Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure by Andrew J. Stershic Department of Civil & Environmental Engineering Duke University Date: Approved: John E. Dolbow, Supervisor Wilkins Aquino Guglielmo Scovazzi Srdjan Simunovic Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Civil & Environmental Engineering in the Graduate School of Duke University 2016 Abstract Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure by Andrew J. Stershic Department of Civil & Environmental Engineering Duke University Date: Approved: John E. Dolbow, Supervisor Wilkins Aquino Guglielmo Scovazzi Srdjan Simunovic An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Civil & Environmental Engineering in the Graduate School of Duke University 2016 Copyright (cid:13)c 2016 by Andrew J. Stershic All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract Thefocusofthisworkistodevelopandemploynumericalmethodsthatprovidechar- acterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies. I first propose the fabric tensor formalism to describe the structure and evolution oflithium-ionelectrodemicrostructureduringthecalendaringprocess. Fabrictensors aredirectionalmeasuresofparticulateassembliesbasedoninter-particleconnectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, I show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode. I then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. I explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, I discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh iv discretizations, I find that employing randomness can improve convergence behavior and provide a computational savings. Next, the Thick Level-Set model is implemented to describe brittle media un- dergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severityofdegradationandusesalengthscaletolimitthedamagegradient. Interms of energy dissipated by fracture and mean fragment size, I find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies. Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the bat- tery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. I find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems. The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing. v Contents Abstract iv List of Tables xi List of Figures xii List of Abbreviations and Symbols xix Acknowledgements xx 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Characterizing active material contact networks . . . . . . . . 6 1.2.2 Evaluating the cohesive element approach for dynamic frag- mentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Applying the Thick Level-Set model for dynamic fragmenta- tion problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.4 Applying the Thick Level-Set model for three-dimensional dy- namic fracture problems . . . . . . . . . . . . . . . . . . . . . 11 2 Modeling the evolution of lithium-ion particle contact distributions using a fabric tensor approach 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Fabric Tensor Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 DEM Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 vi 2.5 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.1 Coordination Number . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Fabric Tensor Comparison . . . . . . . . . . . . . . . . . . . . 32 2.5.3 DEM Model Critique . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Role of numerical precision and stochasticity in dynamic fragmen- tation simulations 40 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Dynamic fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Relaxation and rate dependence . . . . . . . . . . . . . . . . . 43 3.2.2 Fragment size . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . 48 3.4 Numerical precision experiments . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Double-precision vs. quadruple-precision, uniform material . . 50 3.4.2 Double-precision with non-random material defects . . . . . . 51 3.4.3 Double-precisionvs. quadruple-precisionwithnon-randomma- terial defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.4 Simulations with random defects . . . . . . . . . . . . . . . . 55 3.4.5 Simulations with random mesh . . . . . . . . . . . . . . . . . 59 3.4.6 Quantifying contribution of round-off error . . . . . . . . . . . 61 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4 The Thick Level-Set model for dynamic fragmentation 70 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 vii 4.2 Thick Level-Set Model . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Numerical Model and Implementation . . . . . . . . . . . . . . . . . . 79 4.3.1 Finite element framework . . . . . . . . . . . . . . . . . . . . 79 4.3.2 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.3 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.4 Damage evolution . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.5 Nodal status updates . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.6 Element enrichment . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.7 Level-set field reinitialization . . . . . . . . . . . . . . . . . . . 87 4.3.8 Fragment definition . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.9 Dissipation calculation . . . . . . . . . . . . . . . . . . . . . . 90 4.3.10 Pseudo-code for implementation . . . . . . . . . . . . . . . . . 91 4.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.1 Stochastic simulations . . . . . . . . . . . . . . . . . . . . . . 93 4.4.2 Local evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.3 Local/non-local evolution . . . . . . . . . . . . . . . . . . . . 95 4.4.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5.1 Softening function . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5.2 Characteristic length . . . . . . . . . . . . . . . . . . . . . . . 106 4.5.3 Required mesh size . . . . . . . . . . . . . . . . . . . . . . . . 107 4.5.4 Extension to multi-dimensional problems . . . . . . . . . . . . 108 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 viii 5 Application of the Thick Level-Set model for three-dimensional dy- namic fracture 111 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2 Thick Level-Set Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.1 Plate tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.2 Plate shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3.3 Plate dynamic shear test . . . . . . . . . . . . . . . . . . . . . 120 5.3.4 Sphere compression test . . . . . . . . . . . . . . . . . . . . . 120 5.3.5 Particle compression test . . . . . . . . . . . . . . . . . . . . . 123 5.4 Outcomes & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.1 Plate tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.2 Plate shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.3 Plate dynamic shear test . . . . . . . . . . . . . . . . . . . . . 129 5.4.4 Sphere compression test . . . . . . . . . . . . . . . . . . . . . 130 5.4.5 Particle compression test . . . . . . . . . . . . . . . . . . . . . 131 5.4.6 Analytical comparison . . . . . . . . . . . . . . . . . . . . . . 132 5.4.7 Mesh convergence study . . . . . . . . . . . . . . . . . . . . . 133 5.4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6 Conclusions 139 6.1 Engineering significance . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Extensions and future work . . . . . . . . . . . . . . . . . . . . . . . 142 A Modeling the evolution of lithium-ion particle contact distributions using a fabric tensor approach 144 A.1 Particle approximation for contact analysis . . . . . . . . . . . . . . . 144 ix A.1.1 Spherical approximation . . . . . . . . . . . . . . . . . . . . . 144 A.1.2 Ellipsoidal approximation . . . . . . . . . . . . . . . . . . . . 145 A.1.3 Voxel-based method . . . . . . . . . . . . . . . . . . . . . . . 146 A.2 Statistical significance . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B The Thick Level-Set model for dynamic fragmentation 149 B.1 Cohesive element approach comparison . . . . . . . . . . . . . . . . . 149 Bibliography 151 Biography 163 x