Sankara J. Subramanian

Modeling the interaction between densification mechanisms in powder compaction

In this paper a micromechanical model of the interaction between densification mechanisms in powder compaction is presented. It accounts for elastic and power-law creep deformation of the bulk material along with stress-driven diffusion along the interparticle contact areas and curvature-driven diffusion on the pore surfaces. The finite element method is used to obtain the time-dependent deformation of the powder aggregate under plane strain deformation conditions. To reduce the number of case calculations needed to analyze the process, important dimensionless parameters that measure the relative magnitude of the densification mechanisms are identified. The calculated densification rates of the compact are compared with those predicted by analytical models, and conclusions are drawn on the significance of including the interaction between the densifying mechanisms in powder compaction models.

The Eigenfunction Virtual Fields Method

The Virtual Fields Method (VFM, Pierron and Grediac, 2012) is an inverse technique for computing mechanical properties of materials from full-field deformations obtained from techniques such as Digital Image Correlation (DIC). VFM is based on the principle of virtual work, which is a weak statement of the equations of motion. Central to VFM is the appropriate choice of virtual fields, which in prior work, have been assumed to be polynomials that are continuously differentiable, either piece-wise or over the entire domain of interest. In this work, we propose a new method for systematically identifying virtual fields by performing a principal component analysis (PCA) of the strain field measured from experiments. The virtual strain components to be used in VFM are then chosen to be the eigenfunctions so determined. In addition to being a physically meaningful set of virtual fields, such a choice exploits the orthogonality of the computed eigenfunctions while simultaneously eliminating computation of a large number of coefficients that define the virtual fields in prior approaches. In the case of linear elastic behaviour, we show that this new approach, named the Eigenfunction Virtual Fields Method (EVFM), leads to a compact system of equations that can be solved for the unknown material parameters.

Mechanical Modeling of a Solder Thermal Interface Material: Implications for Thermo-Mechanical Reliability

This paper addresses cracking in solder thermal interface materials (STIMs) used in electronic packages under accelerated testing or service conditions. Finite-element models of various packages have been built to study the deformation in the STIM through a few cycles of accelerated testing. Two commonly observed failure modes — center/off-center brittle interfacial cracking, and cohesive corner cracking — were looked at. The success of the modeling approach was evaluated by comparison with thermal impedance data, as well as with CSAM images showing the extent of cracking in the STIM. It is shown that the models agree qualitatively with experimental data, both in terms of failure locations, as well as in terms of rank ordering different packages in terms of STIM degradation.Copyright

Mechanical Design and Analysis of Land Grid Array (LGA) Sockets

Sockets offer a cost effective and high-volume manufacturing friendly interface between CPU packages and motherboards. Land-grid-array (LGA) technology offers avenues to enhance the electrical performance of sockets over its predecessors e.g. pin grid array (PGA) technology. The present paper will describe various technical challenges encountered in the design of Intel’s LGA sockets. The recently launched LGA775 socket will be used as a case study. Methods adopted to overcome these design challenges and successfully implement LGA sockets will be discussed. Design features like direct socket loading (DSL), various issues related to the design of socket housings, LGA contact design optimization and socket reliability enhancement under stresses such as thermal cycling, mechanical shock, vibration and bake will be discussed. DSL is an integrated mechanism that enables application of a compressive mechanical load between the LGA contact pins on the socket and the package LGA pads, so that their interfaces achieve and maintain electrical continuity through the socket design life. Similarly, optimizing the design of the socket contacts significantly impacts the stressing, reliability of the second level interconnect (SLI) ball grid array (BGA) that connects the LGA775 to the motherboard. The successful implementation of these designs is achieved through a combination of geometric tolerance stack analysis, numerical modeling studies, detailed experiments, reliability testing and correlation between these. The details of the same are be discussed in this article.Copyright

Calculation of a constitutive potential for isostatic powder compaction

A macroscopic constitutive potential has been developed for the deformation of a powder compact of cylindrical particles during pressure sintering. The derivation is based on finite-element simulations of the densification process that proceeds under the synergistic action of power-law creep deformation in the particles, evolution of the nonlinearly developing contact area between the particles, and interparticle and pore free-surface diffusional mass transport. Solution to this initial/boundary-value problem as deformation proceeds with time provides all necessary information for the calculation of the constitutive potential. The associated constitutive law predicts the densification rate of the powder compact at a given temperature and pressure in terms of material parameters, such as creep constants and diffusion coefficients, and reflects the role played in the densification process by various micromechanical features at the microscale such as the pore surface curvature. The model predictions are compared with the existing analytical models for plane strain densification and experimental data from sintering of copper wires by grain boundary and curvature-driven pore surface diffusion.

