Showing papers on "Constitutive equation published in 1993"

A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials

Abstract: Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

Thermoelasticity without energy dissipation

Abstract: This paper deals with thermoelastic material behavior without energy dissipation; it deals with both nonlinear and linear theories, although emphasis is placed on the latter. In particular, the linearized theory of thermoelasticity discussed possesses the following properties: (a) the heat flow, in contrast to that in classical thermoelasticity characterized by the Fourier law, does not involve energy dissipation; (b) a constitutive equation for an entropy flux vector is determined by the same potential function which also determines the stress; and (c) it permits the transmission of heat as thermal waves at finite speed. Also, a general uniqueness theorem is proved which is appropriate for linear thermoelasticity without energy dissipation.

One-Dimensional Constitutive Behavior of Shape Memory Alloys: Thermomechanical Derivation with Non-Constant Material Functions and Redefined Martensite Internal Variable

Abstract: A one-dimensional constitutive model for the thermomechanical behavior of shape memory alloys is developed based on previous work by Liang and Tanaka. An internal variable ap proach is used to deri...

One Dimensional Constitutive Behavior of Shape Memory Alloys.

Abstract: A one-dimensional constitutive model for the thermomechanical behavior of shape memory alloys is developed based on previous work by Liang and Tanaka. An internal variable approach is used to derive a comprehensive constitutive law for shape memory alloy materials from first principles without the assumption of constant material functions. This constitutive law is of such a form that it is well suited to further practical engineering applications and calculations. A separation of the martensite fraction internal variable into temperature-induced and stress-induced parts is presented and justified which then allows the derived constitutive law to accurately represent both the pseudoelastic and shape memory effects at all temperatures. Several numerical examples are given which illustrate the ability of the constitutive law to capture the unique thermomechanical behavior of shape memory alloys due to their internal phase transformations with stress and temperature.

Material Inhomogeneities in Elasticity

Abstract: Preface -- 1 Newton's concept of physical force -- 1.1. Newton's viewpoint -- 1.2. D' Alembert's viewpoint -- 1.3. Point particles and continua 7 -- 1.4. The modern point of view: duality -- 1.5. Lagrange versus Euler -- 2 Eshelby's concept of material force -- 2.1. Ideas from solid state physics -- 2.2. Peach-Koehler force -- 2.3. Force on a singularity -- 2.4. Energy-release rate -- 2.5. Pseudomomentum -- 2.6. Relationship with phonon and photon physics -- 3 Essentials of nonlinear elasticity theory -- 3.1. Material continuum in motion -- 3.2. Elastic me sures of strains -- 3.3. Compatibility of strains -- 3.4. Balance laws (Euler-Cauchy) -- 3.5. Balance laws (Piola-Kirchhoff) -- 3.6. Constitutive equations -- 3.7. Concluding remarks -- 4 Material balance laws and inhomogeneity -- 4.1. Fully material balance laws -- 4.2. Material inhomogeneity force and pseudomomentum -- 4.3. Interpretation of pseudomomentum -- 4.4. Four formulations of the balance of linear momentum -- 4.5. Other material balance laws -- 4.6. Comments -- 5 Elasticity as a field theory -- 5.1. Elements of field theory -- 5.2. Noether's theorem -- 5.3. Variational formulation (direct-motion description) -- 5.4. Variational formulation (inverse-motion description) -- 5.5. Other material balance laws -- 5.6. Canonical Hamiltonian formulation -- 5.7. Balance of total pseudomomentum -- 5.8. Nonsimple materals: second-gradient theory -- 5.9. Complementary-energy variational principle -- 5.10. Peach-Koehler force revisited -- 5.11. Concluding remarks -- 6 Geometrical aspects of elasticity theory -- 6.1. Material uniformity and inhomogeneity -- 6.2. Eshelby stress tensor -- 6.3. Covariant material balance law of momentum -- 6.4. Continuous distributions of dislocations -- 6.5. Variational formulation using two variations -- 6.6. Second-gradient theory -- 6.7. Continuous distributions of disclinations -- 6.8. Similarity to Einstein-Cartan gravitation theory -- 7 Material inhomogeneities and brittle fracture -- 7.1. The problem of fracture -- 7.2. Generalized Reynolds and Green-Gauss theorems -- 7.3. Global material force -- 7.4. J-integral in fracture -- 7.5. Dual I-integral in fracture -- 7.6. Variational inequality: fracture propagation criterion -- 7.7. Other material balance laws and related path-independent integrals -- 7.8. Remark on the dynamical case -- 8 Material forces in electromagnetoelasticity -- 8.1. Electromagnetic elastic solids -- 8.2. Reminder of electromagnetic equations -- 8.3. Material electromagnetic fields -- 8.4. Variational principles -- 8.5. Balance of pseudomomentum and material forces -- 8.6. Fracture in electroelasticity and magnetoelasticity -- 8.7. Geometrical aspects: material uniformity -- 8.8. Electric Peach-Koehler force -- 8.9. Example of application: piezoelectric ceramics -- 9 Pseudomomentum and quasi-particles -- 9.1. Pseudomomentum of photons and phonons -- 9.2. Electromagnetic pseudomomentum -- 9.3. Conservation laws in wave theory -- 9.4. Conservation laws in soliton theory -- 9.5. Sine-Gordon systems and topological solitons -- 9.6. Boussinesq crystal equation and pseudomomentum -- 9.7. Sine-Gordon-d'Alembert systems -- 9.8. Nonlinear Schrodinger and Zakharov systems -- 10 Material forces in anelastic materials -- 10.1. Internal variables and dissipation -- 10.2. Balance of pseudomomentum -- 10.3. Global material forces -- Bibliography and references -- Index .

Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys. I: Derivation of general relations

Abstract: A MICROMECHANICS constitutive model has been proposed in this paper to describe the pseudoelastic and shape memory behavior of polycrystalline shape memory alloys under various temperatures The derivation of the model is based on the thermodynamics, micromechanics and microstructural physical mechanism analysis of the material during deformation and it is shown that the inelastic deformation of the material in the mechanical and/or thermal loading processes is associated with some temperature, stress state and loading history dependent yielding surfaces which microscopically correspond to the forward and reverse transformation (or reorientation) processes, respectively

Calculation of viscoelastic flow using molecular models: the connffessit approach

Abstract: A new method for numerical calculation of viscoelastic flow based on simulation of molecular models of polymers is presented. The CONNFFES-SIT ( C alculation o f N on- N ewtonian F low: F inite E lements and S tochastic S imulation T echnique) approach directly combines standard finite element methods as currently used in the calculation of viscoelastic flow with stochastic simulations of polymer dynamics and thus obviates the need for a rheological constitutive equation to describe the fluid. The stresses are obtained from the molecular configurations rather than from constitutive equations. As an illustration of the method, the time development of plane Couette flow is studied for the upper-convected Maxwell, Oldroyd-B, FENE-P and FENE fluids. For the upper-convected Maxwell, Oldroyd-B and FENE-P models comparisons with analytical and existing numerical solutions are presented. Significant deviations between the behavior of the FENE-P and FENE models for the start-up of plane Couette flow are found.

Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys. II: Study of the individual phenomena

Abstract: T he constitutive relation for various phenomena of SMA ( superelasticity, rubber-like elasticity, ferroelasticity, elastic anomaly, shape memory effect ) is studied in detail and compared with the available experimental data. It is shown that the micromechanical model developed in Part I can satisfactorily describe the main peculiarities of the macroscopic thermomechanical constitutive behavior in the course of uniaxial mchanical and/or thermal loadings and that the existing phenomenological models are special cases of the proposed theory under proportional loading conditions. Some theoretical predictions and discussions for complex loading paths are also given which are yet subject to experimental verification.

Cyclic Viscoplastic Constitutive Equations, Part I: A Thermodynamically Consistent Formulation

Abstract: The purpose of the paper is to show how the classical models can be modified in order to follow the general principles of thermodynamics with internal variables. Using the restrictive framework of standard generalized materials, the state variables associated to various kinds of kinematic and isotropic hardening are selected. The evolution equations for these internal variables are then formulated in a slightly less restrictive form

Micromechanical modeling of large plastic deformation and texture evolution in semi-crystalline polymers

Abstract: A micromechanically-based composite model is proposed to study large plastic deformation and texture evolution in semi-crystalline polymers. The microstructure of many semi-crystalline polymers consists of co-existing crystalline and amorphous phases locally associated with each other in a fine plate-like morphological structure. An aggregate of two-phase composite inclusions is used to model these materials. Special consideration is given to molecular chain inextensibility within the crystalline phase. The introduction of a back stress tensor in the constitutive model of the amorphous phase accounts for hardening due to deformation-induced molecular alignment. Interface compatibility and traction equilibrium are enforced within each composite inclusion. A Sachs-like model and two newly-developed self-consistent-like hybrid models are proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. Applications of these composite models arc made to predict stress strain behavior and texture evolution in initially isolropic high density polyethylene (HOPE) under different modes of straining.

