Spherical indentation of incompressible rubber-like materials

In recent years, indentation tests have been proven very useful in probing mechanical properties of small volumes of materials. However, a class of materials that very little has been done in this direction is rubber-like materials (elastomers). The present work investigates the spherical indentation of incompressible rubber-like materials. The analysis is performed in the context of second-order hyperelasticity and is accompanied by finite element computations and an extensive experimental program with spherical indentors of different radii. Uniaxial tensile tests were also performed and it was found that the initial elastic modulus correlates well with the indentation response. The experiments suggest stiffer indentation response than that predicted by linear elasticity, which is somehow counter-intuitive, if the uniaxial material response is to be considered. Regarding the uniqueness of the inverse problem, that is to establish material properties from spherical indentation tests, the answer is disappointing. We prove that the inverse problem does not give unique answer regarding the constitutive relation, except for the initial stiffness.     

一种新的橡胶材料弹性本构模型

橡胶类材料本构关系对于科学研究和工程应用具有重要意义,但已有的橡胶模型的拟合能力和可靠性需要进一步提高.为解决此问题,本文提出了一种新的橡胶材料的各向同性、不可压缩柯西弹性模型.研究了橡胶材料本构关系的模型形式,基于平面应力变形状态,提出了一种以较大的两个伸长率为自变量、适用于一般变形状态的橡胶材料弹性本构模型形式;研究了橡胶材料在侧面受约束条件下的变形规律,分析了橡胶材料本构关系需要满足的约束条件;在此基础上,结合一个可以通过实验确定的描述平面拉伸变形状态下的橡胶材料力学特性函数,提出一种将该函数拓展为平面应力状态一般模型的方法,并给出了一个具体的函数形式,形成了一个新的不可压缩、各向同性的橡胶材料弹性本构模型.使用5组包含3种类型实验的数据和一组较全面的双轴测试数据对该模型进行了参数拟合,结果表明:该模型具有很好的拟合精度和更高的可靠性,仅用一种类型实验数据,如单轴拉伸或者平面拉伸等,也能获得较好的拟合结果.      

Mechanics of indentation of plastically graded materials—II: Experiments on nanocrystalline alloys with grain size gradients

A systematic study of depth-sensing indentation was performed on nanocrystalline (nc) Ni-W alloys specially synthesized with controlled unidirectional gradients in plastic properties. A yield strength gradient and a roughly constant Young's modulus were achieved in the nc alloys, using electrodeposition techniques. The force vs. displacement response from instrumented indentation experiments matched very well with that predicted from the analysis of Part I of this paper. The experiments also revealed that the pile-up of the graded alloy around the indenter is noticeably higher than that for the two homogeneous reference alloys that constitute the bounding conditions for the graded material. These trends are also consistent with the predictions of the indentation analysis.     

On establishing elastic–plastic properties of a sphere by indentation testing

Instrumented indentation is a popular technique for determining mechanical properties of materials. Currently, the evaluation techniques of instrumented indentation are mostly limited to a flat substrate being indented by various shaped indenters (e.g., conical or spherical). This work investigates the possibility of extending instrumented indentation to non-flat surfaces. To this end, conical indentation of a sphere is investigated where two methodologies for establishing mechanical properties are explored. In the first approach, a semi-analytical approach is employed to determine the elastic modulus of the sphere utilizing the elastic unloading response (the “unloading slope”). In the second method, reverse analysis based on finite element analysis is used, where non-dimensional characteristic functions derived from the force–displacement response are utilized to determine the elastic modulus and yield strength. To investigate the accuracies of the proposed methodologies, selected numerical experiments have been performed and excellent agreement was obtained.     

表面残余应力的仪器化压入检测方法研究进展

针对表面残余应力的仪器化压入检测方法,阐释利用压入方式检测残余应力的基本原理及其力学机制,梳理建立残余应力压入检测方法的常用技术路线,结合残余应力压入检测方法的不同分类,着重分析6种代表性压入检测方法的优势与局限,讨论验证压入检测方法可靠性的常用方法,最后总结残余应力压入检测方法的研究进展,展望未来的发展趋势为建立无需参考试样一体化检测材料力学参数和非等轴残余应力的仪器化压入分析方法,探讨建立方法的四个研究要领,即机制清楚、分析可靠、技术可行、结果可信.仪器化压入检测技术是郑哲敏先生晚年的研究兴趣之一,作者在中科院力学所工作期间曾受到郑哲敏先生的热切关注和提问式激励,谨以此文纪念郑哲敏先生逝世周年.     

