Analysis of nanoindentation experiments by means of rheological models

The nanoindentation technique is established in the field of material characterization at small dimensions. It is daily practice to analyze nanoindentation data with an almost “classical” formula based on the publications by Oliver and Pharr and Fischer-Cripps. The procedure works well for elastic-time independent plastic material behavior, for example copper and the calibration material fused silica, even at higher test temperatures. However, low melting solder materials are susceptible to creep behavior. For this reason, additional analysis procedures are required to determine the material parameters more precisely. In this paper the authors want to give an introduction to an “enhanced” analysis of nanoindentation data based on rheological models, which are often used to describe the time-dependence of material response. Two examples of such models are the MAXWELL- and the KELVIN-body. The authors present a rheological model, published by Mencik in 2011 [1], in context with the corresponding equation which is used to extract the material properties from the recorded data. Results of the analysis at the calibration material fused silica are presented and discussed together with the material parameters published in the literature. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

3D FEA of Size Effects in Deformation of Thin Metallic Films

The purpose of this work is to analyze size effects in the deformation occurring during nanoindentation-tests of thin metallic films on ceramic substrates. It is well known that classical phenomenological theories of plasticity are hardly applicable in cases of very small dimensions of a body [1]. Thus, the dependency of the mechanical behavior of thin films on the thickness can not be studied in the framework of classical theories. In order to simulate numerically the deformation, a specific material model has been chosen which is able to account for size effects. It bases on the theory of ”Mechanism Strain Gradient” (MSG) plasticity [2] in conjunction with the deformation theory of plasticity. The material model has been implemented via the user defined element subroutine (UEL) in the commercial FE code ABAQUS/Standard as a ten-node tetrahedron-element. With the developed subroutine the deformation of thin copper films on Si substrates during nanoindentation-tests has been simulated. Different material models of the indentor (rigid and elastic) as well as different friction conditions between the film and the pyramidal indentor were tested. Furthermore, the influence of an additional oxide layer on the films surface has been analysed. In order to verify the numerical investigations, results from nanoindentation experiments have been used for comparison [4]. The FE simulations for different thicknesses in the range of 100-600nm showed a very good agreement with the experiments. In particular, the size dependency of the force-displacement curves, calculated by using the developed subroutine, is in rather good agreement with experiments. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Simulation of Bone Remodeling using Design Space Topology Optimization

The law of bone remodeling, commonly referred to as Wolff's Law, asserts that the internal trabecular bone adapts to external loadings, reorienting with the principal stress trajectories to optimize mechanical efficiency creating a naturally optimum structure. The current study utilized an advanced structural optimization algorithm, called design space toptimization (DSO), to perform a three-dimensional computational bone remodeling simulation on the human proximal femur and analyse the results to determine the validity of Wolff's hypothesis. DSO optimizes the layout of material by iteratively distributing it into the areas of highest loading, while simultaneously changing the design domain to increase computational efficiency. The large-scale simulation utilized a 175 µm mesh resolution with over 23.3 million elements. The resulting anisotropic trabecular architecture was compared to both Wolff's trajectory hypothesis and natural femur samples from literature using radiography. The results qualitatively showed several anisotropic trabecular regions that were comparable to the natural human femur. The realistic simulated trabecular geometry suggests that the DSO method can accurately predict bone adaptation due to mechanical loading and that the proximal femur is an optimum structure as Wolff hypothesized. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determining Material Parameter of Solder Alloys by Nanoindentation

Environmental awareness motivates the replacement of traditional plumbiferous solder joints in microelectronic devices by new material alloys. In our group mechanical properties of some lead-free alloys are determined by nanoindentation experiments. We use a Microsystems-Nanoindenter Machine with Berkovich tip and apply standard techniques to extract the material properties from the measured load-displacement curves. To assess the quality of our experimental results we model and analyze the setup by finite element computations. By comparing the real (input) material data with the results determined from the load-displacement curves we analyze the obtained data in dependence of strength and stiffness of the materials under consideration. For low-strength material we point out deviations. By inverse analysis we adapt numerically elastic modulus and yield stress to experimentally measured load-displacement curves. To obtain information on the material's work hardening we suggest the use of a blunt indenter tip, e.g., a spherical indenter. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of composite plates including damage

