Fractional differential models in finite viscoelasticity

Summary A new class of constitutive models is derived for viscoelastic media with finite strains. The models employ the so-called fractional derivatives of tensor functions.We introduce fractional derivatives for an objective tensor which satisfies some natural assumptions. Afterwards, we construct fractional differential analogs of the Kelvin-Voigt, Maxwell, and Maxwell-Weichert constitutive models. The models are verified by comparison with experimental data for viscoelastic solids and fluids. We consider uniaxial tension of a bar and radial oscillations of a thick-walled spherical shell made of the fractional Kelvin-Voigt incompressible material. Explicit solutions to these problems are derived and compared with experimental data for styrene butadiene rubber and synthetic rubber. It is shown that the fractional Kelvin-Voigt model provides excellent prediction of experimental data. For uniaxial tension of a bar and simple shear of an infinite layer made of the fractional Maxwell compressible material, we develop explicit solutions and compare them with experimental data for polyisobutylene specimens. It is shown that the fractional Maxwell model ensures fair agreement between experimental data and results of numerical simulation. This model allows the number of adjustable parameters to be reduced significantly compared with other models which ensure the same level of accuracy in the prediction of experimental data.     

A variational formulation of constitutive models and updates in non‐linear finite viscoelasticity

The purpose of this article is to present a general framework for constitutive viscoelastic models in finite strain regime. The approach is qualified as variational since the constitutive updates obey a minimum principle within each load increment. The set of internal variables is strain‐based and employs, according to the specific model chosen, a multiplicative decomposition of strain into elastic and viscous components. The present approach shares the same technical procedures used for analogous models of plasticity or viscoplasticity, such as the solution of a minimization problem to identify inelastic updates and the use of exponential mapping for time integration. However, instead of using the classical decomposition of inelastic strains into amplitude and direction, we take advantage of a spectral decomposition that provides additional facilities to accommodate, into simple analytical expressions, a wide set of specific models. Moreover, appropriate choices of the constitutive potentials allow the reproduction of other formulations in the literature. The final part of the paper presents a set of numerical examples in order to explore the characteristics of the formulation as well as its applicability to usual large‐scale FEM analyses. Copyright © 2005 John Wiley & Sons, Ltd.     