Improved reverse analysis for material characterization with dual sharp indenters

It was illustrated by the author in the previous work that combinations between material properties and indentation parameters can be used as mixed parameters in dimensionless functions to capture the indentation response of materials to single and dual sharp indenters. These issues are further extended in the present study. A parametric finite element analysis was performed to investigate the conical indentation response of elasto-plastic solids. Frictional effects are studied. Conical indenters of half-included angles from 50° to 88° are considered to examine several fundamental features of instrumented sharp indentation within the frame work of limit analysis. Regarding dimensional analysis, it is found that a Taylor series expansion according to the elastic indentation work-total indentation work ratio W_e/W_t can be used to improve dimensionless functions. Within this context, a new set of dimensionless functions is explicitly constructed for hardness and indentation parameters of single and dual indenters. Based on formulated functions, a reverse analysis with dual sharp indenters, which was previously proposed by the author, is improved to extract mechanical properties of materials.     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

On a molecular statistical basis for Ogden's model of rubber elasticity

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

Analysis of spherical indentation of materials with plastically graded surface layer

In the present work, a comprehensive parametric study for establishing contact mechanics of instrumented normal spherical indentation on homogeneous materials and materials with plastically graded surface layer (PGSL) was undertaken by dimensional analysis and finite element modeling. The spherical indentation response for homogeneous materials can be described only by two dimensionless parameters: strain hardening exponent and a unified parameter that can describe effects of both the normalized yield strength and the normalized indentation depth. The influences of these two parameters were investigated for a wide range of engineering materials, and the results may be used as an estimate of loading response and pile-up/sink-in behavior when the material properties are known. In the materials with PGSL, a linear gradient in yield strength, and no variation in elastic modulus and strain hardening exponent were explored. The indentation response of the materials with PGSL can be described only by three dimensionless parameters: the normalized indentation depth, the dimensionless strength gradient parameter, and the normalized PGSL thickness. The effects of these three parameters were studied systematically. The normalized pile-up/sink-in parameter is found to be an increasing function of the strength gradient parameter. The normalized pile-up/sink-in parameter increases (decreases) with increasing PGSL thickness for a fixed positive (negative) gradient case at large indentation depth. The results also indicate that the materials with positive PGSL can bear more loads and have significantly more resistance to contact crack formation.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

Rubber-like elasticity by boundary element method: Finite deformation of a circular elastic slice

We briefly review the phenomenological theory of rubber-like elasticity and report a microstructural model that leads us to eventually adopt a particular constitutive equation, which includes the Neo-Hookean and the Mooney materials. A numerical implementation of the Boundary Element method for solving a general two-dimensional or axisymmetric finite deformation problem is described and tested with some simple deformations. The resulting program is used to analyse the finite deformation of a circular elastic slice perfectly bonded to two parallel rigid end plates; the bottom plate is stationary and the top plate is given a constant displacement. The problem has a free surface which must be found as part of the solution. The results indicate that the Boundary Element method can be an efficient tool for stress-strain analyses with rubber-like materials.     

Mechanics of indentation of plastically graded materials—I: Analysis

The introduction of controlled gradients in plastic properties is known to influence the resistance to damage and cracking at contact surfaces in many tribological applications. In order to assess potentially beneficial effects of plastic property gradients in tribological applications, it is essential first to develop a comprehensive and quantitative understanding of the effects of yield strength and strain hardening exponent on contact deformation under the most fundamental contact condition: normal indentation. To date, however, systematic and quantitative studies of plasticity gradient effects on indentation response have not been completed. A comprehensive parametric study of the mechanics of normal indentation of plastically graded materials was therefore undertaken in this work by recourse to finite element method (FEM) computations. On the basis of a large number of computational simulations, a general methodology for assessing instrumented indentation response of plastically graded materials is formulated so that quantitative interpretations of depth-sensing indentation experiments could be performed. The specific case of linear variation in yield strength with depth below the indented surface is explored in detail. Universal dimensionless functions are extracted from FEM simulations so as to predict the indentation load versus depth of penetration curves for a wide variety of plastically graded engineering metals and alloys for interpretation of, and comparisons with, experimental results. Furthermore, the effect of plasticity gradient on the residual indentation pile-up profile is systematically studied. The computations reveal that pile-up of the graded alloy around the indenter, for indentation with increasing yield strength beneath the surface, is noticeably higher than that for the two homogeneous reference materials that constitute the bounding conditions for the graded material. Pile-up is also found to be an increasing function of yield strength gradient and a decreasing function of frictional coefficient. The stress and plastic strain distributions under the indenter tip with and without plasticity gradient are also examined to rationalize the predicted trends. In Part II of this paper, we compare the predictions of depth-sensing indentation and pile-up response with experiments on a specially made, graded model Ni-W alloy with controlled gradients in nanocrystalline grain size.     

