Size Effect Theory Research Articles

Microstructure and size effect of interfacial intermetallic on fracture toughness of Sn3.0Ag0.5Cu solder interconnects

The effects of solid state thermal aging and liquid state reflow on the interfacial fracture toughness of Sn3.0Ag0.5Cu solder interconnects are experimentally investigated. Solid state thermal aging for compact tension specimens was treated at 210 °C up to 48 h, the liquid state reflow was treated at 230 °C up to 48 h as well. Three types of fracture modes are observed including fracture in solder, intermetallic/solder interface and intermetallic layer. In general, the fracture toughness of intermetallic decreases with increasing of treating time under both liquid and solid states. After solid state thermal aging, the fraction of intermetallic fracture decreases while the proportion of solder fracture and interfacial fracture increases. After liquid state reflow, the fracture keeps being dominated by the intermetallic fracture. The morphology evolution and grain size of intermetallic layer are analyzed. In solid state thermal aging, the intermetallic layer tends to transit from the initial scallop-type to lamellar-type. While in liquid state reflow, the intermetallic layer tends to keep scallop-type. Both intermetallic morphology and grain size are found to be essential factors that affect the fracture type and toughness. The effect of intermetallic grain size on fracture toughness is investigated based on the size effect theory. The predicted fracture toughness of solder interconnects after 230 °C reflow agree well with the experimental results.

A resource-rational theory of set size effects in human visual working memory.

Encoding precision in visual working memory decreases with the number of encoded items. Here, we propose a normative theory for such set size effects: the brain minimizes a weighted sum of an error-based behavioral cost and a neural encoding cost. We construct a model from this theory and find that it predicts set size effects. Notably, these effects are mediated by probing probability, which aligns with previous empirical findings. The model accounts well for effects of both set size and probing probability on encoding precision in nine delayed-estimation experiments. Moreover, we find support for the prediction that the total amount of invested resource can vary non-monotonically with set size. Finally, we show that it is sometimes optimal to encode only a subset or even none of the relevant items in a task. Our findings raise the possibility that cognitive "limitations" arise from rational cost minimization rather than from constraints.

Influence of cooling medium on low temperature strength of asphalt binders

The effect of three different cooling media, ethanol, potassium acetate (PA) and air, on the strength of asphalt binder at low temperature is investigated with a modified Bending Beam Rheometer (BBR) and the Direct Tension Tester (DTT). Similar strength values are obtained in air and PA and significantly lower values are obtained in ethanol. The use of fluorescing markers demonstrates that ethanol diffuses in the asphalt binder, while diffusion does not occur when PA is used. Using size effect theory, BBR strengths in PA and in air are found to be equivalent to DTT strength in PA.

The Peer Group in European Market Multiples: The Problem of Size

The aim of this paper is to bring a new perspective to the role of control factors in MARKET-EBITDA, MARKET-SALES and MARKET-BOOK ratios in the European market, which are different from industry. We analyze the role of size and its interactions with the rest of the variables: profitability, risk, and structural asset factors. For this purpose, we have developed a valuation model using panel data methodology for listed European firms from the period 2002-2013. We found that size includes economic aspects of the companies, but also differential investors’ perceptions about these economic aspects. It makes it a complementary explanation factor on the selection of the peer group with the rest of the variables. These findings help to understand the variation of market ratios and share prices depending on size in different economic conditions or economic cycles. This research is in accordance with the theoretical framework of market ratios and with size effect theory.

The Vibronic Theory of Size Effects in CO Oxidation by Gold Nanocatalysts

A theoretical model, based on the vibronic theory of catalysis, has been developed for the CO oxidation reaction catalyzed by gold nanoparticles. The charge fraction transferred between the catalyst and the adsorbed CO molecule depends upon the relative positions of the size-dependent Fermi level of the cluster and the energy levels of the molecular orbitals of CO adsorbed onto the cluster. The charge transferred from the nanocluster to the adsorbed CO molecule increases the absolute values of forces deforming the molecule chemical bond and force constants defining the curvature of potential for nuclear oscillation leading to the decrease in activation energy for CO oxidation. The application of vibronic and jellium theories to CO oxidation provides an explanation of the observed cluster size effects. The relative locations of electronic energy levels of a gold cluster and a CO molecule before and after the adsorption onto the cluster surface

Development of a simple correlation between bending beam rheometer and thermal stress restrained specimen test low-temperature properties based on a simplified size effect approach

In this paper, the possibility of using bending beam rheometer (BBR) experimental results to determine the critical cracking temperature of an asphalt mixture is evaluated. First, thermal stress restrained specimen test (TSRST) together with BBR creep and strength tests is performed on eight asphalt mixtures designed with and without recycled materials. The experimentally determined cracking temperature is initially simply compared and a significant difference is observed between TSRST and BBR values. Based on simple size effect theory, the BBR strength is then converted into the equivalent tensile strength of specimens having the same size of TSRST samples, and the critical cracking temperature is recalculated. The analysis suggests that the TSRST- and BBR-derived cracking temperatures are only in partial agreement.

