An analysis of nanoindentation in elasto-plastic solids

This paper examines the accuracy of the extracted elastic properties using the nanoindentation technique on elasto-plastic materials. The application of the correction factor evaluated in the linearly elastic case [Poon, B., Rittel, D., Ravichandran, G., 2008. An analysis of nanoindentation in linearly elastic solids. Int. J. Solids Structures 45 (24), 6018–6033.] on elastic–plastic materials is critically examined. It is then established that the accurate determination of the projected area of contact is found to be crucial for the accurate determination of elastic material properties. The conventional methods for the accurate determination of contact area are generally limited to ratios of Young’s modulus over yield stress, E/σ_y < 30 for elastic-perfectly plastic materials, which is too stringent for most materials. Thus, a new electrical resistance method is proposed to measure directly the projected contact area. Using numerical simulations, it was found that with the accurate determination of A, the error associated with the extracted elastic material properties is reduced by more than 50% in some cases. Using the newly proposed procedure, the error is also found to be independent of E/σ_y and the tip radius, ρ, and it is only a function of Poisson’s ratio, ν. This suggests that the errors might be due to the residual stresses at the plastic imprint that were found to depend on ν as well.     

A computational study on the instrumented sharp indentations with dual indenters

Finite element analysis was performed to investigate the indentation response of elasto-plastic solids for conical indenters of half included angles of 60° and 70.3°. The interdependence indentation parameters resulting from a single indentation load–depth curve is considered. Regarding dimensional analysis, several dimensionless relationships are constructed as functions of the reduced elastic modulus-loading curvature ratio E¹/C and the strain hardening exponent n. Further, the duality between corresponding parameters with dual indenters is explored. Finally, a new method based on dual indenters is proposed to extract the strain hardening exponent and the reduced elastic modulus of an indented material. The accuracy of this method is verified and discussed with experimental data from the literature and representative materials.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams

The Voronoi tessellation technique and the finite element (FE) method are utilized to investigate the microstructure-property relations of three-dimensional (3-D) cellular solids (foams) that have irregular cell shapes and non-uniform strut cross-sectional areas (SCSAs). Perturbations are introduced to a regular packing of seeds to generate a spatially periodic Voronoi diagram with different degrees of cell shape irregularity (amplitude a), and to the constant SCSA to generate a uniform distribution of SCSAs with different degrees of SCSA non-uniformity (amplitude b). Twenty FE models are constructed, based on the Voronoi diagrams for twenty foam samples (specimens) having the same pair of a and b, to obtain the mean values and standard deviations of the elastic properties. Spatially periodic boundary conditions are applied to each specimen. The simulation results indicate that for low-density imperfect foams, the elastic moduli increase as cell shapes become more irregular, but decrease as SCSAs get less uniform. When the relative density (R) increases, the elastic moduli of imperfect foams increase substantially, while the Poisson's ratios decrease moderately. The effect of the interaction between the two types of imperfections on foam elastic properties appears to be weak. In addition, it is found that the strut cross-sectional shape has a significant effect on the foam properties. Also, the elastic response of foams with the cell shape and SCSA imperfections appears to be isotropic regardless of changes in a, b and R and the strut cross-sectional shape.     

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) ^c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V _SV , V _P ,?V _SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c?=?0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.     

A study on the bending stiffness of single-walled carbon nanotubes and related issues

Single-walled carbon nanotubes (SWCNTs) are usually modeled as elastic tubes and their bending stiffness D is often related to their axial stretching modulus E (Young's modulus) as in mechanics of materials (i.e. D=EI where I is the moment of inertia of the tube). However, recent studies show that large discrepancies may exist when this relationship is used to predict Young's modulus of carbon nanotubes (CNTs) through bending dominated deformations. In the present paper, the bending stiffness of SWCNTs and some related issues are investigated by the combined use of the molecular-mechanics (M-M) model and the deformation mapping technique. Based on the analysis results, the contradictions mentioned above can be explained well. Furthermore, an analytical expression for the bending stiffness of SWCNTs is also presented. It shows that the bending stiffness of a SWCNT is approximately proportional to the cube of its radius which agrees well with the existing molecular dynamics simulation and continuum theory based results.     

On Anisotropic Elastic Materials for which Young''s Modulus E ( n ) is Independent of n or the Shear Modulus G ( n , m ) is Independent of n and m

It is shown that, among anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus E(n) independent of the direction n or the shear modulus G(n,m) independent of n and m. Thus the direction surface for E(n) can be a sphere for certain orthotropic and hexagonal materials. The structure of the elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given. We also study the case in which n of E(n) and n, m of G(n,m) are restricted to a plane. When E(n) is a constant on a plane so are G(n,m) and Poisson's ratio ν(n,m). The converse, however, does not necessarily hold. A plane on which E(n) is a constant can exist for all anisotropic elastic materials. In particular, existence of such a plane is assured for trigonal, hexagonal and cubic materials. In fact there are four such planes for a cubic material. For these materials, not only E(n) is a constant, two other Young's moduli, the three shear moduli and the six Poisson's ratio on the plane are also constant.     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

Impinging rotational stagnation-point flows

A rotational stagnation-point flow of fluid density ρ₁ and kinematic viscosity ν₁ impinges normal to another rotational stagnation-point flow of fluid density ρ₂ and viscosity ν₂. Results are compared with a previous study on the normal impingement of two Homann stagnation-point flows for which the flow in the far field is irrotational.     

