Calculating the elastic modulus from nanoindentation and microindentation reload curves

The Oliver–Pharr method was used to calculate the elastic modulus from the reloading curve and was compared to the traditional unloading curve method. Nanoindentation and microindentation testing instruments were used. This method was applied to load–unload–reload–unload, multistep, and cycle indentation testing procedures at various hold times and force rates. On unloading the reverse plasticity added to the elastic recovery which increased the apparent elastic modulus. During reloading there was mainly elastic deformation making it more reliable for the elastic modulus calculation. It was also found that the metals tested started yielding between 70% and 100% of the reload curve. The reload indentation elastic modulus for fused silica and several metals was equivalent to the tensile test elastic modulus from reference literature.     

On the determination of elastic modulus in very stiff materials by depth sensing indentation

Depth sensing indentation tests were carried out in stiff ceramics like Al₂O₃, AlN, SiC and B₄C, using a diamond Berkovich tip. The experiments show that the accuracy of the data depends on the stiffness ratio between material and indenter. An iterative calibration procedure is proposed to get a reliable estimation of the elastic modulus.     

Localized elastic modulus distribution of nanoclay/epoxy composites by using nanoindentation

The enhancement of mechanical properties by the use of nanoclay platelets in epoxy resin has been extensively investigated through numerous experimental techniques recently. Elastic modulus was obtained mainly from the tensile test of bone-like nanoclay/epoxy specimens. The results from the tensile test have only showed the globalized mechanical properties of composites and their localized elastic modulus distribution has been neglected. Despite the orientation and the degree of exfoliation of nanoclay platelets inside nanoclay/epoxy composites, the localized elastic modulus is important for the understanding of the distribution of agglomerations of nanoclay platelets. The elastic modulus of nanoclay/epoxy composite samples made under different sonication temperatures would be examined by nanoindentation to compare their localized mechanical behaviors. Scanning electron microscopy (SEM) would also be employed to study the distribution of the nanoclay clusters throughout the composites. The results showed that the elastic modulus varied throughout the composites and the nucleation theory of clusters was modified to explain the behavior of nanoclay agglomerations under different sonication temperatures in which the viscosity of the epoxy resin was varied. The gravitational effect was significant to cause the non-uniform distributions of nanoclay clusters at low sonication temperature.     

Some issues on nanoindentation method to measure the elastic modulus of particles in composites

The application of the indentation method to measure the elastic modulus of particles embedded in a composite is theoretically investigated in this paper by finite element simulation. The Oliver–Pharr method, which is widely used in commercial nanoindentation instruments, is used to probe the elastic modulus of the particle from the simulated indentation curve. The predicted elastic modulus is then compared with the inputted value. Two cases are studied, that of a stiff particle embedded in a soft matrix and a soft particle embedded in a stiff matrix. In both of these cases, there exists a particle-dominated depth. If the indentation depth lies within this particle-dominated depth, the Oliver–Pharr method is able to be applied to measure the particle’s elastic modulus with sufficient accuracy if the real contact area is used. This could lead to an experimentally-convenient method of determining the primary properties of individual particle, providing they can be well dispersed in the polymeric matrix.     

Representative indentation elastic modulus evaluated by unloading of nanoindentation made with a point sharp indenter

The conventional method to extract elastic modulus from the nanoindentation on isotropic linearly elastic solids is based on Sneddon’s solution (1965). However, it is known that the solution is valid only for incompressive elastic solids with the Poisson’s ratio ν of 0.5. This paper first proposes the modification of the solution in a wide range of ν from 0 to 0.5 through the numerical analysis on the unloading behavior of a simulated conical nanoindentation with a finite element method. As a result of the modification, the coefficient of linearity between the indentation elastic parameter k_e and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β, where k_e is defined as k_e ≡ P/h² with the indentation load P and penetration depth of the indenter h. According to the linear relationship between k_e and E, it is found that elastic rebound during unloading of a nanoindentation is uniquely characterized by a representative indentation elastic modulus E¹ defined in terms of E, ν and β, and that the value of E¹ can be evaluated from the P–h relationship with k_e and β. For an isotropic elastoplastic solid, the indentation unloading parameter k₂ defined as k₂ ≡ P/(h–h_r)² for a residual depth h_r is different from k_e even though a linearly elastic solid with k_e and elastoplastic solid with k₂ have a common E¹. In order to evaluate E¹ of an elastoplastic solid, the corresponding k_e is estimated from k₂ with an empirical equation as a function of the relative residual depth ξ defined as ξ ≡ h_r/h_max for the maximum penetration depth h_max. A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.     

