The Analytical and Numerical Study on the Nanoindentation of Nonlinear Elastic Materials

In nanoindentation testing of materials, the analytical/numerical models to connect the indentation load, indentation depth and material properties are crucial for the extraction of mechanical properties. This paper studied the methods of extracting the mechanical properties of nonlinear elastic materials and built general relationships of the indentation load and depth of hyperelastic materials combined with the dimensional analysis and finite element method (FEM). Compared with the elastic contact models and other nonlinear elastic contact models, the proposed models can extract the mechanical properties of nonlinear elastic materials under large deformation simply and effectively.     

Effect of Styrene–Butadiene Rubber Latex on Mechanical Properties of Cementitious Materials Highlighted by Means of Nanoindentation

Abstract: In this paper, micro‐mechanical properties of styrene–butadiene rubber (SBR) latex‐modified cement pastes identified by means of the nanoindentation (NI) technique are related to macro‐mechanical properties of SBR latex‐modified mortars obtained from standard test methods, considering an SBR latex/cement ratio varying from 0% to 20%. For this purpose, the average value of the hardness and the so‐called indentation modulus of the different material phases of the cement paste, i.e. calcium–silicate–hydrate (CSH), portlandite, anhydrous cement, etc., obtained from NI are compared with the compressive and flexural strengths, on the one hand, and the dynamic elastic modulus of SBR latex‐modified mortars, on the other hand. This comparison revealed a linear correlation between the dynamic elastic modulus and the indentation modulus and between the compressive strength, flexural strength and hardness. Thus, the obtained results clearly indicate the finer‐scale origin of the macroscopic elastic and strength properties, linking the mechanical properties at the so‐called mortar scale to the cement‐paste scale.     

Effects of Loading Conditions on Deformation Process in Indentation

Static indentation experiments are typically performed to characterize the mechanical properties of a material of interest by a rigid indenter of known geometry to various depths. In contrast, dynamic indentation of materials has not been fully studied. Evaluating material performance under dynamic loading conditions is a challenge and we demonstrate that various modelling schemes may be appropriate for different flavours of dynamic indentation. In order to compare underlying thermo-mechanics and deformation processes in a static and dynamic indentation process, indentation of a rigid indenter into a workpiece to a fixed chosen penetration is extensively studied. A nonlinear strain rate and temperature-sensitive material model is used to characterize the macroscopic response of a titanium-based beta-alloy to indentation.     

Characterization of carbon nanotube/nanofiber-reinforced polymer composites using an instrumented indentation technique

An instrumented indentation technique was tested on three types of carbon nanotube/nanofiber-reinforced composites to investigate its applicability for measuring mechanical properties (elastic modulus and hardness). There was good agreement in the measured elastic modulus between the instrumented indentation and uniaxial tension tests for the case of a nanocomposite with a harder epoxy matrix material. In contrast, there was a considerable difference in elastic modulus between the two tests for the case of a nanocomposite with a softer polystyrene matrix material. A modified area function was then developed for the nanocomposite with the softer polystyrene matrix material, and this eliminated the difference in elastic modulus between the two test techniques. Thus, the instrumented indentation technique can be used for evaluating the mechanical properties of polymer matrix nanocomposites with an added advantage that a small sample size can be used. The instrumented indentation test was also utilized in the case of a patterned nanotube array-reinforced epoxy matrix composite. This clearly showed the modulus of the array nanocomposite improved considerably compared to that of the neat epoxy resin.     

Stress states and nonlinear mechanical behavior associated with nanoindentation testing of polymers

Interest in determining material properties on the nanoscale has promoted use of nanoindentation testing as a measurement technique. Classical elasticity solution of indentation geometry has provided values of the mechanical properties for linear elastic materials. Recent attempts to apply this test technique to polymers have given indications of time dependent response in the early relaxation period. There is corresponding interest in the possibility of obtaining their nonlinear viscoelastic behavior. As a preliminary to analytical study providing a basis for such testing, the first part of this paper examines the initial stress and strain condition in the vicinity of the indenter. Data from recent tests on poly(vinyl acetate) material at load levels typical of current testing indicate that stress magnitudes in the nonlinear and possibly plastic-like range are present near the specimen surface. The second part of this study pursues the examination of how the heavily nonlinear region may be characterized for polymers in analogy with the treatment utilized for metals and other elastic-plastic materials. As an example, analysis of data on PVAc indicates that its behavior in nanoindentation should in several respects correspond to materials exhibiting a relatively low value of the ratio of elastic modulus to yield stress.     