On finite linear viscoelasticity of incompressible isotropic materials

Summary The assumption of incompressibility as a kinematic constraint condition leads to consequences, which are of physical interest in view of a thermodynamically consistent material modelling. Some of these consequences are discussed within the concept of finite linear viscoelasticity. We present two natural possibilities to generalise the familiar Maxwell-model to finite strains; both tensor-valued differential equations are integrated to yield the present Cauchy stress as a functional of the relative Piola or Green strain history. Both types of Maxwell-models are related to a free energy functional in the sense that the dissipation inequality is satisfied. The stress and energy functionals are generalised to incorporate arbitrary kernel functions of relaxation; the only restriction for thermodynamic consistency is that the relaxation functions have a negative slope and a positive curvature. The linear combination of the two types of energy functionals can be understood to be a generalisation of the Mooney-Rivlin model to viscoelasticity. The concrete representation of relaxation functions, motivated from a finite series of Maxwell-elements in parallel, implies a Prony series, corresponding to a discrete relaxation spectrum. A compact notation to express a continuous relaxation spectrum is provided by the concept of a derivative of fractional order. In particular, differential equations of fractional order lead to relaxation functions coming very close to the relaxation behaviour of real materials. As an example, the evolution equation of a fractional Maxwell-model is solved, leading to a relaxation function of the Mittag-Leffler type. An essential advantage is that a rather small number of material constants is involved in this model, while the accuracy of its prediction is quite good. Physically nonlinear effects, such as process-dependent viscosity and relaxation behaviour can be incorporated into finite linear viscoelasticity, if the natural time is replaced with a material-dependent time scale. The rate of the material-dependent time then depends on further internal variables to account for the influence of the process history; this kind of proceeding is fully compatible with the entropy inequality. Temperature-dependence is included, starting from a multiplicative decomposition of the deformation gradient into mechanical and thermal parts. The resulting constitutive theory has a rather simple structure and offers a considerable degree of freedom to allow for physically important phenomena. The numerical simulations illustrate a small selection of these possibilities.Notations 1 Unit tensor - Y^–1, Y ^T Inverse and transpose of a tensor Y - Y·X Scalar product of two tensors X and Y - ab Dyadic tensor-product of two vectors a and b - tr(Y), det (Y) Trace and determinant of a tensor Y - div v Divergence operator applied to a vector field v - I _Y ,II _Y Principal invariants of a tensor Y - ₁, ₂, ₃ Eigenvalues of a tensor - X, x(t) Place of a material particle in the reference and the current configuration - dX,dx(t) Tangent vectors of material lines in the reference and the current configuration - N, n(t) Normal vectors of material surfaces in the reference and the current configuration - F(t), F() Deformation gradients at timest and - F _t () Relative deformation gradient - B, C Right and left Cauchy-Green tensors - E, A, e, a Green, Almansi, Piola and Finger strain tensors - L, D Velocity gradient, symmetric part of the velocity gradient - _R , Mass densities of the reference and the current configuration - _i Stress power per unit mass - Thermodynamic temperature - , Free energy per unit mass, rate of entropy production per unit mass - q, g heat flux vector, temperature gradient - T, Cauchy- and Second Piola-Kirchhoff stress tensor - S, Weighted Cauchy- (or Kirchhoff-) and convected stress tensor - p(X,t) Hydrostatic pressure - S _eq , S _ov Equilibrium- and overstress tensors of the weighted Cauchy type - S _E , Extra stresses of the weighted Cauchy-, Second Piola-Kichhoff and convected type - , Uniaxial engineering stress and strain - Ê, _A , _B Elasticity modulus and shear moduli - z, z _A ,z _B Relaxation times - G _A (t),G _B (t) Relaxation functions - (x) Eulerian Gamma function - e ^x Exponential function - E (x) Mittag-Leffler function - O(x) Landau symbol - h(s) Influence function - F, F _M Thermal and mechanical parts of the deformation gradient - L, L _M Thermal and mechanical velocity gradients - R Mass density of the thermomechanical intermediate configuration - B _M , C _M Mechanical right and left Cauchy-Green tensors - z(t) Process-dependent time scale - F₀ Static predeformation - f(t) Incremental, time-dependent deformation gradient - E _L Linear incremental strain tensor     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Time reversal and symmetry relations in finite linear viscoelasticity

In this paper we postulate the invariance of the work under time reversal in closed infinitesimal strain paths starting from an equilibrium state of finite deformation. This postulate applied to the constitutive equations of finite linear viscoelasticity leads to certain relations which involve the relaxation functions of the constitutive equation and the initial strain. Subsequently we prove certain symmetry relations between the relaxation functions (which do not involve the strain explicitly) for isotropic and transversely isotropic materials.     

Thermodynamic formulation of finite linear viscoelasticity for anisotropic media

Summary The theory of irreversible thermodynamics of continuous media with fading memory is used to formulate general constitutive equations of finite linear viscoelasticity. Specific forms for transversely isotropic and orthotropic media based on both the mechanical and thermodynamic theories are found.

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Zusammenfassung Die Aufstellung allgemeiner Materialgleichungen der endlichen linearen Viskoelastizität wird mit Hilfe der irreversiblen Thermodynamik kontinuierlicher Medien mit Erinnerungsschwund durchgeführt. Spezielle Fassungen für transversal isotrope und orthotrope Medien, basierend auf der mechanischen und der thermodynamischen Theorie, werden angegeben.

     

An efficient finite element scheme for viscoelasticity with moving boundaries

 An efficient finite element scheme is devised for problems in linear viscoelasticity of solids with a moving boundary. Such problems arise, for example, in the burning process of solid fuel (propellant). Since viscoelastic constitutive behavior is inherently associated with a “memory,” the potential need to store and operate on the entire history of the numerical solution has been a source of concern in computational viscoelasticity. A well-known “memory trick” overcomes this difficulty in the fixed-boundary case. Here the “memory trick” is extended to problems involving moving boundaries. The computational aspects of this extended scheme are discussed, and its performance is demonstrated via a numerical example. In addition, a special numerical integration rule is proposed for the viscoelastic integral, which is more accurate than the commonly-used trapezoidal rule and does not require additional computational effort.     

