Showing papers on "Constitutive equation published in 1984"
TL;DR: In this paper, a higher-order shear deformation theory of laminated composite plates is developed, which accounts for parabolic distribution of the transverse shear strains through the thickness of the plate.
Abstract: A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano (1970), but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.
3,504 citations
TL;DR: In this paper, the authors studied the flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles using statistical methods analogous to those used in the kinetic theory of gases.
Abstract: The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.
2,631 citations
TL;DR: In this article, a two-fluid formulation for two-phase flow analyses is presented, where a fully threedimensional model is obtained from the time averaging, whereas the one-dimensional model was developed from the area averaging.
738 citations
TL;DR: In this paper, a quasi-static deformation and fracture analysis for nonlinear viscoelastic media and sample applications are given. But the authors focus on predicting mechanical work available at the crack tip for initiation and continuation of growth.
Abstract: Methods of quasi-static deformation and fracture analysis are developed for a class of nonlinear viscoelastic media and sample applications are given. Selection of the class of media is guided by actual rheological behavior of monolithic and composite materials as well as the need for simplicity to be able to understand the effect of primary material and continuum parameters on crack growth behavior. First, pertinent aspects of J integral and energy release rate theory for nonlinear elastic media are discussed. Nonlinear viscoelastic constitutive equations are then given, and correspondence principles which establish a simple relationship between mechanical states of elastic and viscoelastic media are developed. These principles provide the basis for the subsequent extension of J integral theory to crack growth in viscoelastic materials. Emphasis is on predicting mechanical work available at the crack tip for initiation and continuation of growth; some examples show how viscoelastic properties and the J integral affect growth behavior. Included is the problem of a crack in a thin layer having different viscoelastic properties than the surrounding continuum. The Appendix gives an apparently new constitutive theory for elastic and viscoelastic materials with changing microstructure (e.g. distributed damage) and indicates the conditions under which the fracture theory in the body of the paper is applicable.
710 citations
TL;DR: In this paper, a higher-order shear deformation theory of plates accounting for the von Karman strain was presented, which contains the same dependent unknowns as in the Hencky-Mindlin type first-order deformation theories and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate.
695 citations
Book•
01 Jan 1984
TL;DR: A review of the equations of MECHANICS can be found in this paper, where the Ritz Method and Weighted-Residual Methods are used to approximate the distance between two points.
Abstract: A REVIEW OF THE EQUATIONS OF MECHANICS. Introduction. Kinetics. Kinematics. Thermodynamic Principles. Constitutive Equations. Boundary--Value Problems of Mechanics. Equations of Bars, Beams, Torsion, and Plane Elasticity. ENERGY AND VARIATIONAL PRINCIPLES. Preliminary Concepts. Calculus of Variations. Virtual Work and Energy Principles. Stationary Variational Principles. Hamiltona s Principle. Energy Theorems of Structural Mechanics. VARIATIONAL METHODS OF APPROXIMATION. Some Preliminaries. The Ritz Method. Weighted--Residual Methods. The Finite--Element Method. THEORY AND ANALYSIS OF PLATES AND SHELLS. Classical Theory of Plates. Shear Deformation Theories of Plates. Laminated Composite Plates. Theory of Shells. Finite--Element Analysis of Plates and Shells. Bibliography. Answers to Selected Exercises. Index.
347 citations
TL;DR: In this paper, the bending of a thin plate with rapidly varying thickness was studied and a fourth-order equation for the midplanc displacement was derived using an asymptotic analysis based on 3D linear elasticity.
276 citations
01 Apr 1984
TL;DR: In this article, the physical and mathematical properties of non-local elastic moduli are explored through lattice dynamics and dispersive wave propagations, and the theory is applied to the problems of surface waves, screw dislocation and a crack.
