Showing papers on "Constitutive equation published in 1988"
TL;DR: In this paper, a three-dimensional constitutive model based on the macromolecular structure of glassy polymeric products and the micromechanism of plastic flow is presented. But the model is not suitable for the case of glass polymers.
965 citations
Book•
01 Jan 1988
TL;DR: In this article, the equations of equilibrium and the principle of virtual work for three-dimensional elasticity have been discussed and the boundary value problems of 3D elasticity has been studied.
Abstract: Part A Description of Three-Dimensional Elasticity 1 Geometrical and other preliminaries 2 The equations of equilibrium and the principle of virtual work 3 Elastic materials and their constitutive equations 4 Hyperelasticity 5 The boundary value problems of three-dimensional elasticity Part B Mathematical Methods in Three-Dimensional Elasticity 6 Existence theory based on the implicit function theorem 7 Existence theory based on the minimization of the Energy Bibliography Index
475 citations
TL;DR: In this paper, it was shown that the sintering materials are not linearly viscoelastic, so Laplace transform techniques cannot be applied, however, the relevant deformation of the matrix can be treated as purely viscous flow.
337 citations
TL;DR: A generalized microplane model for brittle-plastic heterogeneous materials such as concrete, which describes not only tensile cracking but also nonlinear triaxial response in compression and shear, is presented in this article.
Abstract: A generalized microplane model for brittle-plastic heterogeneous materials such as concrete, which describes not only tensile cracking but also nonlinear triaxial response in compression and shear, is presented. The constitutive properties are characterized separately on planes of various orientations within the material, called the microplanes, on which there are only few stress and strain components and no tensorial, requirements need to be observed. These requirements are satisfied automatically by integration over all spatial directions. The state of each microplane is characterized by normal deviatoric and volumetric strains and shear strain, which makes it possible to match any Poisson ratio. The microplane strains are assumed to be the resolved components of the macroscopic strain tensor. The central assumption is that on the microplane level the stress-strain diagrams for monotonic loading are path-independent and that all the path dependence on the macrolevel is due to unloading, which happens selectively on microplanes of some orientations. The response on the microplane is assumed to depend on the lateral normal strain which does no work. in consequence, the incremental elastic moduli tensor is nonsymmetric, which is necessary to model friction and dilatancy. This tensor is also generally anistropic and fully populated (i.e., none of its elements can be prescribed as zero). The model involves many fewer free material parameters than the existing comprehensive macroscopic phenomenologic constitutive models for concrete.
316 citations
TL;DR: In this article, an approximate elasticity theory solution for the stress-strain relations of a cracked composite lamina is given, which is written in the familiar form of two dimensional compliances appropriate for use with laminated plate theory, and include the effects of non-mechanical strains.
Abstract: An approximate elasticity theory solution is given for the stress-strain relations of a cracked composite lamina. These relations are written in the familiar form of two- dimensional compliances appropriate for use with laminated plate theory, and include the effects of non-mechanical strains. It is shown explicitly that the cracked lamina com pliances depend upon the overall laminate construction in which the lamina is contained.
223 citations
TL;DR: In this paper, void growth and ductile failure in the non-uniform multiaxial stress fields of notched bars are studied numerically and experimentally, using an elastic-viscoplastic constitutive relation that accounts for strength degradation resulting from the growth of microvoids.
Abstract: Void growth and ductile failure in the nonuniform multiaxial stress fields of notched bars are studied numerically and experimentally. U-notched bars with different notch acuities are made from partially consolidated and sintered iron powder compacts with various residual porosities. The materials are modelled using an elastic-viscoplastic constitutive relation that accounts for strength degradation resulting from the growth of microvoids. The matrix stress-strain relation and the initial void volume fractions used in the calculations are determined experimentally. The remaining parameters in the constitutive equations are evaluated from micromechanical models. Comparisons of the calculations with experimental results indicate that the constitutive model can provide good estimates of the evolution of the void volume fraction and of the strength reduction induced by void growth under a variety of nonuniform stress histories.