Testing protocol to obtain failure properties of asphalt binders at low temperature using creep compliance and stress-controlled strength test

Good fracture properties are an essential requirement for asphalt materials used in the construction of pavements in cold regions. For asphalt binder, two instruments were developed during SHRP research effort to investigate the low-temperature behaviour of these materials: bending beam rheometer (BBR) and direct tension tester (DTT). These two devices are used to obtain the performance grade (PG) of asphalt binders in the US. The DTT is expensive and the results are less repeatable due to a complex sample preparation and difficult-to-achieve strain-controlled loading. For these reasons, many agencies do not use DTT when selecting asphalt binders, and rely entirely on the creep properties obtained with the BBR testing method. In this paper, a modified BBR is used to perform three-point bending strength tests on asphalt binder beams at low temperature, with the final goal of developing a specification for binder selection, similar to the current PG specification. BBR strength and DTT tests are first performed on a common set of binders, and the results are compared using size effect theory. Then, additional tests are performed and a simple and practical strength test procedure based on BBR creep data and on BBR strength data obtained under constant loading rate is proposed.

Megalithic stone beam bridges of ancient China reach the limits of strength and challenge size effect in granite

Abstract In all ancient monuments, stone beams and architraves have unsupported spans that seldom reach 7 m, while ordinary spans are usually much less. These structural elements were and still are believed to be prone to failure, so that several relieving systems (arches, chambers, gaps) were adopted through history to prevent collapse. The perception that stone beams could not exceed a certain span is coherent with the so-called size-effect theory of rock and concrete, which predicts that large elements are proportionally weaker than small ones. While the rest of the world started using architectural design to avoid these problems, in the Fujian region of China (near Xiamen) from the XI to the XII century megalithic stone beam bridges with spans of up to 21 m were being built. These bridges have resisted over the centuries. A spectacular example of these bridges, tending to disprove the size-effect theory and challenging all previous ancient constructions, is the Jiangdong bridge, of which only a part survives, but which should be restored, preserved, and declared human heritage monument.

Size effect on fracture toughness of snow

Abstract: Depending on the scale of observation, many engineered and natural materials show different mechanical behaviour. Thus, size effect theories, based on a multiscale approach, analyse the intrinsic (due to microstructural constraints, e.g., grain size) and extrinsic effects (caused by dimensional constraints), in order to improve the knowledge in materials science and applied mechanics. Nevertheless, several problems regarding Solid Mechanics and Materials Science cannot be solved by conventional approaches, because of the complexity and uncertainty of materials proprieties, especially at different scales. For this reason, a simple model, capable of predicting a fracture toughness at different scale, has been developed and presented in this paper. This model is based on the Golden Ratio, which was firstly defined by Euclide as: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less”. Intimately interconnected with the Fibonacci sequence (1, 2, 3, 5, 8, 13, …), this number controls growth in Nature and recurs in many disciplines, such as art, architecture, design, medicine, etc.., and for man-made and natural brittle materials, the Golden Ratio permits to define the relationship between the average crack spacing and the thickness of quasi-brittle materials. In these cases, the theoretical results provided by the Golden Ratio, used to calibrate a size-effect law of fracture toughness, are in accordance with the experimental measurements taken in several test campaigns carried on different materials (i.e., rocks, ice, and concrete). This paper presents the case of fracture toughness of snow, in which the irrational number 1.61803 recurs when the geometrical dimensions vary. This aspect is confirmed by the results of experimental campaigns performed on snow samples. Thus, we reveals the existence of the size-effect law of fracture toughness of snow and we argue that the centrality of the Golden Ratio in the fracture properties of quasi-brittle materials. Consequently, by means of the proposed model, the Kic of large samples can be simply and rapidly predicted, without knowing the material performances but by testing prototypes of the lower dimensions.

