Fractional Derivative Viscoelasticity at Large Deformations

A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grünwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.     

Amplitude dependence of filler-reinforced rubber: Experiments,constitutive modelling and FEM – Implementation

The focus of the present paper is the experimental investigation, the constitutive representation and the numerical simulation of the amplitude dependence of filler-reinforced elastomers. A standard way to investigate the dynamic properties of viscoelastic materials is via the dynamic modulus which is obtained from stress signals due to harmonic strain excitations. Based on comprehensive experimental data, an amplitude-dependent constitutive model of finite viscoelasticity is developed. The model is based on a modified Maxwell chain with process-dependent viscosities which depend on additional internal state variables. The evaluation of this thermodynamically consistent model is possible in both the time domain, via stress-time signals, and in the frequency domain, via the dynamic modulus. This property is very profitable for the parameter identification process. The implementation of the constitutive model into the commercial finite element code ANSYS with the user-programmable feature (UPF) USERMAT for large deformations in updated Lagrange formulation is presented. This implementation allows simulating the time-dependent behaviour of rubber components under arbitrary transient loading histories. Due to physical and geometrical nonlinearities, these simulations are not possible in the frequency domain. But, transient FEM computations of large loading histories are sometimes not possible in an acceptable time. In the context of the parameter identification the fundamental ideas are presented, how this problem has been solved. Transient FEM simulations of real rubber components are also shown to visualize the properties of the model in the context of the transient material behaviour.     

FE formulation for the viscoelastic body modeled by fractional constitutive law

This paper presents finite element (FE) formulation of the viscoelastic materials described by fractional constitutive law. The time-domain three-dimensional constitutive equation is constructed. The FE equations are set up by equations are solved by numerical integration method. The numerical algorithm developed by the authors for Liouville-Riemann's fractional derivative was adopted to formulate FE procedures and extended to solve the more general case of the hereditary integration. The numerical examples were given to show the correctness and effectiveness of the integration algorithm. The project supported by the Ministry of Education of China for the returned overseas Chinese scholars     

有初始几何缺陷的壳体弹塑性大变形分析

本文在等温小变形弹塑性内时本构方程偏量形式的基础上，导出了适用于大位移、大转动、小应变分析的弹塑性内时本构方程，进一步推导出了带有初始几何缺陷的非线性弹塑性问题的有限元方程，可用于分析缺陷对结构非线性弹塑性反应的影响，也可用于带缺陷的非线性问题求解及稳定性分析．     

Inflation of a bonded non-linear viscoelastic toroidal membrane

The inflation of a bonded viscoelastic toroidal membrane under finite deformations is considered. Three new variables, viz. the two principal stretch ratios and the angle between the normal vector of a deformed membrane and the axis of symmetry are introduced as dependent variables. The governing equations are reduced thereafter to a set of three first-order partial differential integral equations. The constitutive equation developed by Pipkin and Rogers for the non-linear response of a viscoelastic material is used. The creep phenomenon for an inflated viscoelastic toroidal membrane under a constant pressure is presented.     

Finite Element Formulation of Viscoelastic Constitutive Equations Using Fractional Time Derivatives

Fractional time derivatives are used to deduce a generalization ofviscoelastic constitutive equations of differential operator type. Theseso-called fractional constitutive equations result in improvedcurve-fitting properties, especially when experimental data from longtime intervals or spanning several frequency decades need to be fitted.Compared to integer-order time derivative concepts less parameters arerequired. In addition, fractional constitutive equations lead to causalbehavior and the concept of fractional derivatives can be physicallyjustified providing a foundation of fractional constitutive equations.First, three-dimensional fractional constitutive equations based onthe Grünwaldian formulation are derived and their implementationinto an elastic FE code is demonstrated. Then, parameter identificationsfor the fractional 3-parameter model in the time domain as well as inthe frequency domain are carried out and compared to integer-orderderivative constitutive equations. As a result the improved performanceof fractional constitutive equations becomes obvious. Finally, theidentified material model is used to perform an FE time steppinganalysis of a viscoelastic structure.     

A theory of amorphous viscoelastic solids undergoing finite deformations with application to hydrogels

We consider a hydrogel in the framework of a continuum theory for the viscoelastic deformation of amorphous solids developed by Anand and Gurtin [Anand, L., Gurtin, M., 2003. A theory of amorphous solids undergoing large deformations, with application to polymeric glasses. International Journal of Solids and Structures, 40, 1465–1487.] and based on (i) a system of microforces consistent with a microforce balance, (ii) a mechanical version of the second law of thermodynamics and (iii) a constitutive theory that allows the free energy to depend on inelastic strain and the microstress to depend on inelastic strain rate. We adopt a particular (neo-Hookean) form for the free energy and restrict kinematics to one dimension, yielding a classical problem of expansion of a thick-walled cylinder. Considering both Dirichlet and Neumann boundary conditions, we arrive at stress relaxation and creep problems, respectively, which we consider, in turn, locally, at a point, and globally, over the interval. We implement the resulting equations in a finite element code, show analytical and/or numerical solutions to some representative problems, and obtain viscoelastic response, in qualitative agreement with experiment.     