The investigation of internal friction and elastic modulus in surface nanostructured materials

In this paper, 304 stainless steel and pure Fe specimens, which were processed by high-energy shot peening (HESP) and ultrasonic shot peening (USP) respectively, were studied by the internal friction method. Measurements were carried out on a vibrating reed apparatus. The change of internal friction and elastic modulus shows that the treatment duration of specimen is not accompanied by the corresponding persistent increase of internal friction and elastic modulus. There is a transition layer from the top surface to the inside of the materials. Young’s modulus of surface shows obviously a fluctuation along the depth profile. The phenomena have never been shown by other measurement methods. The microstructure change should be related to some basic mechanism of surface layer formation. It may also explain why the improvement of mechanical properties in surface nanocrystallized materials does not simply correspond to the duration time of severe deformation.     

Defect of the elastic modulus of composition materials under ultrasound loading

The effect of the ultrasound loading rate and the amplitude-cyclic and thermal action conditions on the behavior of the Young modulus defect for a Fe – 30% Cu iron-copper pseudo alloy and its components is determined. It is demonstrated that the basic mechanisms of internal energy dissipation in composition materials manifest thermsleves in stages; the characteristics of these mechanisms are analyzed.Translated from Problemy Prochnosti, No. 10, pp. 32–36, October, 1992.     

A new instrument for measuring the elastic modulus of sheet materials

Summary By the use of a photoelectric pickup and an electronic counting circuit, the instrument which has been described allows one to measure infrasonic damped oscillations with great accuracy. Further, the measurements have been automated.In order to attain a high accuracy of measurement of the modulus of elasticity, it is necessary that the amplitudes of oscillation of the specimen be as small as possible with respect to its length. This allows one to determine the modulus of elasticity and its temperature coefficient for flexible sheet materials.     

测量生物组织等效杨氏模量的毫米压痕仪

生物组织的弹性是诊断其是否发生病变的重要依据,杨氏模量是反映组织弹性的重要参数。以Labview为软件平台结合数据采集和运动控制等硬件设备,研制了一套测量弹性模量的毫米压痕弹性仪。以市购冷鲜牛肉、猪肉、猪肝和猪肾为试样,测量了球形压头向试样施加的负载力和对应的试样压痕深度,在对测量数据进行校准的基础上,利用基于赫兹接触力学模型的压头半径R、作用力F、压痕深度δ与试样等效杨氏模量E~*间的解析关系,得到各试样的等效杨氏模量。实验结果表明,测得的组织试样等效杨氏模量数值与文献基本相符,所设计的毫米级压痕法测试装置可用于生物组织等效杨氏模量的检测。     

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.     

Measurement of Young’s relaxation modulus using nanoindentation

In a previous paper (Lu et al., Mechanics of Time-Dependent Materials, 7, 2003, 189–207), we described methods to measure the creep compliance of polymers using Berkovich and spherical indenters by nanoindentation. However, the relaxation modulus is often needed in stress and deformation analysis. It has been well known that the interconversion between creep compliance and relaxation function presents an ill-posed problem, so that converting the creep compliance function to the relaxation function cannot always give accurate results, especially considering that the creep data at short times in nanoindentation are often not reliable, and the overall nanoindentation time is short, typically a few hundred seconds. In this paper, we present methods to measure Young’s relaxation functions directly using nanoindentation. A constant-rate displacement loading history is usually used in nanoindentations. Using viscoelastic contact mechanics, Young’s relaxation modulus is extracted using nanoindentation load-displacement data. Three bulk polymers, Polymethyl Methacrylate (PMMA), Polycarbonate (PC) and Polyurethane (PU), are used in this study. The Young’s relaxation functions measured from the nanoindentation are compared with data measured from conventional tensile and shear tests to evaluate the precision of the methods. A reasonably good agreement has been reached for all these materials for indentation depth higher than a certain value, providing reassurance for these methods for measuring relaxation functions.     

