Showing papers on "Constitutive equation published in 1996"

A New Constitutive Relation for Rubber

Abstract: A simple, two-constant, constitutive relation, applicable over the entire range of strains, is proposed for rubber networks. Behavior in simple extension is derived as an example.

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Abstract: A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

A Comprehensive constitutive equation for granular materials

Abstract: A constitutive equation is proposed for describing changes of states of granular materials which are sufficiently characterized by the void ratio and the stress tensor. It may be considered as an extension of the Critical State concept. It is based on recently published hypoplastic equations and covers a wide range of densities, pressures and deformations. A factorial decomposition allows a rather easy separation and determination of material parameters. Two factors depend on a relative void ratio so that it remains within lower and upper bounds. The bounding void ratios decrease monotonously from maximal values to zero with increasing pressure. The same reduction of the void ratio is proposed for an isotropic compression starting from a suspension. Thus a granulate hardness is defined, and a stiffness factor can be determined. Four material parameters can be estimated from classification tests and determined from the asymptotic behaviour in element tests. Four further parameters are determined by calibration; they are rather constant for wide groups of materials. Strength and stiffness values can be derived and used for the analysis of deformations, stability, and flow. The viscous behaviour is modelled by a rate dependent factor with one further parameter. Limitations and possible extensions of this comprehensive approach are also outlined.

Two-phase binary fluids and immiscible fluids described by an order parameter

Abstract: A unified framework for coupled Navier-Stokes/Cahn-Hilliard equations is developed using, as a basis, a balance law for microforces in conjunction with constitutive equations consistent with a mechanical version of the second law. As a numerical application of the theory, we consider the kinetics of coarsening for a binary fluid in two space dimensions.

Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals

Abstract: The paper addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes. The strong discontinuity analysis is introduced as a basic tool to derive a general framework, in which different families of constitutive equations can be considered, that allows to extract some outstanding aspects of the intended analysis. In particular, a link between continuum and discrete approaches to the strain localization phenomena is obtained. Applications to standard continuum damage and elastoplastic constitutive equations are presented. Relevant aspects to be considered in the numerical simulation of the problem (tackled in Part 2 of the work) are also presented.

Unified constitutive laws of plastic deformation

Abstract: J.L. Chaboche, Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities, and Thermodynamic Framework. Y. Estrin, Dislocation-Density Related Constitutive Modeling. R.W. Evans and B. Wilshire, Constitutive Laws for High-Temperature Creep and Creep Fracture. G.A. Henshall, D.E. Helling, and A.K. Miller, Improvements in the MATMOD Equations for Modeling Solute Effects and Yield-Surface Distortions. A.S. Krausz and K. Krausz,The Constitutive Law of Deformation Kinetics. E. Krempl, A Small-Strain Viscoplasticity Theory Based on Overstress. J. Ning and E.C. Aifantis, Anisotropic and Inhomogenous Plastic Deformation of Polycrystalline Solids. S.V. Raj, I.S. Iskowitz, and A.D. Freed, Modeling the Role of Dislocation Substructure During Class M and Exponential Creep. K. Krausz and A.S. Krausz, Comments and Summary. Subject Index.

Dynamics of entanglements: a nonlinear model consistent with the cox-merz rule

Abstract: It is argued that the flow directly contributes a mechanism by which the topological obstacles are renewed. As a consequence, the chain diffusivity is increased (or, equivalently, the friction coefficient is decreased) in fast flows. Such a nonlinear effect is incorporated in a simple dumbbell model so as to produce a constitutive equation in closed form. An extension to the case of multiple relaxation times is also briefly considered. The predictions compare favourably with the Cox-Merz rule as well as with the behaviour of polymer melts in elongational flows.

Simplifications and Comparisons of Shape Memory Alloy Constitutive Models

Abstract: Here we will present a simplification of the form of one popular shape memory alloy (SMA) constitutive model. This simplification allows a more compact written form and hence easier calculations with the one-dimensional SMA constitutive law first developed by Tanaka, later modified by Liang and Rogers, and again by Brinson. In addition, a new derivation of the model will be given based on micromechanics. In this context, comparisons between the Tanaka and two other models will be presented and implications discussed. It will be shown that the constitutive models are in fact quite similar, and that the important distinction between these SMA models is primarily in the formulation of the transformation kinetics. Several examples will be presented utilizing the models.

