Viscoelastic contact problems with friction

A mechanically rough rigid indentor of abitrary asymmetric curved profile is pressed against the surface of a linear viscoelastic half-space whose behaviour in shear and dilatation is similar. In the first place complete adhesion is assumed and the viscoelastic solution is found to be expressible in terms of corresponding elastic solutions. The shape of the contact area and the tangential component of displacement of a point of the surface of the viscoelastic body inside the contact area are unaffected by the viscoelasticity of the body. Fully adhesive contact between two viscoelastic bodies may be treated similarly. Also, certain frictional elastic contact problem solutions, where the contact area contains regions of slip as well as regions of adhesion, are extended to viscoelasticity.     

Modeling of the contact between a rigid indenter and a heterogeneous viscoelastic material

In this paper, the contact problem between a rigid indenter and a viscoelastic half space containing either isotropic or anisotropic elastic inhomogeneities is solved. The model presented here is 3D and based on semi-analytical methods. To take into account the viscoelastic properties of the matrix, contact and subsurface problem equations are discretized in the spatial and temporal dimensions. A conjugate gradient method and the fast Fourier transform are used to solve the normal problem, contact pressure, subsurface problem and real contact area simultaneously. The Eshelby’s formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on pressure distribution and subsurface stresses. This method can be seen as an enrichment technique where the enrichment fields from heterogeneous solutions are superimposed to the homogeneous viscoelastic problem solution. Note that both problems are fully coupled. The model is validated by comparison with a Finite Element Model. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity is proposed. The model allows to obtain the contact pressure distribution disturbed by the presence of inhomogeneities as well as subsurface and matrix/inhomogeneity interface stresses at every step of the temporal discretization.     

Modelling of the collision of two viscoelastic spherical shells

In the present paper, the collision of two viscoelastic spherical shells is investigated using the wave theory of impact. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. Since the local bearing of the materials of the colliding viscoelastic shells is taken into account, the solution in the contact domain is found via the modified Hertz contact theory involving the operator representation of viscoelastic analogs of Young’s modulus and Poisson’s ratio. The collision of two elastic spherical shells is considered first, and then using Volterra correspondence principle, according to which the elastic constants in the governing equations should be replaced by the corresponding viscoelastic operators, the solution obtained for elastic shells is extended over the case of viscoelastic shells.     

圆形隧道的粘弹性应力分析

本文讨论圆形隧道的粘弹性应力分析问题。文中给出了在隧道围岩自重作用下,衬砌与围岩均为粘弹性材料时,求解衬砌与围岩变形及内力的一般性解答。求解的方法是:将衬砌与围岩相互作用的径向力和切向力表示成三角级数,按弹性问题分别求解衬砌及围岩的变形和内力;然后利用粘弹性——弹性对应原理,由弹性解得到粘弹性问题的解;再利用衬砌与围岩间变形的协调条件,确定衬砌与围岩相互作用之径向力和切向力级数中的系数,进而求得衬砌及围岩内力和变形的粘弹性解。     

Axisymmetric wave propagation in a solid viscoelastic sphere

Two problems of linear wave propagation in a viscoelastic solid sphere are solved. The waves are generated by two types of impact on the surface of the sphere. The deformation has symmetry with respect to an axis through the center of the sphere. The solution is based on a superposition principle which reduces the general solution to a static elastic solution, an elastic solution of an eigenvalue problem and an integral equation of the Volterra type involving time only. The solutions are given in double infinite series involving spherical Bessel functions, Legendre polynomials and Legendre functions of the first kind and order one.     

Circular inclusion in a viscoelastic plate subjected to uniaxial tension

The effects of a circular inclusion in an infinite viscoelastic plate subjected to monotonically increasing uniaxial tension are investigated, in particular the stress distributions around the inclusion. An elastic inclusion is considered and both rough and smooth contact treated. The solution is limited to complete contact around the inclusion circumference. The results hinge on two lemmas concerning the monotonicity and sign of reciprocal and combined viscoelastic response functions.     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.     

