Generation and propagation ofSH-type waves due to stress discontinuity in a linear viscoelastic layered medium

In this paper the generation and propagation ofSH-type waves due to stress discontinuíty in a linear viscoelastic layered medium is studied. Using Fourier transforms and complex contour integration technique, the displacement is evaluated at the free surface in closed form for two special types of stress discontinuity created at the interface. The numerical result for displacement component is evaluated for different values of nondimensional station (distance) and is shown graphically. Graphs are compared with the corresponding graph of classical elastic case.     

Superconvergent biquadratic finite volume element method for two-dimensional Poisson’s equations

In this paper, a kind of biquadratic finite volume element method is presented for two-dimensional Poisson’s equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes. The method can be effectively implemented by alternating direction technique. It is proved that the method has optimal energy norm error estimates. The superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique. Finally, a numerical example verifies the theoretical results and illustrates the effectiveness of the method.     

旋转壳在子午面内整体弯曲的几何非线性摄动有限元法及在波纹管计算中的应用(Ⅰ)

为切实有效地计算波纹管,建立了旋转壳在子午面内整体弯曲几何非线性问题的摄动有限元法。以结构环向应变的均方根为摄动小参数,将有限元节点位移列式和节点力列式直接展开。通过摄动小参数将非线性有限元的载荷分级和迭代过程有机地统一起来,即载荷的分级是有约束的,每一级载荷增量和所对应的位移增量之间的关系是已知的,每一级的计算一步到位。为叙述方便并具实用性,将旋转壳用截锥壳单元进行离散。位移分量和载荷分量沿环向按Fourier级数展开,沿子午线用多项式插值,端面弯矩和横向力化成载荷分量离散到节点上。采用Sanders中小转角非线性几何方程和各向同性广义Hooke定律。对多层材料叠合而成的旋转壳按各层薄膜应变、弯曲应变、扭转应变相等的原则进行处理,该方法能方便有效地计算单层和多层波纹管整体纯弯曲、横向弯曲的几何非线性问题。并为有限元处理非线性问题提供了一条新途径。     

旋转壳在子午面内整体弯曲的几何非线性摄动有限元肖及在波纹管计算中的应用（Ⅰ）

为切实有效地计算波纹管，建立了旋转壳在子午面内整体弯曲几何非线性问题的摄动有限元法。以结构环向应变的均方根为摄动小参数，将有限元节点位移列式和节点力列式直接展开。通过摄动小参数将非线性有限元的载荷分级和迭代过程有机地统一起来，即载荷的分级是有约束的，每一级载荷增量和所对应的位移增量之间的关系是已知的，每一级的计算一步到位。为叙述方便并具实用性，将旋转壳用截锥壳单元进行离散。位移分量和载荷分量沿环向按Fourier级数展开，沿子午线用多项式插值，端面弯矩和横向力化成载荷分量离散到节点上。采用Sanders中小转角非线性几何方程和各向同性广义Hooke定律，对多层材料叠合而成的旋转壳按各层薄膜应变、弯曲应变、扭转应变相等的原则进行处理，该方法能方便有效地计算单层和多层波纹管整体纯弯曲、横向弯曲的几何非线性问题。并为有限元处理非线性问题提供了一条新途径。     

3-D time-dependent analysis of multilayered cross-anisotropic saturated soils based on the fractional viscoelastic model

The fractional Merchant viscoelastic model is introduced to simulate the viscoelasticity of soil skeleton in this study. According to the elastic-viscoelastic correspondence principle, elastic parameters including shear modulus G_v, horizontal elastic modulus E_h and vertical elastic modulus E_v are replaced by the reciprocal of the flexibility coefficient of viscoelastic media in the Laplace transformed domain. Then, based on the precise integration solutions of multilayered cross-anisotropic elastic saturated soils, 3-D solutions of viscoelastic saturated soils are derived. The final solutions in the physical domain are obtained by the Laplace numerical inversion. The correctness of theories and programs is verified by comparing the numerical results with existing references. Sensitivity analyses are conducted to investigate the effects of viscoelastic parameters, cross-anisotropic parameters and stratification of soils on time-dependent displacement and excess pore water pressure.     

Shifted Legendre Polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model

The dynamic analysis of viscoelastic pipes conveying fluid is investigated by the variable fractional order model in this article. The nonlinear variable fractional order integral-differential equation is established by introducing the model into the governing equation. Then the Shifted Legendre Polynomials algorithm is first presented for dealing with this kind of equations. The convergence analysis and numerical example verify that the algorithm is an effective and accurate technique for addressing this type complicated equation. Numerical results for dynamic analysis of viscoelastic pipes conveying fluid show the effect of parameters on displacement, acceleration, strain and stress. It also indicates that how dynamic properties are affected by the variable fractional order and fluid velocity varying. Most of all, the proposed algorithm has enormous potentials for the problem of high precision dynamics under the variable fractional order model.     

