Restrictions placed on constitutive relations by angular momentum balance and Galilean invariance

In this note, we will show that for describing the response of a wide class of bodies, it is sufficient to invoke only the balance of angular momentum to obtain the restrictions on the constitutive functions that one obtains by appealing to frame indifference. While this result is known for hyperelastic materials (although it is not found in any standard text on the subject), we extend this result to classes of elasto-plastic and viscoelastic materials as well as for a class of implicit constitutive equations for viscous fluids. In particular, we show that for a class of bodies capable of instantaneous elastic response that is dictated by a stored energy function, the symmetry of the Cauchy stress alone is enough to obtain all the necessary restrictions. The result is related to Noether’s theorem; if we know that there is a conserved quantity (i.e., angular momentum), we can then show that the energy function must be invariant under a group of transformations. For a class of generalized Newtonian fluids (including the Navier Stokes fluid and the Bingham fluid), the symmetry of the stress and Galilean invariance of the response functions are all that are required to obtain restrictions that are usually obtained by enforcing frame indifference.     

昆虫翼拍动中受载变形的粘弹性本构模型

昆虫翼拍动受载时发生被动变形,被看作为有助于改善飞行性能的机制之一．决定这种被动变形大小的一个关键因素是昆虫翼的材料本构关系,至今缺乏研究．基于蜻蜓翼(离体)的应力松弛实验和模型翼拍动时受载变形的有限元数值分析,揭示了粘弹性本构关系是昆虫翼材料性能的合理描述,并研究了粘弹性参数对被动变形的影响．     

Modelling of wave phenomena in the Zahorski material based on modified library for ADINA software

This paper presents numerical modelling of wave phenomena in simple elastic structures such as rods and shields made of hyperelastic Zahorski material. The main difference between the Zahorski material, which is an elastic material in the Green sense, and the commonly used Mooney–Rivlin material lies in the non-linear term including the constant C₃. Consequently, qualitative and quantitative differences are observed compared to the Mooney–Rivlin material, for example in the values of effective stresses. The extension to the ADINA software developed by the author, which helps create 2D and 3D libraries, significantly facilitates modelling of the Zahorski material. The modification can be used for comparison of wave phenomena that are observed during the propagation of disturbances in the Mooney–Rivlin and Zahorski materials. It should be emphasised that the Zahorski material behaves much better at high strains during the analysis of incompressible rubber and rubber-like hyperelastic materials and can be used in various fields of science wherever the model of Mooney–Rivlin material is successfully applied. The results of numerical computations for both Mooney–Rivlin and Zahorski materials were presented in a graphical form and compared in order to illustrate the differences.     

On Implicit Constitutive Theories

In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).     

Brief note creep-stability analysis of viscoelastic cylindrical shells

The paper presents the creep-stability analysis of viscoelastic cylindrical shells under axial compression. The mechanical properties of the material are described by the constitutive equations of the linear viscoelastic theory in terms of convolution integral operators. The approximate analytical solution to the problem is obtained by means of a modification of the quasi-elastic method. As a result, the instability condition for the shell is formulated. It is shown that for viscoelastic materials with limited creep, there is a safe load limit below which the structure is asymptotically stable. Any load above the safe load limit leads to buckling at the corresponding critical time.     

On time-dependent behavior of cross-ply laminated strips with viscoelastic interfaces

The two-dimensional problem of a simply supported laminated orthotropic strip with viscoelastic interfaces under static loading is studied. State-space formulations are developed based on the exact elasticity equations governing orthotropic media and the Kelvin–Voigt constitutive relation of interfaces. Since the response of the strip is time-dependent, the power series expansion technique is adopted to model the variations of elastic fields with time. Results show that the response of the laminated strip with viscoelastic interfaces changes remarkably with time, which is also significantly different from that of a plate with perfect interfaces or with viscous interfaces. Note that from the present analysis, the response for a laminated plate with spring-like interfaces or with viscous interfaces can be easily obtained because they are just two particular cases of the present Kelvin–Voigt model.     

基于Clough-Penzien谱激励的指数型非黏滞阻尼结构随机地震动响应简明封闭解

非黏滞阻尼模型相比传统黏滞阻尼模型能更准确描述结构材料的耗能行为,其本构关系常用核函数为指数函数的卷积形式表示.针对目前非黏滞阻尼结构的随机地震动响应分析方法所得结果较为复杂,该文提出了一种基于Clough-Penzien(C-P)谱的结构响应0～2阶谱矩分析的简明封闭解法.该方法首先提出非黏滞阻尼结构的精确等效微分本...     

