A simple technique for determining yield strength of thin films

A technique to measure the yield strength of thin films has been developed which combines experimental observations of deflection and plastic deformation with finite element predictions of stress. This technique relies on integrated circuit technology to build bridge and cross beam test structures with a range of dimensions. Each structure is deflected in increments of 1 μm until the structure no longer elastically recovers upon release. In tandem with experimentally verified numerical predictions of force and stress, the yield strength of the thin film can be bounded between the highest elastic stress result and the lowest plastic stress result. For our test material of copper, this method provides a yield strength between 2.80 and 3.09 GPa, a value significantly larger than that for bulk copper, but consistent with thin film theory.     

Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates

A study has been made of the elastic and plastic deformation associated with submicrometer indentation of thin films on substrates using the finite element method. The effects of the elastic and plastic properties of both the film and substrate on the hardness of the film/substrate composite are studied by determining the average pressure under the indenter as a function of the indentation depth. Calculations have been made for film/substrate combinations for which the substrate is either harder or softer than the film and for combinations for which the substrate is either stiffer or more compliant than the film. It is found, as expected, that the hardness increases with indentation depth when either the yield strength or the elastic modulus of the substrate is higher than that of the film. Correspondingly, the hardness decreases with indentation depth when the yield strength or elastic modulus of the substrate is lower than that of the film. Functional equations have been developed to predict the hardness variation with depth under these different conditions. Finite element simulation of the unloading portion of the load displacement curve permits a determination of the elastic compliance of the film/substrate composite as a function of indentation depth. The elastic properties of the film can be separated from those of the substrate using this information. The results are in good agreement with King's analytical treatment of this problem.     

空心颗粒填充复合材料的建模与弹塑性仿真计算

空心玻璃微珠（hollow glass microballon,简称HGM）填充复合材料称为复合泡沫材料,近来已被广泛应用在工业中。利用RSA(Random Sequential Adsorption)方法生成了含不同体积比的HGMs填充代表体元模型,然后用有限元方法计算得出了材料的应力-应变关系,将材料属性简化为双线性随动强化模型,分析了HGM填充比、壁厚对材料的有效弹性常数、屈服极限的影响,并分析了材料内部细观应力场和塑性应变分布情况。结果发现,HGM壁厚比对材料有效弹性模量和屈服极限的增减起着决定性作用,而对于任意填充模式来说,其比模量和比强度总是大于纯树脂,这一点体现了该材料轻质的优良特性。材料基体的应力集中部位分布以及塑性应变区域的分布也取决于HGM的壁厚比。     

Kagome平面格栅结构的屈服面及裂尖塑性区分析

分析了Kagome格栅的等效刚度和屈服面. 其屈服面奇异,由4段直线围成. 利用该屈服面,估算了Kagome具有I型、II型半无限大裂纹的裂尖塑性区,有限元计算验证了解析预测的准确性. 与奇异屈服面相比,由Mises光滑屈服面给出的塑性区误差较大. 因此只有弹性情况,可以将Kagome等效为各向同性;若材料塑性,或应力场奇异性较强,Kagome的强度依赖于主应力方向,不能用各向同性模型来描述.      

Determining the Elastic Modulus of Compliant Thin Films Supported on Substrates from Flat Punch Indentation Measurements

Instrumented indentation is a technique that can be used to measure the elastic properties of soft thin films supported on stiffer substrates, including polymer films, cellulosic sheets, and thin layers of biological materials. When measuring thin film properties using indentation, the effect of the substrate must be considered. Most existing models for determining the properties of thin films from indentation measurements were developed for metal and dielectric films bonded to semiconductor substrates and have been applied to systems with film-substrate modulus ratios between 0.1 and 10. In the present work, flat punch indentation of a thin film either bonded to or in contact with a substrate is examined using finite element modeling. A broad range of film-substrate modulus ratios from 0.0001 to 1 are investigated. As the substrate is effectively rigid compared to the film when the film-substrate modulus ratio is less than 0.0001, the results are also useful for understanding systems with lower film-substrate modulus ratios. The effects of the contact radius, film thickness, elastic properties, and friction between the film and the substrate on the measured stiffness were quantified using finite element modeling in order to understand how the elastic properties of the film can be extracted from indentation measurements. A semi-analytical model was developed to describe the finite element modeling results and facilitate the use of the results to analyze experimental measurements. The model was validated through analysis of indentation measurements of thin polyethylene sheets that were supported on substrates of various stiffness.     