Rheology of concentrated dispersed systems in a low molecular weight matrix

Abstract: The objective of this paper is to provide a complete theoretical and experimental study of shear rheology for concentrated dispersed systems in low molecular weight matrices. A new theoretical approach based on qualitative microstructural physics and thermodynamical arguments is devel- oped which considers flow, elastic deformations, ruptures and restorations of flocs. The approach was developed to study steady and transient phenomena and has predicted very different behaviours for various systems; one of them was a non-monotonic flow curve. The most remarkable feature of the theoretical model is that it contains no mathematical yield criterion but predicts anisotropic yield values with continuous transition from solid state to flow as a bifurcation. A complete set of experiments in simple shearing was carried out for two different dispersed systems: a granular grease and a water-clay platelet mixture. Also some disturbing phenomena were observed and experimental techniques were developed which made it possible to obtain reliable results. Additionally the clay-water system exhibited a minimum in the flow curve, and all the characteristics peculiar to the evolution from solid state to flow, as predicted by the theory. Despite the short time scale of transition phenomena for the materials tested, the general comparison of experimental results with theory showed good agreement.

Numerical study on patterning of shear bands in a Cosserat continuum

Abstract: Numerical simulation of patterns of shear bands in biaxial compression tests using an elasto-plastic Cosserat constitutive equation is presented. Random distribution of the material properties acts as a trigger for the localized deformation. Two types of stress-strain curves, namely strain softening and strain softening followed by strain hardening, are investigated. It is shown that the characteristic of the stress-strain curve is crucial for the patterning of shear bands. While calculations with the stress-strain curve with solely softening yield only one single shear band, a flock of shear bands can be obtained with the stress-strain curve with softening followed by hardening. Benefited from the characteristic length provided by the Cosserat elasto-plastic constitutive equation, the dependence of the calculation on the mesh-size is avoided.

Dynamic Analysis of Generalized Viscoelastic Fluids

Abstract: A general boundary-element formulation is presented for the predic- tion of the dynamic response of fluids with viscoelastic behavior. The fluid is modeled by a generalized constitutive relation that contains either complex-valued parameters and complex-order derivatives or real-valued parameters and fractional- order derivatives, These models are consistent with basic theories and are not arbitrary constructions. The models are valid for linear viscoelastic fluid behavior and are limited to fluid motions with infinitesimally small displacement gradients. The governing equations are transformed into the Laplace domain and the infinite space fundamental solution is derived. The resulting integral equations are then solved by numerical procedures. The method is applied in the prediction of the dynamic mechanical properties of a viscous damper containing a viscoelastic fluid in the form of silicon gel. The fluid is modeled by a fractional derivative Maxwell model. The predicted mechanical properties of the device are found to be in ex- cellent agreement with experimental results.

A justification of nonlinear properly invariant plate theories

Abstract: A single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame-invariance properties of three-dimensional finite elasticity. By a successive scaling of the external loading on the three-dimensional body, the nonlinear membrane theory, the nonlinear inextensional theory and the von Karman equations are derived as the leading-order terms in the asymptotic expansion of finite elasticity. The governing equations of the nonlinear inextensional theory are of particular interest where 1) plane-strain kinematics and plane-stress constitutive equations are derived simultaneously from the asymptotic analysis, 2) the theory can be phrased as a minimization problem over the space of isometric deformations of a surface, and 3) the local equilibrium equations are identical to those arising in the one-director Cosserat shell model. Furthermore, it can be concluded that with a regular, single-scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading-order term of an asymptotic analysis.

On a class of constitutive equations in viscoplasticity : formulation and computational issues

Abstract: The viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton-Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso-error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.

