Showing papers on "Constitutive equation published in 1992"

On undamped heat waves in an elastic solid

Abstract: This paper is concerned with thermoelastic material behavior whose constitutive response functions possess thermal features that are more general than in the usual classical thermoelasticity. After a general development of the constitutive equations in the context of both nonlinear and linear theories, attention is focused on the latter. In particular, the one-dimensional version of the equation for the determination of temperature in the linearized theory provides an easy comparative basis of its predictive capability: In one special case where the Fourier conductivity is dominant, the temperature equation reduces to the classical Fourier law of heat conduction, which does not permit the possibility of undamped thermal waves; however,'in another special case in which the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation.

Crystallographic texture evolution in bulk deformation processing of FCC metals

Abstract: A Taylor-type polycrystalline model, together with a new fully-implicit time-integration scheme has been developed and implemented in a finite element program to simulate the evolution of crystallographic texture during bulk deformation processing of face centered cubic metals deforming by crystallographic slip. The constitutive equations include a new equation for the evolution of slip system deformation resistance which leads to macroscopic strain hardening behavior that is in good accord with experiments performed on OFHC copper. The good predictive capabilities of the constitutive equations and the time-integration procedure for simulating the stress-strain behavior and the evolution of texture under both homogeneous and non-homogeneous deformation conditions are demonstrated by comparing numerical simulations against experimental measurements in simple shear and a simple plane-strain forging experiment on copper.

A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration

Abstract: A constitutive equation for computing particle concentration and velocity fields in concentrated monomodal suspensions is proposed that consists of two parts: a Newtonian constitutive equation in which the viscosity depends on the local particle volume fraction and a diffusion equation that accounts for shear‐induced particle migration. Particle flux expressions used to obtain the diffusion equation are derived by simple scaling arguments. Predictions are made for the particle volume fraction and velocity fields for steady Couette and Poiseuille flow, and for transient start‐up of steady shear flow in a Couette apparatus. Particle concentrations for a monomodal suspension of polymethyl methacrylate spheres in a Newtonian solvent are measured by nuclear magnetic resonance (NMR) imaging in the Couette geometry for two particle sizes and volume fractions. The predictions agree remarkably well with the measurements for both transient and steady‐state experiments as well as for different particle sizes.

Transformation field analysis of inelastic composite materials

Abstract: A new method is proposed for evaluation of local fields and overall properties of composite materials subjected to incremental thermomechanical loads and to transformation strains in the phases. The composite aggregate may consist of many perfectly bonded inelastic phases of arbitrary geometry and elastic material symmetry. In principle, any inviscid or time-dependent inelastic constitutive relation that complies with the additive decomposition of total strains can be admitted in the analysis. The governing system of equations is derived from the representation of local stress and strain fields by novel transformation influence functions and concentration factor tensors, as discussed in the preceding paper by G. J. Dvorak and Y. Benveniste. The concentration factors depend on local and overall thermoelastic moduli, and can be evaluated with a selected micromechanical model. Applications to elastic-plastic, viscoelastic, and viscoplastic systems are discussed. The new approach is contrasted with some presently accepted procedures based on the self-consistent and Mori-Tanaka approximations, which are shown to violate exact relations between local and overall inelastic strains.

A constitutive model for transformation plasticity accompanying strain-induced martensitic transformations in metastable austenitic steels

Abstract: We propose a constitutive model which describes the transformation plasticity accompanying strain-induced martensitic transformation in nonthermoelastic alloys. The model consists of two parts: a transformation kinetics law describing the evolution of the volume fraction of martensite and a constitutive law defining the flow strength of the evolving two-phase composite. The Olson-Cohen model for martensite volume fraction evolution is recast in a generalized rate form so that the extent of martensite nucleation is not only a function of plastic strain and temperature, but also of the stress state. A selfconsistent method is then used for predicting the resultant stress-strain behavior. The model describes both the hardening influence of the transformation product, and the softening influence of the transformation itself, as represented by a spontaneous transformation strain. The model is then implemented in a finite element program suitable for analysis of boundary value problems. Model predictions are compared with existing experimental data for austenitic steels. We present results from a few simple analyses, including tensile necking, illustrating the critical importance of stress state sensitivity in the evolution model.