The paper is concerned with the modelling and numerical simulation of fibre-composite plates in the nonlinear range due to large strains and damage. The layer-wise approach is applied. Each layer is treated as elastic-brittle and assumed to be orthotropic in the local material coordinate system. The appearance of damage is controlled according to the failure criteria [1,2,3,4]. When the failure condition is satisfied, the mechanical properties of the material are modified appropriately, depending on the type of damage (fibre breakage, matrix crack, fibre-matrix shear). We have programmed the model as a user subroutine within the ABAQUS environment and carried out a number of numerical simulations. The obtained numerical results are compared with the experimental data available in the literature [3]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial Random Material Property Model for Vibration of Composites

Investigation of vibration and buckling of thin walled composite structures is very sensitive to parameters like uncertain material properties and thickness imperfections. Because of the manufacturing process and others, thin walled composite and other structures show uncertainties in material properties, and other parameters which cannot be reduced by refined discretization. These parameters are mostly spatial distributed in nature. Here I introduce a semivariogram type material property model to predict the spatial distributed material property (like young's modulus) over the structure. The computation of semivariogram parameters needs the local material properties over a prespecified gird. The material properties at each grid have been obtained by considering a statistically homogeneous representative volume element (RVE) at each gird. According to random nature of the spatial arrangement of fibers, the statistically homogeneous RVE is obtained using image processing. The effective material properties of the RVE have been obtained numerically with the help of periodic boundary condition. The methodology is applied to a composite panel model and modal analysis has been carried. The results of the modal analysis (eigen values and mode shapes) are compared with experimental modal analysis results which are in good agreement. Using the presented material property model we can better predict the vibration characteristics of the thin walled composite structures with the inherent uncertainties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of bone elasticity based on tissue‐independent (‘universal’) phase properties

As candidates for tissue‐independent phase properties of cortical and trabecular bone we consider (i) hydroxyapatite, (ii) collagen, (iii) ultrastructural water and non‐collagenous proteins, and (iv) marrow (water) filling the Haversian canals and the intertrabecular space. From experiments reported in the literature, we assign stiffness properties to these phases (experimental set I). On the basis of these phase definitions, we develop, within the framework of continuum micromechanics, a two step homogenization procedure: (i) At a length scale of 100 – 200 nm, hydroxyapatite (HA) crystals build up a crystal foam ('polycrystal'), and water and non‐collagenous organic matter fill the intercrystalline space (homogenization step I); (ii) At the ultrastructural scale of mineralized tissues, i.e. 5 to 10 microns, collagen assemblies composed of collagen molecules are embedded into the crystal foam, acting mechanically as cylindrical templates. At an enlarged material scale of 5 to 10 mm, the second homogenization step also accommodates the micropore space as cylindrical pore inclusions (Haversian and Volkmann canals, inter‐trabecular space), that are suitable for both trabecular and cortical bone. The input of this micromechanical model are tissue‐specific volume fractions of HA, collagen, and of the micropore space. The output are tissue‐specific ultrastructural and microstructural (=macroscopic=apparent) elasticity tensors. A second independent experimental set (composition data and experimental stiffness values) is employed to validate the proposed model. We report a a good agreement between model predictions and experimentally determined macroscopic stiffness values. The validation suggests that hydroxyapatite, collagen, and water are tissue‐independent phases, which define, through their mechanical interaction, the elasticity of all bones, whether cortical or trabecular.     