Nonlinear Analysis of Reinforced-Concrete Shells

Abstract: Nonlinear finite element procedures are presented for the analysis of reinforced‐concrete shell structures. Cracked concrete is treated as an orthotropic material using a smeared rotating crack approach. The constitutive model adopted for concrete compression response accounts for reductions in strength and stiffness due to the presence of transverse cracks. The model used for concrete in tension represents the tension stiffening effects that significantly influence postcracking response. A heterosis‐type degenerate isoparametric quadrilateral element is developed using a layered‐element formulation, which rigorously considers out‐of‐plane shear response. Selective integration is used to avoid shear‐locking and zero‐energy problems. Good stability and convergence characteristics are provided by the iterative, full‐load secant stiffness solution procedure employed. Simple test elements are used to confirm the analytical procedure's ability to accurately model behavior under conditions of membrane load, fle...

Multiplication of Distributions: A Tool in Mathematics, Numerical Engineering and Theoretical Physics

Abstract: to generalized functions and distributions.- Multiplications of distributions in classical physics.- Elementary introduction.- Jump formulas for systems in nonconservative form. New numerical methods.- The case of several constitutive equations.- Linear wave propagation in a medium with piecewise C? characteristics.- The canonical Hamiltonian formalism of interacting quantum fields.- The abstract theory of generalized functions.

Determination of rock friction constitutive parameters using an iterative least squares inversion method

Abstract: In order to understand the behavior of slip in fault zones that can lead to earthquakes, a detailed description of the constitutive behavior of the slipping rocks is needed. Rate- and state-variable constitutive laws have been very successful in describing results of laboratory studies of rock friction, but the actual determination of constitutive law parameter values has been limited to forward trial-and-error methods only. This paper presents a method of inverting rock friction experimental data to determine the parameters in the Dieterich-Ruina friction constitutive model. We use an iterative least squares method to solve the inverse problem, and we describe the solutions to several difficulties that arose owing to the nonlinearity of both the model and the iterated solution. These solutions include (1) using a finite differences method to estimate the derivatives of the constitutive model, (2) incorporating a vector to weight the relative “importance” of the friction observations, (3) using singular-value decomposition to solve the inverse problem, and (4) creating a damped version of the inverse routine to enhance the ability of the program to converge on a final solution. We explore the effects of machine stiffness and added noise on the covariances and correlations between model parameters. We find that increasing stiffness reduces parameter variances, covariances, and the strong correlations between some model parameters; increasing noise increases parameter variances and covariances without affecting the correlation between model parameters.

Plasticity and Creep Theory Examples and Problems

Abstract: BASIC DEFINITIONS Stress and Strain State Stress tensor Strain tensor Finite Deformations Finite strain tensors in material or spatial coordinates Strain rates tensors Stress tensors in material or spatial descriptions FOUNDATIONS OF PLASTICITY Basic Equations of Perfect Plasticity Uniaxial stress-strain behavior Criteria for yielding in perfect plasticity Stress-strain relations for perfect plasticity Methods of reduction of equations of perfect plasticity Problems Basic Equations of Plastic Hardening Drucker's postulate and the associated flow rule Subsequent yield surfaces for hardening material Theories of plastic hardening Problems Methods of the Theory of Plasticity Analysis of the level of a cross-section Interaction curves on levels of a cross-section or a body Extremum theorems of limit analysis: statically or kinematically admissible solutions Shakedown analysis Integration along characteristics in plane strain problems Problems SOL UTIONS OF ELASTIC-PLASTIC PROBLEMS Elastic-Plastic Torsion and Bending Elastic-plastic torsion of prismatic bars Problems Elastic-plastic bending of prismatic beams and plane frames Problems Elastic-Plastic Analysis of Cylinders, Disks, and Plates Thick-walled tubes, spherical shells and disks Problems Limit analysis of Plates Problems FOUNDATIONS OF CREEP Basic Equations of Uniaxial Creep Models Creep phenomenon Schematizations of creep at constant uniaxial stress Modelling of creep at varying uniaxial stress Linear uniaxial viscoelastic models Modelling of viscoplastic materials Problems Creep Constitutive Equations Under Multiaxial Loading Classical multiaxial creep theories Developed multiaxial creep theories Linear multiaxial viscoelastic equations SOLUTION OF CREEP PROBLEMS Bending, Buckling, and Torsion of Bars Under Creep Conditions Bending and buckling of a prismatic bar made of the linear viscoelastic material Bending of a prismatic be am made of the piece-wise linear elastic/viscoplastic material Bending of a prismatic beam made of the time hardening material Torsion of a circular bar made of the elastic-Bingham material Problems Rotationally Symmetric Creep Problems Creep of a thick-walled tube General formulae for the rotationally-symmetric transient creep problems CREEP RUPTURE Constitutive Equations of Creep Rupture Creep rupture phenomenon Classical creep rupture theories Problems Rotationally Symmetric Creep Rupture Problems Mechanisms of brittle rupture of tubes and disks Design of disks with respect to creep rupture References Author Index Subject Index

