An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

On the Measurement of two Independent Viscoelastic Functions with Instrumented Indentation Tests

In the present paper, a methodology for complete characterization of linear isotropic viscoelastic material with spherical instrumented indentation test is proposed. The developed method allows for measuring two independent viscoelastic functions, shear relaxation modulus and time-dependent Poisson’s ratio, from the indentation test data obtained at non-decreasing loading, but otherwise arbitrary. Finite element modelling (FEM) is relied upon for validating the proposed methodology and for quantifying the influence of experimental variables on the measurements accuracy. Spherical indentation experiments are performed on several viscoelastic materials: polyoxymethylene, bitumen and bitumen-filler mastics. The viscoelastic material functions obtained with the indentation tests are compared with the corresponding results from the standard mechanical tests. Numerical and experimental results presented indicate that the methodology proposed allows mitigating the machine compliance and loading rate effects on the accuracy of the viscoelastic indentation tests.     

用能量法分析层状材料弹性接触问题

利用能量法分析了层状材料（薄膜／基体）弹性接触问题，得到了具有一阶精度的闭合解，给出了求解薄膜弹性模量和泊松比的表达式，并与有限元的数值解进行了比较。二者比较结果表明：在工程材料范围内，理论解与数值解相差在6％以内；同时表明单相材料中剪切模量与弹性模量之间的关系也适用层状材料中的薄膜材料。在数值解的基础上，讨论了薄膜厚度与压头半径的比值对求解精度的影响，发现此比值对精度影响不大。通过对层状材料等效泊松比与等效弹性模量的定义，给出了用压痕实验测定薄膜泊松比与弹性模量的方法。     

负泊松比结构力学设计、抗冲击性能及在车辆工程应用与展望

轻量化多功能负泊松比结构由于具有优异的可设计性、拉胀特性、剪切模量、断裂韧性、抗冲击吸能、减震降噪等特性,在车辆吸能结构设计和多功能优化方面具有巨大的应用潜力.本文详细综述了负泊松比结构的力学设计及其在车辆工程中的典型应用:(1)负泊松比基本概念及其力学特性,以及近几十年来的快速发展趋势;(2)负泊松比材料与结构构型设...     

Mechanics of non-slipping adhesive contact on a power-law graded elastic half-space

In previous work about axisymmetric adhesive contact on power-law graded elastic materials, the contact interface was often assumed to be frictionless, which is, however, not always the case in practical applications. In order to elucidate the effect of friction and the coupling between normal and tangential deformations, in the present paper, the problem of a rigid punch with a parabolic shape in non-slipping adhesive contact with a power-law graded half-space is studied analytically via singular integral equation method. A series of closed-form analytical solutions, which include the frictionless and homogeneous solutions as special cases, are obtained. Our results show that, compared with the frictionless case, the interfacial friction tends to reduce the contact area and the indentation depth during adhesion. The magnitude of the coupling effect depends on both the Poisson ratio and the gradient exponent of the half-space. This effect vanishes for homogeneous incompressible as well as for linearly graded materials but becomes significant for auxetic materials with negative Poisson’s ratio. Furthermore, influence of mode mixity on the adhesive behavior of power-law graded materials, which was seldom touched in literature, is discussed in details.     

Characterization of the Damping Behavior of a Nanoindentation Instrument for Carrying Out Dynamic Experiments

A new displacement modulation based dynamic indentation method is demonstrated and shown to be effective for viscoelastic characterization of a glassy polymer. The analysis of dynamic experiments requires a complete understanding of the measuring system’s dynamic characteristics especially the damping. Accordingly, an improved method, based on the use of a wire spring, is developed for determining the damping characteristics. In general, damping in an indentation instrument is contributed by two elements: the eddy current damping from the electromagnetic loading coil and the squeeze film damping from the capacitive displacement transducer. Therefore, a method to determine the relative contribution from the different damping elements present in the system is demonstrated and the results are compared with the calibration obtained from the wire spring method. Finally, dynamic indentation tests are carried out on a glassy polymer to obtain the complex modulus; the values of which are compared with those obtained from bulk dynamic mechanical analysis (DMA) tests. Storage modulus values are found to be in good agreement with bulk data but some divergence in the case of loss modulus is observed. The calibration procedure of the measuring instrument is critically examined in view of these observations. Overall, displacement modulation based dynamic indentation is shown to be a promising method for viscoelastic characterization at the micron length scales. An erratum to this article can be found at     

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.     