Indentation of plastically graded substrates by sharp indentors

The present paper investigates the indentation of plastically graded substrates by sharp indentors. Contact analysis of plastically graded surfaces can be particularly useful in the design of load-bearing devices such as gears, rollers and electric contacts found in many macro- and micro-electro-mechanical systems. Substrates made of plastically graded materials are often encountered in nature or are artificially produced as a result of chemical and/or physical surface treatments. The variation of the plastic properties depends on micro-structural or compositional changes of the material with depth. The analysis of indentation of plastically graded substrates by sharp indentors provide the load-penetration response, as well as the strains and stresses inside the substrate, at maximum loading and at complete unloading. The parametric analysis of the solutions enables the direct correlation of the plastic properties and the load-penetration curves obtained from instrumented indentation tests. The variation of the plastic properties can subsequently be related to important micro-structural parameters such as particle composition, dislocation density and grain size. The results of this work show how surface modifications can induce plastic graded properties that strengthen substrates against contact-induced damage.     

Determining plastic properties of a material with residual stress by using conical indentation

Instrumented indentation is a popular method for determining mechanical properties in engineering materials. However, there are several shortcomings and challenges involved with correctly interpreting the test results. We propose here a unified method for evaluating instrumented indentation testing conducted on a material that exhibits both strain hardening under yielding and which is subjected to uniform, equi-biaxial residual stresses. The proposed method is based on extensive finite element simulations that relate the parameter-space spanned by Young’s modulus, yield strength, strain hardening and residual stress, to the response from the indentation test. Based on reverse analysis, the proposed method can be used to determine two unknown quantities, such as yield strength and strain hardening. The technique involves utilizing the concept of representative strain and plural indenter-shapes.     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

Experiments and Material Parameter Identification Using Finite Elements. Uniaxial Tests and Validation Using Instrumented Indentation Tests

In this article, we focus our attention on the relation between instrumented indentation tests and the prediction by means of finite element calculations. To this end, a finite strain viscoplasticity model of Perzyna-type with non-linear isotropic and kinematic hardening is calibrated at experimental data of steel S690QL. A particular concept for conducting uniaxial tensile and compression tests is taken up in order to represent the basic rate-dependent material behavior. In this respect, an algorithmic framework of material parameter identification using finite elements is proposed leading to a two-stage procedure in the case of the underlying rate-dependent constitutive model. On the basis of the termination points of relaxation the rate-independent equilibrium stress state can be identified and all viscous parts of the model are obtained using rate-dependent loading paths. Finally, use is made of finite elements for predicting indentation experiments, which results in a critical view on modeling and parameter identification on the basis of experimental results occurring in instrumented indentation tests.     

Local mechanical properties of the 6061-T6 aluminium weld using micro-traction and instrumented indentation

The local mechanical properties of a weld zone, in a 6061-T6 aluminium alloy subjected to the modified indirect electric arc technique have been studied. The mechanical properties of the base metal, the weld metal and the heat affected zone were determined by means of usual and instrumented indentation testing, as well as micro-traction testing. To analyse the heat input effect resulting from the welding process, the evolution of the weld zone size was evaluated by means of classical indentation under a constant applied load. The results were presented using a Vickers hardness map representation. This allows monitoring exact hardness variation while leading to the identification of the different zones of the welded joint. Instrumented indentation testing was carried out to determine the local mechanical properties, such as the yield stress, the bulk modulus and the strain-hardening exponent. Obtained results are compared to those derived from tensile tests conducted on micro-specimen cuts taken from the weld zone. It was observed that yield stress values are directly comparable for indentation and micro-traction experiments. As for the elastic properties, no comparison was possible since the bulk modulus is measured by indentation, whereas it is the Young’s modulus by tensile test. The micro-traction testing seems to be more sensitive to represent the work hardening of a material since the corresponding exponent is found to be constant by instrumented indentation.     

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.