求解一类非线性粘弹性问题的弹性回复对应原理

本文提出了一个全新的粘弹性理论体系,它把传统的线粘弹性理论作为特殊情况包括在内。其主要结果是两个用于求解一类物理非线性粘弹性问题的弹性回复对应原理。利用它,只要知道相应的非线性弹性问题的解,就可以求出非线性粘弹性问题的解答。对应原理不是基于本构关系的相似性,而是基于从粘弹性现时响应到瞬时弹性响应的可回复性^[1]。首先找到非线性粘弹性与非线性弹性本构关系之间的对应,然后导出了两个弹性回复对应原理。通过对改性聚丙烯材料的实验验证了该对应原理的正确性和对此类材料的实用性。     

高分子材料非线性粘弹性问题的解法

尽管实验限制在小变形范围之内，高分子材料仍表现出明显的非线性粘弹性性能。目前由于分析上的困难，通常只好采用线粘弹性假设。基于前文，提出了一个求解高分子材料物理非线性（含线性）粘弹性问题的新原理，通过对改性聚丙烯材料的实验表明了该原理的正确性和对于此类材料的实用性。     

BEM approach to protection of structures from seismic waves by a viscoelastic stratum

In the context of the anti-plane problem, we study harmonic oscillations of an elastic rectangle whose base is isolated from the foundation by a viscoelastic stratum of finite thickness. Such a (viscoelastic) medium is embedded in the ground, which is simulated as a homogeneous isotropic elastic half-space. By using the fundamental representation of solution through single and double layer potentials, the problem is reduced to a set of integral equations. The main practical aspect of this investigation is connected with seismic base isolation. We thus estimate the wave attenuation, which results in the decrease of the vibration amplitude for the rectangular construction, provided by the viscoelastic stratum.     

The band gap of 1D viscoelastic phononic crystal

The band gap of one dimensional (1D) phononic crystal with viscoelastic host material is studied in this paper. The standard solid model is used to simulate the viscoelastic behavior of the host material and the fillers embedded in the host material are still assumed elastic material. The band gap problem in 1D phononic crystal leads to an eigenvalue problem by using the plane wave expansion method and the Bloch–Floquet wave theory in a periodic structure. An iterative algorithm is designed to obtain the band gap structure due to the dependence of elastic constants on frequency for the viscoelastic host material. A numerical example is given for steel/epoxy phononic crystal. The band gap of 1D phononic crystal is evaluated for different viscoelastic constants, namely, relaxation time, initial and final state elastic modulus. It is found that the viscoelastic constants of host material affect not only the location but also the width of band gaps.     

Cold rolling of a laminated composite sheet—A numerical solution

The problem of cold rolling of a composite sheet which consists of alternate layers of viscoelastic and elastic materials is studied by the finite element method. Results computed and presented graphically include the pressure distribution at the contact surface, the normal stress distribution at the middle surface and the asymmetric deformation of the surface of the sheet in contact with the roller.     

粘弹结构动力学分析

本文研究由单一粘弹材料构成的粘弹结构的振动.在时间域和拉氏域内建立了它的动力学方程.给出了粘弹结构的"对偶弹性结构"概念,并用来讨论了粘弹结构的模态特性,证实位移形态可由实模态向量张成的向量空间内加以描述,它就是"对偶弹性结构"的模态向量,无需增广坐标.每个模态自由度含有数个反映蠕变的实特征值和一对反映衰减振动的复共轭特征值.分析了在模态向量空问内的动响应,给出了传递函数(频率响应)和脉冲响应.从而揭示了粘弹结构的动力学特性.     

The transient quasi-static plane viscoelastic moving load problem

The problem of a moving load on a viscoelastic half-plane is considered, in the non-inertial approximation. From the viscoelastic generalization of the Kolosov-Muskhelishvli equations, an integral equation is derived, which applies to the general transient problem. In the case of frictionless contact, explicit transient corrections to the coefficient of hysteretic friction are given for large times—when these corrections are small. Particularly explicit results are given for the case where viscoelastic losses are small. It emerges that the transient corrections to the hysteretic friction coefficient are negative.     