Elastomer bushing response: experiments and finite element modeling

Summary.  Elastomer bushings are essential components in tuning suspension systems since they isolate vibration, reduce noise transmission, accommodate oscillatory motions and accept misalignment of axes. This work presents an experimental study in which bushings are subjected to radial, torsional and coupled radial-torsional modes of deformation. The experimental results show that the relationship between the forces and moments and their corresponding displacements and rotations is nonlinear and viscoelastic due to the nature of the elastomeric material. An interesting feature of the coupling response is that radial force decreases and then increases with torsion. The experimental results were used to assess bushing behavior and to determine the strength of radial-torsional coupling. The experimental results were also compared to finite element simulations of a model bushing. While finite element analysis predicted small displacements at the relaxed state reasonably well, the response to larger radial deformations and coupled deformations was not well captured. Received January 13, 2003 Published online: May 20, 2003 The authors would like to thank Tenneco Automotive and the Great Lakes Truck and Transit Research Center for support of this project.     

Formulation and implementation of three-dimensional viscoelasticity at small and finite strains

Purely elastic material models have a limited validity. Generally, a certain amount of energy absorbing behaviour can be observed experimentally for nearly any material. A large class of dissipative materials is described by a time- and frequency-dependent viscoelastic constitutive model. Typical representatives of this type are polymeric rubber materials. A linear viscoelastic approach at small and large strains is described in detail and this makes a very efficient numerical formulation possible. The underlying constitutive structure is the generalized Maxwell-element. The derivation of the numerical model is given. It will be shown that the developed isotropic algorithmic material tensor is even valid for the current configuration in the case of large strains. Aspects of evaluating experimental investigations as well as parameter identification are considered. Finally, finite element simulations of time-dependent deformations of rubber structures using mixed elements are presented. Communicated by S. N. Atluri, 5 September 1996     

Sor vs. conjugate gradients in a finite element discretization

Sucessive Overrlaxation and Conjugate Gradients are used to solve the linear algebraic system set up with finite elements for the discretization of a plane, linear, head conducting problem. It is numerically shown that even with the optimal overrelaxation factor SOR is hardly superior to CG which is decisively simpler to program.     

3-D schapery representation for non-linear viscoelasticity and finite element implementation

On the basis of the one-dimensional Schapery representation for non-linear viscoelasticity, a three-dimensional constitutive model incorporating the effects of temperature and physical ageing is developed for isotropic non-linear viscoelastic materials. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the contitutive equation is expressed in incremental form for both compressible and incompressible materials, with the hereditary integral updated at the end of each time increment by recursive computation. The proposed model is implemented in the finite element package MARC. Numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms.Laboratory for Engineering Mechanics, Delft University of Technology, P. O. Box 5033, 2600 GA Delft, The Netherlands     

On the introduction of viscoelasticity into one-dimensional models of arterial blood flow

Summary The occurrence of discontinuities in solutions of the equations governing one-dimensional elastic models of the aorta is to be expected. Such shocks are absent if viscoelastic effects are included in the behaviour of the aortic wall. Then a number of characteristic phenomena in the development of the pressure and velocity pulses propagating along the aorta are very similar to features of solutions to Burgers' equation. Accordingly it is shown that one may arrive at Burgers' equation in an approximate theory involving uniform viscoelastic tubes. A numerical technique eminently suited to treat steep gradients at a priori unknown positions is applied to illustrate the occurrence of shocks in an existing elastic model. Using the same technique also viscoelastic solutions are presented and compared with earlier results.     

Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity

The viscoelastic response of hydropolymers, which include glandular breast tissues, may be accurately characterized for some applications with as few as 3 rheological parameters by applying the Kelvin-Voigt fractional derivative (KVFD) modeling approach. We describe a technique for ultrasonic imaging of KVFD parameters in media undergoing unconfined, quasi-static, uniaxial compression. We analyze the KVFD parameter values in simulated and experimental echo data acquired from phantoms and show that the KVFD parameters may concisely characterize the viscoelastic properties of hydropolymers. We then interpret the KVFD parameter values for normal and cancerous breast tissues and hypothesize that this modeling approach may ultimately be applied to tumor differentiation.     