High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials

In this paper, high-order tetrahedral finite elements are employed to analyze structures and solids composed of functionally graded rubber-like materials under finite displacements, finite strains, statically applied forces and isothermal conditions. In order to do so, the following concepts are used: geometrically nonlinear analysis, Green–Lagrange strain tensor, second Piola–Kirchhoff stress tensor, hyperelastic constitutive relations, isoparametric solid tetrahedral finite elements of any order of approximation, and functionally graded materials. The equilibrium of the body is achieved via the Principle of the Stationary Total Potential Energy. The elements are fully integrated via Gaussian quadratures, and the resultant processing time is reduced by means of parallel techniques. To solve the nonlinear system of equations, the Newton–Raphson iterative procedure is employed.The proposed formulation is validated by benchmark problems such as: the Cook’s membrane and the thick cylinder. Other interesting simulation, the Cook’s block is proposed in order to evaluate high strain gradient situations. The results show that, in the context of the present study, locking-free behavior is obtained with simple mesh refinement.     

On phenomenological and micro-mechanical models in finite elasticity and viscoelasticity for rubber-like materials

The structure of rubber-like materials is characterised through chain-like macromolecules which are linked together at certain points. This special structure leads to a completely random three-dimensional network [5]. There are several phenomenological based models and micro-mechanically motivated network models for elastic and viscoelastic polymeric materials which have been proposed in the literature [4]. The aim of the research work is to give an overview of various phenomenological and micro-mechanical models for elastic and viscoelastic processes at large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

Numerical iterative method for Volterra equations of the convolution type

The objective of this paper is to present an algorithm from which a rapidly convergent solution is obtained for Volterra integral equations of Hammerstein type. Such equations are often encountered when describing the response of viscoelastic materials where the time dependency of the material properties is often expressed in the form of a convolution integral. Frequently, singularity is encountered and often ignored when dealing with the constitutive equations of viscoelastic materials. In this paper, the singularity is incorporated in the solution and the iterative scheme used to solve the equation converges within six iterations to a typical toleration error of 10^?5. The algorithm is applied to the strain response of a polymer under impulsive (constant) loading and the results show excellent correlation between the experimental and analytical solution. Copyright © 2010 John Wiley & Sons, Ltd.     

损伤粘弹性力学的广义变分原理及应用

从粘弹性材料的Boltzmann迭加原理和带空洞材料的线弹性本构关系出发,提出了一种损伤粘弹性材料具有广义力场的本构模型．应用变积方法得到了以卷积形式表示的泛函,并建立了损伤粘弹性固体的广义变分原理和广义势能原理．把它们应用于带损伤的粘弹性Timoshenko梁,得到了Timoshenko梁的统一的运动微分方程、初始条件和边界条件． 这些广义变分原理为近似求解带损伤的粘弹性问题提供了一条途径．     

粘弹性Timoshenko梁的动力学普遍方程及其动态特性分析

本文用直接力法在时域内推导了粘弹性Timoshenko梁的控制微分方程,它同时计及了材料的拉伸粘性和剪切粘性.为了测定标准线性固体的复模量和三参数,对有机玻璃(PMMA)和尼龙6(PCL)试件成功地应用了强迫振动梁技术.通过大量数值计算,对粘弹性Timoshenko梁的动力特性,特别是阻尼特性进行了分析.结果表明,材料粘性对结构的动力特性,尤其是对阻尼有较大影响。对于高粘性材料,其动力学性质用标准线性固体模型来描写是合适的.     

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

Nonlinear dynamic equations for isotropic homogeneous hyperelastic materials are considered in the Lagrangian formulation. An explicit criterion of existence of a natural state for a given constitutive law is presented, and is used to derive natural state conditions for some common constitutive relations.For two-dimensional planar motions of Ciarlet–Mooney–Rivlin solids, equivalence transformations are computed that lead to a reduction of the number parameters in the constitutive law. Point symmetries are classified in a general dynamical setting and in traveling wave coordinates. A special value of traveling wave speed is found for which the nonlinear Ciarlet–Mooney–Rivlin equations admit an additional infinite set of point symmetries. A family of essentially two-dimensional traveling wave solutions is derived for that case.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

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.