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.     

A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

On the calculation of predeformation-dependent dynamic modulus tensors in finite nonlinear viscoelasticity

To simulate the frequency-dependent behaviour of nonlinear viscoelastic structures under loadings which consist of a finite predeformation in combination with a superimposed harmonic deformation with small amplitude, frequency-domain formulations of the constitutive models are needed. For this purpose, a recently developed approach of finite viscoelasticity is considered and the corresponding dynamic modulus tensors are derived. The constitutive equations are geometrically linearized in the neighbourhood of the predeformation and evaluated in the frequency-domain. This procedure is applicable to arbitrary constitutive models and can be used to derive their frequency-domain formulations for finite element implementations as proposed by Morman and Nagtegaal [Morman, K.N., Nagtegaal, J.C., 1983. Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. International Journal for Numerical Methods in Engineering, 19, 1079–1103].     

有初始几何缺陷的壳体弹塑性大变形分析

在等温小变形弹塑性内时本构方程偏量形式的基础上，导出了适用于大位移、小应变分析的弹塑性内时本构方程。并导出了带有初始几何缺陷的非线性弹塑性问题的有限元方程。文中给出的算例表明本方法是可行有效的。     

Geometrically nonlinear,steady state vibration of viscoelastic beams

The problem of geometrically non-linear steady state vibrations of beams excited by harmonic forces is considered in this paper. The beams are made of a viscoelastic material defined by the classic Zener rheological model - the simplest model that takes into account all the basic properties of real viscoelastic materials. The constitutive stress-strain relationship for this type of material is given as a differential equation containing derivatives of both stress and strain. This significantly complicates the solution to the problem. The von Karman theory is applied to describe the effects of geometric nonlinearities of beam deformations. The equations of motions are derived using the finite element methodology. A polynomial approximation of bending moments is used. The order of basis functions is set so as to obtain a coherent approximation of moments and displacements. In the steady-state solution of equations of motion, only one harmonic is taken into account. The matrix equations of amplitudes are derived using the harmonic balance method and the continuation method is applied for solving them. The tangent matrix of equations of amplitudes is determined in an explicit form. The stability of steady-state solution is also examined. The resonance curves for beams supported in a different way are shown and the results of calculation are briefly discussed.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

Differential constitutive equations of incompressible media with finite deformations

A method is proposed for constructing a system of constitutive equations of an incompressible medium with nonlinear dissipative properties with finite deformations. A scheme of the mechanical behavior of a material is used, in which the points are connected by horizontally aligned elastic, viscous, plastic, and transmission elements. The properties of each element of the scheme are described with the use of known equations of the nonlinear elasticity theory, the theory of nonlinear viscous fluids, and the theory of plastic flow of the material under conditions of finite deformations of the medium. The system of constitutive equations is closed by equations that express the relation between the deformation rate tensor of the material and the deformation rate tensor of the plastic element. Transmission elements are used to take into account a significant difference between macroscopic deformations of the material and deformations of elements of the medium at the structural level. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 158–170, May–June, 2009.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Modelling and simulation of process-integrated powder coating by radial axial rolling of rings

In this paper, we present finite element (FE) process simulations of a new production method used to coat ring-shaped work pieces with functional layers. Similarly to the conventionally applied hot isostatic pressing (HIP), this new coating method is based on powder metallurgy. It is expected to overcome some important drawbacks by integrating the consolidation of the powdery layer material into the hot rolling of the substrate ring. This makes HIP dispensable. Nevertheless several difficulties arise through the process integration. E.g., the presence of the compactable layer requires a different handling of the rolling stage compared with classical ring rolling. FE simulations shall support the design of this new process in order to investigate critical process parameters. For this purpose, new finite element modules have to be developed. A crucial point is the adequate modelling of the layer material. In this regard, we present a rate-dependent finite strain material model that describes the consolidation of the layer material in a thermodynamically consistent way. Moreover, FE process simulations of the new production method are presented and compared with experimental results.     

Gradient nanoscale polycrystalline elasticity: Intergrain interactions and triple-junction conditions

Using the principle of virtual power, we develop general balance equations, interface conditions, triple-junction conditions, and boundary conditions for second-grade nanocrystalline elastic materials undergoing infinitesimal deformations. We further develop thermodynamically consistent constitutive equations and provide a weak formulation of resulting boundary-value problems that automatically yields internal conditions such those that hold across interfaces and at triple junctions.