A poroelastic-swelling finite element model with application to the intervertebral disc.

Abstract: The swelling process that occurs in soft tissue is incorporated into a poroelastic finite element model. The model is applied to a spinal segment consisting of two vertebrae and a single intervertebral disc. The theory is an extension of the poroelastic theory developed by Biot and the model is an adaptation of an axisymmetric poroelastic finite element model of the intervertebral disc by Simon. The model is completely three-dimensional although the results presented here assume symmetry about the sagittal plane. The theory is presented in two stages. First the development of the poroelastic theory. Following this, the effects of swelling caused by osmotic pressure are developed and expressed as a modification of the constitutive law and initial stresses. In the case of the disc, this pressure is produced mainly by the fixed negative charges on the proteoglycans within the disc. In this development we assume that the number of fixed charges remains constant over time and that the distribution of mobile ions has reached equilibrium. The variations over time in osmotic pressure, and thus in swelling effects are therefore only dependent on the initial state and the change in water content. Variations of the swelling effects caused by changes in mobile ion concentrations will be the subject of a future paper. The results reported in this article illustrate the dramatic effect of swelling on the load carrying mechanisms in the disc. The authors believe it is likely that this will have important useful implications for our understanding not only of normal disc function, but also of abnormal function, such as disc degeneration, herniation, and others.

The mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string

Abstract: This paper presents a study of the stress softening effect encountered in uniaxial extension and explores its effect on the small amplitude transverse vibration of a stretched rubber cord. An idealization of the uniaxial stressstrain behavior of a stress softening material is presented, the importance of the deformation history is emphasized, and parameters are introduced to track the deformation history. An extended investigation of a model proposed by Mullins and Tobin to quantify stress softening by introduction of a strain amplification factor is then presented. A major result derived from this model is shown to be consistent with results reported by others. The uniaxial stress softening theory is used to describe the transverse vibration behavior of a rubber string subjected to repeated stretching. This appears to be the first application of the softening model of Mullins and Tobin to a dynamical problem. Analytical results are compared with uniaxial stress-strain and transverse vibration experiments performed with buna-n, neoprene, and silicone rubber cords. Both types of experiments provide a simple and novel method to evaluate the predictive success of our uniaxial theory without the need for a specific constitutive model. The pseudoelastic response found in biological tissues is discussed in light of results obtained in the transverse vibration experiments.

A taylor-petrov-galerkin algorithm for viscoelastic flow

Abstract: A viscoelastic flow is solved using a generalised Taylor-Galerkin/pressure correction scheme that incorporates consistent Petrov-Galerkin streamline upwinding within the discretisation of the constitutive equations. The numerical approach is indirect, in the sense that fractional equation solution stages are introduced within the framework of a time-stepping scheme. The Oldroyd-B and Phan-Thien-Tanner constitutive models are considered and the proposed Taylor-streamline upwind/Petrov-Galerkin algorithm is used to simulate flow through a 4:1 planar contraction. For the Oldroyd-B model, the algorithm indicates the onset of instability as elasticity is increased and a limiting Weissenberg number is observed, with no lip vortices apparent for values less than five. For a particular class of Phan-Thien-Tanner models, it is found that the algorithm is stable at high Weissenberg numbers. In this case the solutions exhibit a lip-vortex mechanism for the establishment of recirculating regions as has been observed in some experiments. Results presented for various combinations of fluid parameters suggest that the extensional behaviour of the viscoelastic model is the single most important factor governing the stability and convergence of the algorithms for highly elastic fluids.