Civil and Environmental Engineering Faculty Research and PublicationsCopyright (c) 2015 Marquette University All rights reserved.
http://epublications.marquette.edu/civengin_fac
Recent documents in Civil and Environmental Engineering Faculty Research and Publicationsen-usThu, 05 Nov 2015 13:03:14 PST3600Prediction of Solder Joint Geometry
http://epublications.marquette.edu/civengin_fac/105
http://epublications.marquette.edu/civengin_fac/105Tue, 03 Nov 2015 09:59:46 PSTStephen M. HeinrichCantilever-based Resonant Microsensors with Integrated Temperature Modulation for Transient Chemical Analysis
http://epublications.marquette.edu/civengin_fac/104
http://epublications.marquette.edu/civengin_fac/104Wed, 21 Oct 2015 11:30:56 PDT
This work introduces a resonant cantilever platform with integrated temperature modulation for real-time chemical sensing. Embedded heaters allow for rapid thermal cycling of individual sensors, thereby enabling real-time transient signal analysis without the need for a microfluidic setup to switch between analyte and reference gases. Compared to traditional mass-sensitive microsensors operating in steady state, the on-chip generation of signal transients provides additional information for analyte discrimination.
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C. Carron et al.Fundamental Theory of Resonant MEMS Devices
http://epublications.marquette.edu/civengin_fac/103
http://epublications.marquette.edu/civengin_fac/103Wed, 21 Oct 2015 09:21:26 PDTStephen M. Heinrich et al.[Review of] Variational Methods in Mechanics
http://epublications.marquette.edu/civengin_fac/102
http://epublications.marquette.edu/civengin_fac/102Tue, 20 Oct 2015 10:24:50 PDTStephen M. HeinrichForeword to Special Issue on Solder Geometry
http://epublications.marquette.edu/civengin_fac/101
http://epublications.marquette.edu/civengin_fac/101Tue, 20 Oct 2015 10:20:07 PDTStephen M. HeinrichStress Interference in a Transversely Isotropic Body under Axisymmetric Loading
http://epublications.marquette.edu/civengin_fac/100
http://epublications.marquette.edu/civengin_fac/100Mon, 19 Oct 2015 10:05:20 PDT
An unbounded homogeneous transversely isotropic body of revolution containing two twin spheroidal cavities is subjected to an axisymmetric loading. The axis of loading symmetry coincides with the axis of elastic symmetry of the material and the axis of revolution of the body. The elasticity solution is obtained in series form utilizing the displacement potential representation for the equilibrium of transversely isotropic solids, and numerical results are presented for the cases of two spherical cavities perturbing uniaxial and hydrostatic tension fields in certain hexagonal crystals and in isotropic materials. Of primary interest is the influence of cavity spacing and material type on the degree of stress interference between the two perturbations.
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Stephen M. Heinrich et al.Analysis of Constrained Filament Deformation and Stiffness Properties of Brushes
http://epublications.marquette.edu/civengin_fac/99
http://epublications.marquette.edu/civengin_fac/99Mon, 19 Oct 2015 09:58:42 PDT
In this paper, an analytical procedure is developed for evaluating the filament loading, constrained elastic deformation, and overall stiffness of a circular brush, Filament deformation is computed on the basis of a large displacement mechanics analysis in conjunction with kinematic constraints for a flat, rigid workpart with smooth surface finish. Numerical results are reported which examine the relationship between workpart penetration, brush stiffness, and force distribution characteristics of the workpart contact zone.
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R. J. Stango et al.Solder Joint Formation in Surface Mount Technology—Part I: Analysis
http://epublications.marquette.edu/civengin_fac/98
http://epublications.marquette.edu/civengin_fac/98Mon, 19 Oct 2015 09:46:23 PDT
An analytical model of solder joint formation during a surface mount reflow process is developed in the present paper, and the solution is obtained in an explicit integral form. For two limiting cases—infinitesimal and infinite solder areas—the solution is expressed in closed form. Numerical results illustrate the influence of the process parameters (surface tension, density, and cross-sectional area of the molten solder, and the contact angles between the solder andpretinning) on joint shape and overall fillet dimensions. Comparisons between theoretical predictions and laboratory data show excellent agreement.
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Stephen M. Heinrich et al.Solder Joint Formation in Surface Mount Technology—Part II: Design
http://epublications.marquette.edu/civengin_fac/97
http://epublications.marquette.edu/civengin_fac/97Mon, 19 Oct 2015 09:31:59 PDT
In this paper dimensionless design curves relating fillet height and length to joint cross-sectional area are presented for surface-mount solder joints. Based on an analytical surface tension model, the advantage of these dimensionless curves is that they may be used for arbitrary values of solder density and surface tension. The range of applicability of previously developed approximate formulae for predicting joint dimensions is also investigated. A simple example problem is included to illustrate the use of both the design curves and the approximate formulae.
