A non linear integral constitutive equation for viscoelastic fluids

An intégral constitutive equation is written using a particular reference frame, built with unit vectors along the principal axes of the rate-of-deformation tensor, and using the associated intrinsic rate-of-rotation. This equation is easier to handle in calculations than corotational or codeformational models. The material functions for a rheological model including the first six terms of the constitutive equation have been studied in steady and unsteady shear flows, as well as in elongational flows. Material functions are readily written from six memory functions and no inconsistency comes out.     

Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage

Nonequilibrium thermodynamics, rate-process theory, viscoelastic fracture mechanics and various experimentally-motivated simplifications are used to develop constitutive equations that account for effects of viscoelasticity, viscoplasticity, growing damage and aging. Their form is more general than previously developed by the author, and allows for relatively general tensorial effects of damage. Some important special cases are then covered, with emphasis on viscoelasticity. Evolution equations for the damage expressed in terms of internal state variables (ISVs) are discussed, comparing formulations using scalar ISVs and tensor ISVs. Finally, some experimental support for the theory is described. An Appendix illustrates the theory for an aging, linear viscoelastic material with growing cracks. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonlinear constitutive equations for transient creep of viscoelastic polymers including the effect of hydrostatic pressure

The behaviour of polymers is known to be significantly influenced by the hydrostatic pressure in creep deformation or elastic-plastic deformation. The effect of the third stress invariant on the nonlinear viscoelastic deformation is much smaller than that of the hydrostatic pressure. In this paper, a constitutive equation for transient creep is proposed, which includes the effect of the hydrostatic pressure on the yield function. The creep and plastic strains or the creep strain rate converge to zero with increasing hydrostatic pressure. The proposed constitutive equation is in good agreement with the actual creep data of cellulose nitrate and cellulose acetate, under various combinations of superimposed tensile and hydrostatic loadings.     

Single integral representations for nonlinear viscoelastic solids

The finite linear viscoelastic solid and single integral representation with nonlinear dependence on history are investigated in uni-axial stress. Both integral kernels in the stress formulation are determined by single-step constant strain tests, and both kernels in the strain formulation are determined by single-step constant stress tests. Single integral stress and strain formulations are not equivalent. The stress histories required to maintain constant strain-rate for both models are determined from the Volterra integral equations given by the strain formulations once their kernels are determined by constant stress tests. However, known constant strain-rate response does not determine the kernels. Examples are presented to show that variation of the kernel within a given qualitative shape can lead to different shapes of constant strain-rate response, so that both constant stress and constant strain-rate tests may be necessary to deduce the optimum single integral approximation, in preference to multi-step stress tests. It is shown that the apparently simpler finite linear viscoelastic model requires a far lengthier numerical algorithm to solve the Volterra equation, and leads to non-unique and physically unacceptable response, in comparison with the more flexible nonlinear history dependence which yields unique acceptable responses.     

Integral representations for the viscoelastic deformation of ice

Various single integral representations which describe non-linear viscoelastic response are examined with regard to the types of test required to determine the respective kernels. A strain formulation determined by constant uni-axial stress response typical of ice, and its predictions for constant strain-rate response, are reviewed, showing that the latter are sensitive to kernel detail. An alternative stress formulation which is determined by constant strain-rate response is constructed, and it is shown that the predicted strain and strain-rate responses at constant stress are compatible with the typical responses exhibited by ice.     

A three-field formulation for incompressible viscoelastic fluids

This paper presents a new stabilized finite element method for incompressible viscoelastic fluids. A three-field formulation is developed wherein Oldroyd-B model is coupled with the mass and momentum conservation equations for an incompressible viscous fluid. The variational multiscale (VMS) framework is employed to develop a stabilized formulation for the coupled momentum, continuity and stress equations. Based on the new stabilized method a family of linear and higher-order triangle and quadrilateral elements with equal-order velocity-pressure-stress fields is developed. Stability and convergence property of the various elements is studied and optimal rates are attained in the norms considered. The method is applied to some benchmark problems and accuracy and computational economy of the formulation is investigated for various flow conditions.     