Abstract: : Constitutive equations of finite nonlocal elasticity are obtained. Thermodynamic restriction are studied. The linear theory is given for anisotropic and isotropic solids. The physical and mathematical properties of the nonlocal elastic moduli are explored through lattice dynamics and dispersive wave propagations. The theory is applied to the problems of surface waves, screw dislocation and a crack. Excellent agreements with the results known in atomic lattice dynamics and experiments display the power and potential of the theory.
262 citations
01 Jan 1984
TL;DR: In this paper, the central problem of computational plasticity, that is, given a deformation history, find the corresponding stress history by integrating the constitutive equations, is discussed for Mises, Tresca and non-convex yield surfaces.
Abstract: Basic concepts concerning the numerical implementation of rate-independent deviatoric plasticity theories are presented. Both small and finite deformations are dealt with. Emphasis is placed on the central problem of computational plasticity, that is, given a deformation history, find the corresponding stress history by integrating the constitutive equations. Mathematical well-posedness requires non-classical specification of elastic and plastic processes. This problem is discussed for Mises, Tresca and non-convex “starlike” yield surfaces, associative and non-associative flow rules, and strain hardening and softening. The radial-return concept is emphasized in the algorithmic descriptions. A unified treatment of a class of finite-deformation theories is presented. Accurate and efficient procedures for calculating kinematical quantities necessary in finite deformation analysis are described.
261 citations
TL;DR: In this paper, the authors apply the crack band model to the problem of crack shear in concrete, and derive the constitutive law for concrete within the crack bands by the microplane model, where the microstrains on weak planes of various orientations (the microplanes) are assumed to conform to the same macroscopic strain tensor, and the microstresses from all the microplanes are superimposed.
Abstract: The crack band model is applied to the problem of crack shear in concrete. The constitutive law for concrete within the crack band is provided by the microplane model, in which the microstrains on weak planes of various orientations (the microplanes) are assumed to conform to the same macroscopic strain tensor, and the microstresses from all the microplanes are superimposed. Due to the neglect of shear stiffness on individual microplanes, the material behavior is completely characterized by the relation between the normal stress and strain for each microplane. To simulate crack shear, the law for unloading contribution on the microplanes after previous tensile strain‐softening is important, since the shear stresses resisting crack shear, as well as the normal confining stresses and crack dilatancy, result from compression along lines inclined with regard to the crack plane. A satisfactory agreement with the existing results from shearing tests of cracked concrete blocks (i.e., aggregate interlock tests) i...
220 citations
TL;DR: Uniaxial compressive force is applied directly on rabbit thoracic artery in the radial direction to study its constitutive equation under compressive stresses, and the resulting stress-strain curves show that the wall material becomes increasingly stiffer at larger compressive strain, quite similar to the behavior in tension.
TL;DR: In this paper, the authors derived constitutive relations for an incompressible, isotropic power-law matrix material containing a dilute concentration of spherical voids, and derived the overall constitutive relation governing the behavior of the dilutely voided solid.
TL;DR: In this article, the bracket formulation of the Euler fluid mechanics equations is extended to the fluid mechanics equation corresponding to the Navier-Stokes-Fourier and the Edelen constitutive relations.
TL;DR: In this paper, it was shown that the occurrence of such an "anomaly" is not restricted to anisotropic plasticity, but also in the case of hypoelasticity and classical isotropic hardening plasticity theory.
14 Jun 1984-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: Theoretical predictions of lifetimes, expressed as a representative rupture stress, of damage fields and of crack growth are made by using a previously developed finite element system based on the theory of continuum damage mechanics as mentioned in this paper.
