An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

Kinetic wrinkling of an elastic film on a viscoelastic substrate

A compressed elastic film on a compliant substrate can form wrinkles. On an elastic substrate, equilibrium and energetics set the critical condition and select the wrinkle wavelength and amplitude. On a viscous substrate, wrinkle grows over time and the kinetics selects the fastest growing wavelength. More generally, on a viscoelastic substrate, both energetics and kinetics play important roles in determining the critical condition, the growth rate, and the wavelength. This paper studies the wrinkling process of an elastic film on a viscoelastic layer, which in turn lies on a rigid substrate. The film is elastic and modeled by the nonlinear von Karman plate theory. The substrate is linear viscoelastic with a relaxation modulus typical of a cross-linked polymer. Beyond a critical stress, the film wrinkles by the out-of-plane displacement but remains bonded to the substrate. This study considers plane strain wrinkling and neglects the in-plane displacement. A classification of the wrinkling behavior is made based on the critical conditions at the elastic limits, the glassy and rubbery states of the viscoelastic substrate. Linear perturbation analyses are conducted to reveal the kinetics of wrinkling in films subjected to intermediate and large compressive stresses. It is shown that, depending on the stress level, the growth of wrinkles at the initial stage can be exponential, accelerating, linear, or decelerating. In all cases, the wrinkle amplitude saturates at an equilibrium state after a long time. Subsequently, both amplitude and wavelength of the wrinkle evolve, but the process is kinetically constrained and slow compared to the initial growth.     

On a variant of the Maxwell and Oldroyd-B models within the context of a thermodynamic basis

In this paper we develop models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and Oldroyd-B models but differ from them in one important aspect, the manner in which they unload instantaneously from the deformed configuration. As long as the response is not instantaneous, the models that are derived cannot be differentiated from the Maxwell and Oldroyd-B models, respectively. The models can be viewed within the context of materials whose natural configuration evolves, the evolution being determined by the maximization of the rate of entropy production of the material. However, the underpinnings to develop the model are quite different from an earlier development by Rajagopal and Srinivasa [8] in that while the total response of the viscoelastic fluid satisfies the constraint of an incompressible material, the energy storage mechanism associated with the elastic response is allowed to be that for a compressible elastic solid and the dissipative mechanism associated with the viscous response allowed to be that for a compressible fluid, the total deformation however being isochoric. The analysis calls for a careful evaluation of firmly held customs in viscoelasticity wherein it is assumed that it is possible to subject a material to a purely instantaneous elastic response without any dissipation whatsoever. Finally, while the model developed by Rajagopal and Srinivasa [8] arises from the linearization of the non-linear elastic response that they chose and leads to a model wherein the instantaneous elastic response is isochoric, here we develop the model within the context of a different non-linear elastic response that need not be linearized but the instantaneous elastic response not necessarily being isochoric.     

Mode mixity and nonlinear viscous effects on toughness of interfaces

This paper examines steady-state crack growth at interfaces between polymeric materials and hard substrates under quasi-static conditions. The polymeric material is taken to be an elastic nonlinear viscous solid while the substrate is treated as a rigid material. Void growth and coalescence in the rate-dependent fracture process zone is modeled by a nonlinear viscous porous strip of cell elements. In the first part of this paper, the polymeric background material surrounding the process zone is assumed to be purely elastic. Under fixed mode mixity, the computed interface toughness is found to be a monotonically increasing function of crack velocity; toughness also increases rapidly with higher rate sensitivity. This behavior can be explained in terms of voids growing in a strain-rate strengthened process zone. In the second part of the paper, the background material is also treated as an elastic nonlinear viscous solid. The competition between work of separation in the process zone and energy dissipation in the background material leads to a U-shaped toughness–crack velocity curve. Effects of mode mixity, initial porosity, rate sensitivity, as well as the initial yield strain on toughness are studied. The simulations produce trends that agree with interface toughness vs. crack velocity data reported in experimental studies for rubber toughened epoxy-paste adhesive and urethane acrylate adhesive.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

Motion equations of the dynamic theory of consolidation from the point of view of the theory of transient pipe flows

An elastic fluid-saturated porous medium is modeled as a bundle of parallel cylindrical tubes aligned in a direction parallel to the fluid movement. The pore space is filled with viscous compressible liquid. A cell model and the theory of transient pipe flow are used to derive one-dimensional governing equations in such media. All macroscopic constants in these equations are defined by the individual material constants of the fluid and solid. The interaction force includes an additional term not found in Biot's theory.     

