Thermo-mechanical deformations of microstructures in a surface laminar circuit (SLC) substrate are quantified by microscopic moire´ interferometry. Two specimens are analyzed; a bare SLC substrate and a flip chip package assembly. The specimens are subjected to a uniform thermal loading of ΔT=−70°C and the microscopic displacement fields are documented at the identical region of interest. The nano-scale displacement sensitivity and the microscopic spatial resolution obtained from the experiments provide a faithful account of the complex deformation of the surface laminar layer and the embedded microstructures. The displacement fields are analyzed to produce the deformed configuration of the surface laminar layer and the strain distributions in the microstructures. The high modulus of underfill produces a strong coupling between the chip and the surface laminar layer, which produces a DNP-dependent shear deformation of the layer. The effect of the underfill on the deformation of the microstructures is investigated and its implications on the package reliability are discussed. [S1043-7398(00)01304-9]

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A difficulty of the standard Galerkin finite element method has been the ability to accurately resolve oscillating wave solutions at higher frequencies. Many alternative methods have been developed including high-order methods, stabilized Galerkin methods, multi-scale variational methods, and other wave-based discretization methods. In this work, consistent residuals, both in the form of least-squares and gradient least-squares are linearly combined and added to the Galerkin variational Helmholtz equation to form a new generalized Galerkin least-squares method (GGLS). By allowing the stabilization parameters to vary spatially within each element, we are able to select optimal parameters which reduce dispersion error for all wave directions from second-order to fourth-order in nondimensional wavenumber; a substantial improvement over standard Galerkin elements. Furthermore, the stabilization parameters are frequency independent, and thus can be used for both time-harmonic solutions to the Helmholtz equation as well as direct time-integration of the wave equation, and eigenfrequency/eigenmodes analysis. Since the variational framework preserves consistency, high-order accuracy is maintained in the presence of source terms. In the case of homogeneous source terms, we show that our consistent variational framework is equivalent to integrating the underlying stiffness and mass matrices with optimally selected numerical quadrature rules. Optimal GGLS stabilization parameters and equivalent quadrature rules are determined for several element types including: bilinear quadrilateral, linear triangle, and linear tetrahedral elements. Numerical examples on unstructured meshes validate the expected high-order accuracy.

Wood fibers are industrially attractive low-cost natural materials that offer good mechanical properties. It is, however, extremely difficult to experimentally determine the elastic properties of single wood fibers due to the structural complexity and variability of basic properties. We propose a three-step finite element (FE)-modeling algorithm to predict the elastic constants of a single wood fiber. The model is based on calculating the elastic constants of the fiber in three consecutive length scales including nanostructure of cellulose microfibrils (25–30[Formula: see text]nm), ultrastructure in the fiber wall layers (2–3[Formula: see text][Formula: see text]m) and single wood fibers (30–40[Formula: see text][Formula: see text]m). The results for a given set of parameters are compared to previous studies with good agreement. The work exhibits its novelty through the model’s robustness and potential for industrial applications. It merely requires three essential inputs — chemical composition and bulk density of fiber and microfibril angle of [Formula: see text] wall layer, but is capable of predicting reasonably accurately the elastic constants of a wood fiber completely without any required model preprocessing or meshing like common commercial FE method software packages. Furthermore, the validated model is used to perform a parametric study. We have found that cellulose content has positive correlations with almost all the elastic parameters — relatively strong for [Formula: see text] and [Formula: see text], but weaker for [Formula: see text]. Lignin and hemicellulose have the greatest influence on [Formula: see text] and [Formula: see text]. The bulk density of fiber is shown to affect all elastic constants except the longitudinal elastic modulus.

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