Configurational Optimization and Configurational Force in Thermoelasticity: Theory and Computational Procedure

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.

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.

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.

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 J-, L- and M-integrals of fracture mechanics. Several illustrative examples of fracture resistant design are solved with EIGA.

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.

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.