Showing papers on "Constitutive equation published in 1994"

Numerical simulations of fast crack growth in brittle solids

Abstract: Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that, for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29° from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.

Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept

Abstract: Conventional shell formulations, such as 3- or 5-parameter theories or even 6-parameter theories including the thickness change as extra parameter, require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness. This means that the normal stress in thickness direction has to either vanish or be constant. In general, these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions. The main objective of the present study is to introduce directly a complete 3-D constitutive law without modification. Therefore, a 7-parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

Fractal rheological models and fractional differential equations for viscoelastic behavior

Abstract: A constitutive equation for viscoelastic behavior containing time derivatives of stress and strain to fractional order is obtained from a fractal rheological model. Equivalence between tree and ladder fractal models at long times is demonstrated. The fractional differential equation is shown to be equivalent to ordinary differential formulations in the case of a simple power-law response; the adequacy of such formulations to describe non-linearity has been demonstrated previously. The model gives a good description of viscoelastic behavior under all stress modes and will be extended in future to include aging effects.

A new Reynolds stress algebraic equation model

Abstract: A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.

Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities

Abstract: A micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpole's bounds, and Willi's bounds for isotropic or anisotropic elastic multiphase composites are related to the “noninteracting” solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the “noninteracting” micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.

Aspects of the formulation and finite element implementation of large strain isotropic elasticity

Abstract: The paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation. On the theoretical side, an Eulerian setting of isotropic elasticity is discussed exclusively in terms of the Finger tensor as a strain measure. Noval aspects are a direct representation of the Eulerian elastic moduli in terms of the Finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. The isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity. On the computational side, the implementation of the constitutive model in three possible finite element formulations is discussed. After pointing out algorithmic techniques for the treatment of incompressible elasticity, several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity.

Constitutive models for porous materials with evolving microstructure

Abstract: A constitutive model is developed for the effective behavior of nonlinear porous materials which is capable of accounting, approximately, for the evolution of the material's microstructure under large quasi-static deformations. The model is formulated in terms of an effective potential function for the porous material, which depends on appropriate variables characterizing the state of the microstructure, together with evolution equations for these state variables. For the special case of triaxial loading of an initially isotropic porous material, the appropriate state variables are the porosity and the aspect ratios of the typical void; they serve respectively to characterize the evolution of the size and shape of the pores. The implications of the model are studied in the context of two specific examples: axisymmetric and plane strain loading conditions. It is found that the porosity acts as a hardening mechanism when the material is subjected to boundary conditions resulting in overall hydrostatic compression, and as a softening mechanism for overall hydrostatic tension. On the other hand, the change in shape of the voids is found to have a more subtle influence on the overall behavior of the porous material. Thus the change in shape of the voids has a direct effect which may range from strong softening during void collapse to slight hardening during void elongation, but it also has an indirect effect through its concomitant effect on the evolution of the porosity, which may actually be quite significant. Because of the complex interplay between these hardening-softening mechanisms, the new model is found to yield significantly different predictions, in particular for the onset of localization, than the well-known model of Gurson, which neglects the change in shape of the voids, especially, for low-triaxiality loading conditions. The model, in its present form, is not meant to be used for high-triaxialities for which the Gurson and other associated models are considered to be quite accurate.

Peristaltic flow of a second-order fluid in tubes

Abstract: We have analyzed the mechanics of peristaltic pumping of a non-Newtonian fluid through an axisymmetric conduit. The material was represented by the constitutive equation for a second-order fluid. A perturbation series (to second order) in dimensionless wavenumber of an infinite harmonic traveling wave was used to obtain explicit forms for the velocity field and a relation between the flow rate and the pressure gradient, in terms of the Reynolds number, the dimensionless non-Newtonian parameters, and the occlusion. Results were compared with other studies, in both Newtonian and non-Newtonian cases. Also, we have shown that the flow of a Newtonian fluid through a rigid, axisymmetric tube with an axial, sinusoidal variation of radius is a special case of this analysis.

Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering—II. Diffusional densification and creep

Abstract: Constitutive equations for the intermediate and late stages of sintering are derived assuming grain boundary diffusion to be the controlling transport mechanism. The surface of the pore space is assumed to be in equilibrium. Numerical results for the shape of equilibrium surfaces are taken from a companion paper [J. Svoboda, H. Riedel and H. Zipse, Acta metall. mater.42, 435 (1994)]. For open porosity the contacts between grains turn out to be nearly circular in all cases, so that the diffusion problem in the contact areas can be solved readily. The macroscopic response of the powder compact can then be built up of the mechanical and the surface tension forces acting across the contacts. The analysis is carried out for a random, or homogeneous distribution of contacts around each particle, and for simple cubic, body-centered cubic, and face-centered cubic grain arrangements. The cubic viscosities are averaged using various methods, the best of which appears to be the self-consistent method.