Aging, rejuvenation, and thixotropy in yielding magnetorheological fluids

The yielding behavior of dilute magnetorheological (MR) fluids has been investigated using creep–recovery tests. At very low stress levels, MR fluids behave in the linear viscoelastic regime as demonstrated by the fact that the instantaneous strain equals the instantaneous (elastic) recovery. In this region, gap-spanning field-induced structures support the stress levels applied. Upon increasing the stress value, the MR fluid evolves towards a nonlinear viscoelastic response. Here, the retarded elastic and viscous strain decrease, and the plastic contribution to the instantaneous strain grows probably due to the appearance of unattached field-induced structures. A larger stress value results in a viscoplastic solid behavior with negligible retarded and viscous strain and a fully plastic instantaneous strain. Finally, a plastic fluid behavior is found when the stress value is larger than the so-called yield stress. MR fluids exhibit an intermediate behavior between non-thixotropic (simple) and highly thixotropic model yield stress fluids.     

Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers

A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers.Micro-mechanically based constitutive equations are developed for each phase. A viscoplastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains’ elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tie molecules linking the neighboring lamellae is also taken into account.Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress-strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed).Computational algorithms are developed based on fully implicit time-discretization schemes.     

Flow of a mixture of a viscous fluid and a granular solid in an orthogonal rheometer

In this paper, we study the flow of a linearly viscous fluid and a granular solid, consisting of many particles, situated between two parallel plates rotating about different axes. Flow in orthogonal rheometers has been studied for many viscoelastic fluids so that their rheological properties can be measured. The mixture is modeled using the theory of interacting continua, and constitutive relations for the fluid phase, the granular phase, and the interaction forces are provided. For a very special case, an analytical solution to the equations of motion is also provided.     

The “decaying springs” model for irreversibility in polymer melts

In this work, entanglements in a polymer melt are modeled as a system of parallel springs which form and decay spontaneously. The springs are assumed to be nonlinear, and a certain fraction of them is torn apart by a certain strain.Based on these assumptions, a model of behavior in simple shear is developed. This model is shown to predict a behavior comprising that of a Wagner fluid, and is generalized to a tensorial model of single integral type. The integrand depends on a product of a material function, modeling reversible behavior, and a material functional which takes irreversible processes into account.Irreversibility of network disentanglement, which may occur when deformation changes or reverses direction, can be modeled in this way. It is shown that the two well-known Wagner constitutive equations with and without irreversibility assumptions are special cases of the model developed. In case of a deformation which does not change directions, the new material function and the material functional are multiplied to yield Wagner's damping function.When the rate of spring formation is a function of temperature, the developed model is shown to predict thermorheologically simple behavior. A constitutive equation for non-isothermal flow of polymers is developed with this assumption.     

A Dynamic Method for the Residual Stress Measurement During Polymer Crystallization

A new dynamic method based on bilayer system is proposed to characterize the residual stress formation during the crystallization of semi-crystalline polyethylene glycol 10000 (PEG10000). The resin is coated on a solid polymeric film to create a filmsubstrate compound. Its temperature field and dynamic deflection are monitored by synchronized optical and thermography cameras. The crystallization kinetics is first characterized from the former information. Then a simple dynamic model is proposed to relate the dynamic deflection with crystallization process. Residual stresses are established and in the range of 0–2.1 MPa. The generation of residual stresses is due to the edge constraints of the cantilever beam and to the increase of viscosity during solidification that allows the polymer to carry tensile loadings. The spherulite impingement is found to be important for this period from a microscopic view. Boundary condition should be well controlled to steer residual stresses. Such method is promising to measure residual stresses at the micro-scale for polymers to be spread on a flexible substrate and can mimic different mechanical situations of interest.     

Effects of dynamic viscoelastic properties on acoustic diffraction by a solid sphere submerged in a viscous fluid

Summary  This study provides a general analysis for scattering of a planar monochromatic compressional sound wave by a homogeneous, isotropic, viscoelastic, solid sphere immersed in an unbounded viscous, heat-nonconducting, compressible fluid. The dynamic viscoelastic properties of the spherical scatterer and the viscosity of the surrounding fluid are rigorously taken into account in the solution of the acoustic-scattering problem. Havriliak–Negami model for viscoelastic material behaviour along with the appropriate wave-harmonic field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. Subsequently, the associated acoustic quantities such as the scattered far-field pressure directivity pattern, scattered intensity distribution, differential scattering cross section, and the acoustic radiation force are evaluated for given sets of viscoelastic material properties. Numerical results clearly indicate that, in addition to the traditional fluid viscosity-related mechanisms, the dynamic viscoelastic properties of the solid obstacle can be of major significance in sound scattering. Limiting cases are examined and fair agreements with well-known solutions are established. Received 15 January 2002; accepted for publication 2 July 2002 The authors wish to sincerely thank professors Daniel Levesque, Roderic Lakes, Yves Berthelot, S. Temkin, and Andrei Dukhin for valuable and productive consultations on dynamic theory of viscoelasticity and acoustics of (thermo)viscous media.     