Simple Approach to the Investigation of Asphalt Mixture Strength on Small Beam Specimens at Low Temperature

AbstractIn this research, a simplified approach based on size effect theory is used to extrapolate the strength values obtained on small asphalt mixtures specimens at low temperature with a modified bending beam rheometer (BBR). First, nine asphalt mixtures having different mix designs are prepared. Then, Weibull statistics and factorial design are used to evaluate the statistical distribution of the BBR measurements and the effect of conditioning time, loading rate, and testing temperature on material response, respectively. Finally, the BBR strength measured at three different temperatures is compared to the corresponding values obtained from two conventional test methods: direct tension (DT) and indirect tension (IDT) tests. It is observed that BBR strength values are statistically equivalent to those of IDT and DT when the size effect theory and the weakest link model are used to take into account the difference in volume, geometry, and stress field across the three testing methods.

Theory of finite size effects for electronic quantum Monte Carlo calculations of liquids and solids

Concentrating on zero temperature Quantum Monte Carlo calculations of electronic systems, we give a general description of the theory of finite size extrapolations of energies to the thermodynamic limit based on one and two-body correlation functions. We introduce new effective procedures, such as using the potential and wavefunction split-up into long and short range functions to simplify the method and we discuss how to treat backflow wavefunctions. Then we explicitly test the accuracy of our method to correct finite size errors on example hydrogen and helium many-body systems and show that the finite size bias can be drastically reduced for even small systems.

Torque loading tests on the rolling shear strength of cross-laminated timber

In this study , torque loading tests on small shear blocks were performed to evaluate the rolling shear strength of cross-laminated timber (CLT). The CLT plates in the tests were manufactured with Mountain Pine Beetle-afflicted lumber boards and glued with polyurethane adhesive; two types of layups (five-layer and three-layer) with a clamping pressure 0.4 MPa were studied. The small block specimens were sampled from full-size CLT plates and the cross layers were processed to have an annular cross section. These specimens were tested under torque loading until brittle shear failure occurred in the middle cross layers. Based on the test results, the brittle shear failure in the specimens was evaluated by detailed finite element models to confirm the observed failure mode was rolling shear. Furthermore, a Monte Carlo simulation procedure was performed to investigate the occurrence probability of different shear failure modes in the tests considering the randomness of the rolling shear strength and longitudinal shear strength properties in the wood material. The result also suggested the probability of rolling shear failure is very high, which gives more confident proof that the specimens failed dominantly in rolling shear. It was also found that the torque loading test method yielded different rolling shear strength values compared to the previous research from short-span beam bending tests; such a difference may mainly be due to the different stressed volumes of material under different testing methods, which can be further investigated using the size effect theory in the future.

An electrochemical study of the nickel hydroxide-gold modified electrode employing the surface resistance technique

Simultaneous cyclic voltammetry (CV) and surface resistance (SR) measurements were employed to study the electrochemical behaviour of the nickel hydroxide-gold modified electrode. To this end, firstly, gold film electrodes of different thicknesses were synthesized, and their CV and SR responses within the potential regions corresponding to hydroxide ion adsorption (OHads) and gold oxide (AuO) formation in alkaline medium were analysed. Then, these gold film electrodes were modified with different nickel hydroxide surface coverages, and their CV and SR responses were compared with those of bare gold films within the same potential regions and electrolyte solution. While cyclic voltammetric responses of the bare gold film electrode and the nickel hydroxide-gold film modified electrode are practically the same within the whole potential range−0.5V<E<0.35V (SCE), SR responses are the same only within the potential range −0.3V<E<0.35V. Strong differences between the SR responses of the bare gold electrode and the nickel hydroxide-gold modified electrode within the potential range comprised between −0.5V and −0.3V are observed. SR data were interpreted in terms of the field effect and size effect theories formulated to study adsorption processes at thin metallic films. With regard to the bare gold film electrode, the field effect theory allows one to conclude that within the potential region corresponding to hydroxide ion adsorption, 77% of the electron density charge of gold is involved in the hydroxide ion–gold interaction. Besides, a linear relation between the surface resistance change (ΔR) and the degree of oxidation (θox) at fixed gold film thickness (ϕm) is obtained within the potential region corresponding to gold oxide formation. In agreement with the size effect theory, the ΔR/θox slope exhibits a linear dependence on ϕm−2. The same SR responses for the nickel hydroxide-gold modified electrode and the bare gold electrode within the potential range −0.3V<E<0.35V indicate that the nickel hydroxide layer does not directly interact with the gold film surface. However, after the reduction of OHads and AuO layers, at potential values more negative than −0.3V (SCE), an increase of the surface resistance is observed for the nickel hydroxide-gold modified electrode, which was attributed to nickel adatoms that directly interact with the gold film surface. On the basis of the electron conduction scattering model employed to describe the interaction of a metal film surface with a foreign atom layer, one can conclude that nickel film growth involves an initial stage in which nickel islands and then, compact structures are formed.