A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation

Abstract: The stress-strain behavior of carbon black filled rubber is recognized to be nonlinearly elastic in its main part (see e.g. Gent [1]). In addition, inelastic effects occur under monotonic and cyclic processes. The inelastic behavior includes nonlinear rate dependence as well as equilibrium hysteresis. Moreover, the first periods of a stress-strain curve differ significantly from the shape of subsequent cycles; a characteristic feature, which is called the Mullins effect, because it has been pointed out by Mullins [2]. All inelastic phenomena are strongly influenced by the volume fraction of the filler particles (see e.g. Payne [3], So and Chen [4], Meinecke and Taftaf [5]). The aim of the present paper is to design a constitutive model, representing this kind of material behavior as a phenomenological theory of continuum mechanics. In order to motivate the basic structure of the constitutive theory, a series of uniaxial experiments between 100% in tension and 30% in compression are presented and analyzed. First of all, monotonic strain controlled experiments show the nonlinear rate dependence of the stress response. Then, a series of inserted relaxation periods at constant strain yields the monotonic equilibrium stress-strain curve, which is strongly nonlinear and unsymmetric with respect to the origin. Finally, cyclic experiments under strain control display pronounced hysteresis behavior. The hysteresis effects are mainly rate dependent, but there exists also a weak equilibrium hysteresis (compare to similar observations of Orschall and Peeken [6]). The Mullins effect corresponds to a softening phenomenon during the first few cycles. By means of an appropriate preprocess, this effect was excluded during the above experiments. Apart from the Mullins effect, neither hardening nor significant softening phenomena were observed in the context of cyclic loadings. These results motivate the structure of a constitutive model of finite strain viscoplasticity: The total stress is decomposed into an equilibrium stress and an overstress, where the overstress is a rate dependent functional of the strain history. The overstress represents the rate dependence of the material behavior and tends asymptotically to zero during relaxation processes. The nonlinearity of the rate dependence is incorporated by means of a stress dependent relaxation time. The equilibrium stress is assumed to be a rate independent functional of the strain history. For this quantity, we make use of an arclength representation, which was originally introduced by Valanis [7]. In case of vanishing equilibrium hysteresis and vanishing rate dependence our constitutive model reduces to finite strain hyperelasticity, which is the first approximation of the constitutive properties. In more general cases the “main shape” of a stress-strain curve is determined by hyperelasticity, superimposed by rate dependent and equilibrium hysteresis. The representation of the Mullins effect is incorporated by a continuum damage model. Some numerical simulations at the end of the paper demonstrate that the presented theory is able to represent the observed phenomena qualitatively and quantitatively with sufficient approximation.

Modelling Strong Discontinuities in Solid Mechanics via Strain Softening Constitutive Equations. Part 2: Numerical Simulation

Abstract: On the basis of the strong discontinuity analysis of standard local stress–strain constitutive equations, a finite element framework for the simulation of strong discontinuities, which belongs to the family of assumed enhanced strain methods, is presented. Taking the standard linear triangle as the underlying element, an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities. The presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed.

A unified thermodynamic constitutive model for sma and finite element analysis of active metal matrix composites

Abstract: SUMMARY A unified thermodynamic constitutive model for Shape Memory Alloy (SMA) materials, based on the thermodynamic framework proposed by Boyd and Lagoudas,1is presented in this paper. The specific selections for the form of Gibbs free energy associated with the martensitic volume fraction are identified for several earlier constitutive models for SMA. The thermal energy released or absorbed during the forward or reverse transformation predicted by the different models is compared with the experimental data from calorimetric measurements. The unified constitutive model is then implemented in a finite element analysis scheme using a return mapping integration technique for the incremental formulation of the model. Finally, the constitutive model is utilized to analyse the thermomechanical response of an active metal matrix composite with embedded SMA fibres. Both tetragonal and hexagonal periodic arrangements of SMA fibres are considered in the calculation and the results are compared.

On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures

Abstract: This article deals with the continuum formulation of finite strain viscoelasticity and provides its numerical simulation with the finite element method. In particular, elastomeric solids which are of essential engineering interest are discussed. In order to simulate the significant different bulk/shear-response of polymeric media the deformation is decomposed into volumetric elastic and isochoric viscoelastic parts. The constitutive equations are presented within the context of internal variable models and a Lagrangian kinematical description is adopted throughout. For sufficiently slow processes the material responds in a rubbery elastic manner which is assumed to be modelled with an Ogden-type strain energy function well-known from rubber elasticity. The stresses and the symmetric consistent tangent moduli are briefly discussed along with a second-order approximation of the constitutive rate equation. The main thrust of this paper on the computational side is to show the meaningful time-dependent behaviour and the general applicability of the three-dimensional constitutive model. By applying assumed enhanced strain elements which are well-suited for (nearly) incompressible problems three representative numerical examples illustrate relaxation and creeping phenomena at large strains and the equilibrium finite elastic response, which is asymptotically obtained.

Cellular solid structures with unbounded thermal expansion

Abstract: Thermal expansion of materials is of considerable practical interest since materials in service may experience a considerable range of operating temperature. Materials with microstructure, including composites, honeycombs, and foams (which are composites with one void phase) are finding increasing application for many purposes. It is therefore of interest to explore the range of thermal expansion coefficient attainable in materials with microstructure. The constitutive equation for a linear isotropic thermoelastic continuum is [1]:

Constitutive laws for biomechanical modeling of refractive surgery.

