Fractional calculus in viscoelasticity: An experimental study

Viscoelastic properties of soft biological tissues provide information that may be useful in medical diagnosis. Noninvasive elasticity imaging techniques, such as Magnetic Resonance Elastography (MRE), reconstruct viscoelastic material properties from dynamic displacement images. The reconstruction algorithms employed in these techniques assume a certain viscoelastic material model and the results are sensitive to the model chosen. Developing a better model for the viscoelasticity of soft tissue-like materials could improve the diagnostic capability of MRE. The well known “integer derivative” viscoelastic models of Voigt and Kelvin, and variations of them, cannot represent the more complicated rate dependency of material behavior of biological tissues over a broad spectral range. Recently the “fractional derivative” models have been investigated by a number of researchers. Fractional order models approximate the viscoelastic material behavior of materials through the corresponding fractional differential equations. This paper focuses on the tissue mimicking materials CF-11 and gelatin, and compares fractional and integer order models to describe their behavior under harmonic mechanical loading. Specifically, Rayleigh (surface) waves on CF-11 and gelatin phantoms are studied, experimentally and theoretically, in order to develop an independent test bed for assessing viscoelastic material models that will ultimately be used in MRE reconstruction algorithms.     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrical analogous in viscoelasticity

In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these viscoelastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. Moreover, a comparison with numerical experiments based on finite difference time domain method shows that, for long time simulations, the correct time behavior can be obtained only with modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behavior of fractional hereditary materials.     

Computational characterization of ferroelectric solids with a 3D Preisach model based on an orientation distribution function

Ferroelectric or ferromagnetic materials show an interaction between mechanical deformations and polarization or magnetization. A few multiferroic materials possess both ferroic properties and exhibit a magneto-electric (ME) coupling. These ME properties can be achieved in two-phase composites, which combine ferroelectric and ferromagnetic characteristics. To predict a realistic material behavior and a more precise ME coefficient, the application of suitable material models which describe the nonlinear hysteretic behavior is of particular importance. In the present contribution we focus on the characterization of a nonlinear ferroelectric material behavior, in terms of a 3D Preisach model based on an orientation distribution function. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stationary interaction of a Navier-Stokes fluid with an elastic tube wall

We study the stationary problem of a viscous, incompressible Navier-Stokes fluid flowing through a flexible tube with thickness. The behavior of the elastic walls of the tube is described by the equations of nonlinear elasticity for a St.Venant-Kirchhoff material. For smooth enough applied exterior forces we prove the existence of a solution to the coupled problem.     

On the thermodynamics of fluids defined by implicit constitutive relations

In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.     

剪切变形下非晶态高聚物的力学行为

基于非平衡态热力学理论，提出了一个适用于不可压材料的新的热粘弹性本构模型．该模型将橡胶弹性理论中的非高斯分子网络模型推广到计及粘性和热效应的情形．通过引入一组二阶张量形式的内变量，建议了一个新的Helmholtz自由能表达式，从而可以用来合理描述内变量的演化规律．根据以上模型，重点研究了热粘弹性材料在简单剪切变形下的力学行为，考察了由于分子链取向分布的变化而产生的“粘性耗散诱导”各向异性，讨论了应变率效应和由于粘性耗散而导致的热软化效应对剪应力的影响．理论预测结果与G’Sell等人的实验数据的定性比较表明了新的本构模型的有效性。     

Concise formulas for strain analysis of soft biological tissues

We describe a method for the approximate solution of nonlinear elasticity problems in the framework of finite deformation for the case of hyperelastic isotropic materials. This method enables one to write the resulting equations from the finite element method in analytical form, which reduces the amount of computations and simplifies the implementation. This approach is implemented for several types of hyperelastic materials used to describe the mechanical behavior of soft biological tissues.     

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.     

Discussion of Different Model Approaches for the Flow Behavior of Ice

Ice of Antarctic ice shelves is assumed to behave on long-term as an incompressible viscous fluid, which is dominated on short time scales by the elastic response. Hence, a viscoelastic material model is required. The thermodynamic pressure is treated differently in elastic and viscous models. For small deformations, the elastic isometric stress for ν → 0.5 gives similar results to those solving for pressure in an incompressible laminar flow model. A viscous model, in which the thermodynamic pressure is approximated by an elastic isometric stress, can be easily extended to viscoelasticity. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear electro-dynamic analysis of micro-actuators: Effect of material nonlinearity

This paper presents a nonlinear dynamic analysis of a micro-actuator made of nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of mid-plane stretching due to large deformation and material nonlinearity. By employing Linstedt–Poincaré perturbation method, the nonlinear governing equation is transformed into a set of linear differential equations which are then solved using Galerkin’s method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only whereas for large deformation, the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of initial gap and beam length on the nonlinear electro-dynamic behavior of the micro-actuator are also discussed.     

