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.     

Dynamics of an exponentially damped solid rod: Analytic solution and finite element formulations

This paper presents the analytic solution together with two finite element formulations to obtain the dynamic response of an exponentially damped solid rod. The analysis is carried out in time domain, the material being modeled by the standard linear viscoelastic solid model. On the one hand, the analytic solution is reached by means of modal superposition by two different ways: with and without internal variables. On the other hand, two different finite element formulations are proposed: the first one is conceived for direct integration methods, and the other one is developed to apply modal superposition. The interest of modal superposition method lies on investigating the influence of non-viscous modes.     

Response of beams resting on viscoelastically damped foundation to moving oscillators

The response of beams resting on viscoelastically damped foundation under moving SDoF oscillators is scrutinized through a novel state-space formulation, in which a number of internal variables is introduced with the aim of representing the frequency-dependent behaviour of the viscoelastic foundation. A suitable single-step scheme is provided for the numerical integration of the equations of motion, and the Dimensional Analysis is applied in order to define the dimensionless combinations of the design parameters that rule the responses of beam and moving oscillator. The effects of boundary conditions, span length and number of modes of the beam, along with those of the mechanical properties of oscillator and foundation, are investigated in a new dimensionless form, and some interesting trends are highlighted. The inaccuracy associated with the use of effective values of stiffness and damping for the viscoelastic foundation, as usual in the present state-of-practice, is also quantified.     

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.     

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.     

分数阶导数粘弹性模型的有限元法

在分析分数阶导数三元件模型理论的基础上,把分数阶导数三元件模型引入有限元模型中,推导出具有分数阶导数三元件本构关系的粘弹性结构动力学有限元格式。同时,应用分数阶导数型粘弹性结构动力学方程的数值算法求解了该有限元格式的数值解。并以二维沥青路面结构为例进行了路面动态粘弹性响应分析。算例分析表明,该方法能够正确有效地进行路面动态粘弹性分析。     

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     

Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

三种分形和分数阶导数阻尼振动模型的比较研究

标准的整数阶导数方程不能准确描述粘弹性材料的记忆性参考文献[1]和阻尼的分数次幂频率依赖[2],因此分形导数、分数阶导数及正定分数阶导数被用于描述粘弹性介质中的阻尼振动.该文通过分析模型和数值模拟,比较了三种模型描述的振动过程.结果显示,当p小于约O.75或大于约1.9时(p为非整数阶导数的阶数),分形导数模型衰减最快;当P大于约0.75且小于约1.9时,正定分数阶导数模型衰减最快,衰减最慢的分别为分数阶导数模型(p<1)和分形导数模型(p>1).且正定分数阶导数模型衰减快于分数阶导数模型,当p接近2时,两种模型较为相近.     

Constitutive equations in viscoelasticity through the Legendre-Fenchel transformation

A thermodynamic scheme is investigated which allows interesting applications in continuum mechanics. Then the Legendre-Fenchel transformation is considered and it is shown how it can be applied to determine solutions to the Clausius-Duhem inequality as the statement of the second law. Next the attention is confined to linear viscoelastic solids, viewed as materials with internal variables, and a general expression for the stress-strain relation is derived.     

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT,Giesekus and FENE‐P constitutive models

The investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.     

分数算子描述的黏弹性体力学问题数值方法

讨论由黎曼-刘维尔 (Riemann-Liouville)分数导数描述的黏弹性体力学问题的数 值方法. 该方法利用黎曼-刘维尔分数 导数定义中核函数的特性，并结合被积函数在单步中的逼近以及Newmark型数值法，建立 了分数导数计算公式. 算例表明，该算法具有收敛快、精度高、稳定性好和易于 应用和改进的优点. 在对动态系统的瞬态响应分析和有限元分析格式中，算法都获得了 满意的结果.     

气泡在液体中运动过程的数值模拟

本文用数值方法预测气泡在液体中的百定常运动。运用位标函数进行界面的隐含跟踪并且与有限体积法相结合构成一种可行的计算方法。本文方法允许在界面处存在很大的物性差，而且较容易将表面张力引入控制方程。我们对气液两相流中单个气泡的运动进行了计算，得到了与实验结果符合很好的数值结果。     

含随机参数的非线性黏弹性有限元计算

进行了粗粒土与结构接触面单调和循环加载试验，基于宏细观测量结果, 扩展了 损伤概念以 描述该类接触面在受载过程中的物态演化, 及由于物态演化导致的力学特性从初始状态到最终 稳定状态的连续变化过程. 揭示了接触面损伤的细观物理基础主要是接触面内土的颗粒破碎 和剪切压密这两种物态演化；指出接触面的剪胀体应变可以划分为可逆性和不可逆性剪胀体 应变两部分，其中不可逆性剪胀体应变可作为接触面损伤发展的宏观量度，因此其归一化 形式可作为一种损伤因子的定义；提出了建立粗粒土与结构接触面一种损伤本构关系的基本思路.     

Nonlinear transient dynamic response of damped plates using a higher order shear deformation theory

Damped transient dynamic elasto-plastic analysis of plate is investigated. A finite element model based on a C ⁰ higher order shear deformation theory has been developed. Nine noded Lagrangian elements with five degrees of freedom per node are used. Selective Gauss integration is used to evaluate energy terms so as to avoid shear locking and spurious mechanisms. Von Mises and Tresca yield criteria are incorporated along with associated flow rules. Explicit central difference time stepping scheme is employed to integrate temporal equations. The mass matrix is diagonalized by using the efficient proportional mass lumping scheme. A program is developed for damped transient dynamic finite element analysis of elasto-plastic plate. Several numerical examples are studied to unfold different facets of damping of elasto-plastic plates.     

UNIFIED COMPUTATION OF FLOW WITH COMPRESSIBLE AND INCOMPRESSIBLE FLUID BASED ON ROE''''S SCHEME

A unified numerical scheme for the solutions of the compressible and incompressible Navier-Stokes equations is investigated based on a time-derivative preconditioning algorithm. The primitive variables are pressure, velocities and temperature. The time integration scheme is used in conjunction with a finite volume discretization. The preconditioning is coupled with a high order implicit upwind scheme based on the definition of a Roe's type matrix. Computational capabilities are demonstrated through computations of high Mach number, middle Mach number, very low Mach number, and incompressible flow. It has also been demonstrated that the discontinuous surface in flow field can be captured for the implementation Roe's scheme.     

Wave propagation in a fractional viscoelastic Andrade medium: Diffusive approximation and numerical modeling

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.     

Viscoelastic effects in double‐pipe single‐pass counterflow heat exchangers

The conducto‐convective heat loss from a viscoelastic liquid, in the core of a double‐pipe heat exchanger arrangement, to a cooler Newtonian fluid flowing in the outer annulus is investigated with direct numerical simulations. A numerical algorithm based on the finite difference method is implemented in time and space with the Giesekus constitutive model for the viscoelastic liquids. The flow of both the annulus and core‐fluids is considered to be Poiseuille flow, driven by respective pressure gradients. In general, the results show that a viscoelastic core‐fluid leads to slightly lower (albeit comparable) attainable temperatures in the core‐fluid stream as compared with a corresponding Newtonian fluid. Copyright © 2008 John Wiley & Sons, Ltd.