Thickness-dependent electrical and optical properties of Ge8Sb2Te11 thin films

ABSTRACTThe amorphous Ge8Sb2Te11thin films with varying thickness are thermally deposited on well-cleaned glass substrate from its polycrystalline bulk. Absence of any sharp peak confirms the amorphous nature of deposited films. Thickness-dependent electrical and optical properties including dc-activation energy, sheet resistivity, optical band gap, band tailing parameter, etc. of Ge8Sb2Te11thin films have been studied. The optical parameters have been calculated from transmission, reflection and absorbance data in the spectral range of 200–1100 nm. It has been found that optical band gap and band tailing parameter decreases with the increase in Ge8Sb2Te11thin films thickness. The dc-activation energy and sheet resistivity decreases while the crystallization temperature of the amorphous Ge8Sb2Te11 films increases with the increase in thickness of the films. The decrease of the sheet resistivity has been substantiated quantitatively using the classical size-effect theory. These results have been explained on the basis of rearrangements of defects and disorders in the amorphous chalcogenide system.

Theoretical investigation on wave propagation in embedded DWBNNT conveying ferrofluid via stress and strain–inertia gradient elasticity

In the present study, wave propagation characteristics of double-walled boron nitride nanotubes (DWBNNTs) conveying ferrofluid is investigated. Magnetite (Fe3O4) nanofluid is selected as a conveying fluid which reacted in presence of magnetic field. Shear effects of surrounded medium are taken into account using Pasternak model. Stress and strain–inertia gradient elasticity theories are used due to their capability to interpret size effect. Based on Hamilton’s principle and employing Euler–Bernoulli, Timoshenko and Reddy beam models, wave equations of motion in double-walled boron nitride nanotubes are derived and solved by harmonic solution. Regarding the various types of flow regimes in fluid–structure interaction, the upstream and downstream phase velocities of double-walled boron nitride nanotubes conveying ferrofluid are calculated. A detailed parametric study is conducted to clarify the influences of the beam models, size effect theories, magnetic field, surrounding elastic medium and fluid velocity on the wave propagation of double-walled boron nitride nanotubes conveying ferrofluid. The results indicated that in lower wave numbers, the effect of flowing fluid and the difference between the upstream and downstream phase velocities were considerable. The results of this work can be used in design and manufacturing of nanopipes and nanovalves conveying fluid flow to avoid water hammer phenomenon.

On the representative volume element of asphalt concrete at low temperature

The feasibility of characterizing asphalt mixtures’ rheological and failure properties at low temperatures by means of the Bending Beam Rheometer (BBR) is investigated in this paper. The main issue is the use of thin beams of asphalt mixture in experimental procedures that may not capture the true behavior of the material used to construct an asphalt pavement. For the rheological characterization, three-point bending creep tests are performed on beams of different sizes. The beams are also analyzed using digital image analysis to obtain volumetric fraction, average size distribution, and spatial correlation functions. Based on the experimental results and analyses, it is concluded that representative creep stiffness values of asphalt mixtures can be obtained from testing at least three replicates of the thin (BBR) mixture beams. Failure properties are investigated by performing strength tests using a modified Bending Beam Rheometer (BBR), capable of applying loads at different loading rates. Histogram testing of BBR mixture beams and of larger beams is performed and the failure distribution is analyzed based on the size effect theory for quasibrittle materials. Different Weibull moduli are obtained from the two specimens sizes, which indicates that BBR beams do not capture the representative volume element (RVE) of the material.

Modeling Mechanical Behavior of Very Coarse Granular Materials

A novel approach has been developed to predict the mechanical behavior of very coarse granular materials with a constitutive model, which considers both grain breakage and size effect. The behavior of granular assemblies is significantly influenced by particle breakage. A critical-state double-yield-surface model incorporating the change in the critical-state line and in elastic stiffness caused by grain breakage during loading has been adopted. The amount of grain breakage was estimated by extending the size effect theory on individual grains to granular assemblies. The results from earlier studies on granular materials with parallel gradations have been usefully exploited to calibrate and to validate the model. Comparisons between experiments and simulations suggest that this approach can predict the mechanical behavior of very coarse granular materials from test results performed on a finer fraction with a homothetic gradation.