Detecting barely visible impact damage detection on aircraft composites structures

Composites have many advantages as aircraft structural materials and for this reason their use is becoming increasingly widespread. Fragility of composite material to impact loading limits their application in aircraft structures. In particular, low velocity impacts can cause a significant amount of delamination, even though the only external indication of damage may be a very small surface indentation. This type of damage is often referred to as barely visible impact damage (BVID), and it can cause significant degradation of structural properties. If the damaged laminate is subjected to high compressive loading, buckling failure may occur. Therefore, there is the need to develop improved and more efficient means of detecting such damage. In this work a new NDT approach is presented, based on the monitoring of the nonlinear elastic material behaviour of damaged material. Two methods were investigated: a single-mode nonlinear resonance ultrasound (NRUS) and a nonlinear wave modulation spectroscopy (NWMS). The developed methods were tested on different composite plates with unknown mechanical properties and damage size and magnitude.The presence of the nonlinearities introduced by the damage was clearly identified using both techniques. The results showed that the proposed methodology appear to be highly sensitive to the presence of damage with very promising future applications.     

A review of deformation and fatigue of metals at small size scales

Mechanical devices are being introduced whose size scale is well below that of conventional mechanical test specimens. The smallest devices have sizes in the nanometer range, though a good proportion of structural devices are of the micrometer scale. Development of these products raises the question of how their mechanical behaviour and reliability may be predicted. Conventional macroscopic test data can be used, but these are obtained using specimens whose size is much larger than the devices themselves. There is a risk that performance predictions will be inaccurate, due to the existence of size effects. This paper covers small size scale testing in metallic specimens and devices, concentrating on free‐standing specimens. To begin, some examples of micro‐scale devices are given. Fabrication methods for small metallic devices are then briefly described. This is followed by a review of experimental observations of mechanical properties in various metallic materials at the micro‐scale, highlighting the differences in results from different research groups and the gaps in our current knowledge. A section on computational and predictive modelling is included, in recognition of the role of modelling in device design and testing. Overall, the findings are that size effects are common, particularly in crystalline samples when the grain size is similar to one or more of the specimen dimensions. However, observations of size effects differ between studies and mechanical properties can vary widely, even for the same type of material. As a consequence, the relationships between specific device processing methods, specimen size and material properties must be adequately understood to ensure successful performance.     

Mechanical properties of magnetite (Fe3O4), hematite (α-Fe2O3) and goethite (α-FeO·OH) by instrumented indentation and molecular dynamics analysis

Hardness and elastic properties of pure (crystal) and complex (product of corrosion) iron oxides, magnetite (Fe₃O₄), hematite (α-Fe₂O₃) and goethite (α-FeO·OH), were determined by means of molecular dynamics analysis (MDA) and instrumented indentation. To determine local mechanical properties by indentation, multicyclic loading is performed by using incremental mode. Moreover to study the influence of visco-elastoplastic behaviour of the material, various load-dwell-times were applied at each loading/unloading cycle. To support the indentation results, molecular dynamics analysis based on shell model potential is performed for pure oxides to determine Young's modulus, bulk modulus, Poisson's ratio and shear modulus. The comparison between experimental and theoretical values both with the literature data allows the evaluation of the mechanical properties of the pure oxides. Subsequently, this allows the validation of the mechanical properties of complex oxides which can only be deduced from indentation experiments.     