Dynamic stress concentration around a hole in a viscoelastic plate

Summary The transient hoop stresses which are generated at the circumference of a circular hole in a large viscoelastic plate when a radial pressure pulse acts on the hole boundary, are determined. An analytical/numerical approach is employed which is based on the use of Laplace transform and Hankel functions, and the Dubner-Abate-Crump technique for inverting the Laplace-transformed solution. In our formulation of the problem, a general linear-viscoelastic material for the plate is considered, but numerical results are extracted only for a three-parameter model (standard linear solid). The present work accompanies recent efforts by Georgiadis on analogous transient viscoelastodynamic problems involving finite-length cracks in a stress-wave environment. The mathematically simpler problem considered here demonstrates the influence of viscoelastic effects on the dynamic-stress-concentration in marked similarity with the dynamic-stress-intensity behavior of crack-tip viscoelastic stress fields. Besides that, the present problem generalizes the classical Kromm-Selberg-Miklowitz problem in the sense that it considers viscoelastic response. The latter solutions were not utilized here through the usual elastic/viscoelastic correspondence principle but, instead, an independent solution was derived by simple means.     

Measurements of Viscoelastic Functions of Polymers in the Frequency-Domain Using Nanoindentation

A method to measure the complex compliance (or modulus) of linearly viscoelastic materials is presented using nanoindentation with a spherical indenter. The Hertzian solution for an elastic indentation problem, in combination with a hereditary integral operator proposed by Lee and Radok (Journal of Applied Mechanics 27, 1960, 438–444) for the situation of non-decreasing indentation contact area, was used to derive formulas for the complex viscoelastic functions in the frequency-domain. The formulas are most suitable for frequencies lower than a frequency limit such that the condition of non-decreasing contact area holds; they are reasonably good approximation at higher frequencies under which decreasing contact area occurs and the Ting (Journal of Applied Mechanics 33, 1966, 845–854) approach for arbitrary contact area history is needed. Nanoindentation tests were conducted on both polycarbonate and polymethyl methacrylate under a harmonic indentation load superimposed on either step or ramp indentation load, while the resulting displacement under steady state was recorded. The load and displacement data at each frequency were processed using the derived formulas to determine the viscoelastic functions in the frequency-domain. The same materials were also tested using a dynamic mechanical analysis (DMA) apparatus to determine the complex viscoelastic functions. The DMA and nanoindentation results were compared and found in a good agreement, indicating the validity of the new method presented.     

Mode II Macrocrack Initiation in Orthotropic Composite Viscoelastic Plate

Safe loads and initiation time for a straight macrocrack in viscoelastic orthotropic material that is intended to model a fiber composite plate under shear loads is investigated. The composite material is modeled by viscoelastic orthotropic medium. Determination of expression for crack shear displacement as function of time is based on the corresponding elastic solution and the method of operator continued fractions. Initiation time is obtained as a solution of integral equation for the incubation period. Numerical calculations are given for mode II macrocrack initiation.     

Nanorheological analysis of the sphere plane contact problem with interfacial films

The contact between an elastic sphere and plane separated by an elastic interfacial thin layer is analysed under pure squeeze action. This analysis is an important step in the investigation of more realistic problem such as viscoelastic and heterogeneous squeeze problem encountered in surface force experiments. The problem is shown to be governed by two dimensionless parameters coupling the material properties of the layer (shear and bulk modulus) and the substrate (reduced young modulus) with the thinness ratio of the sphere plane distance to the sphere radius. Different regimes are obtained and described according to the value of these two parameters.     

On propagation of small non-stationary disturbances in linear viscoelastic fluids

Summary A representation is given for the solutions of linearized equations describing a flow of compressible viscoelastic fluids, the spherical part of tensor of stresses corresponding to the rheological Voigt body, while the deviator part corresponds to the Maxwell, Oldroyd and Kelvin-Voigt body of any order. Two independent equations for the propagation of longitudinal and transverse disturbances are obtained.The Laplace transform is applied to study the propagation of plane non-stationary longitudinal and transverse waves in linear viscoelastic fluids, where the spherical part of the tensor of stresses corresponds to the elastic body, while the deviator part corresponds to the Maxwell body. The problem of inversion is reduced to the numerical solution of the linear homogeneous Volterra integral equation of the second kind with a discontinuous kernel.