Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

A quasi-correspondence principle for Quasi-Linear viscoelastic solids

In this paper we show that the correspondence principle that allows one to obtain solutions to boundary-initial value problems for Linear viscoelastic solids from solutions to that for a linearized elastic solid can be extended, in many circumstances, to the case of the Quasi-Linear viscoelastic solids introduced by Fung. We illustrate the ability to generalize the correspondence principle by considering a variety of problems including torsion, transverse loading of beams and several problems that involve a single non-zero stress component. This extension is however not possible for certain classes of problems and we present a specific example where the correspondence principle breaks down. The correspondence principle between Linear elasticity and Linear viscoelasticity also breaks down under certain conditions, however the correspondence between the solutions for Linear viscoelasticity and Quasi-Linear viscoelasticity is even more fragile in that it breaks down while the classical correspondence works, and hence we refer to the correspondence as a quasi-correspondence principle.     

Finite element analysis of viscoelastic solids responding to periodic disturbances

A finite element analysis for viscoelastic solids responding to periodic disturbances travelling at constant speed is developed. The disturbance is decomposed into harmonic components using the Discrete Fourier Series, and the viscoelastic material response is determined using the Correspondence principle. The procedure is used to solve two and three dimensional contact problems to demonstrate its effectiveness.     

Variational bounds of the stress intensity factor at the edge of a plane opening mode crack

Criteria for opening mode critical stress intensity factors in three-dimensional elastic or viscoelastic solids are proposed based upon variational bounds similar in principle to those developed by Lavrent'ev for problems in gas dynamics.     

Stress analysis in anisotropic functionally graded materials by the MLPG method

A meshless method based on the local Petrov–Galerkin approach is proposed for stress analysis in two-dimensional (2D), anisotropic and linear elastic/viscoelastic solids with continuously varying material properties. The correspondence principle is applied for non-homogeneous, anisotropic and linear viscoelastic solids where the relaxation moduli are separable in space and time. The inertial dynamic term in the governing equations is considered too. A unit step function is used as the test functions in the local weak-form. It leads to local boundary integral equations (LBIEs). The analyzed domain is divided into small subdomains with a circular shape. The moving least squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. For time-dependent problems, the Laplace-transform technique is utilized. Several numerical examples are given to verify the accuracy and the efficiency of the proposed method.     

Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids

A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of quasi-static and transient dynamic problems in two-dimensional (2-D) nonhomogeneous linear viscoelastic media. A unit step function is used as the test functions in the local weak form. It is leading to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems. The correspondence principle is applied to such nonhomogeneous linear viscoelastic solids where relaxation moduli are separable in space and time variables. Then, the LBIEs are formulated for the Laplace-transformed viscoelastic problem. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-D problems. The moving least squares (MLS) method is used for approximation of physical quantities in LBIEs.     

Computational model for predicting nonlinear viscoelastic damage evolution in materials subjected to dynamic loading

Many inelastic solids accumulate numerous cracks before failure due to impact loading, thus rendering any exact solution of the IBVP untenable. It is therefore useful to construct computational models that can accurately predict the evolution of damage during actual impact/dynamic events in order to develop design tools for assessing performance characteristics. This paper presents a computational model for predicting the evolution of cracking in structures subjected to dynamic loading. Fracture is modeled via a nonlinear viscoelastic cohesive zone model. Two example problems are shown: one for model validation through comparison with a one-dimensional analytical solution for dynamic viscoelastic debonding, and the other demonstrates the applicability of the approach to model dynamic fracture propagation in the double cantilever beam test with a viscoelastic cohesive zone.     

