To more accurately model microcantilever resonant behavior in liquids and to improve lateral-mode sensor performance, a new model is developed to incorporate viscous fluid effects and "Timoshenko beam" effects (shear deformation, rotatory inertia). The model is motivated by studies showing that the most promising geometries for lateral-mode sensing are those for which Timoshenko effects are most pronounced.Analytical solutions for beam response due to harmonic tip force and electrothermal loadings are expressed in terms of total and bending displacements, which correspond to laser and piezoresistive readouts, respectively. The influence of shear deformation, rotatory inertia, fluid properties, and actuation/detection schemes on resonant frequencies (fres) and quality factors (Q) are examined, showing that Timoshenko beam effects may reduce fres and Q by up to 40% and 23%, respectively, but are negligible for width-tolength ratios of 1/10 and lower. Comparisons with measurements (in water) indicate that the model predicts the qualitative data trends but underestimates the softening that occurs in stiffer specimens, indicating that support deformation becomes a factor. For thinner specimens the model estimates Q quite well, but exceeds the observed values for thicker specimens, showing that the Stokes resistance model employed should be extended to include pressure effects for these geometries.

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.

An approximate solution to the equations of elasticity is obtained for the problems of steady-state longitudinal, flexural, and torsional wave propagation in isotropic bars of infinite length and rectangular cross section. Results were obtained for sections with various ratios of width to depth. These results indicate that, in a phase-velocity versus wavenumber plot, the higher branches exhibit a certain minimum feature; i.e., these branches approach their limiting value at bq, = oo from below. This feature has not been reported in the case of bars with a circular cross section. LISTOF SYMBOLS c cs CR Co E s u v w semiwidth of rectangular section eigenvector semidepth of rectangular section phase velocity velocity of distortional wave (,t/o) velocity of Rayleigh surface wave bar velocity (E/o) strain component «(Uk.sq-Us.k) Youngs modulus cosine of angle between xk axis and normal to the boundary ratio of lateral dimensions (a/b) kinetic energy displacement component in xk direction strain energy potential energy (U-W) work done by body and surface forces Cartesian coordinate (x3 is taken in the direction of the bar's axis) element of area on bar's surface element of arc along boundary of cross section wavenumber (2•'/F) b delta operator •ks kronecker delta zX dilatation (us,s), j= 1, ..., 3 0 eigenvalue (c/cs) 2 X, u Lam(•'s constants v Poisson's ratio (taken to be 0.30 in this investigation) • transformed Cartesian coordinate ('•x•) d• element of arc along boundary of transformed cross section o mass density r•s stress component V 2 Laplacian operator I' wavelength Notation (1) Repeated index denotes summation over that index (2) Index occurring only once in a given term is a free index and can stand for any of the numbers 1, 2, 3 (3) Commas denote differentiation with respect to the independent variables; e.g., u•.•s = O•uk/OxiOxi

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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