Thermo-viscoelastic analysis of biological tissue during hyperthermia treatment

Although viscoelastic properties of biological tissue has been reported in many articles, no effort has been made to investigate the coupled thermal and mechanical behavior of biological tissue based on the viscoelastic theory. This provides us a motivation to study the transient thermoelastic coupling response in the context of generalized thermo-viscoelastic model. The dual phase lag thermo-viscoelastic model is established to capture the micro-scale responses of biological tissue. The governing equations are solved by Laplace transformation. The effects of relaxation times and viscoelastic property on the responses of the tumor and normal tissues are discussed and illustrated graphically. According to the numerical results, we can obtain (1) the viscoelastic parameter has a significant effect on the distributions of displacement and stress; (2) the lagging thermo-viscoelastic responses depend not only on the ratio of τ_t/τ_q, but also on the absolute values of τ_t and τ_q.     

A meshfree method with dynamic node reconfiguration for analysis of thermo-elastic problems with moving concentrated heat sources

In this work, a novel approach for efficient analysis of transient thermo-elastic problems including a moving point heat source is presented. This approach is based on a meshfree method with dynamic reconfiguration of the nodal points. In order to accurately capture the large temperature gradients at the location of the concentrated heat source, a fine configuration of nodal points at this location is selected. In contrast, a coarser nodal arrangement is used in other parts of the problem domain. During the problem analysis, the fine nodal arrangement moves with the point heat source. Consequently, the meshfree methods are ideally suited to this approach. In the present work, the meshfree radial point interpolation method (RPIM) is adopted for the numerical analyses. Since the density of the nodal points varies in different parts of the domain, the background decomposition method (BDM) is used for efficient computation of the domain integrals. In the BDM, the density of the integration points conform to that of the nodal points and thus the computational effort is minimized. Some numerical examples are provided to assess the accuracy and usefulness of the proposed approach in computation of the temperature, displacement, and stress fields.     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

3D nonlinear variable strain-rate-dependent-order fractional thermoviscoelastic dynamic stress investigation and vibration of thick transversely graded rotating annular plates/discs

In the present article, the idea of using the variable-order fractional-derivative thermoviscoelastic constitutive laws in dynamic stress and vibration analysis of the engineering structures, the required implementation backgrounds, and the relevant numerical solution procedures are investigated for the first time. In this regard, dynamic 3D stress and displacement fields and radial/transverse vibrations of transversely graded viscoelastic spinning thick plates/discs exposed to sudden thermoelastic loads are investigated. Instead of using the approximate plate theories, the exact thermoviscoelasticity theory is employed in the development of the governing equations. Since the variable fractional order is dependent on the localized deformation rates, the resulting thermoviscoelastic integro-differential equations are nonlinear. These equations are solved by utilizing a combination of the second-order backward/central/forward finite difference discretization of the spatial and time domains, numerical evaluation and updating of the Caputo-type fractional derivatives, updating the growing number of terms of the governing equations, and Picard's iterations. Various edge conditions are considered. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed regarding the effects of the variable fractional order of the constitutive law, time variations of the nonuniformly distributed transverse loads, and edge conditions on the distributions and damping of the resulting displacement and stress components.     

The eigenmode expansion method for oscillations of an elastic body with internal and external friction

A method is proposed for solving dynamical problems for a viscoelastic body (the Kelvin-Voigt model) in a massless viscous medium. Interaction with the external medium produces on the boundary of the body stresses proportional to the rate of displacement. The model of external friction is that used for modelling dynamical processes in elastic media filling an infinite domain [1, 2]. The implementation of numerical methods of solution requires an equivalent restatement of the problem in a finite domain, using external viscous friction to allow for the radiation of energy at infinity.     

A variational formulation of nonlocalmaterials withmicrostructure based on dual macro- and micro-balances

We investigate a variational setting of nonlocal materials with microstructure and outline aspects of its numerical implementation. Thereby, the current state of the evolving microstructure is described by independent global degrees in addition to the macroscopic displacement field, so-called order parameters. Focussing on standard-dissipative materials, the constitutive response is governed by two fundamental functions for the energy storage and the dissipation. Based on these functions, a global dissipation postulate is introduced. Its exploitation constitutes a global variation formulation of nonlocal materials, which can be related to a minimization principle. Following this methodology, we end up with coupled macro- and microscopic field equations and corresponding boundary conditions. On the numerical side, we consider the weak counterpart of these coupled field equations and obtain after linearization a fully coupled system for increments of the displacement and the order parameters. Due to the underlying variational structure, this system of equations is symmetric. In order to show the capability of the proposed setting, we specify the above outlined scenario to a model problem of isotropic damage mechanics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二维黏弹性力学问题的无网格自然单元法

基于无网格自然单元法,建立了求解二维黏弹性力学问题的一条新途径.基于弹性 黏弹性对应原理和Laplace(拉普拉斯)变换技术,首先将黏弹性问题转换成Laplace域内与弹性力学问题相同的形式,然后推导出基于自然单元法分析黏弹性问题的基本公式.作为一种新兴的无网格数值计算方法,自然单元法的实质是一种基于自然邻近插值的Galerkin(伽辽金)法.相对于其他无网格法,自然单元法的形函数具有插值性和支持域各向异性等特点.算例结果证明了所提分析方法的有效性.     