Determination of the elastic modulus of highly porous samples by nanoindentation: a case study on sea urchin spines

Nanoindenation studies were carried out on single crystal calcite and on sea urchin spines from Heterocentrotus mammillatus, Phyllacanthus imperialis, and Prinocidaris baculosa. Unlike dense calcite single crystals resin embedded porous sea urchin spine segments showed a strong dependence of the indentation modulus, but not the indentation hardness, on the local porosity. This implies that the sampled volume for the indentation modulus in nanoindentation with forces down to 15 mN is not nanoscopic but extends approximately 50 μm around the indentation spot. Only for indentation depths ≪100 nm more or less mount-unaffected values of the indentation modulus could be found. The Voigt model for composite materials (calcite/resin) was found to be applicable for the dependency of the indentation modulus on the porosity. This is attributed to the network type of porosity and opens strategies for the control of stiffness in porous networks.     

Method for determining the elastic modulus at grain boundaries for polycrystalline materials

Abstract

On the basis of the model of non-equilibrium grain boundary segregation induced by tensile stress, a set of kinetic equations is derived to formulate this process. These kinetic equations allow excellent simulation of the grain boundary segregation of phosphorus and sulphur observed in steels subjected to low tensile stresses. In the present paper, based on such a widely approved model, a new approach is proposed to quantify the elastic modulus at grain boundaries for polycrystalline materials. Using the observation of Misra, the grain boundary elastic modulus E _gb = 2.03 × 10⁹ Pa at 883 K for Cr-Mo-V-2.6Ni steel is obtained for the first time. This result shows excellent agreement with the local elastic constants simulated theoretically by Kluge et al., and indicates that the grain boundary elastic modulus for a polycrystalline material is much lower than the commonly assumed value.     

Contact problems involving forces and moments for incompressible linearly elastic materials

Aclass of rigid punch problems for an incompressible linearly elastic body involving forces and moments is considered by the theory of variational inequalities. After showing a proof of existence of the solution under a compatibility condition of the applied force and moment, extensive discussions about the reduced constraint of incompressibility in the finite element approximation are given. The reduced constraint is an explanation of the so-called reduced integration technique to resolve the incompressibility. Finally, the contact condition is controlled pointwise. Some justifications of the pointwise control are given by the idea of numerical integration.     

Modelling and determination of dynamic elastic modulus of magnesium based metal matrix composites

Abstract

The aim of the present study was to determine the elastic modulus of magnesium based composites containing different volume fractions of SiC particulate using an innovative suspended beam type impact based technique. This applies classical vibration theory, which relates the resonant frequency of the test specimens to the geometry and material properties of the metal matrix composites. The elastic modulus values were determined from the funda mental resonant frequency obtained from the experiment and density measurements. In addition, a finite element model was proposed for determining the dynamic elastic modulus of MMCs with different SiC reinforcement content using the first natural frequency corresponding to the flexural mode. The elastic modulus values obtained from the finite element model were in close agreement with the values obtained from the impact based experiments and in better agreement than those from theoretical methods such as the shear lag, Eshelby, and Halpin–Tsai models.     