Grid indentation analysis of composite microstructure and mechanics: Principles and validation

Several composites comprise material phases that cannot be recapitulated ex situ, including calcium silicate hydrates in cementitous materials, hydroxyapatite in bone, and clay agglomerates in geomaterials. This requirement for in situ synthesis and characterization of chemically complex phases obviates conventional mechanical testing of large specimens representative of these material components. Current advances in experimental micro and nanomechanics have afforded new opportunities to explore and understand the effect of thermochemical environments on the microstructural and mechanical characteristics of naturally occurring material composites. Here, we propose a straightforward application of instrumented indentation to extract the in situ elastic properties of individual components and to image the connectivity among these phases in composites. This approach relies on a large array of nano to microscale contact experiments and the statistical analysis of the resulting data. Provided that the maximum indentation depth is chosen carefully, this method has the potential of extracting elastic properties of the indented phase which are minimally affected by the surrounding medium. An estimate of the limiting indentation depth is provided by asssuming a layered, thin film geometry. The proposed methodology is tested on a “model” composite material, a titanium-titanium monoboride (Ti–TiB) of various volumetric proportions. The elastic properties, volume fractions, and morphological arrangement of the two phases are recovered. These results demonstrate the information required for any micromechanical model that would predict composition-based mechanical performance of a given composite material.     

多孔材料代表单元的性质

为了弄清多孔材料代表单元的基本性质,对泡沫镍的力学性能进行了实验研究和计算机模拟,两者结果的变化趋势吻合较好。在此基础上,用离散的弹性梁构成代表单元,结合连续介质力学的方法,建立了多孔材料的理想力学模型,导出了其宏观本构关系,讨论了其代表单元各向异性性质和材料常数之间的关系。结果表明由代表单元周期性排列构成的多孔材料,在宏观上呈各向异性,只有当代表单元无序地随机排列时,多孔材料才在宏观上出现统计各向同性。同时指出了一些文献中存在的错误。     

Review Micromechanical testing of bone trabeculae - potentials and limitations

The mechanical properties of bone are studied mostly for reasons related to skeletal pathology. However, bone is also very interesting from a material science perspective because it is a natural hierarchical composite material. The mechanical properties of bone depend on both the structural arrangement and the properties of the constituting materials, namely the organic polymer collagen and the inorganic salt apatite. While the mechanical properties of bone samples at the macroscopic scale are measured routinely, mechanical tests on micrometer-sized specimens are still at development stage. In this paper, protocols for measuring the elasticity of cancellous bone trabeculae are reviewed. The published values for the elastic modulus of trabeculae vary between 1 GPa and 15 GPa. Reasons for this broad range of values may be located in the intrinsic difficulties of preparing, handling, and testing inhomogeneous, anisotropic and asymmetric micro-samples. We discuss the major error sources in existing testing procedures and suggest potential strategies to enhance their performance.     

Micromechanics-based investigation of the elastic properties of polymer-modified cementitious materials using nanoindentation and semi-analytical modeling

Cementitious materials are modified by the addition of polymers in order to improve the durability and the adhesive strength. However, polymer-modified mortars and concretes exhibit lower elastic moduli in comparison to unmodified systems. The macroscopic properties are governed by microstructural changes in the binder matrix, which consists of both cementitious and polymer components. Herein, different polymer-modified cement pastes were characterized using nanoindentation to better understand the microscopic origin of the macroscopic elastic modulus. By means of the statistical nanoindentation technique, the existence of three micromechanical phases in plain and polymer-modified cement pastes with a water-to-cement mass ratio of 0.40 is evidenced, illustrating that the polymer modification does not induce the formation of additional reaction products. Instead, the polymers adsorb on the hydration products as well as on unhydrated clinker grains and decrease the indentation moduli of the micromechanical phases. The link between the microscopic and macroscopic mechanical properties is established by means of a continuum micromechanics approach. A multiscale model aimed at the prediction of the elastic moduli of polymer-modified cementitious materials is developed with input parameters that are partially obtained from the nanoindentation tests. The comparison of the modeling results with the experimentally determined elastic (macroscopic) moduli at the scales of cement paste, mortar, and concrete is satisfactorily good, underlining the predictive capability of the modeling approach. The improvement of prediction models is essential for the application of polymer-modified cementitious materials in construction and will encourage their integration into design guidelines.     

