Showing papers on "Constitutive equation published in 1983"

Slip instability and state variable friction laws

Abstract: The dependence of the friction force on slip history is described by an experimentally motivated constitutive law where the friction force is dependent on slip rate and state variables The state variables are defined macroscopically by evolution equations for their rates of change in terms of their present values and slip rate Experiments may strongly suggest that one state variable is adequate or prove that one is inadequate Analysis of steady slip governed by a single state variable in a spring and (massless) slider predict oscillations at a critical spring stiffness k = kcrit The critical stiffness kcrit is given by a simple formula and steady slip is stable for k > kcrit and unstable for k < kcrit State variable friction laws may superficially appear as a simple slip rate dependence, slip distance dependence, or time dependent static friction, depending on experiment and testing machinery Truly complicated motion is possible in a spring-slider model if more than one state variable is used Further consequences of state variable friction laws can include creep waves and apparent rate independence for some phenomena

On the Plastic and Viscoplastic Constitutive Equations—Part I: Rules Developed With Internal Variable Concept

Abstract: Mise au point d'un modele mathematique de l'ecoulement plastique ou viscoplastique. Application a un alliage refractaire IN 100

An Integral Constitutive Equation for Mixed Flows: Viscoelastic Characterization

Abstract: A viscoelasticconstitutive equation of the single‐integral form has been designed. Its memory function is factored into a time‐dependent part and a strain‐dependent part. The time function is the usual series of exponential relaxations. Its relaxation times and weighting coefficients are determined by nonlinear regression on linear viscoelastic data: stress relaxation after small‐strain and small‐amplitude sinusoidal oscillations. The relative accuracy of linear and nonlinear regression fitting is compared. The strain‐dependent function is new. It is of a simple sigmoidal form with only two parameters: one determined from shear and the other from extensional data. Its sigmoidal form provides a finite linear viscoelastic region, a steady viscosity in uniaxial extension, and a well‐behaved power‐law shear viscosity at high shear rate. An efficient strategy for collecting sufficient data to determine the parameters of the equation is described. Predictions of the equation are tested against shear and extension data collected on the Rheometrics System Four for two polydimethylsiloxanes and against data for other polymer melts from the literature. Both uniaxial and biaxial extension as well as shear data are described. Transient shear normal stresses are somewhat underpredicted. The constitutive equation has the potential for modeling mixed shear and extensional flows as encountered in processing operations and is simple enough to be attractive for efficient computer‐aided analysis by modern finite element methods.

Constitutive Equation for Nematic Liquid Crystals under Weak Velocity Gradient Derived from a Molecular Kinetic Equation

Abstract: Constitutive equation for the nematic liquid crystals under weak velocity gradient is derived from the kinetic equation presented by Doi. The constitutive equation is consistent with the phenomenological equation proposed by Ericksen and Leslie. The six viscosity parameters (Leslie coefficients) appearing in the phenomenological theory are expressed by the molecular parameters.

The Boundary Element Method Applied to Inelastic Problems

Abstract: 1 Introduction and Motivation.- 1.1 Introduction.- 1.2 Literature Survey-Nonlinear Applications.- 1.3 Layout of Notes.- 2 Basic Theory.- 2.1 Introduction.- 2.2 Theory of Elasticity.- 2.3 Inelastic Behaviour of Materials.- 2.4 Governing Equations.- 3 Boundary Element Formulation for Elastic Problems.- 3.1 Introduction.- 3.2 Somigliana's Identity.- 3.3 Fundamental Solutions.- 3.4 Stresses at Internal Points.- 3.5 Boundary Integral Equation.- 3.6 Infinite and Semi-Infinite Regions.- 3.7 Numerical Implementation.- 3.8 Examples - Half-Plane Formulation.- 4 Boundary Element Equations for Inelastic Problems.- 4.1 Introduction.- 4.2 Somigliana's Identity for Inelastic Problems.- 4.3 Internal Stresses.- 4.4 Alternative Boundary Element Formulations.- 4.4.1 Initial Strain.- 4.4.2 Initial Stress.- 4.4.3 Fictitious Tractions and Body Forces.- 4.5 Half-Plane Formulations.- 4.6 Spatial Discretization.- 4.7 Internal Cells.- 5 Elastoplastic Boundary Element Analysis.- 5.1 Introduction.- 5.2 Some Simple Elastoplastic Relations.- 5.3 Initial Strain - Numerical Solution Technique.- 5.4 Examples - Initial Strain Formulation.- 5.4.1 Perforated Aluminium Strip.- 5.4.2 Polystyrene Crazing Problem.- 5.4.3 Plane Strain Punch.- 5.4.4 Thick Cylinder.- 5.5 General Elastoplastic Stress-Strain Relations.- 5.6 Initial Stress-Outline of Solution Techniques.- 5.7 Examples - Kelvin Implementation.- 5.7.1 Notched Tensile Specimen.- 5.7.2 Deep Circular Tunnel.- 5.7.3 Rough Punch.- 5.8 Examples - Half-Plane Implementation.- 5.8.1 Strip Footing.- 5.8.2 Shallow Tunnel.- 6 Viscoplasticity and Creep Using Boundary Elements.- 6.1 Introduction.- 6.2 Rate Dependent Constitutive Equations.- 6.3 Solution Technique.- 6.4 Examples.- 6.4.1 Deep Beam.- 6.4.2 Thin Disc.- 6.4.3 Plate Under Thermal Shrinkage.- 7 General Discussion and Conclusions.- References.- Appendix A Indirect Computation of Principal Values.- Appendix B Stress Rates at Boundary Nodes.- Appendix C Displacements Due to Constant Inelastic Strain Fields.- Appendix D Some Particular Expressions for 2-D Inelastic Problems.