Scaling laws of nanoporous gold under uniaxial compression: Effects of structural disorder on the solid fraction,elastic Poisson's ratio,Young's modulus and yield strength

In this work the relationship between the structural disorder and the macroscopic mechanical behavior of nanoporous gold under uniaxial compression was investigated, using the finite element method. A recently proposed model based on a microstructure consisting of four-coordinated spherical nodes interconnected by cylindrical struts, whose node positions are randomly displaced from the lattice points of a diamond cubic lattice, was extended. This was done by including the increased density as result of the introduced structural disorder. Scaling equations for the elastic Poisson's ratio, the Young's modulus and the yield strength were determined as functions of the structural disorder and the solid fraction. The extended model was applied to identify the elastic–plastic behavior of the solid phase of nanoporous gold. It was found, that the elastic Poisson's ratio provides a robust basis for the calibration of the structural disorder. Based on this approach, a systematic study of the size effect on the yield strength was performed and the results were compared to experimental data provided in literature. An excellent agreement with recently published results for polymer infiltrated samples of nanoporous gold with varying ligament size was found.     

Non-linear and failure behaviour of spotwelds: a “global” finite element and experiments in pure and mixed modes I/II

The paper deals with joint element model used in crashworthiness simulations. The first part of the paper is dedicated to the formulation of a new “global” finite element for spotweld modelling. The mechanical behaviour of the joint is elastic–plastic type and damage is taken into account to model the failure of the welded area. The second part of the paper concerns a new experimental procedure for joint strength analysis in pure and mixed modes I/II and for joint model characterisation. Experiment is based on Arcan principle and results are compared to open literature. In the last part of the paper, the parameters of the new joint model are identified using experiments and used for several shapes of spot-welded specimens. The model predicts reasonably the elastic–plastic part of the response but is unable to predict the post-peak response observed especially in the case of pure shear.     

A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices

A macroscopic yield function for porous solids with pressure-sensitive matrices modeled by Coulomb's yield function was obtained by generalizing Gurson's yield function with consideration of the hydrostatic yield stress of a spherical thick-walled shell and by fitting the finite element results of the yield stresses of a voided cube. The macroscopic yield function is valid for the negative hydrostatic stress as well as for the positive hydrostatic stress. From the yield function, a plastic potential function for the porous solids was derived either for plastic normality flow or for plastic non-normality flow of the pressure-sensitive matrices. In addition, void nucleation was modeled by a normal distribution function with the macroscopic hydrostatic stress regarded as a controlling stress. This set of constitutive relations was implemented into a finite element code abaqus as a user material subroutine to analyze the cavitation and the deformation behavior of a rubber-modified epoxy around a crack tip under the Mode I plane strain conditions. By comparing the cavitation zone and the plastic zone obtained in the analysis with those observed in an experiment, the mean stress and the standard deviation for the void nucleation model could be determined. The cavitation and the deformation behavior of the rubber-modified epoxy were also analyzed around notches under four-point bending. The size and shape of the cavitation zone and the plastic zone were shown to be in good agreement with those observed in an experiment.     

A numerical approach to indentation technique to evaluate material properties of film-on-substrate systems

Spherical indentation approach (Lee et al., 2005, Lee et al., 2010) for the evaluation of bulk material properties is extended to that for elastic–plastic properties of film-on-substrate systems. Our interest focuses on single isotropic, metallic, and elastic–plastic film on a substrate, and we do not consider the size effects in plasticity behavior. We first determine the optimal data acquisition location, where the strain gradient is the least and the effect of friction is negligible. Dimensional analysis affords the mapping parameters as functions of normalized indentation variables. An efficient way is further introduced to reduce both the number of analyses and the regression order of mapping functions. The new numerical approach to the film indentation technique is then proposed by examining the finite element solutions at the optimal point. With the new approach, the values of elastic modulus, yield strength, and strain-hardening exponent of film materials are successfully obtained from the spherical indentation tests. We have shown that the effective property ranges such as indenter properties, substrate modulus, and E/E_s ratio can be extended without additional simulations and even loss of accuracy. For other ranges of variables or other properties, which are not dealt with in this study, this methodology is applicable through resetting FEA variables and finding proper normalized parameters.     