Size effects in foams: Experiments and modeling

Mechanical properties of cellular solids depend on the ratio of the sample size to the cell size at length scales where the two are of the same order of magnitude. Considering that the cell size of many cellular solids used in engineering applications is between 1 and 10 mm, it is not uncommon to have components with dimensions of only a few cell sizes. Therefore, both for mechanical testing and for design, it is important to understand the link between the cellular morphology and size effects, which is the aim of this study. In order to represent random foams, two-dimensional (2D) Voronoi tessellations are used, and four representative boundary value problems - compression, shear, indentation, and bending - are solved by the finite element (FE) method. Effective elastic and plastic mechanical properties of Voronoi samples are calculated as a function of the sample size, and deformation mechanisms triggering the size effects are traced through strain maps. The modeling results are systematically compared with experimental results from the literature. As a rule, with decreasing sample size, the effective macroscopic stiffness and strength of Voronoi samples decrease under compression and bending, and increase under shear and indentation. The physical mechanisms responsible for these trends are identified.     

Conditions of applying Oliver–Pharr method to the nanoindentation of particles in composites

The indentation test is a popular experimental method to measure a material’s mechanical properties such as elastic modulus and hardness, and the Oliver–Pharr method is commonly used in commercial indentation instruments to obtain these two quantities. To apply the Oliver–Pharr method correctly in all of these cases, it is essential to know the limitations of this method. The present study focuses on the applicability of the Oliver–Pharr method to measure the mechanical properties of particles in composites. The finite element method is used to undertake virtual indentation tests on a particle embedded in a matrix. In our numerical studies, the indentation “pile-up” phenomenon is generally observed in our numerical case studies, which indicates that the contact area used for predicting the elastic modulus should be measured directly, not be estimated from the indentation curve. The Oliver–Pharr method based on the real contact area is applied to estimate the elastic modulus of the particles by using the indentation curve from the numerical simulation, with the estimated elastic modulus being compared with the input value. Applying the real contact area value (not the one predicted from the indentation curve) we show that the Oliver–Pharr method can still be applied to measure the elastic modulus of the particle with sufficient accuracy if the indentation depth is smaller than the particle-dominated depth, a value defined in this work. The influences of the matrix and particle properties on the particle-dominated depth are studied using a dimensional analysis and parametric study. Our results provide guidelines to allow the practical application of the Oliver–Pharr method to measure the elastic modulus of particles in composites. This could be particularly important where particles are formed in situ in a matrix (as opposed to being preformed and subsequently incorporated in a matrix), or when the modulus of individual performed particles is required such as for subsequent modelling, but the modulus of individual material particles (or its material) cannot readily be determined.     

Mechanical and thermal properties of physical vapour deposited alumina films Part II Elastic, plastic, fracture, and adhesive behaviour

Two mechanical characterization techniques were used to deduce the elastic, plastic, fracture, and adhesive properties of non-reactive physical vapour deposited alumina films of varying thickness on Al₂O₃-TiC substrates deposited at two different substrate biases. Depth-sensing indentation at both nano- and macroscopic load scales was used to determine the elastic and plastic properties of the films. Gravity-loaded Vickers indentation was performed to examine the fracture properties of the film and of the interface. Novel fracture mechanics models were developed to describe indentation-induced film fracture by channel cracks and indentation-induced interface delamination. The former model was used to determine the film toughness and the latter model was used to deduce the interfacial fracture resistance of the films and correctly predicted the effect of changing film thickness. Both models described the measured crack lengths with indentation load well and were used to identify the transition from radial and lateral cracking to channel and interfacial cracking.     

Multiscale modelling of nanoindentation test in copper crystal

The nanoindentation test in the dislocation free crystal of copper is simulated by a nonlinear elastic finite element analysis coupled with both ab initio calculations of the ideal shear strength and crystallographic considerations. The onset of microplasticity, associated with the pop-in effect identified in experimental nanoindentation tests (creation of first dislocations), is assumed to be related to the moment of achieving the value of the ideal shear strength for the copper crystal. Calculated values of the critical indentation depth lie within the range of experimentally observed pop-ins in the copper crystal. The related indentation load is somewhat lower than that observed in the experiment.     