211 citations
TL;DR: In this article, the usefulness of cylinder impact test data to determine constants for various computational constitutive models was examined, including the Johnson-Cook and Zerilli-Armstrong models.
Abstract: This paper examines the usefulness of cylinder‐impact test data to determine constants for various computational constitutive models. The Johnson–Cook and Zerilli–Armstrong constitutive models are evaluated by comparing model predictions to tension, torsion, and cylinder‐impact test data. Then the cylinder‐impact test data are used to determine constitutive model constants for various forms of these models. Under bounded conditions of strains and strain rates, this approach can produce useful results. It can also produce very erroneous results if not used properly.
201 citations
TL;DR: In this article, the authors provide a critical examination of these models and show that some of them imply a negative Poisson's ratio, in contradiction to experimental evidence, and propose constitutive equations to describe body sinter under constraint.
187 citations
TL;DR: In this article, the role played by irreducible even rank tensors in the representation of internal structure is discussed and clarified, and the restrictions placed by thermodynamics on constitutive equations are considered and the use of potentials in these equations is examined and criticized.
Abstract: The paper is concerned with the representation of the relationship that exists, for a given material and temperature and for small deformations, between histories of applied stress and the observed strain and the accompanying changes in internal structure of the material. Emphasis is given to creep damage in metals as a vehicle for illustration of the main ideas introduced in the paper. In particular, the role played by irreducible even rank tensors in the representation of internal structure is discussed and clarified. The restrictions placed by thermodynamics on constitutive equations are considered and the use of potentials in these equations is examined and criticized.
185 citations
TL;DR: In this paper, a general three-dimensional plastic potential function expressed in terms of stress invariants is proposed based on review and evaluation of a large number of test data, which includes tensile behavior of materials with effective cohesion.
TL;DR: In this article, the yield surfaces for biaxial macroscopic stresses are obtained by accounting for the interaction between geometrically similar elastic and plastic cell crushing modes.
TL;DR: In this article, a yield function is proposed such that yield surfaces are equivalent to plastic work contours, and two parameters are required in this function: the yield surface resembles an asymmetric tear drop in the principal stress space.
TL;DR: In this article, a restructured version of the constitutive equation of motion, called the Explicity Elliptic Momentum Equation (EEME), is used for viscoelastic flow analysis.
Abstract: Finite element calculations for viscoelastic flows are reported that use a restructured form of the equation of motion that makes explicit the elliptic character of this equation. We call this restructured equation the Explicity Elliptic Momentum Equation, and its use is illustrated for flow of an upper convected Maxwell (UCM) model between eccentric and concentric rotating cylinders and also for a modified upper convected Maxwell (MUCM) model in the stick-slip problem. Sets of mixed-order approximations for velocity, stress, and a modified pressure are used to test the algorithm in both problems. Both sets of calculations are shown to converge with mesh refinement and are limited at high values of Deborah number by the formation of elastic boundary layers that are identified in the momentum equation by the growth of low-order derivative terms that involve the local velocity gradient and divergence of stress. Similar convergence properties are observed for bilinear and biquadratic Lagrangian approximations to the stress components. However, calculations with the more accurate basis for stress converge to higher values of De and are sensitive to the weighted residual method used for the constitutive equation, particularly for the eccentric cylinder problem. Streamline-upwind Petroy-Galerkin (SUPG) and artificial diffusivity (AD) formulations of the constitutive equation are tested for solution of both problems by calculations of the stress fields with fixed kinematics and by solution of the coupled problem. The SUPG method improves the performance of the calculations with the biquadratic basis set for the eccentric cylinder problem. For the UCM model, adding artificial diffusion to the constitutive equation in the stick-slip problem changes the dominant balance for the stress field near the singularity, making it appear as an integrable stress approximation for fixed mesh. For the MUCM model the Newtonian-like behavior of the stress near this point is unaffected by the AD method and calculations converge to moderate De .