Stress Integration of the Hoek-Brown Material Model Using Incremental Plasticity Theory

This paper presents the formulation of the stress integration procedure for the Hoek-Brown (HB) material model with non-associative yielding condition by using the incremental plasticity method. Main idea of this method is to reach the solution by calculating the plastic matrix according to the method of incremental plasticity (used for elastic constitutive matrix corrections), and with the use of the total strain increment. Computational procedure is implemented within the PAK program package. Results of this procedure were compared with the solutions obtained by other program that contains this material model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microplane modeling of cyclic behavior of concrete: a gradient plasticity-damage formulation

A microplane model is developed to simulate the behavior of concrete under cyclic loading conditions. Pure damage mechanics or pure plasticity models yield satisfactory results for concrete under monotonic loading but cannot capture correctly the unloading and reloading response. Therefore, coupling damage and plasticity is necessary for accurate constitutive modeling of concrete. The microplane model offers a straightforward approach to simulate induced anisotropy by formulating the material laws on many randomly oriented planes. Distinguishing between compression and tension response using the proper plastic yield function and damage laws is considered. Furthermore, gradient enhancement is employed to handle the pathological mesh sensitivity related to strain softening. The new formulation is implemented within a 3D finite element code and a numerical example is simulated and compared to experiments in order to evaluate the capabilities of the model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nanoindentation testing and modeling of chromium-carbide-based composites

High-resolution measurements of mechanical properties are of immense importance in the design of new composite materials. Measuring the intrinsic properties of each phase separately in multiphase composites gives information on the spatial heterogeneity of their local properties and serves as a guide to process engineering and to the design of advanced materials. In this study, the nanoindentation, X-ray analysis, and microstructural SEM investigations have been used to reveal the properties and structural features of ceramic-metal composites — chromium-carbide-based cermets. The semiellipse method for the account of pileups has been applied to this multiphase material to determine the hardness and elastic modulus of the constituent phases. After reconsideration of the contact area, the properties of the phases showed a good agreement with published data. Finally, the measured local elastic properties were used as inputs for modeling the effective elastic response of these materials, and a very good agreement with experimental results was found.     

Micromechanics-Supported Conversion of Computer Tomographic (CT) Images into Anisotropic and Inhomogeneous Finite Element Models of Organs: the Case of a Human Mandible

We here present a strategy for reliable prediction of elastic properties from X-ray attenuation coefficients visualized in Computer Tomographic images, as basis for Finite Element models. By example, we show the distribution of the axial normal stress throughout a human mandible, due to a bite on the leftmost premolar. Remarkably, this distribution is not heavily altered if we replace the inhomogeneous material distribution by one discerning merely cortical and trabecular bone, but it is strongly affected by the consideration of material anisotropy. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plastic flow in a conical channel: Qualitative features of the solutions under different yield conditions

The problem of the convergence of the solutions of problems of plasticity theory, with a yield condition which depends on the hydrostatic stress, to solutions based on classical plasticity theory with von Mises or Tresea conditions is considered, with a particular choice of the parameters of the material model. For the case of axisymmetric flow of material in a channel with converging and diverging walls, solutions according to two plasticity theories with a yield condition which depends on the hydrostatic stress are compared with the classical solution. It is shown that only the solution using Spencer's model shows all the main features of the classical solution. As the internal criterion of the choice of the preferred plasticity theory when examining a special class of problems, it is suggested that the criterion of the convergence of the solutions to the solutions of classical plasticity theory should be used.     

Micro-macro modelling of fiber reinforced composites exhibiting elastoplastic deformation

Fiber reinforced plastics such as carbon fiber-reinforced composites are typically characterized by their high siffness to weight ratio making them particularly attractive in lightweight construction. In addition, the architecture of these materials means that the correct modelling of their orthotropy is very important. In this work, volume averaged stress-strain responses are generated from a micro representative volume element (RVE). A nonlinear macro constitutive material model accounting for anisotropic plasticity is proposed. The model is fitted and compared to the micro stress-strain response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The anisotropy of compact bone material

From the measurements of the main elements of the microstructure of compact bone material, it is concluded that the theoretical model of a transtropic material can be applied to bone tissue. This conclusion is confirmed by the experimental data obtained on compression. The correlation connection between the ultimate strength of compact bone material and the elasticity modulus has been found. It is shown that the anisotropy of the compact material is satisfactorily described by the tensor formula.Scientific-Research Institute of Medical Radiology, Academy of Medical Sciences of the USSR, Obninsk. S. M. Kirov Leningrad Order of Lenin Wood Technology Academy. Translated from Mekhanika Polimerov, No. 4, pp. 711–716, July–August, 1972.     

