What is indentation hardness?

Using dimensional analysis and finite element calculations, we derive simple scaling relationships for loading and unloading curve, contact depth, and hardness. The relationship between hardness and the basic mechanical properties of solids, such as Young's modulus, initial yield strength, and work-hardening exponent, is then obtained. The conditions for 'piling-up' and 'sinking-in' of surface profiles during indentation are determined. A method for estimating contact depth from initial unloading slope is examined. The work done during indentation is also studied. A relationship between the ratio of hardness to elastic modulus and the ratio of irreversible work to total work is discovered. This relationship offers a new method for obtaining hardness and elastic modulus. Finally, a scaling theory for indentation in power-law creep solids using self-similar indenters is developed. A connection between creep and 'indentation size effect' is established.

Scaling, Dimensional Analysis, and Indentation Measurements

We provide an overview of the basic concepts of scaling and dimensional analysis, followed by a review of some of the recent work on applying these concepts to modeling instrumented indentation measurements. Specifically, we examine conical and pyramidal indentation in elastic-plastic solids with power-law work-hardening, in power-law creep solids, and in linear viscoelastic materials. We show that the scaling approach to indentation modeling provides new insights into several basic questions in instrumented indentation, including, what information is contained in the indentation load-displacement curves? How does hardness depend on the mechanical properties and indenter geometry? What are the factors determining piling-up and sinking-in of surface profiles around indents? Can stress-strain relationships be obtained from indentation load-displacement curves? How to measure time dependent mechanical properties from indentation? How to detect or confirm indentation size effects? The scaling approach also helps organize knowledge and provides a framework for bridging micro- and macroscales. We hope that this review will accomplish two purposes: (1) introducing the basic concepts of scaling and dimensional analysis to materials scientists and engineers, and (2) providing a better understanding of instrumented indentation measurements.

Size Dependent Nanoindentation of a Soft Film on a Hard Substrate

Nanoindentation experiments on Al/glass systems show that, as the indentation depth increases, the hardness decreases during a shallow indentation, and increases when the indenter tip approaches the film–substrate interface. We associate the rise in hardness during two stages with the strong strain gradient effects, the first stage is related with the small scale effects and the second stage with the strain gradient between the indenter and the hard substrate. Using the strain gradient theory proposed by Chen and Wang and the classical plasticity theory, the observed nanoindentation behavior is modeled and analyzed by means of the finite element method, and it is found that the classical plasticity cannot explain the experiment results but the strain gradient theory can describe the experiment data at both shallow and deep indentation depths very well. The results prove that both the strain gradient effects and substrate effects exist in the nanoindentation of the film–substrate system.

Fracture Mechanisms And Size Effects Of Brittle Metallic Foams: In Situ Compression Tests Inside Sem

In situ compressive tests on specially designed small samples made from brittle metallic foams were accomplished in a loading device equipped in the scanning electron microscopy (SEM). Each of the small samples comprises only several cells in the effective test zone (ETZ), with one major cell in the middle. In such a system one can not only obtain sequential collapse-process images of a single cell and its cell walls with high resolution, but also correlate the detailed failure behaviour of the cell walls with the stress-strain response, therefore reveal the mechanisms of energy absorption in the mesoscopic scale. Meanwhile, the stress-strain behaviour is quite different from that of bulk foams in dimensions of enough large, indicating a strong size effect. According to the in situ observations, four failure modes in the cell-wall level were summarized, and these modes account for the mesoscopic mechanisms of energy absorption. Paralleled compression tests on bulk samples were also carried out, and it is found that both fracturing of a single cell and developing of fracture bands are defect-directed or weakness-directed processes. The mechanical properties of the brittle aluminum foams obtained from the present tests agree well with the size effect model for ductile cellular solids proposed by Onck et al. (C) 2008 Elsevier Ltd. All rights reserved.

Scaling Functions In Conical Indentation Of Elastic-Plastic Solids

The finite element method was used to simulate the conical indentation of elastic-plastic solids with work hardening. The ratio of the initial yield strength to the Young's modulus Y/E ranged from 0 to 0.02. Based on the calculation results, two sets of scaling functions for non-dimensional hardness H/K and indenter penetration h are presented in the paper, which have closed simple mathematical form and can be used easily for engineering application. Using the present scaling functions, indentation hardness and indentation loading curves can be easily obtained for a given set of material properties. Meanwhile one can use these scaling functions to obtain material parameters by an instrumented indentation load-displacement curve for loading and unloading if Young's modulus E and Poisson's ratio nu are known.