沥青混合料黏弹性主曲线模拟及换算

基于时间-温度等效原理将不同温度下沥青混合料的储存模量和蠕变柔量进行平移形成主曲线,采用广义Maxwell和Kelvin模型分别模拟沥青混合料的松弛和蠕变特性,利用Prony级数表达式和Laplace变换实现黏弹性主曲线间的相互换算,并将换算结果与试验结果进行了比较。结果表明,广义Maxwell和Kelvin模型可以较好地模拟沥青混合料黏弹性主曲线,储存模量和蠕变柔量主曲线相互换算结果与试验结果规律基本一致,松弛模量实际结果应介于由储存模量和蠕变柔量换算的松弛模量主曲线之间,这可为沥青混合料黏弹性力学研究及黏弹性参数的获取提供有益参考。     

Analysis of nanocontact problems of layered viscoelastic solids with surface energy effects under different loading patterns

Surface stresses have a remarkable effect on nanocontact response of layered viscoelastic solids, especially under specific loading patterns. In the framework of nonlinear viscoelastic contact mechanics, a numerical model is developed to investigate the quasistatic nanocontact response of elastically layered viscoelastic solids under different loading patterns. The developed model accounts for surface energy effects by adopting the complete Gurtin–Murdoch surface elasticity model. The Schapery’s constitutive viscoelastic creep model is used for the stress, strain, and time relationships. The transient term in the creep compliance is expressed by Prony’s series. Frictionless contact condition is assumed throughout the contact interface. The equilibrium contact configuration, in which the contact constraints are exactly satisfied without any need for an appropriate value for the penalty parameter, is obtained by using the Lagrange multiplier method in the framework of the Newton–Raphson procedure. The developed model is applied to study and analyze the quasistatic nanocontact response of two different problems under different loading patterns. Results show the significant effect of the type of loading pattern and its rate on the nanocontact response of elastically layered viscoelastic solids.     

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.     

Modeling failure of heterogeneous viscoelastic solids under dynamic/impact loading due to multiple evolving cracks using a two-way coupled multiscale model

This paper presents a model for predicting damage evolution in heterogeneous viscoelastic solids under dynamic/impact loading. Some theoretical developments associated with the model have been previously reported. These are reviewed briefly, with the main focus of this paper on new developments and applications. A two-way coupled multiscale approach is employed and damage is considered in the form of multiple cracks evolving in the local (micro) scale. The objective of such a model is to develop the ability to consider energy dissipation due to both bulk dissipation and the development of multiple cracks occurring on multiple length and time scales. While predictions of these events may seem extraordinarily costly and complex, there are multiple structural applications where effective models would save considerable expense. In some applications, such as protective devices, viscoelastic materials may be preferred because of the considerable amount of energy dissipated in the bulk as well as in the fracture process. In such applications, experimentally based design methodologies are extremely costly, therefore suggesting the need for improved models. In this paper, the authors focus on the application of the newly developed multiscale model to the solution of some example problems involving dynamic and impact loading of viscoelastic heterogeneous materials with growing cracks at the local scale.     

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.     

Micromechanical modelling of the viscoelastic behaviour of an amorphous poly(ethylene)terephthalate (PET) reinforced by spherical glass beads

Homogenization micromechanical models are applied to predict the linear viscoelastic properties of an amorphous poly(ethylene)terephthalate (PET) composite in the range of glass transition by using the elastic-viscoelastic superposition principle defined by Hashin [1]. An amorphous PET is reinforced by glass beads and is submitted to dynamic mechanical tests at temperatures surrounding its quasi-static glass transition temperature. The viscoelastic properties of the matrix and the elastic properties of the beads are measured experimentally. The micromechanical models predictions of the linear viscoelastic behaviour in the glassy state are acceptable. In the rubbery state, the beads seem to reduce the molecular mobility of the matrix driving to a large change in the viscoelastic properties of the materials. Thus, this paper aims to emphasize that classical homogenization micromechanical models, which depend only on the constituent behaviour, shape and distribution, cannot predict this change in the linear viscoelastic behaviour of the beads/PET composites.     