Modeling of microscopic mechanisms in granular material

Abstract: Absract The computer program BALL has been used for the past several years to model and study the behavior of two – dimensional assemblies of discs. The main result of the study thus far, a qualitative description of the material behavior, is discussed. An attempt has been made to introduce measures that may be used in a future constitutive model. The main measures, a partitioned stress tensor, and a constraint ratio, which is related to the stability of the assembly, are introduced and illustrated for some numerical experiments with BALL.

Peristaltic Flow of Viscoelastic Liquids

Abstract: Peristaltic pumping is investigated, generated by means of an infinite train of waves travelling along the wall of a cylindrical tube. The theory is based on the general second order integral constitutive equation for viscoelastic (simple) liquids. Analytical and closed form solutions are presented for the first order and the stationary part of the second order flow field approximations with respect to the amplitude ratio. It turns out that the zero—shear viscosity η 0 and the complex viscosity η*(ω) are the only relevant fluid properties.

Comment on Orthotropic Models for Concrete and Geomaterials

Abstract: Incrementally linear constitutive equations that are characterized by an orthotropic tangential stiffness or compliance matrix have recently become widely used in finite element analysis of concrete structures and soils. It does not seem to be, however, widely appreciated that such constitutive equations are limited to loading histories in which the prinicipal stress directions do not rotate, and that a violation of this condition can sometimes have serious consequences. It is demonstrated that in such a case the orthotropic models do not satisfy the form‐invariance condition for initially isotropic solids, i.e., the condition that the response predicted by the model must be the same for any choice of coordinate axes in the initial stress‐free state. An example shows that the results obtained for various such choices can be rather different. The problem cannot be avoided by rotating the axes of orthotropy during the loading process so as to keep them parallel to the principal stress axes, first, because t...

Visco-elasto-plastic constitutive law for a bituminous mixture under repeated loading

Abstract: A constitutive law for a bituminous mixture subjected to repeated loading is presented. The elastic, plastic, viscoelastic, and viscoplastic strain components are incorporated into the model as they are simultaneously present in the loading process. The model parameters are extracted from a series of repeated uniaxial creep and creep recovery experiments conducted under constant compression stress. The experiments were performed at constant temperature for various stress levels, time periods, and numbers of cycles. The elastic strain is found to depend solely and linearly on the stress. The plastic strain is linearly proportional to stress and exhibits a power-law dependence on the number of loading cycles. The viscoelastic strain is nonlinear with respect to stress and is governed by a power law of time. The viscoplastic strain component is nonlinear with respect to stress and thus can be represented by the product of a second-order polynomial of stress and two power laws of time and number of cycles, respectively. The reliability of this constitutive equation was evaluated by means of two verification tests. Good agreement was found between the predicted and measured strains. (Author)

A micromechanical theory of grain-size dependence in metal plasticity

Abstract: T he effect of grain-size on the elastoplastic behavior of metals is investigated from the micromechanics standpoint. First, based on the observations that dislocation pile-ups, formation of cell structures, and other inelastic activities influenced by the presence of grain boundary actually take place transcrystallinely, a grain-size dependent constitutive equation is proposed for the slip deformation of slip systems. By means of a modified Hill's self-consistent relation the local stress of a grain is calculated, and used in conjunction with this constitutive equation to evaluate the plastic strain of each constituent grain. The grain-size effect on the plastic flow of polycrystals then can be determined by an averaging process. To check the validity of the proposed theory it was finally applied to predict the stress-strain curves and flow stresses of a copper at various grain-sizes. The obtained results were found to be in good agreement with experimental data.

An encapsulted dumbbell model for concentrated polymer solutions and melts I. Theoretical development and constitutive equation

Abstract: The finitely extensible nonlinear elastic (FENE) dumbbell model used earlier for dilute polymer solutions may also be used for concentrated solutions and melts if the Brownian motion and hydrodynamic forces acting on the beads are made nonisotropic. This corresponds roughly to imagining that a dumbbell is constrained to move within a capsule-shaped region embedded in the surrounding fluid. The constitutive equation obtained (a modified convected Maxwell model) contains four parameters: σ and β specify the extent of anisotropy in Stokes' law and in the Brownian motion forces respectively; λ is a time constant; and b describes the finite extensibility of the dumbbell. When σ and β are set equal to unity, the dilute-solution constitutive equation is recovered. In this sense the molecular theory given here permits a description of polymeric liquids over the entire concentration range.