Experimental and numerical rolling contact fatigue study on the 32CrMoV13 steel

The aim of this work is to study pure rolling contact fatigue in 32CrMoV13 quenching and tempering steel. The study involves both experimental and numerical work. The influence of the roughness and the residual stresses on the mechanisms and zones of cracking were studied. The results show a rapid reduction in roughness during the first minute of test but even so there will be specimen deterioration. The residual stress profile after rolling contact tests have high compression values in the surface and at a depth of approximately 240 μm, which is related with the Hertzian maximal shear stress. The numerical simulation of the Hertzian loading was used both to determine the elastic shakedown of the material and to apply a high‐cycle multiaxial fatigue criterion. The three‐dimensional finite element analysis used in the numerical calculation includes elastic‐linear kinematic hardening plastic material and allows the introduction of an initial residual stress state. Taking into account the elastoplastic load induced by the Hertz pressure, low‐cycle fatigue tests were used to characterize the mechanical properties of the material. In order to validate the numerical simulation, the results of the calculation after elastic shakedown were compared with the values measured by X‐ray diffraction after rolling contact tests. The results showed a reasonable agreement between calculated and measured stresses. The Dang Van high‐cycle multiaxial fatigue criterion showed a good relationship with the experimental findings.     

Effective behavior of porous elastomers containing aligned spheroidal voids

The theoretical need to recognize the link between the basic microstructure of nonlinear porous materials and their macroscopic mechanical behavior is continuously rising owing to the existing engineering applications. In this regard, a semi-analytical homogenization model is proposed to establish an overall, continuum-level constitutive law for nonlinear elastic materials containing prolate/oblate spheroidal voids undergoing finite axisymmetric deformations. The microgeometry of the porous materials is taken to be voided spheroid assemblage consisting of confocally voided spheroids of all sizes having the same orientation. Following a kinematically admissible deformation field for a confocally voided spheroid, which is the basic constituent of the microstructure, we make use of an energy-averaging procedure to obtain a constitutive relation between the macroscopic nominal stress and deformation gradient. In this work, both prolate and oblate voids are considered. As a numerical example, we study macroscopic nominal stress components for a hyperelastic porous material consisting of a neo-Hookean matrix and prolate/oblate voids subjected to 3-D and plane strain dilatational loadings. In this numerical study, the relation between the relevant microstructural variables (i.e., initial porosity and void aspect ratio) for a rather large range of applied stretch is put into evidence for two types of loading. Finally, a finite element (FE) simulation is presented, and the homogenization model is assessed through comparison of its predictions with the corresponding FE results. The illustrated agreement between the results demonstrates a good accuracy of the model up to rather large deformations.     

The mechanical response of Rohacell foams at different length scales

The quasi-static mechanical response of polymethacrylimide (PMI) foams of density ranging from 50 to 200 kg m⁻³ is investigated in order to provide experimental data to inspire and validate numerical constitutive models for the response of polymer foams. The macroscopic mechanical response is characterised by conducting quasi-static compression, tension, shear and indentation experiments, whereas microscopic deformation mechanisms are identified by conducting in situ SEM observations during static compression and tension tests; it is shown that foams of low density collapse by cell wall buckling while foams of high density undergo plastic cell-wall bending. As a result, both the elastic and plastic macroscopic response of the foam display a tension/compression asymmetry.     

钛合金锥形压痕约束因子和代表应变的确定

目的 研究钛合金性能参数与约束因子和代表应变的关系。方法 对不同性能参数条件下压痕变形过程进行有限元仿真,根据模拟结果建立约束因子和代表应变与材料性能间的定量关系。结果 发现钛合金屈服强度与压痕硬度之间存在线性关系。约束因子随着弹性模量的增加而增加,代表应变随着弹性模量的增加而减小。当弹性模量固定不变时,约束因子随屈服强度线性减小,代表应变固定不变。在材料性能范围内,使用代表应变求解的代表应力,与通过压痕硬度和约束因子求解的代表应力两者误差小于±3%。结论 材料性能不同,约束因子和代表应变的值也会变化,这两个参数并不存在统一的值。