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Stephen M. Heinrich et al.Effect of Workpart Curvature on the Stiffness Properties of Circular Filamentary Brushes
http://epublications.marquette.edu/civengin_fac/96
http://epublications.marquette.edu/civengin_fac/96Mon, 19 Oct 2015 09:19:21 PDT
In this paper the role which workpart geometry plays in the stiffness properties and contact zone characteristics of circular filamentary brushes is investigated. The problem is examined by employing a recently developed mechanics-based procedure for evaluating large filament deformation, in conjunction with appropriate kinematic constraints for filament tip contact with a smooth, rigid workpart of prescribed constant curvature. Numerical examples for a circular filamentary brush system are presented and discussed for the purpose of illustrating the basic nature of brush stiffness response characteristics for a range of prescribed constant-curvature workpart surfaces
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Stephen M. Heinrich et al.Torsional Stress Interference in Transverse Isotropy
http://epublications.marquette.edu/civengin_fac/95
http://epublications.marquette.edu/civengin_fac/95Mon, 19 Oct 2015 09:11:21 PDT
An unbounded homogeneous transversely isotropic body of revolution containing two twin spheroidal cavities is subjected to a torsional loading at infinity. The axis of twist coincides with the axis of elastic symmetry of the material and the axis of revolution of the body. The linear elasticity solution is obtained in series form by employing the Lekhnitskii stress function approach, and numerical results are presented for the case of two spherical cavities perturbing the Saint‐Venant torsion field for a circular cylinder. Of particular interest is the influence of cavity spacing and material constants on the degree of stress interference between the two perturbations.
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Stephen M. HeinrichFinite Element Method for Predicting Equilibrium Shapes of Solder Joints
http://epublications.marquette.edu/civengin_fac/94
http://epublications.marquette.edu/civengin_fac/94Mon, 19 Oct 2015 09:01:38 PDT
This paper discusses the development and application of a finite element method for determining the equilibrium shapes of solder joints which are formed during a surface mount reflow process. The potential energy governing the joint formation problem is developed in the form of integrals over the joint surface, which is discretized with the use of finite elements. The spatial variables which define the shape of the surface are expressed in a parametric form involving products of interpolation (blending) functions and element nodal coordinates. The nodal coordinates are determined by employing the minimum potential energy theorem. The method described in this paper is very general and can be employed for those problems involving the formation of three dimensional joints with complex shapes. It is well suited for problems in which the boundary region is not known a priori (e.g., “infinite tinning” problems). Moreover, it enables the user to determine the shape of the joint in parametric form which facilitates meshing for subsequent finite element stress and thermal analyses.
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Nicholas J. Nigro et al.Prediction of Solder Joint Geometries in Array-Type Interconnects
http://epublications.marquette.edu/civengin_fac/93
http://epublications.marquette.edu/civengin_fac/93Mon, 19 Oct 2015 07:56:52 PDT
An approximate mathematical model is developed for predicting the shapes of solder joints in an array-type interconnect (e.g., a ball-grid array or flip-chip interconnect). The model is based on the assumption that the geometry of each joint may be represented by a surface of revolution whose generating meridian is a circular arc. This leads to simple, closed-form expressions relating stand-off height, solder volume, contact pad radii, molten joint reaction force (exerted on the component), meridian curvature, and solder surface tension. The qualitative joint shapes predicted by the model include concave (hourglass-shaped), convex (barrel-shaped, with a truncated sphere as a special case), and truncated-cone geometries. Theoretical results include formulas for determining the maximum and minimum solder volumes that can be supported by a particular pair of contact pads. The model is used to create dimensionless plots which summarize the general solution in the case of a uniform array (i.e., one comprising geometrically identical joints) for which the contact pads on the component and substrate are of the same size. These results relate the values of joint height and width (after reflow) to the solder joint volume and the molten-joint force for arbitrary values of the pad radius and solder surface tension. The graphs may be applied to both upright and inverted reflow, and can be used to control stand-off for higher reliability or to reduce bridging and necking problems causing low yields. A major advantage of the model is that it is numerically efficient (involving only simple, closed-form expressions), yet generates results that are in excellent agreement with experimental data and more complex models. Thus, the model is ideally suited to performing parametric studies, the results of which may be cast in a convenient form for use by practicing engineers. Although in the present paper the array is assumed to be doubly-symmetric, i.e., possess two orthogonal planes of symmetry, the model may be extended to analyze arrays of arbitrary layout. The motivation for predicting joint geometries in array-type interconnects is two-fold: (1) to achieve optimal joint geometries from the standpoint of improved yield and better reliability under thermal cycling and (2) to take full advantage of the flexibility of new methods of dispensing solder, such as solder-jet and solder-injection technologies, which enable the volume of each individual joint to be controlled in a precise manner. Use of dispensing methods of these types permits the solder volumes in the array to be distributed in a non-uniform manner. Results such as those presented here (in combination with appropriate fatigue studies) can be used to determine the optimal arrangement of solder volumes.