Compressive response of the Y-shaped sandwich core

The compressive response of the Y-shaped sandwich core has been investigated by analytical and finite element calculations. The effect of geometry, parent material properties and boundary conditions on the compressive response is explored in detail and models are developed for the stiffness, plastic collapse and elastic buckling strength of the Y-core. Practical Y-core geometries made from common structural alloys deform in an elastic-plastic manner; the elastic contribution to the collapse response can only be neglected for unrealistically low values of the parent material yield strain. Most practical Y-core designs have an almost constant collapse stress with increasing compressive strain implying that this geometry is an efficient energy absorbing structure.     

考虑塑性和刚度不匹配的焊接残余应力估算

现有残余应力计算方法未能考虑材料塑性变形和焊接接头刚度不匹配的影响,使得焊接残余应力计算结果和实际残余应力存在较大偏差.在2219-T87铝合金钨极氩弧焊焊接头残余应力测试基础上,提出一种基于非线性有限元和材料弹性模量分区的残余应力—释放应变曲线的残余应力计算方法,研究了材料塑性变形和接头刚度不匹配对焊接残余应力计算的影响.结果表明,焊接接头中非均质材料塑性不匹配可以引起对于残余应力计算的较大误差;材料塑性变形对残余应力的影响大于接头刚度不匹配对残余应力的影响.所提出方法修正了传统方法在焊接接头的残余应力计算中由于未考虑接头非均质材料塑性不匹配而引起的误差.     

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

Numerical study of the plastic behaviour in tension of welds in high strength steels

The influence of the mismatch between material properties and constraint on the plastic deformation behaviour of the heat affected zone of welds in high strength steels is investigated in this study, using finite element simulations. An elastoplastic implicit three-dimensional finite element code (EPIM3D) was used in the analysis. The paper presents the mechanical model of the code and the methodology used for the numerical simulation of the tensile test of welded joints. Numerical results of the tensile test of welded samples with different hypothetical widths for the Heat Affected Zone and various material mismatch levels are shown. The analysis concerns the overall strength and ductility of the joint and in relation to the plastic behaviour of the heat affected zone. The influence of the yield stress, tensile strength and constraint on the stress and plastic strain distribution in the soft heat affected zone is also discussed.     

A study on robust indentation techniques to evaluate elastic–plastic properties of metals

Spherical indentation is studied based on numerical analysis and experiment, to develop robust testing techniques to evaluate isotropic elastic–plastic material properties of metals. The representative stress and plastic strain concept is critically investigated via finite element analysis, and some conditions for the representative values are suggested. The representative values should also be a function of material properties, not only indenter angle for sharp indenter and indentation depth for spherical indenter. The pros and cons of shallow and deep spherical indentation techniques are also discussed. For an indentation depth of 20% of an indenter diameter, the relationships between normalized indentation parameters and load–depth data are characterized, and then numerical algorithm to estimate material elastic–plastic curve is presented. From the indentation load–depth curve, the new approach provides stress–strain curve and the values of elastic modulus, yield strength, and strain-hardening exponent with an average error of less than 5%. The method is confirmed to be valid for various elastic properties of indenter. Experimental validation of the approach then is performed by using developed micro-indentation system. For the material severely disobeying power law hardening, a modified method to reduce errors of predicted material properties is contrived. It is found that our method is robust enough to get ideal power law properties, and applicable to input of more complex physics.     