Multicycle Indentation for Evaluation of Polymer Material Viscoelastic Characteristics

The evaluation of viscoelastic characteristics by an indentation test is required in order to separate the viscoelastic deformation and plastic deformation from the measured penetration depth. In this study, a multicycle indentation test is performed in order to separate these deformations and to evaluate the viscoelastic characteristics of acrylonitrile butadiene styrene (ABS) and polybutylene terephthalate (PBT). In the multicycle indentation test, two or more indentation tests are conducted at the same specimen position. After the first cycle, the indentation test is performed on the impression; thus, an evaluation formula for obtaining the viscoelastic characteristics is proposed, and the validity of this formula is confirmed by finite element analysis. As a result, the evaluated creep compliances of both materials are close the value evaluated by tensile testing. Therefore, it is concluded that the proposed method is applicable to the evaluation of viscoelastic characteristics.     

Viscoelastic solutions for conical indentation

The aim of indentation analysis is to link indentation data, typically an indentation force vs. indentation depth curve, P–h, to meaningful mechanical properties of the indented material. While well established for time independent behavior, the presence of a time dependent behavior can strongly affect both the loading and the unloading responses. The paper presents a framework of viscoelastic indentation analysis based on the method of functional equations, developed by Lee and Radok [1960, The contact problem for viscoelastic bodies, J. Appl. Mech. 27, 438–444]. While the method is restricted to monotonically increasing contact areas, we show that it remains valid at the very beginning of the unloading phase as well. Based on this result, it is possible to derive closed form solutions following the classical procedure of functional formulations of viscoelasticity: (1) the identification of the indentation creep function, which is the indentation response to a Heaviside load; and (2) a convolution integral of the load history over the indentation creep function. This is shown here for a trapezoidal loading by a conical indenter on three linear isotropic viscoelastic materials with deviator creep: the 3-parameter Maxwell model, the 4-parameter Kelvin–Voigt model and the 5-parameter combined Kelvin–Voigt–Maxwell model. For these models, we derive closed form solutions that can be employed for the back-analysis of indentation results from the loading and holding period and for the definition of unloading time criteria that ensure that viscous effects are negligible in the unloading response.     

Second-order two-scale computational method for ageing linear viscoelastic problem in composite materials with periodic structure

The correspondence principle is an important mathematical technique to compute the non-ageing linear viscoelastic problem as it allows to take advantage of the computational methods originally developed for the elastic case. However, the correspondence principle becomes invalid when the materials exhibit ageing. To deal with this problem, a second-order two-scale (SOTS) computational method in the time domain is presented to predict the ageing linear viscoelastic performance of composite materials with a periodic structure. First, in the time domain, the SOTS formulation for calculating the effective relaxation modulus and displacement approximate solutions of the ageing viscoelastic problem is formally derived. Error estimates of the displacement approximate solutions for SOTS method are then given. Numerical results obtained by the SOTS method are shown and compared with those by the finite element method in a very fine mesh. Both the analytical and numerical results show that the SOTS computational method is feasible and efficient to predict the ageing linear viscoelastic performance of composite materials with a periodic structure.     

Higher-order theory for periodic multiphase materials with inelastic phases

An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material's periodic microstructure. The model's analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The model's predictive accuracy in generating both the effective inelastic stress-strain response and the local stress and inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model and with finite-element results for transverse loading.     

Size-dependent sharp indentation—I: a closed-form expression of the indentation loading curve

In this paper, a closed-form expression of the size-dependent sharp indentation loading curve has been proposed based on dimensional analysis and the finite deformation Taylor-based nonlocal theory (TNT) of plasticity (Int. J. Plasticity 20 (2004) 831). The key issue is to link the results of FEM based on TNT plasticity with those obtained using conventional FEM by taking as the effective strain gradient, η, that presented in the work of Nix and Gao (J. Mech. Phys. Solids 46 (1998) 411), thus avoiding large-scale finite element computations using strain gradient plasticity theories. Two experiments carried out on 316 stainless-steel and pure titanium have been used to verify the effectiveness of the present analytical model; the results demonstrate that the present analytical expression of the size-dependent indentation loading curve corresponds very well to the experimental indentation loading curve. The empirical constant, α, in the Taylor model estimated from the experimental data has the correct order of magnitude. Also, the results presented in this part can be further applied to establish an analytical framework to extract the plastic properties of metallic materials with sharp indentation on a small scale where the size effect caused by geometrically necessary dislocations is significant. This will be discussed in detail in the second part of the paper.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Thermo-electromechanical behavior of functionally graded piezoelectric hollow cylinder under non-axisymmetric loads

This paper presents an analytical solution of a thick walled cylinder com- posed of a functionally graded piezoelectric material (FGPM) and subjected to a uniform electric field and non-axisymmetric thermo-mechanical loads. All material properties, except Poisson's ratio that is assumed to be constant, obey the same power law. An exact solution for the resulting Navier equations is developed by the separation of variables and complex Fourier series. Stress and strain distributions and a displacement field through the cylinder are obtained by this technique. To examine the analytical approach, different examples are solved by this method, and the results are discussed.     