Isotropic linear constitutive relations for nonsimple fluids

We investigate the general constitutive relation of an isotropic linear fluid when the stress tensor can depend on higher-order spatial gradients of the velocity. We apply the results to the case of second-grade and third-grade fluids, be they compressible or not. However, the expression of the general isotropic tensor can be a matter of interest also for other classes of nonsimple material.     

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.     

Asymptotic constitutive approximations for rapid deformations of viscoelastic materials

Summary Asymptotic constitutive approximations for the rapid finite deformation of general viscoelastic materials are developed. The zero-order approximation is an elastic-type constitutive equation, although different from the elastic equation for the slow-deformation approximation. The higher-order terms are multiple integrals of the departure of the deformation history from the step-function history. The approximations are shown to be form-invariant to change of deformation measure. The approximations are specialized to isotropic solids and to fluids. As observed byMetzner, White andDenn [3] andPipkin [4], fluids undergoing rapid deformations exhibit a solid-like behavior.

---

Zusammenfassung Es werden asymptotische Materialgleichungen für die rasche endliche Deformation eines allgemeinen viskoelastischen Materials entwickelt. Die nullte Näherung ist eine Materialgleichung vom elastischen Typ, wenn auch verschieden von der elastischen Näherungsgleichung für langsame Deformationen. Die Glieder höherer Ordnung sind Vielfachintegrale der Abweichung der Deformationsgeschichte von der Stufenfunktionsgeschichte. Es wird gezeigt, daß die Näherungen invariant sind gegen eine Änderung des Deformationsmaßes. Sie werden für isotrope Festkörper und für Flüssigkeiten spezialisiert. WieMetzner, White undDenn [3] sowiePipkin [4] beobachtet haben, zeigen Flüssigkeiten, die raschen Deformationen unterworfen werden, ein festkörperähnliches Verhalten.

     

The SPH equations for fluids

Existing smoothed particle hydrodynamics (SPH) formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the SPH equations do not satisfy low‐order polynomial completeness conditions. Here SPH equations are derived that have convergent error terms and a correction method is presented for enforcing low‐order polynomial completeness irrespective of how many completeness conditions are required. Discretization is achieved through division of the model domain, in its initial state, into sub‐domains that have Lagrangian boundaries. It is shown that boundary integrals appearing in one derivation of the SPH equations may be treated as a convergent error. In simulations of basic fluid flows convergence and zeroth‐order completeness are demonstrated, but significant instabilities and a failure to conserve energy are observed. Copyright © 2009 John Wiley & Sons, Ltd.     

Fundamental solutions for micropolar fluids

New fundamental solutions for micropolar fluids are derived in explicit form for two- and three-dimensional steady unbounded Stokes and Oseen flows due to a point force and a point couple, including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a low-Reynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid.     

Non-local constitutive equations for functionally graded materials

In this paper, composites with a graded distribution of heterogeneities are considered. The heterogeneities vary in statistically non-uniform fashion since in a finite layer (or region) properties such as local volume fraction vary gradually. In order to study this class of composites, a procedure of analysis which leads to the effective constitutive non-local operator of the medium is proposed. For two-phase composites, an approximation of Hashin–Shtrikman type for this operator has been obtained in real space and this has been developed explicitly in the case of laminates.     

Perturbed hard-core equations of state for fluids

The abilities of several equations of state in fitting p-V-T data of pure fluids have been compared. Also the temperature dependence of the characteristic parameters of each substance has been studied. All the equations of state discussed are of the form: p = p (ref) + p (att) where p(ref) accounts for the repulsive contributions to the pressure, while p(att) contains the contributions from the attractive forces. The best results correspond to an equation derived by Miyano and Masuoka from a perturbation theory, though Deiters equation gives results which are almost comparable to Miyano's one and it seems that Deiters equation is more suitable for further improvements and extension to more complex fluids.     

Rate-type constitutive equations and elastic-plastic materials

This paper is concerned with a class of rate-type constitutive equations, which characterize the behavior of elastic-plastic solids under large deformation. The theory is developed without à priori introduction of the concept of plastic strain and involves only the stress and the total strain as basic ingredients, apart from the thermodynamic variables. The relationship between the results obtained here and those of a more familiar form of a theory of elastic-plastic solids is indicated.