Homogenized Constitutive Equations for Linear Viscoelastic Unidirectional Composites with Growing Transverse Cracks

Three-dimensional, thermomechanical constitutive equations and an associated correspondence principle are developed for a linear viscoelastic, unidirectional composite with growing damage. The type of damage modeled is transverse cracking, which is commonly observed in laminates consisting of unidirectional fiber layers. The equations allow for aging. First, the constitutive equations without damage are summarized. They are then modified to allow for time-dependent damage by means of a correspondence principle. This principle relates elastic and viscoelastic solutions in the time-plane; the familiar correspondence principle based on the Laplace transform is not applicable because of the aging and crack growth. Some existing, simplified micromechanical models for elastic media are extended to viscoelasticity and then used to demonstrate that the requirements of the correspondence principle are quite-well satisfied for realistic material behavior. The correspondence principle is developed in the Appendix. It generalizes a previously-developed principle for plane stress in fiber-composites to three-dimensional conditions, and is not limited to the specific type of damage covered in the body of the paper.     

基于不可逆内变量热力学的岩石材料粘弹-粘塑性本构方程

岩石材料的粘弹性和粘塑性变形是与时间相关的能量耗散行为。在Rice不可逆内变量热力学框架下,引入两组内变量分别用来描述在粘弹性和粘塑性变形过程中材料的内部结构调整。通过给定比余能的具体形式和内变量的演化方程,推导出内变量粘弹-粘塑性本构方程。粘弹性本构方程具有普遍性,能涵盖Kelvin-Voigt和Poynting-Thomson在内的经典粘弹性模型的本构方程。并指出热力学力与应力呈线性关系是组合元件模型为线性模型的根本原因。粘塑性本构方程能较好地刻画岩石材料在粘塑性变形过程中的硬化现象。对模拟岩石的模型相似材料进行单轴加卸载蠕变试验,将蠕变过程中的粘弹性和粘塑性变形分离并根据试验数据对本构方程的材料参数进行辨识。试验数据和理论曲线对比结果表明该文提出的本构方程能很好地模拟材料的蠕变行为。该类型的本构方程能为岩石工程的长期稳定性的预测、评价以及加固分析提供基础。     

A simultaneous solution procedure for strong interactions of generalized Newtonian fluids and viscoelastic solids at large strains

The paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space–time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress‐deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non‐Newtonian fluid models like the power‐law. The consideration of non‐linear constitutive equations for solid and fluid documents the capability of the monolithic space–time finite element formulation to deal with complex material models. The methodology is applied to fluid‐conveying cantilevered pipes in order to determine the influence of material non‐linearities on stability characteristics of coupled systems. Copyright © 2005 John Wiley & Sons, Ltd.     

分数指数模型的热力学分析及其应用

本文论证了两种经典粘弹性固体模型的等价性并指出了其存在的问题。给出了热力学对分数指数模型 [1]参数的限制条件。计算与实验结果比较表明:因为该模型具有适当多的参数,采用同一组参数可以做到同时与同一材料的蠕变和松弛试验结果很好吻合;并能做到松弛模量和蠕变柔量的Stieltjes卷积近似等于单位阶跃函数;在很宽广的频率范围内能同时很好地模拟真实材料的存储模量和损耗模量。由于其计算速度快,能与大多数真实材料的性能实验结果相拟合,可以广泛应用于工程实际中的粘弹性静力和动力问题的计算。     

Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiple holes,cracks, and inclusions in anisotropic viscoelastic solids

By using the elastic–viscoelastic correspondence principle, the problems with multiple holes, cracks, and inclusions in two-dimensional anisotropic viscoelastic solids are solved for the cases with time-invariant boundaries. Based upon this principle and the existing methods for the problems with anisotropic elastic materials, two different approaches are proposed in this paper. One is concerned with an analytical solution for certain specific cases such as two collinear cracks, collinear periodic cracks, and interaction between inclusion and crack, and the other is a boundary-based finite element method for the general cases with multiple holes, cracks, and inclusions. The former considers only specific cases in infinite domain and can be used as a reference for any other numerical methods, and the latter is applicable to any combination of holes, cracks and inclusions in finite domain, whose number, size and orientation are not restricted. Unlike the conventional finite element method or boundary element method which usually needs very fine meshes to get convergence solutions, in the proposed boundary-based finite element method no meshes are needed along the boundaries of holes, cracks and inclusions. To show the accuracy and efficiency of these two proposed approaches, several representative examples are implemented analytically and numerically, and they are compared with each other or with the solutions obtained by the finite element method.