on stress–strain behavior obtained from macroscopic mechanical tests, but should also draw information from micromechanical, gradient-dominant tests such as micro-indentation or nano-indentation. According to this viewpoint, we explore an alternative formulation of strain gradient plasticity in which the Taylor model is adopted as a founding principle. We distinguish the microscale at which dislocation interaction is considered from the mesoscale at which the plasticity theory is formulated. On the microscale, we assume that higher order stresses do not exist, that the square of flow stress increases linearly with the density of geometrically necessary dislocations, strictly following the Taylor model, and that the plastic flow retains the associative structure of conventional plasticity. On the mesoscale, the constitutive equations are constructed by averaging microscale plasticity laws over a representative cell. An expression for the effective strain gradient is obtained by considering models of geometrically necessary dislocations associated with bending, torsion and 2-D axisymmetric void growth. The new theory differs from all existing phenomenological theories in its mechanism-based guiding principles, although the mathematical structure is quite similar to the theory proposed by Fleck and Hutchinson. A detailed analysis of the new theory is presented in Part II of this paper.

Viscous fluid flow

Abstract: * Presents the principles of control volume early for use throughout the book * Emphasizes the constitutive equation that relates deformation to stress. * Examines in detail the flow of Newtonian fluids-which can often be easily generalized to non-Newtonian fluids mechanics. * Includes many applications on topics like liquid droplets, gas bubbles, microelectronics processing, injection molding, painting and coatings.

Configurational Forces as Basic Concepts of Continuum Physics

Abstract: Configurational Forces within a Classical Context.- Kinematics.- Standard Forces. Working.- Migrating Control Volumes. Stationary and Time-Dependent Changes in Reference Configuration.- Configurational Forces.- Thermodynamics. Relation Between Bulk Tension and Energy. Eshelby Identity.- Inertia and Kinetic Energy. Alternative Versions of the Second Law.- Change in Reference Configuration.- Elastic and Thermoelastic Materials.- The Use of Configurational Forces to Characterize Coherent Phase Interfaces.- Interface Kinematics.- Interface Forces. Second Law.- Inertia. Basic Equations for the Interface.- An Equivalent Formulation of the Theory. Infinitesimal Deformations.- Formulation within a Classical Context.- Coherent Phase Interfaces.- Evolving Interfaces Neglecting Bulk Behavior.- Evolving Surfaces.- Configurational Force System. Working.- Second Law.- Constitutive Equations. Evolution Equation for the Interface.- Two-Dimensional Theory.- Coherent Phase Interfaces wtih Interfacial Energy and Deformation.- Theory Neglecting Standard Interfacial Stress.- General Theory with Standard and Configurational Stress within the Interface.- Two-Dimensional Theory with Standard and Configurational Stress within the Interface.- Solidification.- Solidification. The Stefan Condition as a Consequence of the Configurational Force Balance.- Solidification with Interfacial Energy and Entropy.- Fracture.- Cracked Bodies.- Motions.- Forces. Working.- The Second Law.- Basic Results for the Crack Tip.- Constitutive Theory for Growing Cracks.- Kinking and Curving of Cracks. Maximum Dissipation Criterion.- Fracture in Three Space Dimensions (Results).- Two-Dimensional Theory of Corners and Junctions Neglecting Inertia.- Preliminaries. Transport Theorems.- Thermomechanical Theory of Junctions and Corners.

Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies

Abstract: The stress–strain behaviour of granular materials can be modelled with hypoplastic constitutive relations. A hypoplastic model is briefly introduced for the axially symmetric case, and a procedure for the determination of its parameters is described in detail. It is shown, for several sands and one gravel, that all parameters of the hypoplastic model are closely related to the granulometric properties of grain assemblies. Recalculations of some element tests are presented in order to verify the proposed procedure. Copyright © 1999 John Wiley & Sons, Ltd.

A constitutive model for ferroelectric polycrystals

Abstract: A constitutive model is developed for the non-linear switching of ferroelectric polycrystals under a combination of mechanical stress and electric field. It is envisaged that the polycrystal consists of a set of bonded crystals and that each crystal comprises a set of distinct crystal variants. Within each crystal the switching event, which converts one crystal variant into another, gives rise to a progressive change in remanent strain and polarisation and to a change in the average linear electromechanical properties. It is further assumed that switching is resisted by the dissipative motion of domain walls. The constitutive model for the progressive switching of each crystal draws upon elastic–plastic crystal plasticity theory, and a prescription is given for the tangent moduli of the crystal, for any assumed set of potentially active transformation systems. A self-consistent analysis is used to estimate the macroscopic response of tetragonal crystals (representative of lead titanate) under a variety of loading paths. Also, the evolution of the switching surface in stress-electric field space is calculated. Many of the qualitative features of ferroelectric switching, such as butterfly hysteresis loops, are predicted by the analysis.

Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature

Abstract: A computational analysis of hygro-thermal and mechanical behaviour of concrete structures at high temperature is presented. The evaluation of thermal, hygral and mechanical performance of this material, including damage effects, needs the knowledge of the heat and mass transfer processes. These are simulated within the framework of a coupled model where non-linearities due to high temperatures are accounted for. The constitutive equations are discussed in some detail. The discretization of the governing equations is carried out by Finite Elements in space and Finite Differences in time. Copyright © 1999 John Wiley & Sons, Ltd.

Formulation of a unified constitutive model for clays and sands

Abstract: This paper presents a new generalized effective stress model, referred to as MIT-S1, which is capable of predicting the rate independent, effective stress–strain–strength behaviour of uncemented soils over a wide range of confining pressures and densities. Freshly deposited sand specimens compressed from different initial formation densities approach a unique condition at high stress levels, referred to as the limiting compression curve (LCC), which is linear in a double logarithmic void ratio, e, mean effective stress space, p′. The model describes irrecoverable, plastic strains which develop throughout first loading using a simple four-parameter elasto-plastic model. The shear stiffness and strength properties of sands in the LCC regime can be normalized by the effective confining pressure and hence can be unified qualitatively, with the well-known behaviour of clays that are normally consolidated from a slurry condition along the virgin consolidation line (VCL). At lower confining pressures, the model characterizes the effects of formation density and fabric on the shear behaviour of sands through a number of key features: (a) void ratio is treated as a separate state variable in the incrementally linearized elasto-plastic formulation: (b) kinematic hardening describing the evolution of anisotropic stress–strain properties: (c) an aperture hardening function controls dilation as a function of ‘formation density’; and (d) the use of a single lemniscate-shaped yield surface with non-associated flow. These features enable the model to describe characteristic transitions from dilative to contractive shear response of sands as the confining pressure increases. This paper summarizes the procedures used to select input parameters for clays and sands, while a companion paper compares model predictions with measured data to illustrate the model capability for describing the shear behaviour of clays and sands. Copyright © 1999 John Wiley & Sons, Ltd.

Severn-Trent sand: a kinematic hardening constitutive model for sands: the q-p formulation

Abstract: A simple model is described which is able to represent the mechanical behaviour of granular soils over a wide range of void ratios and mean stresses (neglecting grain crushing). The model combines Mohr–Coulomb failure, critical states, dependence of strength and stiffness on the state parameter, a hyperbolic law for plastic stiffness degradation and a flow rule similar to that of Cam Clay. The model is formulated within the framework of kinematic hardening and bounding-surface plasticity. The model requires two elastic and eight plastic parameters which are linked to clear physical features of the mechanical response. The formulation and numerical implementation are eased by making use of a 'normalized' stress space in which the stress– strain response shows no strain softening. The model is validated by comparison with experimental results obtained from triaxial tests on Hostun sand covering a wide range of density and stress level. Some proposals are made for possible future developments of the model wi...

Response of boron carbide subjected to large strains, high strain rates, and high pressures

Abstract: This article presents an analysis of the response of boron carbide (B4C) to severe loading conditions that produce large strains, high strain rates, and high pressures. Experimental data from the literature are used to determine and/or estimate constants for the JH-2 constitutive model for brittle materials. Because B4C is a very strong material, it is not always possible to determine the constants explicitly. Instead they must sometimes be inferred from the limited experimental data that are available. The process of determining constants provides insight into the constitutive behavior for some loading conditions, but it also raises questions regarding the response under other loading conditions. Several Lagrangian finite element and Eulerian finite difference computations are provided to illustrate responses for a variety of impact and penetration problems.

Constitutive equations of symmetrical triple layer piezoelectric benders

Abstract: Piezoelectric triple layer benders, with a structure of two piezoelectric top and bottom layers sandwiched by a non-piezoelectric elastic central layer, are one of the most commonly used piezoelectric devices. In this paper, we present the derivation of the constitutive equations of a symmetrical triple layer piezoelectric bender under different excitation conditions. The constitutive equations are presented by a 4/spl times/4 matrix with an external moment M, an external tip force F, a uniform load p, and an applied electric voltage V as the extensive parameters, with the generated tip angular deflection (slope) /spl alpha/, tip deflection /spl delta/, volume displacement v, and electric charge Q as the internal parameters. Further analysis on the electromechanical behavior of the triple layer piezoelectric bender can be made on the constitutive equations.

Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage

Abstract: Nonequilibrium thermodynamics, rate-process theory, viscoelastic fracture mechanics and various experimentally-motivated simplifications are used to develop constitutive equations that account for effects of viscoelasticity, viscoplasticity, growing damage and aging Their form is more general than previously developed by the author, and allows for relatively general tensorial effects of damage Some important special cases are then covered, with emphasis on viscoelasticity Evolution equations for the damage expressed in terms of internal state variables (ISVs) are discussed, comparing formulations using scalar ISVs and tensor ISVs Finally, some experimental support for the theory is described An Appendix illustrates the theory for an aging, linear viscoelastic material with growing cracks

A two scale damage concept applied to fatigue

Abstract: The ductile type of damage is a phenomenon now well understood. Once the fully coupled set of constitutive equations is identified, Damage Mechanics is a powerful tool to predict failure. Brittle materials do not exhibit such a damageable macroscopic behavior. Nevertheless, they still fail. On the idea that damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), we propose a three-dimensional failure modeling for monotonic as well as for fatigue loading. We develop a two scale model of what we shall call brittle damage: at the microscopic scale, micro-cracks or micro-voids exhibit a damageable plastic-like behavior with no effect on the global (mesoscopic) elastic behavior. Microscopic failure is assumed to coincide with the RVE failure. This model turns out to represent quite well physical phenomena related to high cycle fatigue such as the mean stress effect, the nonlinear accumulation of damage, initial strain hardening or damage effect and the nonproportional loading effect for bi-axial fatigue. The model has been implemented as a post-processor computer code. A simplified identification procedure for the determination of the material properties is given.

Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations

Abstract: A recent constitutive equation derived from molecular considerations on a model architecture containing two branch points a “pom-pom” captures the qualitative rheological behavior of low density polyethylene (LDPE) in shear and extension for the first time [, J. Rheol. 42, 82 (1998)]. We use a hypothetical melt of pom-poms with different numbers of arms to model the behavior of LDPE. The linear relaxation spectra for various LDPE samples are mapped to the backbone relaxation times of the pom-pom modes. Data from start-up flow in uniaxial extension fixes the nonlinear parameters of each mode giving predictions for shear and planar extension with no free parameters. This process was carried out for data in the literature and for our own measurements. We find that multimode versions of the pom-pom equation, with physically reasonable distributions of branching, are able to account quantitatively for LDPE rheology over four decades in the deformation rate in three different geometries of flows. The method suggests a concise and functional method of characterizing long chain branching in polymer melts.

A cyclic elasto-plastic constitutive model for sand considering a plastic-strain dependence of the shear modulus

Abstract: A cyclic elasto-plastic constitutive model based on a non-linear kinematic hardening rule for sand is proposed. Three points are incorporated into the model: a new flow rule, a cumulative strain-dependent characteristic of the plastic shear modulus and a fading memory characteristic of the initial anisotropy of the constitutive model. In order to verify its effectiveness, the proposed model was evaluated by means of the results of a series of hollow-cylinder torsional shear tests with and without an initial shear stress after isotropic and anisotropic consolidation. The liquefaction strength curve, the effective stress path and the stress—strain relation during cyclic loading are well reproduced by the proposed model. KEYWORDS: constitutive relations; liquefaction; plasticity; repeated loading; sands. Nous proposons ici un modele constitutif elastoplastique cyclique base sur une regle de durcissement cinematique non lineaire du sable. Trois points sont incorpores dans le modele: une nouvelle regle d'ecoul...

A kinematic hardening constitutive model for sands: the multiaxial formulation

Abstract: This paper explores the possibility of using well-accepted concepts—Mohr-Coulomb-like strength criterion, critical state, existence of a small strain elastic region, hyperbolic relationship for representing global plastic stress–strain behaviour, dependence of strength on state parameter and flow rules derived from the Cam-Clay Model—to represent the general multiaxial stress–strain behaviour of granular materials over the full range of void ratios and stress level (neglecting grain crushing). The result is a simple model based on bounding surface and kinematic hardening plasticity, which is based on a single set of constitutive parameters, namely two for the elastic behaviour plus eight for the plastic behaviour, which all have a clear and easily understandable physical meaning. In order to assist the convenience of the numerical implementation, the model is defined in a ‘normalized’ stress space in which the stress–strain behaviour does not undergo any strain softening and so certain potential numerical difficulties are avoided. In the first part the multiaxial formulation of the model is described in detail, using appropriate mixed invariants, which rationally combine stress history and stress. The model simulations are compared with some experimental results for tests on granular soils along stress paths lying outside the triaxial plane over a wide range of densities and mean stresses, using constitutive parameters calibrated using triaxial tests. Furthermore, the study is extended to the analysis of the effects induced by the different shapes of the yield and bounding surfaces, revealing the different role played by the size and the curvature of the bounding surface on the simulated behaviour of completely stress- and partly strain-driven tests. Copyright © 1999 John Wiley & Sons, Ltd.

Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

Abstract: A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.

Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method : DAVSS-G/DG

Abstract: Accurate and robust finite element methods for computing flows with differential constitutive equations require approximation methods that numerically preserve the ellipticity of the saddle point problem formed by the momentum and continuity equations and give numerically stable and accurate solutions to the hyperbolic constitutive equation. We present a new finite element formulation based on the synthesis of three ideas: the discrete adaptive splitting method for preserving the ellipticity of the momentum/continuity pair (the DAVSS formulation), independent interpolation of the components of the velocity gradient tensor (DAVSS-G), and application of the discontinuous Galerkin (DG) method for solving the constitutive equation. We call the method DAVSS-G/DG. The DAVSS-G/DG method is compared with several other methods for flow past a cylinder in a channel with the Oldroyd-B and Giesekus constitutive models. Results using the Streamline Upwind Petrov–Galerkin method (SUPG) show that introducing the adaptive splitting increases considerably the range of Deborah number (De) for convergence of the calculations over the well established EVSS-G formulation. When both formulations converge, the DAVSS-G and DEVSS-G methods give comparable results. Introducing the DG method for solution of the constitutive equation extends further the region of convergence without sacrificing accuracy. Calculations with the Oldroyd-B model are only limited by approximation of the almost singular gradients of the axial normal stress that develop near the rear stagnation point on the cylinder. These gradients are reduced in calculations with the Giesekus model. Calculations using the Giesekus model with the DAVSS-G/DG method can be continued to extremely large De and converge with mesh refinement.

On the thermomechanics of shape memory wires

Abstract: The thermomechanical behavior of a shape memory wire is modeled based on a theory that takes cognizance of the fact that the body can possess multiple natural configurations [1]. The constitutive equations are developed by first constructing the form of the Helmholtz potential (based on different modes of energy storage), and dissipation mechanisms. The internal energy includes contributions from the strain energy, the latent energy, the interfacial energy and thermal energy. The entropy of the system includes the"entropy jump" associated with the phase transition.¶The role of the rate of mechanical dissipation as a mechanism for entropy generation and its importance in describing the hysteretic behavior is brought out by considering the difference between hysteretic and non-hysteretic (dissipation-less) behavior.¶Finally, simple linear or quadratic forms are assumed for the various constitutive functions and the full shape memory response is modeled. A procedure for the determination of the constants is also indicated and the constants for two systems (CuZnAl and NiTi) are calculated from published experimental data (see [2, 3]). The predictions of the theory show remarkable agreement with the experimental data. However, some of the results predicted by the theory are different from the experimental results reported in Huo and Muller [2] We discuss some of the issues regarding this discrepancy and show that there appears to be some internal inconsistency between the experimental data reported in Figure 6 and Figure 9 of Huo and Muller [2] (provided they represent the same sample).

Determination of constitutive properties fromspherical indentation data using neural networks. Part i:the case of pure kinematic hardening in plasticity laws

Abstract: In this paper the power of neural networks in identifying material parameters fromdata obtained by spherical indentation is demonstrated for an academic problem (pure kinematichardening, given Youngs modulus) . To obtain a data basis for the training and validation of theneural network, numerous finite element simulations were carried out for various sets of materialparameters. The constitutive model describing finite deformation plasticity is formulated withnonlinear kinematic hardening of Armstrong–Frederick type. It was shown by Huber and Tsakmakis, 1998a that the depth–load response of a cyclic indentation process, consisting ofloading, unloading and reloading of the indenter displays a typical hysteresis loop for givenmaterial parameters. The inverse problem of leading the depth–load response back to the relatedparameters in the constitutive equations is solved using a neutral network.

A domain wall model for hysteresis in piezoelectric materials

Abstract: This paper addresses the modeling of hysteresis and nonlinear constitutive relations in piezoelectric materials at moderate to high drive levels. Hysteresis and nonlinearities are due to the domain structure inherent to the materials and both aspects must be addressed to attain the full potential of the materials as sensors and actuators in high performance applications. The model employed here is based on previously developed theory for hysteresis in general ferroelectric materials. This theory is based on the quantification of the reversible and irreversible motion of domain walls pinned at inclusions in the material. This yields an ODE model having five parameters. The relationship of the parameters to physical attributes of the materials is detailed and algorithms for determining estimates of the parameters using measured values of the coercive field, differential susceptibility and saturation properties of the materials are detailed. The accuracy of the model and its capability for the prediction of measured polarization at various drive levels is illustrated through a comparison with experimental data from PZT5A, PZT5H and PZT4 compounds. Finally, the ODE model formulation is amenable to inversion which facilitates the construction of an inverse compensator for linear control design.