Multiphase Poroelastic Finite Element Models for Soft Tissue Structures

Abstract: During the last two decades, biological structures with soft tissue components have been modeled using poroelastic or mixture-based constitutive laws, i.e., the material is viewed as a deformable (porous) solid matrix that is saturated by mobile tissue fluid. These structures exhibit a highly nonlinear, history-dependent material behavior; undergo finite strains; and may swell or shrink when tissue ionic concentrations are altered. Give the geometric and material complexity of soft tissue structures and that they are subjected to complicated initial and boundary conditions, finite element models (FEMs) have been very useful for quantitative structural analyses. This paper surveys recent applications of poroelastic and mixture-based theories and the associated FEMs for the study of the biomechanics of soft tissues, and indicates future directions for research in this area. Equivalent finite-strain poroelastic and mixture continuum biomechanical models are presented. Special attention is given to the identification of material properties using a porohyperelastic constitutive law ans a total Lagrangian view for the formulation. The associated FEMs are then formulated to include this porohyperelastic material response and finite strains. Extensions of the theory are suggested in order to include inherent viscoelasticity, transport phenomena, and swelling in soft tissue structures. A number of biomechanical research areasmore » are identified, and possible applications of the porohyperelastic and mixture-based FEMs are suggested. 62 refs., 11 figs., 3 tabs.« less

A model for finite strain elasto-plasticity based on logarithmic strains: computational issues

Abstract: In the context of general isothermal processes, issues related to the constitutive modelling and computational treatment of finite deformation elasto-plasticity are examined employing logarithmic stretches as strain measures. A strain-energy function for isotropic elastic materials is proposed, which leads to a linear stress-strain relation and constant and isotropic elastic modulus in material setting. It is assumed that isotropy is maintained in the intermediate configuration which necessitates a representation of the plastic flow based on the scalar internal variables. By exploiting the main features of the present approach, expressed through the simple hyperelastic constitutive model in conjunction with notions of multiplicative decomposition of the deformation gradient and unstressed configuration, a computationally effective framework is formulated. It is pointed out that in this context, an algorithm could be proposed for rate-independent finite strain elasto-plasticity, which is exact for elastic processes and in the limit of non-hardening, deviatoric elasto-plasticity is in accordance with physical observations. Large elasto-plastic deformations at moderate elastic strains are examined within the approximation theory and displacement based finite element formulation of the boundary value problem proposed. Numerical analysis is performed for a realistic example capturing shear band localisation and the results are compared with experimental data.

Non-linear constitutive relations for piezoceramic materials

Abstract: Piezoelectric materials produce electric charges when mechanically deformed and an electric potential causes a mechanical deformation. This property makes them suitable for sensor and transducer applications. Understanding of the electroelastic constitutive behavior is critical for predicting the response of a structure with embedded piezoelectric material. A concise formulation of relevant non-linear constitutive relations is presented in this paper.

A multi-dimensional constitutive model for shape memory alloys

Abstract: This paper presents a multi-dimensional thermomechanical constitutive model for shape memory alloys (SMAs). This constitutive relation is based upon a combination of both micromechanics and macromechanics. The martensite fraction is introduced as a variable in this model to reflect the martensitic transformation that determines the unique characteristics of shape memory alloys. This constitutive relation can be used to study the complex behavior associated with 2-D and 3-D SMA structures. A simple example using this constitutive model is also presented, which reveals a new and interesting phenomenon of 3-D SMA structures.

