Maximum Entropy Methods in the Mechanics of Quasi-Static Deformation of Granular Materials

In statistical physics of dilute gases maximum entropy methods are widely used for theoretical predictions of macroscopic quantities in terms of microscopic quantities. In this study an analogous approach to the mechanics of quasi-static deformation of granular materials is proposed. The reasoning is presented that leads to the definition of an entropy that is appropriate to quasi-static deformation of granular materials. This entropy is formulated in terms of contact quantities, since contacts constitute the relevant microscopic level for granular materials that consist of semirigid particles. The proposed maximum entropy approach is then applied to two cases. The first case deals with the probability density functions of contact forces in a two-dimensional assembly with frictional contacts under prescribed hydrostatic stress. The second case deals with the elastic behaviour of two-dimensional assemblies of non-rotating particles with bonded contacts. For both cases the probability density functions of contact forces are determined from the proposed maximum entropy method, under the constraints appropriate to the case. These constraints form the macroscopic information available about the system. With the probability density functions for contact forces thus determined, theoretical predictions of macroscopic quantities can be made. These theoretical predictions are then compared with results obtained from two-dimensional Discrete Element simulations and from experiments.

Analysis of the Elasto-Plastic Response of a Polygonal Packing

We investigate the constitutive response of two-dimensional packed samples of polygons using molecular dynamics simulation. The incremental elasto-plastic response is examined in the pre-failure regime. Besides the Young modulus and the Poisson ratio, an additional parameter must be included, which takes into account the anisotropy of the elastic response. The plastic deformations are described by the introduction of the yield and the flow directions. These directions do not agree, which reproduces the non-associated feature of realistic soils. In order to detect the yield surface, different loading-unloading-reloading tests were performed. During the reload path, it is found that the yielding develops continuously with the amplitude of loading, which does not allow to identify a purely elastic regime.

A Comparative Study of the Response of Double Shearing and Hypoplastic Models

The principal objective of this paper is to compare the mechanical response of a double shearing model with that of a hypoplastic model under biaxial compression and under cyclic shear loading. As the origins and nature of these two models are completely different, it is interesting to compare the predictions of these two models. The constitutive relations of the double shearing and the hypoplastic models are implemented in the finite element program ABACUS/Explicit. It is found that the hypoplastic and the double shearing constitutive models both show strong capability in capturing the essential behavior of granular materials. In particular, under the condition of non-cyclic loading, the stress ratio and void ratio predictions of the double shearing and the hypoplastic models are relatively close, while under the condition of cyclic loading, the predictions of these models are quite different. It is important to note that in the double shearing model employed in this comparison the shear rates on the two slip systems are assumed to be equal. Hence, the conclusions derived in this comparison pertain only to this particular double shearing model. Similarly, the hypoplasticity model considered here is that proposed by Wu, et al. [30] and the conclusions reached here pertain only to this particular hypoplasticity model.

Effect of Voids on Thermal Fatigue Reliability of Lead-Free Solder Joints

There have been serious debates about whether Pb should be removed from solder joints, in view of environmental problems. These debates have now developed to the extent that a remarkable movement to establish regulations for the removal of Pb has emerged, especially in European countries and Japan. Therefore, many studies are aggressively being undertaken to develop technologies for replacing Sn-Pb solder with a lead-free alternative. From the results obtained so far, it has been proven that the fatigue strength in lead-free solder joints is almost equivalent to the fatigue strength of Sn-Pb eutectic solder joints. However, the other problem is that voids are easily formed in lead-free solder joints during the reflow process, and the effects of void formation on the fatigue strength of solder joints has attracted attention. In this study, the relationship between formation of voids and fatigue fracture mode and fatigue strength of solder joints was examined using FEM (finite element method) analysis and mechanical shear fatigue test. From the results of FEM analysis, it has been found that the equivalent plastic strain and shear strain of solder joints with voids are not always larger than those of solder joints without voids and the magnitude of the strains relate to the position and size of voids in solder joints. However, the difference of the strains is not so great as to affect the fatigue strength of solder joints. It has also been proven from the mechanical shear fatigue test that the fatigue fracture mode of solder joints with voids is similar to that of solder joints without voids and fatigue strength in both cases is also almost equivalent.