A Proposed Three-Dimensional Constitutive Model for Shape Memory Alloys

Abstract: The shape memory alloy (SMA) stress-strain response associated with martensitic twinning hysteresis and austenite-martensite/martensite-austenite superelasticity is modeled using constitutive equations. Compared to the modeling work done on viscoelastic and viscoplastic be havior, this has been an area of limited study. The equations which are presented here express the growth of inelastic strain in a rate-type formulation similar to viscoplastic laws. This constitutive model is obtained by extending a one-dimensional evolutionary model of SMA behavior to three dimensions. The resulting model is then reduced to meet the loading conditions of three special cases: uniaxial loading, shear loading, and non-proportional biaxial loading (combined axial- torsional loading). The model which is being considered, although nonlinear, is relatively simple in that only two evolutionary equations are required to model inelastic strain and a generalized back stress at a material point. Thus the model being presented use...

Effects of temperature on friction: Constitutive equations and experiments with quartz gouge

Abstract: A state-variable constitutive relation that can describe the dependence of friction on temperature at or near steady state conditions is presented. The relation is derived from existing state-variable relations used to describe velocity dependence and the assumption that the micro-mechanisms of friction are thermally activated and follow an Arrhenius relationship. The relation adequately describes the transient and steady friction behavior displayed in sliding experiments on 1.5-mm-thick layers of fine-grained (less than 100 micron diameter) quartz gouge. Gouge layers were sheared up to 10 mm between rough steel surfaces at a constant effective confining pressure of 20 MPa and under room-dry or water-saturated conditions in a servo-controlled triaxial apparatus. In each experiment, velocity was stepped between 4, 0.4, and 0.04 microns/s at constant temperature, and temperature was stepped between 24, 57, and 82 C at a constant velocity of 0.04 microns/s. Coefficient of friction was calculated from measurements of sample strength using corrections for all apparatus effects including the temperature and velocity dependent strength of the metal sleeves used to isolate the sample and of the graphite-lubricated interface that allowed lateral slip of the sample halves. Experiments indicate that (1) an abrupt increase in temperature induces a transient friction response similarmore » to that induced by a step decrease in velocity, (2) the transient friction response is relatively symmetric for steps up and steps down in temperature, and (3) the characteristic slip distance for friction to evolve to steady state after step changes in temperature is the same as after step changes in velocity. The apparent activation energy determined for wet quartz gouge at the test conditions is 89 +/- 23 kJ/mol. (Abstract Truncated)« less

Mechanisms-based creep constitutive equations for an aluminium alloy

Abstract: A number of mechanisms-based constitutive equations were assessed in an effort to describe the creep behaviour of an aluminium alloy at 150°C. It was found that a sinh function of stress, r...

FEM-FDM coupled liquefaction analysis of a porous soil using an elasto-plastic model

Abstract: The phenomenon of liquefaction is one of the most important subjects in Earthquake Engineering and Coastal Engineering. In the present study, the governing equations of such coupling problems as soil skeleton and pore water are obtained through application of the two-phase mixture theory. Using au-p (displacement of the solid phase-pore water pressure) formulation, a simple and practical numerical method for the liquefaction analysis is formulated. The finite difference method (FDM) is used for the spatial discretization of the continuity equation to define the pore water pressure at the center of the element, while the finite element method (FEM) is used for the spatial discretization of the equilibrium equation. FEM-FDM coupled analysis succeeds in reducing the degrees of freedom in the descretized equations. The accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids. An elasto-plastic constitutive model based on the non-linear kinematic hardening rule is formulated to describe the stress-strain behavior of granular materials under cyclic loading. Finally, the applicability of the proposed numerical method is examined. The following two numerical examples are analyzed in this study: (1) the behavior of seabed deposits under wave action, and (2) a numerical simulation of shaking table test of coal fly ash deposit.

Fracture Energy Formulation for Inelastic Behavior of Plain Concrete

Abstract: A constitutive formulation is presented that covers the triaxial load‐response spectrum of plain concrete in tension as well as in shear. The elastoplastic concrete model resorts to an isotropic‐hardening description of the prepeak behavior and to a fracture energy‐based isotropic‐softening description of the post‐peak response. To control inelastic dilatancy, a nonassociated plastic flow rule is adopted in regard to the inelastic volume change. The constitutive parameters are calibrated from a series of stroke‐controlled laboratory experiments that include the direct‐tension test and triaxial compression tests at three different levels of confinement. The predictive capabilities of the proposed model are assessed with a broad range of experimental data. The issue of failure on the constitutive level is addressed with the aid of the instability indicator for continuous material branching and the localization indicator for the formation of weak discontinuities. The relevance of these two failure diagnostic...