Abstract: Theories that have been used to predict the rate of growth of cracks due to creep are reviewed and assessed. The need is expressed for a sounder understanding of the mechanisms by which creep crack growth takes place. The aim of this paper is to answer the question: can continuum damage mechanics provide the mechanism by which cracks grow by creep? The paper reports the results of theoretical and experimental studies on internally and externally cracked, plane strain, tension members, in an aluminium alloy, in copper and in 316 stainless steel, all of which undergo high temperature creep rupture under steady loads. Theoretical predictions of lifetimes, expressed as a representative rupture stress, of damage fields and of crack growth are made by using a previously developed finite element system (Hayhurst, Dimmer & Morrison, Phil. Trans. R. Soc. Lond . 311, 103 (1984)) based on the theory of continuum damage mechanics. The theoretical predictions are shown to be in close agreement with experimental observations. The effect of the growth of continuum damage is to produce considerable stress redistribution and to cause the nullification of stress singularities. The multi-axial stress rupture criterion of the material plays an important role in the determination of lifetimes and of the planes upon which crack propagation takes place. The numerical solution procedure is automatic but requires that the constitutive equations model the elastic response, the creep strain rates, including tertiary behaviour, and the multiaxial stress rupture criterion of the material at the appropriate stress levels. Continuum damage mechanics theory is shown to be capable of modelling the propagation of cracks through material which has suffered relatively low damage.
TL;DR: In this paper, the basic kinematics and rate constitutive equations are briefly presented within the framework of a macroscopic formulation of finite plastic transformations for structured media, employing the concept of tensorial structure variables.
TL;DR: In this article, a finite element algorithm was developed for analysis of nonlinear viscoelastic materials, based on Schapery's integral constitutive law, which is used to describe viscoels behavior.
TL;DR: In this paper, the problem of formulation of a model for polythermal glaciers, focussing attention in particular on the temperate zone where ice and water can coexist at the melting temperature, is considered.
Abstract: We reconsider the problem of formulation of a model for polythermal glaciers, focussing attention in particular on the temperate zone where ice and water can coexist at the melting temperature. The energy equation for the ice-water mixture in this zone introduces a moisture flux, and a constitutive law for this flux is required. By analogy with the flow through a porous medium, we use Darcy's law (i.e. the second momentum equation of a two-phase flow model with “porous” geometry), and then require a mechanical constitutive relation relating the water pressure p w to the average ice pressure p i . Experience in two phase flows suggests that p w =p i may be problematical, and experience in soil mechanics suggests it is inaccurate. A constitutive relation is therefore presented based on work of Nye (1976), and its effect on the well-posedness of the model is examined. Considerations of the sort presented here have clear relevance in the formulation of similar problems in other geophysical situations...
TL;DR: In this article, a one-dimensional constitutive relation is presented which describes quantitatively the multistep creep and recovery behavior of this material in the case where the specimens are not mechanically preconditioned.
TL;DR: In this article, the Leslie coefficients of the nematic phase of concentrated solutions of rod-like polymers are calculated based on the molecular theory given in the previous paper, and the coefficients are obtained in the form of an asymptotic expansion with respect to the concentration.
Abstract: The Leslie coefficients of the nematic phase of concentrated solutions of rodlike polymers are calculated based on the molecular theory given in the previous paper (Kuzuu and Doi: J. Phys. Soc. Jpn. 52 (1983)). The Leslie coefficients are obtained in the form of an asymptotic expansion with respect to the concentration.
TL;DR: In this paper, the authors provided an existence result for slow flows with no in-and outflow boundaries, where the fluid is assumed to be described by constitutive equations of a differential nature.
Abstract: : Questions of existence and uniqueness of steady flows of viscoelastic fluids have thus far not been understood, even for slow flows perturbing rest. This paper provides an existence result for slow flows with no in- and outflow boundaries. The fluid is assumed to be described by constitutive equations of a differential nature. The method used to prove existence is constructive and in fact very close to procedures used in numerical calculations.
TL;DR: In this paper, the authors considered saturated-unsaturated flow of an incompressible fluid through a porous medium in the case of time-dependent water levels and proved an existence result for the corresponding weak formulation.
Abstract: Saturated-unsaturated flow of an incompressible fluid through a porous medium is considered in the case of time-dependent water levels. This corresponds to coupling the mass conservation law with a continuous constitutive condition between water content and pressure. An existence result for the corresponding weak formulation is proved. Finally we study the limit as the constitutive relation degenerates into a maximal monotone graph.