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Stephen M. Heinrich et al.Membrane Analogy for Saint-Venant Torsion: New Results
http://epublications.marquette.edu/civengin_fac/92
http://epublications.marquette.edu/civengin_fac/92Mon, 19 Oct 2015 07:51:41 PDT
The Saint-Venant torsion problem for a twisted shaft may be formulated in terms of the Prandtl stress function (PSF) or the conjugate warping function (CWF), each of which may be interpreted as the transverse deflection of a thin membrane via the well-known membrane (soap-film) analogies. The deflection of the PSF membrane is associated with a uniform transverse pressure loading, whereas the CWF membrane is nonpressurized—its deformation is due solely to imposed deflections along its edge. The utility of the PSF analogy stems from the simple relationships that exist between the geometry of the deformed membrane (slope, enclosed volume) and the shaft quantities (shear stress, rigidity). However, despite the fact that the CWF analogy involves a simpler loading than its PSF counterpart, similar relationships relating the deformed CWF membrane to the shaft quantities do not appear in the literature. The purpose of the present note is to complete the CWF analogy by deriving such expressions. The use of the new relationships is illustrated in a purely geometric derivation of the solution for a circular cylindrical shaft. The results may also furnish a new means of visualizing the torsional behavior of shafts of other cross-sectional shapes.
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Stephen M. HeinrichPrediction of Solder Geometry for an Axisymmetric Through-Hole Joint
http://epublications.marquette.edu/civengin_fac/91
http://epublications.marquette.edu/civengin_fac/91Mon, 19 Oct 2015 07:47:38 PDT
Solutions for axisymmetric profiles of solder joints formed between a cylindrical pin and a printed circuit board (PCB) are presented. The dimensionless differential equation governing the formation of the solder joints is developed and then solved numerically for the cases of single upright joints, single inverted joints, and through-hole joints. Results are presented in terms of the following dimensionless parameters: bond number, solder volume, board thickness, and tinning radius. The dimensionless approach makes the results from this study suitable for use in a broader range of applications.
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A. F. Elkouh et al.Improved Analytical Estimate of Global CTE Mismatch Displacement in Areal-Array Solder Joints
http://epublications.marquette.edu/civengin_fac/90
http://epublications.marquette.edu/civengin_fac/90Mon, 19 Oct 2015 07:41:31 PDT
An analytical expression is derived for determining the maximum solder joint shearing displacement occurring in an a real-array interconnect under global CTE mismatch loading. The result may be viewed as a “load correction factor” to be applied to the commonly used estimate which is based on the free thermal expansion of component and substrate. The new expression for the correction factor includes the following parameters: (a) dimensions and material properties of component and substrate; (b) array size and population; (c) material properties of solder; and (d) geometric parameters of the individual joints. The theoretical result is based on modeling the assembly as two circular elastic disks connected by a shear-type “elastic foundation” whose distributed shear stiffness is related to the joint/array characteristics. The analytical expression and the graphical aids presented herein may provide convenient alternatives to performing time-consuming and expensive finite element “macro-analyses” on the assembly for the purpose of specifying boundary conditions for a subsequent “micro-analysis” on a single joint.