Nonlinear micromechanical model of filament-wound composites considering fiber undulation

Flexible-matrix composites with highly anisotropic properties have successfully been used in numerous fields to improve the performance of conventional structures or to facilitate new innovations. Many of them are designed on the basis of tubes which are produced efficiently by the filament winding process. To predict the elastic behavior of filament-wound flexible-matrix composites, aspects of the nonlinear behavior of the flexible material have to be considered, as well as the features of the distinct fiber undulation geometry inherent to the filament winding process. The present study considers these characteristics in the micromechanical modeling of the elastic behavior by including a nonlinear material model to represent the strain-dependent moduli and manufacturing-dependent geometries. The structure is characterized by a unit cell and subcells, analyzed separately and combined based on different sets of isostress and isostrain assumptions that depend on the winding angle. On the basis of experimentally obtained nonlinear lamina properties, an iterative method of solution is chosen to calculate the axial stress–strain behavior of tubes with various winding parameters. The resulting predictions are validated by testing tubes in tension and compression. The model shows good agreement with the experiments. Predictions made using the model show a strong influence of filament winding parameters on the axial modulus of flexible-matrix composite tubes.     

Theoretical study and numerical simulation of textiles

We propose a new approach for developing continuum models fit to describe the mechanical behavior of textiles. We develop a physically motivated model, based on the properties of the yarns, which can predict and simulate the textile behavior. The approach relies on the selection of a suitable topological model for the patch of the textile, coupled with constitutive models for the yarn behavior. The textile structural configuration is related to the deformation through an energy functional, which depends on both the macroscopic deformation and the distribution of internal nodes. We determine the equilibrium positions of these latter, constrained to an assigned macroscopic deformation. As a result, we derive a macroscopic strain energy function, which reflects the possibly nonlinear character of the yarns as well as the anisotropy induced by the microscopic topological pattern. By means of both analytical estimates and numerical experiments, we show that our model is well suited for both academic test cases and real industrial textiles, with particular emphasis on the tricot textile.     

带空洞损伤的古典压力容器问题

本文根据微弹性结构线性理论研究了带空洞损伤的压力容器问题．解答是准静态的，其应力场为古典弹性力学关于球体对称压力容流问题应力解答，位移场和损伤场具有由于考虑损伤而表现出体积粘弹性特点．     

Finite Element Models of Hyperelastic Materials Based on a New Strain Measure

To construct constitutive equations for hyperelastic materials, one increasingly often proposes new strain measures, which result in significant simplifications and error reduction in experimental data processing. One such strain measure is based on the upper triangular (QR) decomposition of the deformation gradient. We describe a finite element method for solving nonlinear elasticity problems in the framework of finite strains for the case in which the constitutive equations are written with the use of the QR-decomposition of the deformation gradient. The method permits developing an efficient, easy-to-implement tool for modeling the stress–strain state of any hyperelastic material.     

Dynamic analysis of size-dependent micro-beams with nonlinear elasticity under electrical actuation

This paper presents a nonlinear electro-dynamic analysis for a size dependent micro-beam made of materials with nonlinear elasticity by employing the modified couple stress theory based on Euler–Bernoulli beam model. By employing Hamilton's principle, the nonlinear partial differential governing equation is derived then solved by using Galerkin method in the space domain and backward differential method in the time domain to obtain the nonlinear electro-dynamic response and phase portrait of the micro-beam. A parametric study is conducted, with a particular focus on the influences of nonlinear elasticity and size-dependency of the micro-beam on its nonlinear dynamic behavior. Numerical results show that a micro-beam exhibiting nonlinear elastic stress–strain relationship has reduced effective stiffness while the size effect has the opposite effect.     

On the theory of orthotropic porous elastic rods

We employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. In this purpose, we consider the direct approach to the theory of rods, in which the thin body is modeled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. For orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches (directed curves and three-dimensional rods). (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel nonlinear viscoelastic solid model

The standard linear solid (SLS) model has been widely used to describe linear viscoelastic materials. Although the SLS can be extended to describe more complex phenomena through the inclusion of additional components, a compact model of nonlinear viscoelasticity remains elusive. Here, using the framework of the three component SLS model, the stress-strain relationships of the individual components have been generalised. This work describes the derivation of the new generalised model, termed nonlinear viscoelastic solid model, or NVS, that can potentially describe nonlinear viscoelastic phenomena in a compact differential form and reduces to the familiar SLS model if linear components are selected. As a proof of concept, exponential functions of strain and strain rate were selected for the three components and major viscoelastic phenomena such as stress relaxation, creep and sawtooth strain loading were simulated. Finally, to demonstrate its efficacy in describing biological tissue, the NVS model was used to simulate the cyclic loading of mammalian stomach and cardiac muscle tissues.