A bioactive interface approach for the simulation of mechanically stimulated osseointegration of hip joint endoprostheses

In this contribution, an approach on the mechanically stimulated osseointegration of hip joint endoprostheses considering load cases of daily movements is presented. The mechanical behavior of the bone-implant interface is modeled with a material model adopted from computational plasticity. Under consideration of a detailed analysis of the interface conditions, limiting factors for the osseointegration process are identified and the bone ingrowth prediction is performed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite thermo-viscoplasticity of amorphous glassy polymers: Experiments,modeling and simulations

The contribution is concerned with experimental procedures, constitutive modeling and the numerical simulations of finite thermo-viscoplastic behavior of glassy polymers. The experimental study involves both homogeneous and inhomogeneous tests at different temperatures under isothermal conditions. The true stress-true strain curves obtained from compressive homogeneous uniaxial and plane strain experiments are employed in the identification of adjustable material parameters. In contrast to the existing kinematic approaches to finite plasticity of glassy polymers, we propose a distinct kinematic framework constructed in the logarithmic strain space. This leads us to an algorithmically very attractive, additive kinematic structure in R⁶ similar to the geometrically linear theory. The proposed three-dimensional model is implemented into a finite element code. The load-displacement curves acquired from inhomogeneous experiments are compared against the results obtained from finite element analyses where the material parameters identified from homogeneous experiments are used. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Neural network prediction of load from the morphology of trabecular bone

Bone adaptation models are often solved in the forward direction, meaning that the response of bone to a given set of loads is determined by running a bone tissue adaptation model. The model is generally solved using a numerical technique such as the finite element model. Conversely, one may be interested in the loads that have resulted in a given state of bone. This is the inverse of the former problem. Even though the inverse problem has several applications, it has not received as much attention as the forward problem, partly because solving the inverse problem is more difficult. A nonlinear system identification technique is needed for solving the inverse problem. In this study, we use artificial neural networks for prediction of tissue adaptation loads from a given density distribution of trabecular bone. It is shown that the proposed method can successfully identify the loading parameters from the density distribution of the tissue. Two important challenges for all load prediction algorithms are the non-uniqueness of the solution of the inverse problem and the inaccuracies in the measurement of the morphology of the tissue. Both challenges are studied, and it is shown that the load prediction technique proposed in this paper can overcome both.     

A Unified Approach to Second Order Optimality Criteria in Nonlinear Design of Experiments

In the nonlinear regression model we consider the optimal design problem with a second order design D-criterion. Our purpose is to present a general approach to this problem, which includes the asymptotic second order bias and variance criterion of the least squares estimator and criteria using the volume of confidence regions based on different statistics. Under assumptions of regularity for these statistics a second order approximation of the volume of these regions is derived which is proposed as a quadratic optimality criterion. These criteria include volumes of confidence regions based on the u _n - representable statistics. An important difference between the criteria presented in this paper and the second order criteria commonly employed in the recent literature is that the former criteria are independent of the vector of residuals. Moreover, a refined version of the commonly applied criteria is obtained, which also includes effects of nonlinearity caused by third derivatives of the response function.     

Empirical model selection in generalized linear mixed effects models

This paper focuses on model selection in generalized linear mixed models using an information criterion approach. In these models in general, the response marginal distribution cannot be analytically derived. Thus, for parameter estimation, two approximations are revisited both leading to iterative model linearizations. We propose simple model selection criteria adapted from information criteria and based on the linearized model obtained at convergence of the algorithm. The quality of derived criteria are evaluated through simulations.