Book•
01 Jan 1984
TL;DR: In this article, the authors present an analysis of the relationship between velocity gradient tensors and spin tensors, and show that the latter is a function of the acceleration of the tensors.
Abstract: 1 Kinematics of Flow.- 1.1 Introduction.- 1.2 Velocity Gradient Tensor.- 1.3 Rate of Deformation Tensor.- 1.4 Analysis of Strain Rates.- 1.5 Spin Tensor.- 1.6 Curvature-Twist Rate Tensor.- 1.7 Objective Tensors.- 1.8 Balance of Mass.- 1.9 Concluding Remarks.- 1.10 References.- 2 Field Equations.- 2.1 Introduction.- 2.2 Measures for Mechanical Interactions.- 2.3 Euler's Laws of Motion.- 2.4 Stress and Couple Stress Vectors.- 2.5 Stress and Couple Stress Tensors.- 2.6 Cauchy's Laws of Motion.- 2.7 Analysis of Stress.- 2.8 Energy Balance Equation.- 2.9 Entropy Inequality.- 2.10 Concluding Remarks.- 2.11 References.- 3 Couple Stresses in Fluids.- 3.1 Introduction.- 3.2 Constitutive Equations.- 3.3 Equations of Motion.- 3.4 Boundary Conditions.- 3.5 Steady Flow Between Parallel Plates.- 3.6 Steady Tangential Flow Between Two Coaxial Cylinders.- 3.7 Poiseuille Flow Through Circular Pipes.- 3.8 Creeping Flow Past a Sphere.- 3.9 Some Time-Dependent Flows.- 3.10 Stability of Plane Poiseuille Flow.- 3.11 Hydromagnetic Channel Flows.- 3.12 Some Effects on Heat Transfer.- 3.13 Concluding Remarks.- 3.14 References.- 4 Anisotropic Fluids.- 4.1 Introduction.- 4.2 Balance Laws.- 4.3 Microstructure of a Dumbbell-Shaped Particle.- 4.4 Field Equations.- 4.5 Constitutive Equations.- 4.6 Implications of the Second Law of Thermodynamics.- 4.7 Incompressible Fluids.- 4.8 Simple Shearing Motion.- 4.9 Orientation Induced by Flow.- 4.10 Poiseuille Flow Through Circular Pipes.- 4.11 Cylindrical Couette Flow.- 4.12 Concluding Remarks.- 4.13 References.- 5 Micro Fluids.- 5.1 Introduction.- 5.2 Description of Micromotion.- 5.3 Kinematics of Deformation.- 5.4 Conservation of Mass.- 5.5 Balance of Momenta.- 5.6 Microinertia Moments.- 5.7 Balance of Energy.- 5.8 Entropy Inequality.- 5.9 Constitutive Equations for Micro Fluids.- 5.10 Linear Theory of Micro Fluids.- 5.11 Equations of Motion.- 5.12 Concluding Remarks.- 5.13 References.- 6 Micropolar Fluids.- 6.1 Introduction.- 6.2 Skew-Symmetry of the Gyration Tensor and Microisotropy.- 6.3 Micropolar Fluids.- 6.4 Thermodynamics of Micropolar Fluids.- 6.5 Equations of Motion.- 6.6 Boundary and Initial Conditions.- 6.7 Two Limiting Cases.- 6.8 Steady Flow Between Parallel Plates.- 6.9 Steady Couette Flow Between Two Coaxial Cylinders.- 6.10 Pipe Poiseuille Flow.- 6.11 Micropolar Fluids with Stretch.- 6.12 Concluding Remarks.- 6.13 References.- Notation.
TL;DR: In this paper, a general discussion of terms in an energy functional which might be the basis from which equations governing stress, stability, and vibration analyses are derived is given, including thermal effects, moderately large rotations, boundary conditions, and distributed and concentrated loads.
TL;DR: In this paper, a general procedure based on polynomial expansion of yield function in terms of invariants of the stress tensor is proposed in the context of associated plasticity for isotropic materials undergoing isotropically hardening.