Growth and dynamics of the Nordre Strømfjord Shear Zone

Abstract: The Nordre Stromfjord shear zone is exposed for a length of 170 km within the Precambrian Nagssugtoqidian Mobile Belt of West Greenland. Its western 100 km consists of a 15–20 km wide zone of syndeformational granulite facies rocks. The movement was sinistral with a displacement of 120 km across the zone. The strain across the shear zone margin increases exponentially but levels off to a near constant value over major sections of the inner zone. This profile suggests a growth model, in which the marginal zones are focii of instantaneous deformation that moves outward as hardening occurs along the inner boundaries and weakening occurs along the outer boundaries. This growth model allows the minimum possible viscosity change associated with the zone to be calculated from the logarithmic strain gradient and from the width across which weakening occurs. When applied to an area within the granulite grade part of the shear zone, this model indicates a decrease in viscosity by a factor 10 to 50, and the functional form of this weakening shows that the parameter undergoing change enters the exponential part of the appropriate constitutive equation. The available evidence seems to exclude shear heating as a viable weakening mechanism but supports chemical weakening. It is hypothesized that ‘hydrolytic weakening' was responsible. At higher crustal levels a fundamental change in geometry of the Nordre Stromfjord zone occurs. The applicability of the present model is thus restricted to deep crustal levels. In conclusion it is illustrated that the model is of relevance to small scale shear zones as well and may also be representative of mantle deformation.

Perturbation theory for viscoelastic fluids between eccentric rotating cylinders

Abstract: The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.

A Constitutive Formulation for High-Elongation Propellants

Abstract: A constitutive model is presented that appears to successfully represent the stress-strain behavior of a highelongation propellant over a wide range of conditions. The model uses a Lagrangian formulation and separates the propellant response into shearing and volumetric components. Particular features of the model are the use of a strain-softening function multiplying a convolution integral that is analogous to that of linear viscoelasticity. Within the integral is an additional term that modifies the history effects. This term appears to correlate behavior at changing strain rate with that at changing temperature. Comparisons with experimental data are given.

Finite element simulation of die swell for a Maxwell fluid

Abstract: The finite element formulation recently developed [1] for viscoelastic fluids of singly memory type has been extended to handle problems involving free surfaces. Previously [1], flows without free surfaces were demonstrated by solving the die entry and reverse entry flows. The single integral Maxwell fluid is retained here as the constitutive model for this further test of the technique. New work in this paper includes an algorithm to track particles on the free surface and a new free surface updating scheme. For the present die swell problem, the method successfully converged up to Deborah number of about 1. As in the previous problems, convergence appears to be limited by numerically induced incompatibility of large strains. The resulting errors in the computed stress field have a more pronounced impact on free surface flows.

A viscoplastic analysis of rapid crack propagation experiments in steel

Abstract: I n order to investigate rapid Mode I crack propagation in steel under large scale yielding conditions a finite element program was developed in which the material is treated as elastic-viscoplastic, based on a theory by P. P erzyna . In this model, rate effects in the plastic deformation process are accounted for. The parameters of the constitutive model were determined from ordinary tensile tests on small specimens by varying the loading rate. It was found that the strain rate can satisfactorily be related to the increase of the inelastic stress-strain relation raised to some power n . The FEM-program was applied to a number of crack propagation experiments on highly loaded SEN specimens of a high strength steel. For comparison some experiments were also analysed assuming linear elastic and simulated elastic-plastic material behaviour with no rate sensitivity. It was found that combined plastic and viscous effects are of great importance for an adequate description of rapid crack growth when large scale yielding is at hand. The energy flow to the crack tip region in the viscoplastic case, which is used as a fracture parameter, seems to converge to finite and non-trivial values for sufficiently small values of the element size along the crack tip. This may be due to the asymptotic dominance of the elastic field at the crack-tip for the particular rate-dependent model used. Geometry effects, which are present in a linear elastic analysis, are shown to be negligible in the present elastic-viscoplastic analysis with respect to different specimen heights.

Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. Part I: Theoretical development

Abstract: A general method is presented for the isothermal mechanical analysis of incompressible material solids in which a small-amplitude time harmonic oscillation is superposed on a static finite deformation field. Material behaviour is assumed to be of the ‘fading memory’ type describable by the finite linear viscoelasticity theory of Coleman and Noll.4 Existing finite element codes, that treat finite elasticity material behaviour, can be extended with minor modifications to treat the subject problem. A specialized form of the finite linear viscoelasticity constitutive equations proposed by Morman8 has been implemented in the MARC nonlinear finite element program for predicting the response of statically deformed elastomeric components to small amplitude vibrations. Numerical results obtained with MARC for the static force-deflection behaviour and dynamic (complex) stiffness for a viscoelastic cylinder subject to combined axial and twisting pre-loads, are in excellent agreement with corresponding analytical results.

A constitutive model for sand with anisotropic hardening rule

Abstract: An anisotropic hardening model for sand or other granular materials is presented. The state of initial densification is represented by a configuration surface and the state of loading by a loading surface in the stress space. The domain of elastic response may not exist and the irreversible strain is assumed to occur for both active and reverse loading. The derived constitutive equations are applicable both for dense and loose sands. Some model predictions are compared with available experimental data for triaxial compression and extension under undrained condition.