Improved accuracy in finite element analysis of Biot's consolidation problem

Abstract: Numerical analysis and error estimates of finite element approximations of Biot's consolidation problem are presented. Initially different orders of interpolation are employed, leading to lower orders of convergence for the pore pressure compared to the displacements of the porous medium. Lower accuracy also occurs in the approximation of the effective stress tensor, whether it is calculated directly from the constitutive equation, or through a primal mixed stress formulation. To improve the rates of convergence of the pore pressure and effective stresses, a sequential Galerkin Petrov-Galerkin post-processing technique is proposed.

The constitutive law of nonlinear viscous and porous materials

Abstract: This Paper examines with the help of micromechanics the overall behaviour of nonlinear viscous materials containing voids. In complement to variational bounds derived here and to several models proposed in the recent literature we derive a simple model which meets exactly a closed-form solution of a hollow sphere under hydrostatic tension. More generally we consider the problem of a hollow sphere under hydrostatic tension when the constitutive material of the sphere is already porous. The solution is used in a self-consistent scheme and a differential scheme to derive a constitutive law for a porous material containing different populations of micro-voids with distributed sizes. These schemes predict a higher damage effect for the same porosity than the models based on a single size of voids. All the models considered in this paper make use of a strain-rate potential and most of them assume a simple “quadratic” form for it.

Simple constitutive equations for steel at high temperature

Abstract: This work develops and investigates simple unified constitutive equations to model the mechanical behavior of plain carbon steel in the austenite temperature region for use in finite element stress analysis of processes such as continuous casting. Four different forms of constitutive relations are considered: constant structure, time-hardening, strain-hardening, and simultaneous time- and strain-hardening models. Each relation is judged on its ability to reproduce experimental data from both tensile and creep tests and its ability to exhibit reasonable behavior under complex loading conditions. Three of the equations appear suitable for small strain monotonic loading conditions for a wide range of low strain rates (10−3 to 10−6 s−1), high temperatures (950 °C to 1400 °C), and varying carbon contents (0.005 to 1.54 wt pct C).

Flow-deformation response of dual-porosity media

Abstract: A constitutive model for the coupled flow-deformation response of dual-porosity media is presented in this paper. The formulation is stated in component terms, where behavior is controlled by mechanical and hydraulic response of the individual porous and fractured phases. Representation in this form allows the potential importance of dual-porosity effects to be evaluated in controlling the coupled flow-deformation behavior of fractured geologic media.

Analyses of Plastic Flow Localization in Metals

Abstract: The continuum mechanics framework for analyzing plastic flow localization is reviewed. The prediction of the localization of deformation into shear bands is sensitive to the constitutive description. The classical isotropic hardening elastic-plastic solid with a smooth yield surface and normality is very resistant to localization, but deviations from these idealizations have a strong effect. Thus, a material that forms a sharp vertex on the yield surface, as predicted by crystal plasticity, shows flow localization at quite realistic levels of strain, and even the formation of a rounded vertex on the yield surface has an important influence. Also softening induced by material damage or by the heating due to plastic dissipation have significant influence in promoting the onset of flow localization. In a practical situation one effect, such as thermal softening under high deformation rates, may be the dominant cause of localization, but often the interaction of different effects appears to be the more realistic explanation of observed flow localization. Some relevant constitutive models are reviewed and the effect of the different material models on localization predictions is illustrated. Important information on localization behavior in uniformly strained solids is obtained by a relatively simple material stability analysis, but often failure bymore » flow localization occurs in nonuniformly strained regions, where numerical solution procedures are necessary to obtain theoretical predictions. The numerical results reviewed cover localization under dynamic as well as quasi-static loading conditions. 81 refs., 16 figs.« less

Finite-difference time-domain analysis of gyrotropic media. I. Magnetized plasma

Abstract: When subjected to a constant magnetic field, both plasmas and ferrites exhibit anisotropic constitutive parameters. For electronic plasmas this anisotropy must be described by using a permittivity tensor in place of the usual scalar permittivity. Each member of this tensor is also very frequency dependent. A finite-difference time-domain formulation which incorporates both anisotropy and frequency dispersion, enabling the wideband transient analysis of magnetoactive plasma, is described. Results are shown for the reflection and transmission through a magnetized plasma layer, with the direction of propagation parallel to the direction of the biasing field. A comparison to frequency-domain analytic results is included. >

Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain

Abstract: The main goal of this paper is to establish general constitutive relations for the electromagnetic fields EBAR, DBAR, BBAR, and HBAR in a time domain setting. The four basic assumptions of the medium are linearity, invariance to time translations, causality, and continuity. These four assumptions imply that the constitutive relations are convolutions of the Riemann-Stieltjes type. A review of the classification of media in bianisotropic, biisotropic, anisotropic, and isotropic media, respectively, is made. Dissipation and reciprocity are defined and the constraints these concepts make on the constitutive relations are analyzed. Furthermore, an appropriate form of time reversal and functions of positive type are introduced and some consequences of these concepts are showed. (Less)

Generalized Cole-Cole behavior and its rheological relevance

Abstract: Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.

A Visco-Plastic Constitutive Model for 60/40 Tin-Lead Solder Used in IC Package Joints

Abstract: A new unified visco-plastic constitutive model for the 60 Sn-40 Pb alloy used in solder joints of surface-mount IC packages and semiconductor devices is proposed. The model accounts for the measured stress-dependence of the activation energy and for the strong Bauschinger effect exhibited by the solder. The latter is represented by a back stress state variable which, in turn, evolves according to a hardening-recovery equation. Based on the observed hardening behavior, it is assumed that the isotropic resistance to plastic flow does not evolve within the deformation range covered in this study (e< 3 percent). The deformation phenomena associated with the solder’s monotonic and steady-state cyclic responses are accurately predicted for −55°C≦T≦150°C and 8 x 10−2 s−1 ≦ e ≦ 8 x 10−5 s−1 . The model also predicts well the overall trend of steady-state creep behavior. The constitutive model is formulated within a continuum mechanics framework and is therefore well suited for implementation into finite element or other structural codes.

Constitutive models for the sintering of ceramic components—I. Material models

Abstract: A number of possible two state variable material models for the sintering of fine grained ceramic compacts are described. The two state variables employed in the models relate to the relative density of the material and the mean grain size. Constitutive relationships for the strain-rate of a material element and evolution laws for the two state variables are presented which reflect the different stages of sintering. The predictions of the model are in good agreement with experimental studies involving pressureless sintering and HIPing of green compacts.

Low-Velocity Impact Response of Foam-Core Sandwich Composites

Abstract: The low-velocity impact response of foam-core sandwich composites with fiberglass/epoxy face sheets is treated by a combination of computational and experimental methods. Linear elastic constitutive models are used for the face sheets and epoxy bond layer in conjunction with a foam constitutive model that includes nonlinear hardening plasticity and coupling between volumetric and deviatoric deformation. A transient finite- element code, utilizing four-noded uniform strain quadrilaterals, is used to explicitly solve the equations for balance of mass and momentum. The resulting deformation histories are compared to the experimental results and show qualitative agreement. The computed transverse shear stresses are used to correlate ultrasonic measurement of damage in the core/epoxy interface. Comparison of the plate stiffness prior to and after impact illustrates the effect of damage on subsequent behavior.

An Anatomical Heart Model with Applications to Myocardial Activation and Ventricular Mechanics