Thermodynamic functions of the heated electron subsystem in the field of cold nuclei

The paper presents electronic heat capacities and thermal pressures calculated for aluminum and tungsten at densities ρ₀ ≤ ρ ≤ 2ρ₀ in the case when the temperature of electrons is finite (a few electron-volts) and nuclei are cold. Calculations were done with the all-electron full-potential linear muffin-tin orbital method (FP-LMTO) and compared with data obtained with the Liberman's average-atom model and the Vienna Ab-initio Simulation Package code (VASP) which uses pseudopotentials. It is shown that results obtained with different approaches qualitatively agree within the ranges of electron temperatures and densities under consideration, and quantitatively agree within 10% in most cases.     

Three-dimensional modeling of the mechanical property of linearly elastic open cell foams

Three-dimensional Voronoi models are developed to investigate the mechanical behavior of linearly elastic open cell foams. Dependence of the Young’s modulus, Poisson’s ratio and bulk modulus of the foams on the relative density is evaluated through finite element analysis. Obtained results show that in the low density regime the Young’s modulus and bulk modulus of random Voronoi foams can be well represented by those of Kelvin foams, and are sensitive to the geometric imperfections inherent in the microstructure of foams. In contrast, the compressive plateau stress of the foams is less sensitive to the imperfections. Failure surface of the foams subject to multi-axial compression is determined and is found to comply with the maximum compressive principal stress criterion, consistent with available experimental observations on polymer foams. Numerical results also show that elastic buckling of cell edges at microscopic level is the dominant mechanism responsible for the compressive failure of elastic open cell foams.     

Influence of nanoprecipitates on the creep strength and ductility of a Fe–Ni–Al alloy

Creep studies of a duplex Fe–Ni–Al intermetallic alloy, in two microstructural states, have been carried out at temperatures between 725 and 800 °C (about 0.6 T_m). In the as-cast state, the alloy contains a large volume fraction of nanoprecipitates (50–100 nm) which confer a very high creep strength with a stress exponent of 3 and an activation energy of 280 kJ/mol. The different microstructure obtained in the second state of the alloy, obtained after annealing at 1000 °C for 24 h, leads to a much lower creep strength with a higher stress exponent as well as a large value of the apparent activation energy. While volume diffusion appears to control creep in the as-cast state, both thermal and athermal processes seem to contribute to the different creep rate of material in the annealed state. The latter also exhibits a much larger ductility (12%) relative to that observed in the as-cast material (3%), due to the presence of large numbers of interfaces between the two phases present where strain incompatibilities can be accommodated.     

Gelation behavior of polysaccharide-based interpenetrating polymer network (IPN) hydrogels

We report the preparation and rheological characterization of interpenetrating polymer network (IPN) hydrogels made from alginate and hydrophobically modified ethyl hydroxyl ethyl cellulose (HMEHEC). To our knowledge, there have been no studies of the gelation behavior of IPNs. We found that the rheology of these systems can be easily tuned, with the elastic modulus of the IPN strongly dependent on the relative ratio of HMEHEC to alginate. The sol–gel transition of these systems was found to satisfy the Winter–Chambon criterion for gelation at various crosslinker densities. From the power law relationship of the dynamic moduli (G ^′ ~G ^″ ~ω ⁿ), the exponent n appears to be dependent on both the crosslinker density and relative amount of two polymers. The value of n was found to be ~0.5 for all samples for stoichiometric amounts of crosslinker. The effect of molecular weight of HMEHEC on the gel point and viscoelastic exponent has also been reported. Alginate seems to dominate the kinetics of the process but the effect of high molecular weight HMEHEC on the gel point, especially at lower proportion was also evident.     