TL;DR: In this paper, a modification of the bead-and-spring dumbbell model for non-Newtonian flow has been proposed, which allows the numerical solution of (some) flow problems to proceed to large Deborah numbers.
Abstract: The failure of some careful attemps to provide numerical solutions of the equations for non-Newtonian flow suggests to us some inadequacies of the constitutive equations. (After all no one would doubt the validity of the conservation of mass and momentum.) To understand the physics in the constitutive equation, and thence to correct its undesirable features, it is helpful to look at a micro-structural model which leads to the constitutive equation. The bead-and-spring dumbbell model for a dilute polymer solution leads to an Oldroyd-like equation. The simplest version of the bead-and-spring model has a linear spring and a constant friction coefficient for the beads. While this model is simple and usefully combines viscous and elastic behaviour, it has the very unphysical feature of blowing up in strong straining flows (i.e. at a Deborah number in excess of unity), with the spring lengthening indefinitely in time and the steady extensional viscosity becoming unbounded at a critical flow strength. The hope that the corresponding large stresses would not occur in a flow calculation seems to have been misguided: some simple examples show that the large stresses may not act through the momentum equation to inhibit the flow. To cure this unphysical behaviour one clearly needs to use a non-linear spring force which gives a finite limit to the extension. Incorporating this modification into the constitutive equation enables the numerical solution of (some) flow problems to proceed to large Deborah numbers. Care is of course still needed in the numerical calculations, for example in resolving thin layers of high stress. (A boundary layer theory needs to be developed for the nonlinearity introduced by the non-Newtonianness.) A further modification of the bead-and-spring model may be necessary if argument is sought between numerical calculations and experiments. Many flows of interest subject the fluid to a sudden strong strain. In such circumstances the polymer chains will not be in thermodynamic equilibrium and so will not give the standard entropic spring. It may be possible to model this behaviour by a large temporary internal viscosity.
TL;DR: In this article, a constitutive model for concrete, built within the framework of rate-independent theory of plasticity, is presented, which invokes the concept of a failure locus which is introduced a priori as a path-independent criterion.
TL;DR: In this article, several aspects of electromagnetic wave propagation and scattering in isotropic chiral media (D = ∊E + β∊▽ × E, B = μH + βμ▽× H) are explored.
Abstract: Several aspects of electromagnetic wave propagation and scattering in isotropic chiral media (D = ∊E + β∊▽ × E, B = μH + βμ▽× H) are explored here. All four field vectors, E, H, D, and B, satisfy the same governing differential equation, which reduces to the vector Helmholtz equation when β = 0. Vector and scalar potentials have been postulated. Conservation of energy and momentum are examined. Some properties, consequences, and computationally attractive forms of the applicable infinite-medium Green’s function have been explored. Finally, the mathematical expression of Huygens’s principle, as applicable to chiral media, has also been derived and employed to set up a scattering formalism and to establish the forward plane-wave-scattering amplitude theorems. Several of the results given that pertain to the field equations and Green’s dyadic are available for constitutive equations other than those mentioned above; these results, along with some others, have been given now for the above-mentioned constitutive equations. The derivations of Huygens’s principle and other developments described here have not been given earlier, to our knowledge, for any pertinent set of constitutive equations. With advances in polymer science, the formalisms developed here may be useful in the utilization of artificial chiral dielectrics at suboptical and microwave frequencies; application to vision research is also anticipated.
TL;DR: In this article, a finite streamline element method for use with integral type constitutive equations in viscoelastic flow simulation, including thermal effects, is presented and some new results using this method in modelling the IUPAC extrusion experiments for low-density polyethylene (LDPE), involving both long and short circular dies, are reported and discussed.