Abstract: 4A three-dimensional finite element model of the mechanical and electrical behavior of the heart is being developed in a collaboration among Auckland University, New Zealand; the University of California at San Diego, U.S.; and McGill University, Canada. The equations of continuum mechanics from the theory of finite deformation elasticity are formulated in a prolate spheroidal coordinate system and solved using a combination of Galerkin and collocation techniques. The finite element basis functions used for the dependent and independent variables range from linear Lagrange to cubic Hermite, depending on the degree of spatial variation and continuity required for each variable. Orthotropic constitutive equations derived from biaxial testing of myocardial sheets are defined with respect to the microstructural axes of the tissue at the Gaussian quadrature points of the model. In particular, we define the muscle fiber orientation and the newly identified myocardial sheet axis orientation throughout the myocardium using finite element fields with nodal parameters fitted by least-squares to comprehensive measurements of these variables. Electrical activation of the model is achieved by solving the FitzHugh–Nagumo equations with collocation at fixed material points of the anatomical finite element model. Electrical propagation relies on an orthotropic conductivity tensor defined with respect to the local material axes. The mechanical constitutive laws for the Galerkin continuum mechanics model are (1) an orthotropic “pole–zero” law for the passive mechanical properties of myocardium and (2) a Wiener cascade model of the active mechanical properties of the muscle fibers. This chapter concentrates on two aspects of the model: first, grid generation, including both the generation of nodal coordinates for the finite element mesh and the generation of orthotropic material axes at each computational point, and, second, the formulation of constitutive laws suitable for numerically intensive finite element computations. Extensions to this model and applications to the mechanical and electrical function of the heart are described in Chapter 2 by McCulloch and co-workers.

Constitutive modeling and simulation of energy absorbing polyurethane foam under impact loading

Abstract: The compressive-stress strain response of polyurethane foam under uniaxial compressive impact loading has been studied. The development of a uniaxial constitutive model from strain rate controlled compression tests is detailed. Density and temperature functions have been added to the integral power model proposed by Schwaber, Meincke, and Nagy. The model assumes that the effects of density, temperature, strain and strain rate on stress are separable functions. The model correlated well with actual static compression tests and was used successfully to predict the impact response of energy absorbing polyurethane foam under uniaxial compressive loading.

Calculation of steady-state viscoelastic flow through axisymmetric contractions with the EEME formulation

Abstract: The EEME/finite-element method is used to compute steady flows of fluids with constitutive behavior described by the Modified Upper Convected Maxwell Model of Apelian and a modified form of the dumbbell model of Chilcott and Rallison through abrupt, axisymmetric contractions. Both constitutive models predict constant viscosity and shear thinning first normal stress coefficient, but differ qualitatively in the behavior of the elongational viscosity. Asymptotic analysis for both models predicts that the solution has Newtonian-like spatial structure near the reentrant corner and integrable stresses and velocity gradients there. With the Newtonian-like asymptotics, the stress field can be approximated by conventional Lagrangian finite elements and computed by the streamline upwind Petrov Galerkin (SUPG) method. The finite element calculations are stable and convergent: higher values of Deborah number are reached with increasing mesh refinement. Moreover, the predicted asymptotic structure of the stress and velocity fields is recovered near the corner in the calculations. Calculations with both constitutive equations show the stretching of the Newtonian corner vortex toward the reentrant corner and its growth upstream with increasing Deborah number for 4:1 and 8:1 contraction ratios. The characteristics of the computed vortex are in semi-quantitative agreement with experiments for Boger fluids for which the flow is axisymmetric and steady.

Constitutive Modeling and Simulation of Energy Absorbing Polyurethane Foam Under Impact Loading

Abstract: The compressive-stress strain response of polyurethane foam under uniaxial compressive impact loading has been studied. The development of a uniaxial constitutive model from strain rate controlled compression tests is detailed. Density and temperature functions have been added to the integral power model proposed by Schwaber, Meincke and Nagy. The model assumes that the effects of density, temperature, strain and strain rate on stress are separable functions. The model correlated well with actual static compression tests and was used successfully to predict the impact response of EA polyurethane foam under uniaxial compressive loading.

Analysis of simple constitutive equations for viscoelastic liquids

Abstract: To construct viscoelastic constitutive equations properly one needs to know, apart from their predictability, how robust they are and what possible underwater reefs one may meet that could ruin a weak rheological ship in the course of its difficult route through the ocean of solving complicated problems.