Estimation of surface curvature from full-field shape data using principal component analysis

Three-dimensional digital image correlation (3D-DIC) is a popular image-based experimental technique for estimating surface shape, displacements and strains of deforming objects. In this technique, a calibrated stereo rig is used to obtain and stereo-match pairs of images of the object of interest from which the shapes of the imaged surface are then computed using the calibration parameters of the rig. Displacements are obtained by performing an additional temporal correlation of the shapes obtained at various stages of deformation and strains by smoothing and numerically differentiating the displacement data. Since strains are of primary importance in solid mechanics, significant efforts have been put into computation of strains from the measured displacement fields; however, much less attention has been paid to date to computation of curvature from the measured 3D surfaces. In this work, we address this gap by proposing a new method of computing curvature from full-field shape measurements using principal component analysis (PCA) along the lines of a similar work recently proposed to measure strains (Grama and Subramanian 2014 Exp. Mech. 54 913–33). PCA is a multivariate analysis tool that is widely used to reveal relationships between a large number of variables, reduce dimensionality and achieve significant denoising. This technique is applied here to identify dominant principal components in the shape fields measured by 3D-DIC and these principal components are then differentiated systematically to obtain the first and second fundamental forms used in the curvature calculation. The proposed method is first verified using synthetically generated noisy surfaces and then validated experimentally on some real world objects with known ground-truth curvatures.

Fast, Sub-pixel Accurate Digital Image Correlation Algorithm Powered by Heterogeneous (CPU-GPU) Framework

Digital Image Correlation (DIC) is a popular non-contact image-based full-field deformation measurement tool widely used in mechanics. In spite of its significant advantages, it is still primarily used as a post-processing tool due to its computational cost. In recent years, parallel computing platforms such as multi-core processors and Graphics Processing Units (GPUs) have been used to improve the speed of the DIC algorithm, with GPUs being well-suited for implementing data-parallel operations. Previous works have performed GPU-based DIC wherein each sub-image (i.e. a collection of a few pixels in the local neighborhood of a point of interest) is allocated to a single thread on the GPU, thus achieving parallelism across sub-images. However, this is not the only type of parallelism that is possible: one can also achieve parallelism within a sub-image as well as across whole images. The aim of this work is to efficiently implement 2D-DIC such that parallelism within a sub-image as well as across sub-images leads to considerable reduction in computation time. We use a heterogeneous framework consisting of an Intel Xeon octa-core CPU and an Nvidia Tesla K20C GPU card in this work. The CPU is used to handle image pre-processing, whereas the GPU is used to process four compute-intensive tasks: affine shape function computation, B-Spline interpolation, residual vector calculation and deformation vector update. Parallelization within and across sub-images is achieved in this work by efficient thread handling and use of pre-compiled BLAS libraries. In order to estimate the speedup provided by the GPU, the same four tasks were also evaluated on the octa-core CPU; a speedup of approximately 7 to 5 times was observed for a single sub-image whose size varies from 21×21 to 61×61 respectively. However, it is expected that for a larger number of sub-images, the GPU speedup will be higher and this is indeed the case: when the affine shape function computation and B-Spline interpolation steps were evaluated on 1869 21×21 pixel sub-images, the speedup was around a more impressive 453 times. Further GPU optimization as well as parallelization across image pairs is currently underway and even faster GPU-assisted DIC seems achievable.

Spatio-Temporal Principal Component Analysis of Full-Field Deformation Data

Full-field displacements are the output of several non-contact experimental mechanics techniques such as the Grid method or Digital Image Correlation (DIC). Although it appears that an enormous amount of data is available from such measurements, such data are often highly redundant. In the past, orthogonal shape descriptors such as Zernike moments, Fourier-Zernike moments (Patki and Patterson, Exp Mech 1:1–13, 2011) and Tchebicheff moments (Sebastian et al., Appl Mech Mater 70:63–68, 2011) have been proposed to reduce dimensionality. We recently proposed the use of Principal Components Analysis (PCA) to reduce the dimensionality of full-field displacement data, identify primary spatial variations and compute strains without any a priori assumptions on the form the shape descriptors. In this work, we extend this idea to time-dependent problems and investigate spatio-temporal PCA to identify evolution of the primary displacement patterns with time in a deforming solid. The proposed method is applied to synthetic data obtained from a finite-element analysis of a thin visco-plastic solder specimen subjected to cyclic shear loading. Progressive damage is introduced into the specimen through the reduction of element stiffness at a specific location after pre-determined number of cycles. Displacement fields collected at periodic intervals are analysed using spatio-temporal PCA and the possibility of inferring local damage from the time-evolution of the eigenfunctions and their singular values is demonstrated.

Strain Evolution during Lap Shear Testing of SnCu Solder

The lap-shear test is frequently used in the microelectronics industry to obtain mechanical properties of solder joints. In these tests, solder joints formed between slender metallic substrates are pulled apart in a simple shear configuration. Although it is known that calculation of stress-strain curves from lap shear tests is not straightforward due to rotation of the joints and strain inhomogeneity within the joint, these tests still find widespread use due to their simplicity and apparent ease of use. Chawla and co-workers [1, 2] show that the state of strain near the solder-substrate interfaces is significantly different from that in the interior of the joint and that this effect is only minimized for large joints. In the present work, we offer experimental evidence for these conclusions by presenting full-field strain measurements on solder joints in double-lap shear configuration, obtained using Digital Image Correlation (DIC). While confirming that significant strain gradients exist within the joint, the present work also indicates that a simple calculation of shear strain as axial displacement of the joint divided by joint thickness is misleading due to the presence of a significant gradient of the transverse displacement along the loading direction. This gradient persists through the course of the deformation and results in the actual average shear strain in the joint being smaller than that computed from the axial displacement alone.