Investigation on mechanical properties and size effect of nanocrystalline twin copper

The stress-strain relations of nanocrystalline twin copper with variously sized grains and twins are studied by using FEM simulations based on the conventional theory of mechanism-based strain gradient plasticity (CMSG). A model of twin lamellae strengthening zone is proposed and a cohesive interface model is used to simulate grain-boundary sliding and separation. Effects of material parameters on stress-strain curves of polycrystalline twin copper are studied in detail. Furthermore, the effects of both twin lamellar spacing and twin lamellar distribution on the stress-strain relations are investigated under tension loading. The numerical simulations show that both the strain gradient effect and the material hardening increase with decreasing the grain size and twin lamellar spacing. The distribution of twin lamellae has a significant influence on the overall mechanical properties, and the effect is reduced as both the grain size and twin lamellar spacing decrease. Finally, the FEM prediction results are compared with the experimental data.

Small scale, grain size and substrate effects in nano-indentation experiment of film-substrate systems

The mechanical properties of film-substrate systems have been investigated through nano-indentation experiments in our former paper (Chen, S.H., Liu, L., Wang, T.C., 2005. Investigation of the mechanical properties of thin films by nano-indentation, considering the effects of thickness and different coating-substrate combinations. Surf. Coat. Technol., 191, 25-32), in which Al-Glass with three different film thicknesses are adopted and it is found that the relation between the hardness H and normalized indentation depth h/t, where t denotes the film thickness, exhibits three different regimes: (i) the hardness decreases obviously with increasing indentation depth; (ii) then, the hardness keeps an almost constant value in the range of 0.1-0.7 of the normalized indentation depth h/t; (iii) after that, the hardness increases with increasing indentation depth. In this paper, the indentation image is further investigated and finite element method is used to analyze the nano-indentation phenomena with both classical plasticity and strain gradient plasticity theories. Not only the case with an ideal sharp indenter tip but also that with a round one is considered in both theories. Finally, we find that the classical plasticity theory can not predict the experimental results, even considering the indenter tip curvature. However, the strain gradient plasticity theory can describe the experimental data very well not only at a shallow indentation depth but also at a deep depth. Strain gradient and substrate effects are proved to coexist in film-substrate nano-indentation experiments. (c) 2006 Elsevier Ltd. All rights reserved.

Three-dimensional coupled wave study for finite-sized anisotropic volume holographic gratings under ultrashort pulsed beam readout

The three-dimensional coupled wave theory is extended to systematically investigate the diffraction properties of finite-sized anisotropic volume holographic gratings (VHGs) under ultrashort pulsed beam (UPB) readout. The effects of the grating geometrical size and the polarizations of the recording and readout beams on the diffraction properties are presented, in particular under the influence of grating material dispersion. The wavelength selectivity of the finite-sized VHG is analyzed. The wavelength selectivity determines the intensity distributions of the transmitted and diffracted pulsed beams along the output face of the VHG. The distortion and widening of the diffracted pulsed beams are different for different points on the output face, as is numerically shown for a VHG recorded in a LiNbO3 crystal. The beam quality is analyzed, and the variations of the total diffraction efficiency are shown in relation to the geometrical size of the grating and the temporal width of the readout UPB. In addition, the diffraction properties of the finite-sized and one-dimensional VHG for pulsed and continuous-wave readout are compared. The study shows the potential application of VHGs in controlling spatial and temporal features of UPBs simultaneously. (C) 2007 Optical Society of America

SCALING OF LIMB PROPORTIONS AND LIMB BONE DIAMETERS IN 3 SPECIES OF CHINESE SNUB-NOSED LANGURS (GENUS RHINOPITHECUS)

In this study, aspects of the structural mechanics of the upper and lower limbs of the three Chinese species of Rhinopithecus were examined. Linear regression and reduced major axis (RMA) analyses of natural log-transformed data were used to examine the dimensions of limb bones and other relationships to body size and locomotion. The results of this study suggest that: (1) the allometry exponents of the lengths of long limbs deviate from isometry, being moderately negative, while the shaft diameters (both sagittal and transverse) show significantly positive allometry; (2) the sagittal diameters of the tibia and ulna show extremely significantly positive allometry - the relative enlargement of the sagittal, as opposed to transverse, diameters of these bones suggests that the distal segments of the fore- and hindlimbs of Rhinopithecus experience high bending stresses during locomotion; (3) observations of Rhinopithecus species in the field indicate that all species engage in energetic leaping during arboreal locomotion. The limbs experience rapid and dramatic decelerations upon completion of a leap. We suggest that these occasional decelerations produce high bending stresses in the distal limb segments and so account for the hypertrophy of the sagittal diameters of the ulna and tibia.