Manufacturability of Large SMT Connectors

With the advent of higher signaling frequencies on today’s motherboard, evolution suggests that the electronic industry is near a phase transition of going from through hole mount to surface mount connectors. Current computer infrastructure supports assembly of motherboards using through hole mount connectors from a historical standpoint. This legacy has led placement equipment manufacturers to develop fabrication machines that do not consider long, thin devices such as a PCI or DIMM connector. The reassessment of the world’s method of motherboard manufacturing presents a challenge. Hence the question arises, what is the realistic lifetime of a through hole mount connector and what are the assembly and reliability issues associated with a proposed replacement such as a surface mount connector. This paper addresses the basic reliability concerns of a long, thin, surface mount PCI connector both with, and without retention features.

Processing Characteristics of Layered Silicate Nanocomposites With Application to Blown Film

The aim of this study is to investigate the film blowing processing of various polyamide 6-based layered silicate nanocomposites (LSN’s) and to correlate their processing behaviors to the underlying rheology and structure. In-situ polymerized nanocomposites were found to possess a wider processing window compared to the base PA6. Shear and dynamic rheological measurements were employed to correlate the rheological behavior of the nanocomposites to the bubble formation and stability in the film blowing process. The in-situ plymerized LSN melts having higher elastic modulus (G′) were found to perform better in the film blowing process. DSC measurements indicated that nanoclays induce the γ type of crystallinity, which may also play a role in film blowing behavior.

Product Strength and Orientation Manipulation via Vibration-Assisted Injection Molding

This investigation focuses on determining why polystyrene ASTM specimens exhibit an increase in tensile strength when processed by vibration assisted injection molding (VAIM) while polycarbonate parts do not. VAIM is one of several polymer processing methods that attempt to improve product properties via manipulation of the polymer melt. Observation of birefringence patterns in VAIM processed polystyrene samples show a significant impact on molecular orientation. The same studies were conducted on opaque polycarbonate and were unable to determine the degree of molecular orientation via birefringence measurement. It was theorized that VAIM did not produce significant orientation due to its higher thermal conductivity and stiffer backbone. It has been determined by this investigation that VAIM processing does impart significant molecular orientation in polycarbonate specimens but still does not increase its UTS. It is proposed that increased molecular orientation induced by VAIM processing inhibits crazes from growing into cracks. VAIM therefore favors polymers that fail by crazing (e.g., polystyrene) rather than those that fail by shear yielding (e.g., polycarbonate).

A Multi-Level Superelement Technique for Damage Analysis

A multi-level superelement technique is applied to model the effects of circular voids on the effective elastic properties of a material. A two-dimensional representative volume element with a circular void in its center is initially modeled by a superelement. Using this superelement, a thin planar material with circular voids is constructed. The finite element computation is then conducted to estimate the effective Young’s modulus, Poisson’s ratio and the shear modulus of the material using the ABAQUS code for different void sizes. The values of the isotropic damage variables, DE and DG, under various degree of damage are hence determined. These values are compared with those calculated by using a conventional micromechanics damage model. A new isotropic damage model is proposed based on the results of this analysis. To demonstrate the applicability of this damage model, an example case of a notched cylindrical bar under tensile loading is investigated.

An Interfacial Debonding Model for Particle-Reinforced Composites

To simulate the evolution process of interfacial debonding between particle and matrix, and to further estimate its effect on the overall elastic behavior of particle-reinforced composites, a two-level microstructural-effective damaged model is developed. The microstructural damage mechanism is governed by the interfacial debonding of reinforcement and matrix. The progressive damage process is represented by the debonding angles that are dependent on the external loads. For those debonded particles, the elastic equivalency is constructed in terms of the stiffness tensor. Namely, the isotropic yet debonded particles are replaced by the orthotropic perfect particles. The volume fraction evolution of debonded particles is characterized by the Weibull’s statistical approach. Mori-Tanaka’s method is utilized to determine the effective stiffness tensor of the resultant multi-phase composites. The proposed constitutive framework is developed under the general three-dimensional loading condition. Examples are conducted to demonstrate the capability of the proposed model.

Transport Properties in Sheared Granular Flows

This paper tends to review the experiments performed in a shear cell device at National Central University. The influences of wall conditions and particle solid fractions were investigated. The glass spheres with a mean diameter of 3 mm were used as granular materials. Three bi-directional stress gages were installed to measure the normal and shear stresses along the upper wall. The image processing technology and particle tracking method were employed to measure the average and fluctuation velocities in the streamwise and the transverse directions. By tracking the movements of particles continually, the variation in the mean-square diffusive displacements with time was plotted and the self-diffusion coefficient was determined. The self-diffusion coefficients in the streamwise direction were much higher than those in the transverse direction. Besides the fundamental observations in a shear device, this paper will briefly describe the technology of “moving granular bed filter” developed at National Central University. This technology is mainly developed for high temperature gas cleanup.