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Stephen M. Heinrich et al.Shearing Deformation in Partial Areal Arrays: Analytical Results
http://epublications.marquette.edu/civengin_fac/89
http://epublications.marquette.edu/civengin_fac/89Mon, 19 Oct 2015 07:34:40 PDT
Simple, closed-form expressions, based on elasticity theory, are derived for determining the location and magnitude of the maximum shearing displacement in a partial areal array of solder joints. Both uniform and nonuniform thermal loadings are considered, as is the heterogeneity of the component, which often arises due to different values of coefficient of thermal expansion (CTE) and elastic properties among the module’s constituent materials. The model is based on the following assumptions: (a) the square geometry of the array and component may be replaced with an equivalent axisymmetric geometry; and (b) the stiffness of the solder joints is negligible with respect to that of the component and substrate. The “soft joint” assumption corresponds to low-modulus solders or to thermal excursions occurring at high temperatures or low frequencies, for which significant stress relaxation occurs in the solder. For arrays exhibiting higher stiffness characteristics, the model yields conservative estimates of shearing displacement. Results indicate that, unlike homogeneous-component models under uniform temperature changes, the critical joints are not necessarily at the outer corners of the array. Other candidate locations predicted by the model (and observed in experimental and numerical studies) include the inner corner joints and any joints positioned beneath the die corners. The analytical results, also presented graphically, are found to depend on only three dimensionless parameters: the ratio of inner to outer array dimension, the ratio of die size to outer array dimension, and a “mismatch parameter,” which depends on the material, geometry, and loading characteristics of the problem. The results can be used to quickly determine the location and magnitude of peak shearing displacement in the array, possibly minimizing or eliminating the need to perform expensive and time-consuming finite element macroanalyses on entire assemblies involving hundreds of joints. Thus, the analyst may proceed directly to a detailed finite element microanalysis of the critical joint for fatigue life estimation, using the calculated shearing displacement as a required boundary condition in the finite element model.
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Stephen M. Heinrich et al.Effect of Component Heterogeneity on Global CTE Mismatch Displacement in Areal-Array Solder Interconnects
http://epublications.marquette.edu/civengin_fac/88
http://epublications.marquette.edu/civengin_fac/88Mon, 19 Oct 2015 07:20:20 PDT
An analytical model is developed for the problem of a thermally loaded electronic assembly consisting of a nonhomogeneous component attached to a substrate by means of an areal array of solder joints. The component is heterogeneous in the sense that the interior, or “die”, portion and the exterior, or “carrier”, portion may have different elastic moduli, different coefficients of thermal expansion, and/or different temperature excursions. Analytical results are presented for determining (a) the location of the critical joint(s) in the array, defined as those experiencing the maximum shearing displacement due to global CTE mismatch, and (b) the magnitude of the maximum shearing displacement. The critical joint location and the (dimensionless) peak displacement are shown to depend on only two parameters: one involving the material, geometric, and loading characteristics of the component and substrate, the other being the ratio of the die dimension to the array dimension. The stiffness of the solder array is neglected in the model; thus, the results should be valid for (a) thermal/power cycling of low-modulus solders (relative to component and substrate materials), and (b) high-temperature, low-frequency thermal/power cycling, for which a large degree of stress relaxation occurs in the solder. The results may also provide conservative estimates for situations involving stiffer arrays.
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Stephen M. Heinrich et al.Analysis of Contact Mechanics for a Circular Filamentary Brush/Workpart System
http://epublications.marquette.edu/civengin_fac/87
http://epublications.marquette.edu/civengin_fac/87Mon, 19 Oct 2015 07:13:02 PDT
This paper addresses the contact problem associated with the filament/workpart interaction that arises during brushing processes. A discretized model of a filament within the brushing tool is developed by employing Lagrange’s equations in conjunction with special constraint equations that are appropriate for the impact and impending large displacement of a flexible fiber whose tip traverses a flat, rigid surface. This formulation leads to the identification of five nondimensional parameters which fully characterize the filament/workpart contact problem. A damping mechanism is also included which can be used for modeling complex filament interactions that arise during the actual brushing operation. Special consideration is given to examining the initial filament/workpart impact and the subsequent forces that are generated along the contact region. Initial velocity of the filament is determined by employing an inelastic impact mechanics analysis. Time-varying transient response of the filament is then obtained by employing a predictor-corrector technique in conjunction with a finite difference method. Overall brush force is computed by a superposition of filament contact forces exerted onto the workpart surface. Numerical results are reported and compared with experimentally obtained data for an actual brush/workpart system.
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Chih-Yuan Shia et al.Parametric Finite Element Method for Predicting Shapes of Three-Dimensional Solder Joints
http://epublications.marquette.edu/civengin_fac/86
http://epublications.marquette.edu/civengin_fac/86Fri, 16 Oct 2015 14:01:18 PDT
This paper discusses the application of the parametric finite element method for predicting shapes of three-dimensional solder joints. With this method, the surface of the joint is meshed (discretized) with finite elements. The spatial variables (x, y, z) are expanded over each element in terms of products of interpolation (blending) functions expressed in parametric form and element nodal coordinates. The element nodal coordinates which are not constrained by the boundary conditions are determined by minimizing the potential energy function which governs the joint formation problem. This method has been employed successfully in the past to predict the shapes of two dimensional fillet and axisymmetric joints. In this paper, the method is extended to three dimensional problems involving sessile drops formed on a rectangular pad and solder columns formed between two horizontal planes and subject to a vertical force.
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Nicholas J. Nigro et al.