Abstract: Membrane inflation tests were performed on fresh, intact human corneas using a fiber optic displacement probe to measure the apical displacements. Finite element models of each test were used to identify the material properties for four different constitutive laws commonly used to model corneal refractive surgery. Finite element models of radial keratotomy using the different best-fit constitutive laws were then compared. The results suggest that the nonlinearity in the response of the cornea is material rather than geometric, and that material nonlinearity is important for modeling refractive surgery. It was also found that linear transverse isotropy is incapable of representing the anisotropy that has been experimentally measured by others, and that a hyperelastic law is not suitable for modeling the stiffening response of the cornea.

Crack tip fields in strain gradient plasticity

Abstract: Solutions are presented for mode I and mode II crack tip fields for plane strain deformations of an elastic-plastic solid whose constitutive behavior depends on both strains and strain gradients. The constitutive law is the simplest generalization of the J2 deformation theory of plasticity to include strain gradient effects. Only one new constitutive parameter enters, a length parameter characterizing the scale over which gradient effects become important. The formulation is cast within the framework of coupled stress theory. Crack tip solutions are obtained which display the transition from the HRR fields, governing behavior in an intermediate region with the plastic zone, to the dominant fields at the tip. The dominant fields are obtained in closed form, and finite element methods have been used to produce the solution over the entire field. Some of the difficulties encountered in arriving at an accurate numerical scheme are detailed. Implications of the solutions for fracture are discussed, as are avenues for further research.

Fixed angle softened truss model for reinforced concrete

Abstract: A softened truss model has previously been developed for reinforced and prestressed concrete membrane elements subjected to in-plane shear and normal stresses. This existing model satisfies the three principles of mechanics of materials : two-dimensional stress equilibrium, Mohr's circular strain compatibility, and the softened biaxial constitutive laws of concrete. That is, the model can predict the strength of a membrane element as well as its load-deformation history. However, this model cannot predict the contribution of concrete observed in tests, because it is based on the assumption that the direction of the cracks (and thus, the concrete struts) is inclined at the rotating angle following the postcracking principal stresses of the concrete. This paper presents a new and more general softened truss model in which the direction of the cracks is assumed to incline at the fixed angle following the principal stresses of the applied loading. This new model, although more complex, is capable of predicting the contribution of concrete. The fixed angle softened truss model requires four constitutive laws of materials. Three have been established previously for the rotating angle softened truss model (concrete in compression, concrete in tension, and steel embedded in concrete). This paper presents the fourth constitutive law relating the average shear stress of concrete to the average shear strain.

Micromechanical Modelling of Superelasticity in Shape Memory Alloys

Abstract: Micromechanical methods developed to describe the thermomechanical behavior of solids are applied to phase transition related problem. Results obtained are compared with those obtained using a macroscopic phenomenological approach. This micromechanical analysis is based on a kinematical description of the physical strain mechanisms and a definition of a local thermodynamical potential. Volume fractions of the different variants of martensite are chosen as internal variables to describe the evolution of the microstructural state of the material. This analysis determines local constitutive equations for the behavior. Global relationships are obtained using a self consistent scheme. This approach gives results in good agreement with experimental observations performed on Cu-based Shape Memory alloys.

A multi‐mode approach to finite, three‐dimensional, nonlinear viscoelastic behavior of polymer glasses

Abstract: In this study a phenomenological constitutive model is proposed to describe the finite, nonlinear, viscoelastic behavior of glassy polymers up to the yield point. It is assumed that the deformation behavior of a glassy polymer up to the yield point is completely determined by the linear relaxation time spectrum and that the nonlinear effect of stress is to alter the intrinsic time scale of the material. A quantitative three‐dimensional constitutive equation for polycarbonate as a model polymer was obtained by approximating the linear relaxation time spectrum by eighteen Leonov modes, all exhibiting the same stress dependence. A single Leonov mode is a Maxwell model employing a relaxation time that is dependent on an equivalent stress proportional to the Von Mises stress. Furthermore, a Leonov mode separates the (elastic) hydrostatic and (viscoelastic) deviatoric stress response and accounts for the geometrical complexities associated with simultaneous elastic and plastic deformation. Using a single set of...

3-D schapery representation for non-linear viscoelasticity and finite element implementation

Abstract: On the basis of the one-dimensional Schapery representation for non-linear viscoelasticity, a three-dimensional constitutive model incorporating the effects of temperature and physical ageing is developed for isotropic non-linear viscoelastic materials. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the contitutive equation is expressed in incremental form for both compressible and incompressible materials, with the hereditary integral updated at the end of each time increment by recursive computation. The proposed model is implemented in the finite element package MARC. Numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms.

Micromechanical Definition of the Strain Tensor for Granular Materials

Abstract: In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.