Abstract: A finite streamline element method for use with integral type constitutive equations in viscoelastic flow simulation, including thermal effects, is presented and some new results using this method in modelling the IUPAC extrusion experiments for low-density polyethylene (LDPE), involving both long and short circular dies, are reported and discussed. It will be shown that the special streamline-based elements provide a convenient way of particle tracking and strain tensor calculation. The extrusion calculation when using a relatively realistic KBKZ type integral model with multiple relaxation times is stable, and practically important Weissenberg numbers and swelling ratios have been reached without difficulty. The agreement with experimental swelling ratios is on the whole satisfactory. In order to make the model a good fit to the Trouton viscosity experimental data of the LDPE sample, a spectrum of elongational parameters has been introduced. The second normal stress term has also been added to the integral model to examine its effect on extrusion swelling. The non-isothermal effects and the possibility of a wall slip near the die exit at high apparent shear rates are also discussed.
TL;DR: In this paper, the authors examined constitutive equations that have been used primarily to analyze constrained sintering problems, and they concluded that either the internal stresses are small (as predicted by the constitutive laws given in Part I) or the basic assumptions of linearity and isotropy used in all of the analyses are incorrect.
TL;DR: In this article, the authors examined the behavior of viscoelastic fluid models which exhibit local extrema of the steady shear stress and constructed a variety of steady singular solutions with jumps in shear rate and their stability to one dimensional disturbances analyzed.
Abstract: We examine the behavior of viscoelastic fluid models which exhibit local extrema of the steady shear stress. For the Johnson-Segalman and Giesekus models, a variety of steady singular solutions with jumps in shear rate are constructed and their stability to one dimensional disturbances analyzed. It is found that flow-rate versus imposed stress curves in slit-die flow fit experimental observation of the “spurt” phenomenon with some precision. The flow curves involve linearly stable singular solutions, but some assumptions on the dynamics of the spurt process are required. These assumptions are tested by a semi-implicit finite element solution technique which allows solutions to be efficiently integrated over the very long time-scale involved. The Johnson-Segalman model with added Newtonian viscosity is used in the calculations. It is found that the assumptions required to model spurf are satisfied by the dynamic model. The dynamic model also displays a characteristic “latency time” before the spurt ensues and a characteristic “shape memory” hysteresis in load/unload cycles. These as well as other features of the computed solutions should be observable experimentally. We conclude that constitutive equations with shear stress extrema are not necessarily flawed, that their predicted behavior may appear to be arrested “wall slip”, and that such behavior may actually have been observed already.
TL;DR: In this paper, the authors present experimental results and constitutive models for single phase ceramics, transforming ceramic and ceramic composites with the aim of developing the framework for a general theory of cyclic compression fracture in notched brittle solids.
TL;DR: In this paper, a unified constitutive model for the behavior of frictional materials is described, which is based on concepts from elasticity and plasticity theories, including Hooke's law for the elastic behavior, a nonassociated flow rule, a yield criterion that describes contours of equal plastic work, and a workhardening/softening law.
01 Apr 1988
TL;DR: In this article, the authors define a set of equations constitutives decrivant chacun des trois mecanismes independant deformation: glissement des dislocations, glissements des joints de grains, and flux diffusionnel directionnel.
Abstract: L'ecoulement plastique des solides polycristallins a lieu par l'un des trois mecanismes independants de deformation: glissement des dislocations, glissement des joints de grains, et flux diffusionnel directionnel. Developpement des equations constitutives decrivant chacun des trois mecanismes. Les equations se fondent sur une dependance exponentielle de la vitesse de deformation. Determination de l'exposant de contrainte. Influence de la microstructure: taille des grains, taille des sous-grains, densite des dislocations. Description quantitative des competitions entre ces differents mecanismes. Analyse du processus dominant
TL;DR: In this article, the concept of relative spinW D/S is introduced to facilitate the formulation of large deformation stress-rate models of inelasticity, which is a measure of non-coaxiality between stress and deformation rate of the formWD/S =W−W S withW S andW signifying the angular velocities or spins of the two material frames corresponding to the stress and strain rate respectively.