Size distributions of quantum islands on stepped substrates

The size distributions of self-assembled quantum islands on stepped substrates are studied using kinetic Monte Carlo simulations. It is found that the energy barrier E-SW between the step and the terrace region is the key factor in affecting the size distribution of islands. With small E-SW (<= 0.1 eV), lines of uniform islands can be obtained at relative low surface coverage. As the surface coverage is increased, wirelike islands can be obtained. Scaling behavior is obeyed for the size distributions of the wirelike islands. When the size distributions are separated into their width and length components, however, scaling is only observed in the length distribution of the wirelike islands. With larger E-SW, the size distribution of islands shows a clear bimodal size distribution and anomalous growth temperature dependent island size evolutions are observed. The simulation results reproduce qualitatively the phenomena observed in the cases of InAs islands grown on stepped GaAs substrates. (c) 2009 American Institute of Physics. [doi:10.1063/1.3248367]

Investigation of mode characteristics for a square cavity with a pedestal by a three-dimensional finite-difference time-domain technique

Mode characteristics of a strongly confined square cavity suspended in air via a pedestal on the substrate are investigated by a three-dimensional finite-difference time-domain technique. The mode wavelengths and mode quality factors (Q factors) are calculated as the functions of the size of the pedestal and the slope angle 0 of the sidewalls of the square slab, respectively For the square slab with side length of 2 mu m, thickness of 0.2 mu m, and refractive index of 3.4, on a square pedestal with refractive index of 3.17, the Q factor of the whispering-gallery (WG)-like mode transverse-electric TE(3.5)o first increases with the side length b of the square pedestal and then quickly decreases as b > 0.4 mu m, but the Q factor of the WG-like mode TE(4.6)o drops down quickly as b > 0.2 mu m, owing to their different symmetries. The results indicate that the pedestal can also result in mode selection in the WG-like modes. In addition, the numerical results show that the Q factors decrease 50% as the slope angle of the sidewalls varies from 90 degrees to 80 degrees. The mode characteristics of WG-like modes in the square cavity with a rectangular pedestal are also discussed. The results show that the nonsquare pedestal largely degrades the WG-like modes. (c) 2006 Optical Society of America

Size tolerance analysis of a 4X4 tapered multimode interference coupler in a silicon-on-insulator structure

Size tolerance of a 4X4 general interference tapered multimode interference (MMI) coupler in a silicon-on-insulator (SOI) structure is investigated by means of a 2-D finite difference beam propagation method (2D-FDBPM), together with an effective refractive index method (EIM). The results show that the tapered multimode interference coupler exhibits relatively larger size tolerance when light is launched from the edgeport than from midport, though it has much better output power uniformity when light is launched from midport. Besides that, it can reduce the device length greatly. The 4X4 general interference tapered MMI coupler has a slightly larger size tolerance compared with a conventional straight multimode interference coupler. (C) 2003 Society of Photo-Optical Instrumentation Engineers.

constructing finite algebras with falcon

The generation of models and counterexamples is an important form of reasoning. In this paper, we give a formal account of a system, called FALCON, for constructing finite algebras from given equational axioms. The abstract algorithms, as well as some implementation details and sample applications, are presented. The generation of finite models is viewed as a constraint satisfaction problem, with ground instances of the axioms as constraints. One feature of the system is that it employs a very simple technique, called the least number heuristic, to eliminate isomorphic (partial) models, thus reducing the size of the search space. The correctness of the heuristic is proved. Some experimental data are given to show the performance and applications of the system.

A finite-element algorithm for modeling variably saturated flows

A general numerical algorithm in the context of finite element scheme is developed to solve Richards’ equation, in which a mass-conservative, modified head based scheme (MHB) is proposed to approximate the governing equation, and mass-lumping techniques are used to keep the numerical simulation stable. The MHB scheme is compared with the modified Picard iteration scheme (MPI) in a ponding infiltration example. Although the MHB scheme is a little inferior to the MPI scheme in respect of mass balance, it is superior in convergence character and simplicity. Fully implicit, explicit and geometric average conductivity methods are performed and compared, the first one is superior in simulation accuracy and can use large time-step size, but the others are superior in iteration efficiency. The algorithm works well over a wide variety of problems, such as infiltration fronts, steady-state and transient water tables, and transient seepage faces, as demonstrated by its performance against published experimental data. The algorithm is presented in sufficient detail to facilitate its implementation.

Longitudinal scaling property of the charge balance function in Au plus Au collisions at root s(NN)=200 GeV

We present measurements of the charge balance function, from the charged particles, for diverse pseudorapidity and transverse momentum ranges in Au + Au collisions at root S-NN = 200 GeV using the STAR detector at RHIC. We observe that the balance function is boost-invariant within the pseudorapidity coverage vertical bar-1.3, 1.3 vertical bar. The balance function properly scaled by the width of the observed pseudorapidity window does not depend on the position or size of the pseudorapidity window. This scaling property also holds for particles in different transverse momentum ranges. In addition, we find that the width of the balance function decreases monotonically with increasing transverse momentum for all centrality classes. (c) 2010 Elsevier B.V. All rights reserved.