Propagation of Transient Hydroelastic Waves in a Semiinfinite, Piecewise-Constant Cylindrical Shell Containing a Fluid

A solution is formulated for a new problem of wave propagation in a semiinfinite cylindrical shell with a junction connecting two shells of different radii. The material of the shell is assumed to be viscoelastic, and the fluid is assumed to be viscous. The motion of the shell is described by Kirchhoff–Love theory, and the motions of the fluid are described by equations averaged over the cross section. The problem is solved by means of the time Laplace transform and subsequent numerical inversion. The numerical results for the pressure and radial displacement of the shell are analyzed for various values of the parameters.     

A boundary element methodology for viscoelastic analysis: Part II without cells

In this study Kelvin and Boltzmann viscoelastic models are implemented in a two-dimensional boundary element atmosphere. This general methodology is based on differential constitutive relations for viscoelasticity, avoiding the use of relaxation functions. In this part of the study, important algebraic operations are introduced into the formulation allowing analysing viscoelastic problems without using internal cells. This improvement is very important to model infinite and semi-infinite regions. The formulation is verified comparing the numerical results with analytical solutions. An extension of the formulation to consider soil–structure interaction is presented in order to demonstrate the vast applicability of the technique.     

On subcritical development of a shear crack in a composite with viscoelastic components

We present results of an investigation of the development of a transverse shear crack in a composite material with linearly viscoelastic components under external shear load. The solution is divided into the following two main stages: determination of the time dependence of the crack tip opening displacement and determination of the crack-growth kinetics as a result of the solution of integral equations. In the first stage, we use the solution of the corresponding elastic problem of determination of the crack opening displacement and the problem of determination of the effective moduli of the composite reinforced with unidirectional discrete fibers. Using the theoretically proved principle of elasto-viscoelastic analogy and the method of Laplace inverse transformation, we obtain a solution in a time domain. In the second stage, using the criterion of critical crack opening displacement for a transverse shear crack and an equation for the viscoelastic crack opening displacement of this crack, we construct an equation of crack growth. We present results of the numerical solution, which illustrate the influence of relations between the relaxation parameters of the materials of the components on the durability of the body with a crack.     

Numerical analysis of a non-clamped dynamic thermoviscoelastic contact problem

In this work, we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a non-monotone relation between the tangential stress and the tangential velocity. This leads to a system of second-order inclusion for displacement and a parabolic equation for temperature. We provide a fully discrete approximation of the problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated by numerical simulations.     

Finite element method for viscoelastic fluids flows: approximation of the Maxwell model

We discuss about the finite element approximation of viscoelastic fluid flow. We consider a fluid obeying the Oldroyd model and particularly we study the purely viscoelastic case, the so-called Maxwell model, important in practice for the applications. We examine two kinds of methods used for the approximation of the Maxwell model: method using a splitting technique and finite element method satisfying inf-sup conditions relating tensor and velocity. We present numerical results for these methods and we discuss about their stability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation and computational aspects of a 3D elastic wave modelling by PUFEM

This paper presents an enriched finite element model for three dimensional elastic wave problems, in the frequency domain, capable of containing many wavelengths per nodal spacing. This is achieved by applying the plane wave basis decomposition to the three-dimensional (3D) elastic wave equation and expressing the displacement field as a sum of both pressure (P) and shear (S) plane waves. The implementation of this model in 3D presents a number of issues in comparison to its 2D counterpart, especially regarding how S-waves are used in the basis at each node and how to choose the balance between P and S-waves in the approximation space. Various proposed techniques that could be used for the selection of wave directions in 3D are also summarised and used. The developed elements allow us to relax the traditional requirement which consists to consider many nodal points per wavelength, used with low order polynomial based finite elements, and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The effectiveness of the proposed technique is determined by comparing solutions for selected problems with available analytical models or to high resolution numerical results using conventional finite elements, by considering the effect of the mesh size and the number of enriching 3D plane waves. Both balanced and unbalanced choices of plane wave directions in space on structured mesh grids are investigated for assessing the accuracy and conditioning of this 3D PUFEM model for elastic waves.