Size Effect in Crack Spacing of Quasi-brittle Materials

Depending on the scale of observation, many engineered and natural materials show different mechanical behaviour. Thus, size effect theories, based on a multiscale approach, analyse the intrinsic (due to microstructural constraints, e.g., grain size) and extrinsic effects (caused by dimensional constraints), in order to improve the knowledge in materials science and applied mechanics. Nevertheless, several problems regarding Solid Mechanics and Materials Science cannot be solved by conventional approaches, because of the complexity and uncertainty of materials proprieties, especially at different scales. It is sufficient to think, for instance, to the concept of void nucleation, or to localisation of deformations at microscopic level produced by the non-linear phenomena occurring at mesoscopic level.For this reason, a simple model, capable of predicting crack patterns at different scale, has been developed and presented in this paper. This model is based on the Golden Ratio, which was firstly defined by Euclide as: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less”. Intimately interconnected with the Fibonacci sequence (1, 2, 3, 5, 8, 13,.), this number controls growth in Nature and recurs in many disciplines, such as art, architecture, design, medicine, etc. As for the classical tension-stiffening approach, the Golden Ratio permits to define the relationship between the average crack spacing and the thickness of quasi-brittle materials. In these cases, the theoretical results provided by the Golden Ratio, used to calibrate a size-effect law of crack spacing, are in accordance with the experimental measurements taken in several test campaigns carried on different materials (i.e., rocks, ice, and concrete).Thus, we argue that the centrality of the Golden Ratio in the crack pattern of quasi-brittle materials has profound physical meanings and reveals the existence of the size-effect law of crack spacing. As a consequence, by means of the proposed model, the crack pattern of large structures can be simply and rapidly predicted, without knowing the material performances but by testing prototypes of the lower dimensions.

Molecular dynamics simulation of the thermophysical properties and phase change behaviors of aluminum nanoparticles

With the development of energy storage technology, phase change materials which can be used to store thermal energy have received much attention in recent years. The nano-metallic materials are universally used as phase change materials due to their many desirable thermophysical properites. In this paper, the molecular dynamics simulation method is adopted to simulate the variations of melting point, density and phonon thermal conductivity of the nano aluminum with grain size ranging from 0.8 nm to 3.2 nm. The variations of density, specific heat capacity and phonon thermal conductivity with temperature of aluminum nanoparticles at a grain size of 1.6 nm are also studied. By using the embedded-atom potential, the thermophysical properties and phase change behaviors of aluminum nanoparticles are stimulated. The phase transition temperature of aluminum nanoparticles is studied based on the energy-temperature curve and the specific heat capacity-temperature curve. The surface energy theory and the size effect theory are applied to the analysis of the variation of the melting point of the aluminum nanoparticles, and the results show that the melting point increases as grain size augments, and it increases slowly when its grain size is between 2.2 nm and 3.2 nm but still holds the trend of increase. In order to obtain accurate thermal conductivity, the Green-Kubo method is adopted to calculate the phonon thermal conductivity of aluminum nanoparticle. As the grain size of aluminum nanoparticles increases, its density monotonically decreases, and the thermal conductivity monotonically increases linearly, which is in line with the theory of phonon. Similarly, with the increase of temperature, the density and thermal conductivity of aluminum nanoparticles of 1.6 nm in grain size both decrease. Moreover, the density of aluminum nanoparticle is generally lower than that of its bulk material. The study also shows that the heat transfer manner of aluminum nanoparticle is based on ballistic-diffusive heat conduction instead of the traditional diffusive heat conduction when it is in a nanoscale. The simulation studies the thermophysical properties of nanoparticles from the atomic perspective, and is of significance for guiding the design of the phase change materials based on the aluminum nanoparticles for thermal energy storage.

Investigation of the Effect of Cutting Conditions and Tool Edge Radius on Micromachining with the Use of the Finite Elements Method

Micromachining comprises manufacturing processes that are in the forefront of contemporary industry. The need for high efficiency, high precision, better quality and lower cost makes the study of these processes and the phenomena of the micro regime that accompany, e.g. the size effect, of great importance. The quite popular for modeling manufacturing processes Finite Element Method is applicable in micromachining, too. However, assumptions and simplifications need to be made in order to provide a realistic simulation. In the present paper a numerical simulation using the Third Wave AdvantEdge® software is presented. A FEM model of micromachining of AISI 1045 is used for a parametric analysis of the simulation of micro-cutting. The effect of cutting conditions and tool geometry are investigated and size-effect theories are tested with the aid of the numerical model. From the analysis several useful conclusions are drawn.