Abstract: The concept of relative spinW D/S is introduced to facilitate the formulation of large deformation stress-rate models of inelasticity Roughly speaking, it is a measure of non-coaxiality between stress and deformation rate of the formW D/S =W−W S withW S andW signifying the angular velocities or spins of the two material frames corresponding to the stress and strain rate respectively It is suggested that an objective stress rate be defined with respect toW S for use in the constitutive equations and this requires explicit representations forW D/S reflecting the aforementioned non-coaxiality It is shown that this practice leads conveniently to an elegant generalization of previous proposals resorting to either use of a variety of different spins or considerably complex constitutive equations, in order to dispense with undesirable oscillatory solutions of simple shear problems
TL;DR: A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium.
Abstract: A three-dimensional constitutive law is proposed for the myocardium. Its formulation is based on a structural approach in which the total strain energy of the tissue is the sum of the strain energies of its constituents: the muscle fibers, the collagen fibers and the fluid matrix which embeds them. The ensuing material law expresses the specific structural and mechanical properties of the tissue, namely, the spatial orientation of the comprising fibers, their waviness in the unstressed state and their stress-strain behavior when stretched. Having assumed specific functional forms for the distribution of the fibers spatial orientation and waviness, the results of biaxial mechanical tests serve for the estimation of the material constants appearing in the constitutive equations. A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium. Moreover, the results provide general conclusions concerning the structural basis for the tissue overall mechanical properties, the main of which is that the collagen matrix, though comprising a relatively small fraction of the whole tissue volume, is the dominant component accounting for its stiffness.
TL;DR: In this paper, the theoretical and experimental consequences of slip phenomena are reviewed, and it is maintained that future progress will depend heavily on productive interplay between fluid mechanics and materials science.
Abstract: Non-Newtonian fluid mechanics is often distinguished from its Newtonian counterpart by the additional requirement that a constitutive equation be specified as part of the problem statement. For some modeling problems important to polymer processing, a wall boundary condition in which provision is made for slip is also a necessary ingredient. This requires that attention be directed to microscopic processes influencing flow behavior on a macroscopic scale. In this paper theoretical and experimental consequences of slip phenomena are reviewed. It is maintained that future progress will depend heavily on productive interplay between fluid mechanics and materials science.
TL;DR: In this paper, the implementation in finite element calculations of complex elastoplastic constitutive equations exhibiting non-linear kinematic and isotropic hardening, and fully coupled to a continuous ductile damage evolution model is described.
Abstract: This paper deals with the implementation in finite element calculations of complex elastoplastic constitutive equations exhibiting non-linear kinematic and isotropic hardening, and fully coupled to a continuous ductile damage evolution model. The Newton method is used to solve the non-linear global equilibrium equations as well as the non-linear local equations obtained by fully implicit integration of the constitutive equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. The procedure described has been implemented in the general purpose code ABAQUS.
TL;DR: In this paper, a new mixed algorithm for calculating viscoelastic flow is proposed, where the coupling between velocity and stress components is solved by means of multiple bilinear stress sub-elements embedded in the Lagrangian element for the velocity field while streamline upwinding is used for solving the advection dominated constitutive equations.
Abstract: In a recent paper, Marchal and Crochet have proposed a new mixed algorithm for calculating viscoelastic flow. The coupling between velocity and stress components is solved by means of multiple bilinear stress sub-elements embedded in the Lagrangian element for the velocity field while streamline upwinding is used for solving the advection dominated constitutive equations. The paper reviews the motivation and the contents of the new method and explores in detail the flow of Oldroyd-B, Phan Thien-Tanner and Giesekus-Leonov fluids through a circular abrupt contraction. No limitation based on the value of the Weissenberg number has been found for the calculation of such flows. The sensitivity of the macroscopic flow features upon a variation of the material parameters is also investigated.
TL;DR: In this paper, the authors present a general theoretical analysis of the problem of mass transfer during the drying of shrinking multiphase systems, and derive a complete theory for drying of cellular materials in the stage during which the cellular structure prevails.