<p dir="ltr">Stress concentration represents a considerable challenge in 2.5D/ 3D integration systems, primarily due to the inherent structural and material heterogeneities. Discontinuities at interfaces between different materials and at corners lead to the formation of localized stress regions. Furthermore, coefficients of thermal expansion mismatch among materials such as silicon, copper, and polymers exacerbate these stress concentrations, particularly at critical locations including edges, corners, and interfaces. The presence of these localized stresses from both thermal and mechanical loadings heightens the risk of mechanical failure such as crack initiation or crack propagation, thereby underscoring the necessity for meticulous design and analysis to ensure the reliability of these advanced systems.</p><p dir="ltr">Numerical modeling provides insights into the relationship between design, loading, and material behavior leading to failure. Isogeometric analysis (IGA) offers a significant advantage over traditional finite element methods (FEM) by integrating Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE) using NURBS-based approximations. Enriched Isogeometric Analysis (EIGA) enhances this framework by incorporating known behaviors at critical features like crack tips or interfaces to facilitate accurate modeling of flux/stress singularities.</p><p dir="ltr">Asymptotic analysis of flux singularity is systematically studied in order to capture the local thermal behavior around crack tips or junction of multi-material wedges. The general expressions of temperature and flux in polar coordinates are derived. Formulation of EIGA around crack tips or junction of multi-material wedges then presented. A bi-material wedge model is demonstrated to show that singular flux/stress can be obtained in EIGA with a very coarse discretization compared with FEM.</p><p dir="ltr">Configurational force, a key concept in fracture mechanics, describes the energy-driven force that dictates crack propagation and helps predict crack paths and material failure under varying loads and conditions. To develop configurational force for thermoelasticity, configurational optimization problem is introduced. Configurational optimization problem is proposed for determining the optimal location, orientation, and the scaling of a finite-sized heterogeneity inserted into a homogeneous domain. The derivation leads to some important results: a generalized Eshelby energy-momentum tensor, path-independent integral forms for sensitivity, and representation of <i>J-</i>, <i>L-</i> and <i>M-</i>integrals of fracture mechanics. Several illustrative examples of fracture resistant design are solved with EIGA. </p><p dir="ltr">The generalized configurational force for thermoelasticity is derived by solving the configurational optimization problem. Using the general form of Helmholtz free energy potential for thermoelasticity, the generalized configurational force and generalized Eshelby energy-momentum tensor for thermoelasticity are obtained practically without needing the assumption of thermal displacement made in prior literature. </p><p dir="ltr"><br></p><p dir="ltr">Finally, a multiscale modeling for 2.5D/3D integration is demonstrated with all the developed techniques: asymptotic analysis of flux/stress singularities, enriched isogeometric analysis as well as configurational force in thermoelasticity. The one-way coupled multiscale modeling is applied to solve the length scale spanning of package to line. By transferring the global nodal value such as displacement or temperature to the local model as boundary conditions, the one-way coupled is achieved. In local model, a fined-mesh model with EIGA provides more details, and post-processing of configurational force computation leads to a prediction of the direction of crack driving force.</p>

<div>As physical testing does not always yield insight into the mechanistic cause of failures, computational modeling is often used to develop an understanding of the goodness of a design and to shorten the product development time. One common, and widely used analysis technique is the Finite Element Method. A significant difficulty with the finite element method is the effort required to generate an analysis-suitable mesh due to the difference in the mathematical representation of geometry CAD and CAE systems. CAD systems commonly use Non-Uniform Rational B-Splines (NURBS) while the CAE tools rely on the finite element mesh. Efforts to unify CAD and CAE by carrying out analysis directly using NURBS models termed Isogeometric Analysis reduces the gap between CAD and CAE phases of product development. However, several challenges still remain in the field of isogeometric analysis. A critical challenge relates to the output of commercial CAD systems. B-rep CAD models generated by commercial CAD systems contain uncoupled NURBS patches and are therefore not suitable for analysis directly. Existing literature is largely missing methods to smoothly couple NURBS patches. This is the first topic of research in this thesis. Fracture-caused failures are a critical concern for the reliability of engineered structures in general and semiconductor chips in particular. The back-end of the line structures in modern semiconductor chips contain multi-material junctions that are sites of singular stress, and locations where cracks originate during fabrication or testing. Techniques to accurately model the singular stress fields at interfacial corners are relatively limited. This is the second topic addressed in this thesis. Thus, the overall objective of this dissertation is to develop an isogeometric framework for parametric domain decomposition and analysis of singular stresses using enriched isogeometric analysis.