A hierarchical framework for the multiscale modeling of microstructure evolution in heterogeneous materials.

Abstract

ACKNOWLEDGEMENTSAs far as I am concerned, there is no better way to celebrate a success than to share the credit with those that have contributed to the effort. Here I share credit with those people that have had tremendous impact to the successful completion of requirements for my PhD degree at Georgia Tech, including this dissertation.Of primary importance in this acknowledgement is to convey how substantial I consider my wife's sacrifices over the last five years. With the sole exception of unmarried PhD candidates, Ido not imagine that there exists a PhD thesis whose acknowledgement section does not make reference to the author's spouse. So I strive to make this acknowledgement of my wife's contributions to my success so compelling and unique that it escapes the status of 'formality'. To attempt this, I will provide a single anecdote selected from five years of worthy material.Stephanie Luscher endured months alone keeping our house in Los Alamos while I was in Atlanta attending classes and fulfilling residency requirements. During that time, she continued working as an engineering design review coordinator at Los Alamos National Lab. While in Oregon to visit family during the winter of 2004-5, a water pipe inside our Los Alamos house froze and burst. Over the next week, 90,000 gallons of water flowed throughout our house ruining just about everything. We returned home and opened soggy Christmas presents from the relative comfort of our car parked in the driveway while contemplating our next course of action.Our house required extensive repairs including a new heating system, kitchen cabinets, flooring, and complete refinishing of the interior walls. We lived in a motor home, a rented house, and a hotel over the course of six months of these repairs. Within two weeks of this happening, I was back in Atlanta and Stephanie was left to manage the reconstruction effort by herself. Of course, this is a single example of how Stephanie has sacrificed, and tended to the many details of life, in order to allow me to focus attention on the work presented within this thesis. Stephanie, I thank iv you for the support and sacrifice you have made to help me accomplish this goal. I'm glad you embraced "my goal" as "our goal."Well in advance of thoughts regarding graduate school, my parents instilled a few helpful traits in me. As a young boy, during hot summers spent roofing with my Dad, Mark Luscher, I learned to work hard and to play hard. My Dad taught me that sometimes "getting the job done" is not elegant; it simply requires you to roll up your sleeves and get to work. As a complement to this, my Mom taught me the importance of precision and attention to detail. Together, my parents have taught me the importance of family, which supersedes all else. Engineering division leader (W-DO), gave me opportunities to excel within the weapons program. Ryan Maupin, group leader for Advanced Engineering Analysis (W-13) has enthusiastically supported and trusted me, both as my manager and friend, to pursue a course of research th...

A fully second‐order homogenization formulation for the multi‐scale modeling of heterogeneous materials

“…Suitable algebraic manipulation of Equation (1) allows rewriting itself in a format that, by standard variational arguments, leads to the corresponding strong equilibrium equations: 1,52,54,56,58…”

“…The mixed finite element proposed by Matsushima et al, 76 referred to here as Q8F4L1, has been tested and successfully used in the literature; 1,4,52,54 thus it is employed to solve bi-dimensional problems in this contribution. It has eight nodes for displacements and geometry interpolation, with quadratic shape functions, a bi-linear interpolation for the relaxed deformation gradient, with four nodes, and a constant Lagrange multiplier field in the element domain, resulting in a total of 36 DOFs (Figure 4A).…”

“…These equations define both the first Piola-Kirchhoff and higher-order stress tensors (P and Q) as a function of the deformation gradient and the second gradient of the displacement field (F and G). Kouznetsova 1 and Luscher 54 employed simplified energy functions based on the proposal of Mindlin 19 to derive a second-order constitutive model, referred to as the Mindlin elastic constitutive model. The elastic strain energy function per unit reference volume proposed by Luscher 54 is used here, being expressed by:…”

“…Kouznetsova 1 and Luscher 54 employed simplified energy functions based on the proposal of Mindlin 19 to derive a second-order constitutive model, referred to as the Mindlin elastic constitutive model. The elastic strain energy function per unit reference volume proposed by Luscher 54 is used here, being expressed by:…”

Multiscale numerical modeling of deformation responses of heterogeneous materials

“…In order to verify the accuracy and convergence of the solutions obtained by the developed higher-order triangular finite element, a simple elastic shear layer problem, usually used as benchmark test in the higher-order formulations e.g. in [4,5,36], is analyzed. The schematic presentations of the geometry, boundary conditions and finite element mesh are given in Fig.…”