Abstract: A general procedure based on polynomial expansion of yield function in terms of invariants of the stress tensor is proposed in the context of associated plasticity for isotropic materials undergoing isotropic hardening. The procedure can be used to evolve one or more models for a material by using appropriate laboratory test results. One of the functions showing invariance at ultimate and a single function to describe continuous yield and ultimate yield behavior is investigated in detail. Based on comprehensive series of bests on cubical specimens for different (geological) materials, a hardening or growth function is defined in terms of the trajectory of plastic strain and the ratio of deviatoric to total plastic strain. The predictions of the proposed model are verified with respect to the observed results from tests with different stress paths. The model provides highly satisfactory predictions for both stress‐strain and volumetric strain responses from various stress paths. The proposed model shows po...
TL;DR: In this article, the authors presented a method to extend an elastoplastic constitutive model valid only in the triaxial compression condition to one applicable in general stress conditions.
TL;DR: In this paper, a numerical method to simulate discharging processes in mass flow silos is presented, which provides transient velocity and stress fields within the bulk material for a first period of discharging.
Abstract: A numerical method to simulate discharging processes in mass‐flow silos is presented. The essential point is to formulate the appropriate constitutive law for a granular bulk material, which covers solid‐like as well as fluid‐like behavior during discharging. An elastic‐plastic law is chosen for the former one, which is completed with a simple first approach for fluid‐like behavior. As large and fast deformations occur, geometric nonlinearities and mass properties of the bulk material are considered with respect to an Eulerian frame of reference. The complete set of field equations is numerically solved by the finite element method spatially and by the finite difference method in time. Due to the nature of the finite element method a broad variety of boundary conditions can be studied. The method provides transient velocity and stress fields within the bulk material for a first period of discharging. Remarkable stress redistributions with strong increases of wall pressures are computed.
Abstract: The hardening model proposed by Z. Mroz based on the uniaxial fatigue behavior of many metals is adopted to derive an incremental constitutive equation for general three-dimensional problems. This constitutive law is then employed in the analysis of metal forming problems to assess the influence of loading cycles, of the types involved in standard forming processes, on the ultimate formability of sheet metals. The predicted forming limit curves differ quantitatively from results obtained via an isotropie hardening model and differ qualitatively from those obtained via a kinematic model. Also investigated are the effects of such loading cycles on material response to simple tensile loading, which is often used to characterize a material. Significant differences between the present model and the other two models considered are observed in such characterizers of simple tensile behavior as the stress-strain curve, the anisotropy parameter and the uniform elongation. These differences suggest a rather simple experiment to identify the proper material model to be used in analyses of problems which involve loading cycles. Comparisons with some experimental results reveal that the employment of an anisotropic hardening model, such as the generalized Mroz model derived herein, is indeed crucial in accurately predicting material response to complicated loading histories.
TL;DR: In this paper, a time and temperature dependent plasticity model is formulated in a Lagrangian system to describe finite deformation, where history dependence and large strain behavior are incorporated through the introduction of one tensor internal variable.
TL;DR: In this article, finite element methods for five viscoelastic constitutive equations each derived as a limit of the Giesekus model were used to calculate the two-dimensional flows between two slightly eccentric cylinders with the inner one rotating.
Abstract: The two-dimensional flows between two slightly eccentric cylinders with the inner one rotating are calculated by finite-element methods for five viscoelastic constitutive equations each derived as a limit of the Giesekus model. Comparisons with exact results for Newtonian, second-order, and corotational Maxwell-like fluids set the accuracy of the calculations as a function of eccentricity and Deborah number (De). Computer-implemented perturbation methods are used to demonstrate bifurcation and turning points in De for an upper-convected Maxwell fluid. The locations of the limit points are moderately stable to extensive mesh refinement and, therefore, seem to be an intrinsic property of this constitutive equation. Similar solution pathology is demonstrated for the three-constant Oldroyd-B model, but no limiting value of De is found for calculations with the Leonov-like version of the Giesekus fluid.