Calibration of a class of non-linear viscoelasticity models with adaptive error control

The calibration of constitutive models is considered as an optimization problem where parameter values are sought to minimize the discrepancy between measured and simulated response. Since a finite element method is used to solve an underlying state equation, discretization errors arise, which induce errors in the calibrated parameter values. In this paper, adaptive mesh refinement based on the pertinent dual solution is used in order to reduce discretization errors in the calibrated material parameters. By a sensitivity assessment, the influence from uncertainties in experimental data is estimated, which serves as a threshold under which there is no need to further reduce the discretization error. The adaptive strategy is employed to calibrate a viscoelasticity model with observed data from uniaxial compression (i.e., homogeneous stress state), where the FE-discretization in time is studied. The a posteriori error estimations show an acceptable quality in terms of effectivity measures.     

Bulk vs nanoscale WS2: finite size effects and solid-state lubrication

To investigate phonon confinement in nanoscale metal dichalcogenides, we measured the low-temperature specific heat of layered and nanoparticle WS2. Below 9 K, the specific heat of the nanoparticles deviates from that of the bulk counterpart. Further, it deviates from the usual T 3 dependence below 4 K due to finite size effects that eliminate long wavelength acoustic phonons and interparticle-motion entropy. This separation of nanoscale effects from T 3 dependence can be modeled by assuming that the phonon density of states is flexible, changing with size and shape. We invoke relationships between the low-temperature T 3 phonon term, Young's modulus, and friction coefficient to assess the difference in the tribological properties. On the basis of this analysis, we conclude that the improved lubrication properties of the nanoparticles are extrinsic.     

Smooth interfacing of finite element models

A technique for smoothing the transition between structural components modelled with dissimilar finite element meshes is presented. The interface mismatch is assigned a potential of dislocation, which is minimized with respect to selected kinematic parametes of the mating surface. This leads to linear constraints between the caididate representations of the mating surface which essentially dictate a least-square error in dislocaiton. The constraints are enforced by the Lagrange multiplier or reduced stiffness technique. Numerical results indicate gradual transition in the predicted stresses neighbouring the interface. Existing computer codes which permit multipoint constraint specifications are readily available to implement the technique.     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

Dynamic behavior of a tensile crack: finite difference simulation of fracture experiments

In an idealized fracture model a bilateral brittle tensile crack is assumed to propagate with constant fracture velocity up to its final length. The initial and boundary conditions are plane strain and uniaxial stress perpendicular to the crack surface, which is stress-free. The problem is tackled by solving the elastic wave equation with a finite difference technique. The time function of the displacement at a given point can be interpreted in terms of the arrival of various types of waves, originating from both the initiation and the termination of the fracture. The influence of the fracture velocity is discussed. The model is then modified to allow for a variable fracture velocity and a non-homogeneous prestress field. The results of these calculations are compared with experiments, which were performed with Araldite B. It was found that the finite difference technique can satisfactorily describe the propagation of a crack. The relationship between the fracture velocity and the dynamic stress intensity factor K _I is discussed. The critical K _I value for crack propagation is K _Ic20 to 30 N/mm^3/2. The fracture velocity seems to be limited by branching at about 550 m/s, which occurs at K _I values K _{I branch}150 to 250 N/mm^3/2.

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Résumé On suppose, dans un modèle idéalisé, une rupture fragile soumise à traction et se propageant de manière bilatérale avec une vitesse constante jusqu'à atteindre sa longueur finale. Les conditions initiales et les conditions aux limites sont celles d'un état plan de déformations et d'une tension normale à la surface de la fissure, laquelle est libre de tension. Le problème est attaqué en résolvant léquation d'onde élastique par une technique de différences finies. Le déplacement en fonction du temps d'un point donné peut être interprété comme l'aboutissement de divers types d'ondes, qui prennent naissance à la fois du point d'amorçage et du point d'arrêt de la rupture. On discute l'influence de la vitesse de la rupture, et le modèle subit une modification pour tenir compte d'une vitesse variable et d'un champ de précontrainte non homogène. Les résultats des calculs sont comparés à ceux d'expériences effectuées sur de l'Araldite B. On trouve que la technique des différences finies peut décrire la propagation d'une fissure de manière satisfaisante. On discute la relation entre la vitesse de la rupture et le facteur d'intensité des contraintes dynamiques K _I. La valeur critique de K _I correspondant à la propagation d'une fissure est K _Ic20 à 30 N/mm^3/2. La vitesse de la rupture paraît être limitée par une arborescence qui se produit à environ 550 m/sec et qui correspond à une valeur K _I de l'ordre de 150 à 250 N/mm^3/2.