Finite elements for viscoelastic flows with change of type

We compare several mixed finite-element methods for calculating viscoelastic flows where the vorticity equation changes type from elliptic to hyperbolic whenever inertia is taken into account. The flows are perturbed viscometric flows with slightly wavy walls. The perturbed uniform flow gives rise to a closed-form analytical solution. We examine five different finite-element algorithms; it is found that the so-called SU4×4 and EVSS methods perform much better than the other three. We also examine a number of features proper to flows with change of type, such as the propagation of disturbances along characteristic lines of the vorticity equation.This author acknowledges financial support from IRSIA (Institut pour l'Encouragement de la Recherche Scientifique dans l'Industrie et l'Agriculture) through a research grant.     

Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation

The large deformation of an elastic axisymmetric membrane in adhesive contact with a rigid flat punch is studied. Detachment of membrane is analyzed using a critical energy release rate criterion. Two types of incompressible hyperelastic material models are considered: neo-Hookean and a class of materials whose elastic energy density functions are independent of the trace of the Cauchy–Green tensor (I₂-based material). We also include pre-stretch in our formulation and study the stability of detachment process. Closed form analytical solutions for the membrane stresses, deformed profiles and energy release rate are obtained in the regime of large longitudinal stretch. For the I₂-based material, we discover an interesting “pinching” instability where the contact angle suddenly increases in a displacement controlled test. The region of validity of our analytical solutions is determined by comparing them with numerical solutions of the governing equations. We found that the accuracy of our solution improves with pre-stretch; for pre-stretch ratios greater than 1.3, our analytical solution also works well in the small deformation regime.     

A new finite element scheme using the Lagrangian framework for simulation of viscoelastic fluid flows

A new numerical scheme for simulation of viscoelastic fluid flows was designed, making use of finite element algorithms generally regarded as advantageous for tackling the problem. This includes the Lagrangian approach for the solution of viscoelastic constitutive equation using the co-deformational frame of reference with a possibility of analytically solving the equation along the particles trajectories, which in turn allowed eluding the solution of any system of linear equations for the stress. Then, the full ellipticity of the momentum conservation equation was utilised thanks to a possibility of accurate determination of the stress tensor independently of the velocity field at the current stage of computation. The needed independent stress was calculated at each time step on the basis of the past deformation history, which in turn was determined on the basis of the past velocity fields, all incorporated into a modified Euler time stepping algorithm. Owing to explicit inclusion of the full viscous term from the viscoelastic model into the momentum conservation equation, no stress splitting was necessary. The trajectory feet tracking was done accurately using a semi-analytic solution of the displacement gradient evolution equation and a weak formulation of the kinematics equation, the latter at the expense of solving an extra symmetric system of linear equations.The error expressed in the form of the Sobolev norms was determined using a comparison with available analytical solution for UCM fluid in the transient regime or numerically obtained steady-state stress values for the PTT fluid in Couette flow. The implementation of the PTT fluid model was done by modifying the relative displacement gradient tensor so that a new convective frame was defined.The stability of the algorithm was assessed using the well-known benchmark problem of a sphere sedimenting in a tube with viscoelastic fluid. The stable numerical results were obtained at high Weissenberg numbers, with the limit of convergence Wi=6.6, exceeding any previously reported values. The robustness of the code was proven by simulation of the Weissenberg effect (the rod-climbing phenomenon) with the use of PTT fluid.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Measurements of Two Independent Viscoelastic Functions by Nanoindentation

Current nanoindentation measurement techniques normally assume that one material function (such as the Poisson's function) is a constant, and measures just one material function, such as the creep compliance in shear. For materials with significant viscoelastic effects and unknown viscoelastic functions, assuming a constant for one material function is not satisfactory. Accurate measurements require simultaneously determining two independent material functions. This paper provides a method to use nanoindentation to measure both bulk and shear relaxation functions. Two different nanoindenter tips, namely Berkovich and spherical indenters, are used for nanoindentation on polymers. Any two independent viscoelastic functions, such as bulk relaxation modulus and shear relaxation modulus, have different representations in the load–displacement curves obtained with these two indenters so that the two independent viscoelastic functions can be separated and determined. Two polymers, poly(vinyl acetate) (PVAc) and poly(methyl methacrylate) (PMMA) were used in nanoindentation. Nanoindentation measurements were conducted on PVAc above glass transition temperature (Tg) and on PMMA below Tg. Both shear and bulk relaxation functions determined from nanoindentation were found in a reasonably good agreement with data obtained from conventional tests, providing validation of the method presented. The new method can be applied in measurements of two independent viscoelastic functions at sub-micron scale of very small amounts of materials such as polymeric films on a substrate, heterogeneous materials such as bones, tissues, and nanocomposites.