Separation strain rate dependence on the failure of polymer adhesion with mobile promoters

Comprehensive molecular dynamics simulations, employing a coarse-grained bead-spring model, are conducted to study the failure of adhesion between two immiscible polymers stitched together via mobile promoters. A realistic model under separating tension is constructed that enables both chain pulling out viscously and bulk dissipation in two dissimilar glassy polymers that one is dense melt and another is loose. The contributions to the adhesion energy from thermodynamics and chain suction are studied for dependence of the strain rate at fixed basic molecular parameters. With low density of connectors, either adhesion toughness or strength changes slightly with separation strain rate as viscous loss is negligible. But rate effects become evident for long connectors with high density, viscoelastic sliding friction and reptation of chains dominate and the fracture energy increases with strain rate. The results provide insights into the evolution of adhesion surfaces coupled with promoter molecular slipping out of bulk melts, which are useful for future developments of continuum models for failure of polymeric interfaces.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Numerical modeling and investigation of viscoelastic fluid–structure interaction applying an implicit partitioned coupling algorithm

We investigate the interaction between a viscoelastic Oldroyd-B fluid and an elastic structure via simulations applying an implicit partitioned coupling algorithm. Simulations are done for a flow through a channel with a flexible wall and a lid-driven cavity flow with flexible bottom. In addition, we make use of a mass–spring–dashpot prototype model to study the dynamic interaction problem. Both the simulation results and the analysis of the prototype model show that there are obvious differences in the fluid–structure interaction if the fluids are viscoelastic instead of purely viscous. These differences appear in the deformation of the solid at stationary state and in the equilibrium position, amplitude, frequency as well as phase shift of the oscillation. Moreover, we investigate the influence of numerical and physical parameters on the implicit partitioned coupling algorithm for simulation of viscoelastic fluid–structure interaction problems.     

Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber

Through direct comparisons with experiments, Lefèvre et al. (Int. J. Frac. 192:1–23, 2015) have recently confirmed the prevailing belief that the nonlinear elastic properties of rubber play a significant role in the so-called phenomenon of cavitation—that is, the sudden growth of inherent defects in rubber into large enclosed cavities/cracks in response to external stimuli. These comparisons have also made it plain that cavitation in rubber is first and foremost a fracture process that may possibly depend, in addition to the nonlinear elastic properties of the rubber, on inertial effects and/or on the viscous dissipation innate to rubber. This is because the growth of defects into large cavities/cracks is locally in time an extremely fast process.The purpose of this Note is to provide insight into the relevance of inertial and viscous dissipation effects on the onset of cavitation in rubber. To this end, leaving fracture properties aside, we consider the basic problem of the radially symmetric dynamic deformation of a spherical defect embedded at the center of a ball made up of an isotropic incompressible nonlinear viscoelastic solid that is subjected to external hydrostatic loading. Specifically, the defect is taken to be vacuous and the viscoelastic behavior of the solid is characterized by a fairly general class of constitutive relations given in terms of two thermodynamic potentials—namely, a free energy function describing the nonlinear elasticity of the solid and a dissipation potential describing its viscous dissipation—which has been shown to be capable to describe the mechanical response of a broad variety of rubbers over wide ranges of deformations and deformations rates. It is found that the onset of cavitation is not affected by inertial effects so long as the external loads are not applied at a high rate. On the other hand, even when the external loads are applied quasi-statically, viscous dissipation can greatly affect the critical values of the applied loads at which cavitation ensues.     

Cooling of a viscoelastic film during unsteady extrusion from an annular die

The unsteady extrusion of a viscoelastic film from an annular and axisymmetric die is examined. External, elastic, viscous and inertia forces deform the film, which is simultaneously cooled via forced convection to the ambient air. This moving boundary problem is solved by mapping the liquid/air interfaces onto fixed ones and by employing a regular perturbation expansion for all the dependent variables. The ratio of the film thickness to its inner radius at the exit of the die is used as the small parameter in the perturbation expansion. The fluid mechanical aspects of the process depend on the Stokes, Deborah, Reynolds, and Capillary numbers. The heat transfer in the film and to the environment gives rise to four additional dimensionless groups: the Peclet, Biot and Brinkman numbers and the activation energy, which determines the temperature dependence of fluid viscosity and elasticity. A variable heat transfer coefficient is also considered. For typical fluid properties and process conditions, the Peclet number is very large. In this case it is the ratio of the Biot to the Peclet number, the Stanton number, which arises in the energy conservation equation. It is shown that film cooling becomes important when the Stanton number and/or the activation energy are in the high-end of their typical values. In such cases, the cooling of the parison leads to a more uniform flow and shape for the film. The influence on the process of a variable heat transfer coefficient and the Brinkman number is small. Received: 7 April 1999/Accepted: 10 August 1999     

Constitutive model for a semi-crystalline polymer under dynamic loading

This paper deals with a viscoelastic–viscoplastic model for semi-crystalline polymers in crash applications. A polymer behaviour model is implemented in the commercial PAM CRASH© code thanks to a user material card. Global variables (load, displacement) and local variables (strain) are validated on flat and notched tensile specimens by comparing the numerical responses with data obtained by digital image correlation.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.