The determination of the elastic modulus of microcantilever beams using atomic force microscopy

An investigation into the determination of the micromechanical properties of thin film materials has been performed. Thin metal and ceramic films are used extensively in the computer microprocessor industry and in the field of micro-electromechanical systems (MEMS). The demand for miniaturization and increased performance has resulted in the use of materials without a clear understanding of their mechanical properties on this scale. Micromechanical properties are difficult to obtain due to the lack of adequate testing equipment. The atomic force microscope (AFM), most commonly used as an imaging tool, lends itself to mechanical interaction with the sample surface utilizing a cantilever probe. An array of aluminum microcantilever beams were fabricated using standard IC processing techniques. The microbeams were deflected by the AFM cantilever probe and from this, the micromechanical properties of stiffness and elastic modulus were determined. Initial results indicate that this technique reliably determines the micromechanical properties of thin films.     

Elastic constants determination of thin cold-rolled stainless steels by dynamic elastic modulus measurements

Temperature dependence of elastic constants of thin cold-rolled stainless steel has been measured by using the acoustic resonance method. Identification of the vibration mode has been examined numerically and experimentally. The elastic constants at room temperature have also been measured by the pulse echo method. In addition, the texture effect on the elastic constants has been analysed by assuming the specimen has orthorhombic structure.     

Influence of interface energy and grain boundary on the elastic modulus of nanocrystalline materials

With reducing the grain size into nanometer scale for polycrystalline materials, the influence of nonlocal interactions in grain boundaries on the mechanical properties of the material is reinforced as well as the interface energy stemming from the surfaces of grains is increased, resulting in that the mechanical properties of the polycrystalline represent size-dependence significantly. In this work, the influence of the interface energy and grain boundaries on the elastic properties of nanocrystalline materials is investigated in the framework of continuum mechanics. An analytical expression of the elastic modulus is addressed to describe the grain size effects on the Young’s modulus of nanocrystalline materials. The numerical results illustrate that the elastic modulus of nanocrystalline materials decreases with the reduction of the grain size to nanometer scale. The grain size effects become remarkable when the grain size lowers down to several tens nanometers, and the influence of the interface energy and grain boundary must be taken into account. The contribution of the density on the mechanical properties in nanocrystalline materials is analyzed by discussing the influence of the grain boundary thickness on the elastic modulus. The comparison between the proposed theoretical results and the present measurement shows that the proposed model can predict the experiments quite well.     

Evolution of elastic modulus in roll forming

Roll forming is a continuous process in which a flat strip is incrementally bent to a desired profile. This process is increasingly used in automotive industry to form High Strength Steel (HSS) and Advanced High Strength Steel (AHSS) for structural components. Because of the large variety of applications of roll forming in the industry, Finite Element Analysis (FEA) is increasingly employed for roll forming process design. Formability and springback are two major concerns in the roll forming AHSS materials. Previous studies have shown that the elastic modulus (Young’s modulus) of AHSS materials can change when the material undergoes plastic deformation and the main goal of this study is to investigate the effect of a change in elastic modulus during forming on springback in roll forming. FEA has been applied for the roll forming simulation of a V-section using material data determined by experimental loading-unloading tests performed on mild, XF400, and DP780 steel. The results show that the reduction of the elastic modulus with pre-strain significantly influences springback in the roll forming of high strength steel while its effect is less when a softer steel is formed.     

Weibull modulus of nano-hardness and elastic modulus of hydroxyapatite coating

Here we report the microstructural dependence of nano-hardness (H) and elastic modulus (E) of microplasma sprayed (MIPS) 230 μm thick highly porous, heterogeneous hydroxyapatite (HAP) coating on SS316L. The nano-hardness and Young’s modulus data were measured on polished plan section (PS) of the coating by the nanoindentation technique with a Berkovich indenter. The characteristic values of nano-hardness and Young’s modulus were calculated through the application of Weibull statistics. Both nano-hardness and the Young’s modulus data showed an apparent indentation size effect. In addition, there was an increasing trend of Weibull moduli values for both the nano-hardness and the Young’s modulus data of the MIPS-HAP coating as the indentation load was enhanced from 10 to 1,000 mN. An attempt was made in the present work, to provide a qualitative model that can explain such behavior.