The large strain compression, tension, and simple shear of polycarbonate

Abstract: Polymeric materials subjected to large strains undergo an evolution in molecular orientation. The developing orientation and corresponding strengthening are highly dependent on the state of strain. In this paper, we examine and compare the very different stress-strain results of polycarbonate produced from four types of mechanical testing: uniaxial compression, plane strain compression, uniaxial tension, and simple shear. These tests produce different states of orientation within the material and, in the case of simple shear, the principle axes of orientation rotate during the deformation. The ability of the recent constitutive model of Anuda and Boyce (1992) to predict the observed behavior is evaluated. The model has been incorporated into a finite element code in order to properly simulate the material behavior during the inhomogeneous deformations of tension (cold drawing) and simple shear. The material properties of the model are obtained from the uniaxial compression test and the model is then found to be truly predictive of the other states of deformation demonstrating its fully three dimensional capability. The disadvantages of the tensile and simple shear tests for obtaining the data needed to accurately quantify the large strain material behavior of polymers are shown and discussed.

A simple hypoplastic constitutive model for sand

Abstract: The paper presents a hypoplastic constitutive model for the three-dimensional non-linear stress-strain and dilatant volume change behaviour of sand. The model is developed without recourse to the concept in elastoplasticity theory such as yield surface, plastic potential and decomposition into elastic and plastic parts. Benefited from the non-linear tensorial functions available from the representation theorem the model possesses simple mathematical formulation and contains only four material parameters, which can be easily identified with triaxial compression tests. Comparison of the predictions with the experimental results shows that the model is capable of capturing the salient behaviour of sand under monotonic loading and is applicable to both drained and undrained conditions.

A nonproportionality parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and memory effects of isotropic hardening.

Abstract: The present paper is concerned with the formultion of a viscoplastic constitutive model describing both cyclic hardening and cyclic softening under proportional and nonproportional loading conditions. First a nonproportionality parameter and the relevant internal structural tensor to describe the nonproportional hardening is discussed in detail. Internal variables and the related evolution equations to describe the amplitude dependence of cyclic hardening/softening are also examined. Then the history effects of cyclic hardening and softening are considered in the evolution equations of isotropic hardening variable. The proposed model is established by incorporating these equations into Chaboche model

Methodology for Relating the Tensile Constitutive Behavior of Ceramic‐Matrix Composites to Constituent Properties

Abstract: A methodology for the straightforward and consistent evaluation of the constituent properties of ceramic-matrix composites (CMCs) is summarized, based on analyses from the literature. The results provide a constitutive law capable of simulating the stress/strain behavior of these materials. The approach is illustrated using data for two CMCs: SiC/CAS and SiC/SiC. The constituent properties are also used as input to mechanics procedures that characterize stress redistribution and predict the effect of strain concentrations on macroscopic performance.

Implementation of the transformation field analysis for inelastic composite materials

Abstract: The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (62); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.

Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates

Abstract: One objective of the present study is to use arbitrary complete 3-dimensional constitutive equations without reduction or manipulation in nonlinear plate and shell analysis. The obvious consequence, namely the extension of a conventional 5-parameter shell formulation with Reissner-Mindlin kinematics to a 6-parameter formulation including the full set of stress and strain state does not solve the problem because a significant error in bending dominated cases occurs. To avoid this error the transverse normal strain is allowed to vary linearly across the thickness. This so-called 7-parameter theory recently proposed in the group of the authors resorts to the Enhanced Assumed Strain concept and preserves the basic features of a displacement formulation.

A Fully Coupled Constitutive Model for Electrostrictive Ceramic Materials

Abstract: A three-dimensional, electromechanical constitutive law has been formulated for electrostrictive ceramic materials. This fully coupled, phenomenological model relates the key state variables of str...

Overview no. 117 The structure of constitutive laws for the sintering of fine grained materials

Abstract: This paper examines the general structure of constitutive laws for the sintering of fine grained materials for situations where power-law creep and grain-boundary diffusion are the dominant mechanisms of deformation and densification. The description of grain-boundary diffusion accepts an interface reaction. Constitutive laws for general multiaxial stress states are expressed in terms of scalar potential functions. Bounds to the potential are obtained which take into account the coupling between power-law creep and diffusion.