利用纳米硬度试验法确定Cr/Cu梯度材料的变形特性

利用纳米硬度仪研究了在Cu基底上的Cu／Cr梯度膜的机械性能。梯度膜是通过将Cu靶和Cr靶同时溅射到Cu基底材料上，但两个靶的相对溅射功率随溅射时间变化而制备。利用Oliver and Pharr方法得到了膜随其厚度变化的硬度和弹性模量。然后利用加载／卸载／再加载的方法得到了在不同深度(即膜的厚度)压头平均压力与相对压人深度之间的关系曲线，在此曲线上可以明显反映出材料的屈服特性。     

Orientation dependence of stress distributions in polycrystals deforming elastoplastically under biaxial loadings

The influence of biaxiality of the loading on the crystallographic orientation dependence of crystal stress distributions is examined for polycrystalline solids deformed well into the elastoplastic regime. The examination is couched in terms of two decompositions of the stress. The first is a split of the tensor into its hydrostatic and deviatoric components; the second is a spectral decomposition of the deviatoric stress from which we express the relative values of the principal components as a function of the biaxiality of the stress. Using the framework provided by these decompositions, we investigate trends observed in the lattice strains in polycrystals subjected to biaxial loadings, comparing strains measured by neutron diffraction with finite element simulations. We conclude by showing how the orientation dependence of the stress distributions is influenced by the load biaxiality and by connecting features of the distributions to the elastic and plastic properties of the crystals. Implications of the results are discussed relative to the modeling of strain hardening and defect initiation.     

A framework for numerical integration of crystal elasto-plastic constitutive equations compatible with explicit finite element codes

In this paper we summarize the elements of a numerical integration scheme for elasto-plastic response of single crystals. This is intended to be compatible with large-scale explicit finite element codes and therefore can be used for problems involving multiple crystals and also overall behavior of polycrystalline materials. The steps described here are general for anisotropic elastic and plastic response of crystals. The crystallographic axes of the lattice are explicitly stored and updated at each time step. A plastic predictor–elastic corrector scheme is used to calculate the plastic strain rates on all active slip systems based on a rate-dependent physics-based constitutive model without the need of further auxiliary assumptions. Finally we present the results of numerous calculations using a physics-based rate- and temperature-dependent model of copper and the effect of elastic unloading, elastic crystal anisotropy, and deformation-induced lattice rotation are emphasized.     

Two new expanding cavity models for indentation deformations of elastic strain-hardening materials

Two expanding cavity models (ECMs) are developed for describing indentation deformations of elastic power-law hardening and elastic linear-hardening materials. The derivations are based on two elastic–plastic solutions for internally pressurized thick-walled spherical shells of strain-hardening materials. Closed-form formulas are provided for both conical and spherical indentations, which explicitly show that for a given indenter geometry indentation hardness depends on Young’s modulus, yield stress and strain-hardening index of the indented material. The two new models reduce to Johnson’s ECM for elastic-perfectly plastic materials when the strain-hardening effect is not considered. The sample numerical results obtained using the two newly developed models reveal that the indentation hardness increases with the Young’s modulus and strain-hardening level of the indented material. For conical indentations the values of the indentation hardness are found to depend on the sharpness of the indenter: the sharper the indenter, the larger the hardness. For spherical indentations it is shown that the hardness is significantly affected by the strain-hardening level when the indented material is stiff (i.e., with a large ratio of Young’s modulus to yield stress) and/or the indentation depth is large. When the indentation depth is small such that little or no plastic deformation is induced by the spherical indenter, the hardness appears to be independent of the strain-hardening level. These predicted trends for spherical indentations are in fairly good agreement with the recent finite element results of Park and Pharr.     

Effect of strain hardening in elastic–plastic transition behavior in a hemisphere in contact with a rigid flat

An axisymmetrical hemispherical asperity in contact with a rigid flat is modeled for an elastic–plastic material on the lines of the Kogut–Etsion Model (KE Model) and the Jackson–Green Model (JG Model). The present work extends the previous KE and JG works, accounting for the effect of realistic material behavior in terms of the varying yield strengths and the isotropic strain hardening behavior. The predicted results show that the transition behavior of the materials from the elastic–plastic to the fully plastic case is influenced by the yield strength and the tangent modulus (E_t) and such transition do not take place at specific values of interference ratios as suggested by the KE model. New empirical relations are proposed to determine the contact load and the contact area based on the analysis. Numerical results from the finite element modeling are also validated with an experimental ball on flat configuration approach.