The interlayer shear effect on graphene multilayer resonators

Graphene nanostrips with single or few layers can be used as bending resonators with extremely high sensitivity to environmental changes. In this paper we report molecular dynamics (MD) simulation results on the fundamental and secondary resonant frequencies f of cantilever graphene nanostrips with different layer number n and different nanostrip length L. The results deviate significantly from the prediction of not only the Euler-Bernoulli beam theory (f∝nL⁻²), but also the Timoshenko's model. Since graphene nanostrips have extremely high intralayer Young's modulus and ultralow interlayer shear modulus, we propose a multibeam shear model (MBSM) that neglects the intralayer stretch but accounts for the interlayer shear. The MBSM prediction of the fundamental and secondary resonant frequencies f can be well expressed in the form f−f_mono∝[(n-1)/n]^bL^−2(1−b), where f_mono denotes the corresponding resonant frequency as the layer number is 1, with b=0.61 and 0.77 for the fundamental and secondary resonant modes. Without any additional parameters fitting, the prediction from MBSM agrees excellently with the MD simulation results. The model is thus of importance for designing multilayer graphene nanostrips based applications, such as resonators, sensors and actuators, where interlayer shear has apparent impacts on the mechanical deformation, vibration and energy dissipation processes therein.     

Effective elastic properties of porous materials: Homogenization schemes vs experimental data

In this paper, we focus on the prediction of elastic moduli of isotropic porous materials made of a solid matrix having a Poisson's ratio vm of 0.2. We derive simple analytical formulae for these effective moduli based on well-known Mean-Field Eshelby-based Homogenization schemes. For each scheme, we find that the normalized bulk, shear and Young's moduli are given by the same form depending only on the porosity p. The various predictions are then confronted with experimental results for the Young's modulus of expanded polystyrene (EPS) concrete. The latter can be seen as an idealized porous material since it is made of a bulk cement matrix, with Poisson's ratio 0.2, containing spherical mono dispersed EPS beads. The Differential method predictions are found to give a very good agreement with experimental results. Thus, we conclude that when vm=0.2, the normalized effective bulk, shear and Young's modulus of isotropic porous materials can be well predicted by the simple form (1 − p)² for a large range of porosity p ranging between 0 and 0.56.     

Dynamic compressive strength properties of aluminium foams. Part II—‘shock’ theory and comparison with experimental data and numerical models

One-dimensional ‘steady-shock’ models based on a rate-independent, rigid, perfectly-plastic, locking (r-p-p-l) idealisation of the quasi-static stress-strain curves for aluminium foams are proposed for two different impact scenarios to provide a first-order understanding of the dynamic compaction process. A thermo-mechanical approach is used in the formulation of their governing equations. Predictions by the models are compared with experimental data presented in the companion paper (Part I) and with the results of finite-element simulations of two-dimensional Voronoi honeycombs.A kinematic existence condition for continuing ‘shock’ propagation in aluminium foams is established using thermodynamics arguments and its predictions compare well with the experimental data. The thermodynamics highlight the incorrect application of the global energy balance approach to describe ‘shock’ propagation in cellular solids which appears in some current literature.     

Spherically isotropic,elastic spheres subject to diametral point load strength test

This paper presents an analytic solution for the stress concentrations within a spherically isotropic, elastic sphere of radius R subject to diametral point load strength test. The method of solution uses the displacement potential approach together with the Fourier–Legendre expansion for the boundary loads. For the case of isotropic sphere, our solution reduces to the solution by Hiramatsu and Oka, 1966 and agrees well with the published experimental observations by Frocht and Guernsey (1953) . A zone of higher tensile stress concentration is developed near the point loads, and the difference between this maximum tensile stress and the uniform tensile stress in the central part of the sphere increases with E/E′ (where E and E′ are the Youngs moduli governing axial deformations along directions parallel and normal to the planes of isotropy, respectively) , G′/G (where G and G′ are the moduli governing shear deformations in the planes of isotropy and the planes parallel to the radial direction) , and ν̄/ν′ (where ν̄ and ν′ are the Poissons ratios characterizing transverse reduction in the planes of isotropy under tension in the same plane and under radial tension, respectively) . This stress difference, in general, decreases with the size of loading area and the Poissons ratio.     

Youngs modulus interpreted from plane compressions of geomaterials between rough end blocks

This note derives an approximate expression of the true Youngs modulus of a rectangular solid under plane compression between two rough end blocks, provided that the Poissons ratio ν of the solid is known. The friction between the loading platens and the ends of the specimen is assumed to be large enough to restrain slippage at the contact. By using the function space concept of Prager and Synge (1947) , a correction factor λ with calculable error is obtained which can be multiplied to the apparent Youngs modulus (i.e., the one obtained by assuming uniform stress field) to yield the true Youngs modulus; it is evaluated numerically for 0 ⩽ ν ⩽ 0.49 and 0 ⩽ η ⩽ 3 (where η = b⧸h with b and h being the half width and half length of the specimen) . In general, λ increases with ν and η for both plane strain and plane stress compressions. Within this range of ν and η, λ may vary from 0.37–1.0 for the plane strain case and from 0.84–1.0 for the plane stress case. Thus, the assumption of uniform stress field may lead to erroneous interpretation of the Youngs modulus. When the special case of ν = 1⧸3 and η = 1 is considered, we obtain λ = 0.9356, which compares well with 0.9359 obtained by Greenberg and Truell, 1948 .