</div><div><br></div><div>Geometrically speaking, multi-material junctions, sub-domain interfaces and crack surfaces are lower-dimensional features relative to the two- or three-dimensional domain. The enriched isogeometric analysis described in this research builds enriching approximations directly on the lower-dimensional geometric features that then couple sub-domains or describe cracks. Since the interface or crack geometry is explicitly represented, it is easy to apply boundary conditions in a strong sense and to directly calculate geometric quantities such as normals or curvatures at any point on the geometry. These advantages contrast against those of implicit geometry methods including level set or phase-field methods. In the enriched isogeometric analysis, the base approximations in the domain/subdomains are enriched by the interfacial fields constructed as a function of distance from the interfaces. To circumvent the challenges of measuring distance and point of influence from the interface using iterative operations, algebraic level sets and algebraic point projection are utilized. The developed techniques are implemented as a program in the MATLAB environment named as <i>Hierarchical Design and Analysis Code</i>. The code is carefully designed to ensure simplicity and maintainability, to facilitate geometry creation, pre-processing, analysis and post-processing with optimal efficiency. </div><div><br></div><div>To couple NURBS patches, a parametric stitching strategy that assures arbitrary smoothness across subdomains with non-matching discretization is developed. The key concept used to accomplish the coupling is the insertion of a “parametric stitching” or p-stitching interface between the incompatible patches. In the present work, NURBS is chosen for discretizing the parametric subdomains. The developed procedure though is valid for other representations of subdomains whose basis functions obey partition of unity. The proposed method is validated through patch tests from which near-optimal rate of convergence is demonstrated. Several two- and three-dimensional elastostatic as well as heat conduction numerical examples are presented.</div><div><br></div><div>An enriched field approximation is then developed for characterizing stress singularities at junctions of general multi-material corners including crack tips. Using enriched isogeometric analysis, the developed method explicitly tracks the singular points and interfaces embedded in a non-conforming mesh. Solution convergence to those of linear elastic fracture mechanics is verified through several examples. More importantly, the proposed method enables direct extraction of generalized stress intensity factors upon solution of the problems without the need to use <i>a posteriori</i> path-independent integral such as the J-integral. Next, the analysis of crack initiation and propagation is carried out using the alternative concept of configurational force. The configurational force is first shown to result from a configurational optimization problem, which yields a configurational derivative as a necessary condition. For specific velocities imposed on the heterogeneities corresponding to translation, rotation or scaling, the configurational derivative is shown to yield the configurational force. The use of configurational force to analyze crack propagation is demonstrated through examples.</div><div><br></div><div>The developed methods are lastly applied to investigate the risk of ratcheting-induced fracture in the back end of line structure during thermal cycle test of a epoxy molded microelectronic package. The first principal stress and the opening mode stress intensity factor are proposed as the failure descriptors. A finite element analysis sub-modeling and load decomposition procedure is proposed to study the accumulation of plastic deformation in the metal line and to identify the critical loading mode. Enriched isogeometric analysis with singular stress enrichment is carried out to identify the interfacial corners most vulnerable to stress concentration and crack initiation. Correlation is made between the failure descriptors and the design parameters of the structure. Crack path from the identified critical corner is predicted using both linear elastic fracture mechanics criterion and configurational force criterion. </div>

<p dir="ltr">This thesis explores the thermodynamics and mechanics of reaction-diffusion interfaces in solid materials, focusing on configurational forces for bulks and surfaces, which are essential in understanding phenomena like electromigration, phase separation, and void evolution. The work is divided into four themes: bulk and surface configurational mechanics, electromigration in solder joints, and solid mixture theory. The thesis develops theories based on continuum mechanics and configurational forces, deriving Eshelby stress tensors and balance laws for interfaces. Experimental work on electromigration in SnBi solder joints is used to validate the theory. The research contributes to advancing the understanding of solid-state diffusion and phase evolution in engineering materials.</p>