Lattice to continuum: A theoretical framework to calculate elastic fields in lattice scale

Lattice scale defects dictate the mechanical and functional properties of crystalline materials such as mechanical strength, elastic stiffness, fatigue resistance, and electrical properties. For example, hydrogen interstitials cause embrittlement in metals, while NV centers in 4H-SiC possess promising quantum spin properties for quantum computing applications. Atomistic simulation techniques, such as Density Function Theory (DFT) and Molecular Dynamics (MD), as well as classical field theories, such as continuum elasticity, are utilized extensively to determine the effect of defects on the functional properties. While continuum elasticity solutions provide information about physical fields such as displacement fields at the macroscopic scale, atomistic calculations can only provide such information at discrete points in space, i.e., atomic sites, in the lattice scale. This gap has persisted over the past decades and precluded the seamless integration of continuum and atomistic insights about material properties. This thesis aims to bridge this gap by developing methods to determine continuum elastic descriptions of lattice defects and mechanical properties of materials. We first introduce Regularized Green’s Function Method (RGFM), which uses the combination of the force equilibrium equation and Quantum Mechanical Force Density (QMFD) to determine elastic fields generated by point defects. QMFD describes the force variations in space and is related to the electron wavefunctions and densities obtained by DFT calculations. The force equilibrium equation is then solved by a spherical harmonic expansion in a computationally efficient way to calculate Regularized Green’s Functions. RGFM eliminates the reliance on the point forces that are employed in existing methodologies and generate divergent elastic fields near the defect core. Unlike the existing techniques for elastic modeling of point defects, RGFM provides elastic models that are accurate in long- and short-range regimes. As opposed to point defects, dislocations generate elastic fields that are periodic along the dislocation line, i.e., quasiperiodic. To account for this effect, we propose an extension to RGFM, Regularized Green’s Function Method for Dislocations (RGFMD). This method accounts for quasiperiodicity by taking QMFD as periodic functions that repeat along the dislocation line. The corresponding force equilibrium equations are solved by the spherical harmonic expansion method to determine the Regularized Green’s Functions for the dislocation. RGFMD offers significant improvements upon the classical elasticity solutions of dislocations. Eshelby-Stroh (ES) formalism, a classical anisotropic elasticity solution for dislocations, predicts elastic fields that tend to infinity at the core of the dislocation line due to the implied use of point forces. RGFMD removes this singularity by employing QMFD, which are bounded and finite, as forcing functions. Moreover, ES formalism relies on the assumption of plane strain conditions where the elastic fields do not vary along the dislocation line. This assumption does not hold in the lattice scales since the local atomic configuration along the dislocation line is not equivalent at every point in space. RGFMD does not rely on plane strain conditions; hence, able to predict the variations of elastic fields along the dislocation line. Furthermore, this thesis derives the theoretical formulas for the Quantum Mechanical Moduli Field (QMMF), which represents the local elastic moduli in the lattice scale using principles of quantum mechanics. QMMF provides valuable insights into the interplay between local elastic moduli and lattice defects. We identify three terms contributing to QMMF: kinetic, exchange-correlation, and electrostatic. We provide novel mathematical expressions for these terms with the numerical techniques to evaluate them. This thesis demonstrates the potential uses of QMMF formulation by four insightful examples. Firstly, macroscopic elastic moduli of pure Ni and B2 NiTi are accurately determined as the volumetric average of QMMF. Secondly, the effect of a H interstitial on the elastic properties of Ni crystal is considered. QMMF provides a computationally efficient way of capturing the effect of H concentration on the softening of the Ni crystal. Thirdly, the effect of volumetric strain on the elastic moduli is examined. The predictions of QMMF are shown to agree with experiments. Fourthly, local elastic moduli of Ni crystal with W solute are calculated. The addition of W solutes increases the shear modulus of Ni, which is in agreement with experiments. The increase in the macroscopic shear modulus is thought to be due to the hard spots that form around the W solutes. However, QMMF revealed that the macroscopic increase in the shear modulus is due to the hardening of the Ni matrix rather than W solutes forming hard spots. Such surprising phenomena had been unobservable before the advent of QMMF.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Gengor, Gorkem

Contributors dc:contributor

• Sehitoglu, Huseyin
• Sofronis, Petros
• Wharry, Janelle
• Schleife, Andre

• Atomistics
• Density Functional Theory
• Mechanics of Materials
• Quantum Mechanics
• Point Defects
• Dislocations
• SiC
• NV centers
• Solute Hardening
• Material Property Prediction
• Alloying
• Molecular Dynamics
• Continuum Mechanics
• Solid Mechanics
• Green's Function
• Quantum Materials
• Elastic Moduli
• Local Elastic Moduli
• Ab-initio Calculation
• Hydrogen Effect
• Hydrogen Embrittlement

Statement dc:rights

• Copyright 2025 Gorkem Gengor

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129684

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129684