{"product_id":"dynamics-of-lattice-materials-isbn-9781118729595","title":"Dynamics of Lattice Materials","description":"\u003cul\u003e \u003cli\u003eProvides a comprehensive introduction to the dynamic response of lattice materials, covering the fundamental theory and applications in engineering practice\u003c\/li\u003e \u003cli\u003eOffers comprehensive treatment of dynamics of lattice materials and periodic materials in general, including phononic crystals and elastic metamaterials\u003c\/li\u003e \u003cli\u003eProvides an in depth introduction to elastostatics and elastodynamics of lattice materials\u003c\/li\u003e \u003cli\u003eCovers advanced topics such as damping, nonlinearity, instability, impact and nanoscale systems\u003c\/li\u003e \u003cli\u003eIntroduces contemporary concepts including pentamodes, local resonance and inertial amplification\u003c\/li\u003e \u003cli\u003eIncludes chapters on fast computation and design optimization tools\u003c\/li\u003e \u003cli\u003eTopics are introduced using simple systems and generalized to more complex structures with a focus on dispersion characteristics\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eList of Contributors xiii\u003c\/p\u003e \u003cp\u003eForeword xv\u003c\/p\u003e \u003cp\u003ePreface xxv\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction to Lattice Materials 1\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eA. Srikantha Phani andMahmoud I. Hussein\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introduction 1\u003c\/p\u003e \u003cp\u003e1.2 Lattice Materials and Structures 2\u003c\/p\u003e \u003cp\u003e1.2.1 Material versus Structure 3\u003c\/p\u003e \u003cp\u003e1.2.2 Motivation 3\u003c\/p\u003e \u003cp\u003e1.2.3 Classification of Lattices and Maxwell’s Rule 4\u003c\/p\u003e \u003cp\u003e1.2.4 ManufacturingMethods 6\u003c\/p\u003e \u003cp\u003e1.2.5 Applications 7\u003c\/p\u003e \u003cp\u003e1.3 Overview of Chapters 8\u003c\/p\u003e \u003cp\u003eAcknowledgment 10\u003c\/p\u003e \u003cp\u003eReferences 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Elastostatics of Lattice Materials 19\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eD. Pasini and S. Arabnejad\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e2.1 Introduction 19\u003c\/p\u003e \u003cp\u003e2.2 The RVE 21\u003c\/p\u003e \u003cp\u003e2.3 Surface Average Approach 22\u003c\/p\u003e \u003cp\u003e2.4 Volume Average Approach 25\u003c\/p\u003e \u003cp\u003e2.5 Force-based Approach 25\u003c\/p\u003e \u003cp\u003e2.6 Asymptotic Homogenization Method 26\u003c\/p\u003e \u003cp\u003e2.7 Generalized Continuum Theory 29\u003c\/p\u003e \u003cp\u003e2.8 Homogenization via BlochWave Analysis and the Cauchy–Born Hypothesis 32\u003c\/p\u003e \u003cp\u003e2.9 Multiscale Matrix-based Computational Technique 34\u003c\/p\u003e \u003cp\u003e2.10 Homogenization based on the Equation of Motion 36\u003c\/p\u003e \u003cp\u003e2.11 Case Study: Property Predictions for a Hexagonal Lattice 38\u003c\/p\u003e \u003cp\u003e2.12 Conclusions 42\u003c\/p\u003e \u003cp\u003eReferences 43\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Elastodynamics of Lattice Materials 53\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eA. Srikantha Phani\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e3.1 Introduction 53\u003c\/p\u003e \u003cp\u003e3.2 One-dimensional Lattices 55\u003c\/p\u003e \u003cp\u003e3.2.1 Bloch’s Theorem 57\u003c\/p\u003e \u003cp\u003e3.2.2 Application of Bloch’s Theorem 59\u003c\/p\u003e \u003cp\u003e3.2.3 Dispersion Curves and Unit-cell Resonances 59\u003c\/p\u003e \u003cp\u003e3.2.4 Continuous Lattices: Local Resonance and sub-Bragg Band Gaps 61\u003c\/p\u003e \u003cp\u003e3.2.5 Dispersion Curves of a Beam Lattice 62\u003c\/p\u003e \u003cp\u003e3.2.6 Receptance Method 64\u003c\/p\u003e \u003cp\u003e3.2.7 Synopsis of 1D Lattices 67\u003c\/p\u003e \u003cp\u003e3.3 Two-dimensional Lattice Materials 67\u003c\/p\u003e \u003cp\u003e3.3.1 Application of Bloch’s Theorem to 2D Lattices 67\u003c\/p\u003e \u003cp\u003e3.3.2 Discrete Square Lattice 70\u003c\/p\u003e \u003cp\u003e3.4 Lattice Materials 72\u003c\/p\u003e \u003cp\u003e3.4.1 Finite Element Modelling of the Unit Cell 75\u003c\/p\u003e \u003cp\u003e3.4.2 Band Structure of Lattice Topologies 77\u003c\/p\u003e \u003cp\u003e3.4.3 Directionality ofWave Propagation 84\u003c\/p\u003e \u003cp\u003e3.5 Tunneling and EvanescentWaves 85\u003c\/p\u003e \u003cp\u003e3.6 Concluding Remarks 87\u003c\/p\u003e \u003cp\u003e3.7 Acknowledgments 87\u003c\/p\u003e \u003cp\u003eReferences 87\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Wave Propagation in Damped Lattice Materials 93\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eDimitri Krattiger, A. Srikantha Phani andMahmoud I. Hussein\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introduction 93\u003c\/p\u003e \u003cp\u003e4.2 One-dimensionalMass–Spring–DamperModel 95\u003c\/p\u003e \u003cp\u003e4.2.1 1D Model Description 95\u003c\/p\u003e \u003cp\u003e4.2.2 Free-wave Solution 96\u003c\/p\u003e \u003cp\u003eState-spaceWave Calculation 97\u003c\/p\u003e \u003cp\u003eBloch–Rayleigh Perturbation Method 97\u003c\/p\u003e \u003cp\u003e4.2.3 Driven-wave Solution 98\u003c\/p\u003e \u003cp\u003e4.2.4 1D Damped Band Structures 98\u003c\/p\u003e \u003cp\u003e4.3 Two-dimensional Plate–Plate Lattice Model 99\u003c\/p\u003e \u003cp\u003e4.3.1 2D Model Description 99\u003c\/p\u003e \u003cp\u003e4.3.2 Extension of Driven-wave Calculations to 2D Domains 100\u003c\/p\u003e \u003cp\u003e4.3.3 2D Damped Band Structures 101\u003c\/p\u003e \u003cp\u003eReferences 104\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Wave Propagation in Nonlinear Lattice Materials 107\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eKevin L.Manktelow,Massimo Ruzzene andMichael J. Leamy\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e5.1 Overview 107\u003c\/p\u003e \u003cp\u003e5.2 Weakly Nonlinear Dispersion Analysis 108\u003c\/p\u003e \u003cp\u003e5.3 Application to a 1D Monoatomic Chain 114\u003c\/p\u003e \u003cp\u003e5.3.1 Overview 114\u003c\/p\u003e \u003cp\u003e5.3.2 Model Description and Nonlinear Governing Equation 114\u003c\/p\u003e \u003cp\u003e5.3.3 Single-wave Dispersion Analysis 115\u003c\/p\u003e \u003cp\u003e5.3.4 Multi-wave Dispersion Analysis 116\u003c\/p\u003e \u003cp\u003eCase 1. GeneralWave–Wave Interactions 117\u003c\/p\u003e \u003cp\u003eCase 2. Long-wavelength LimitWave–Wave Interactions 119\u003c\/p\u003e \u003cp\u003e5.3.5 Numerical Verification and Discussion 122\u003c\/p\u003e \u003cp\u003e5.4 Application to a 2D Monoatomic Lattice 123\u003c\/p\u003e \u003cp\u003e5.4.1 Overview 123\u003c\/p\u003e \u003cp\u003e5.4.2 Model Description and Nonlinear Governing Equation 124\u003c\/p\u003e \u003cp\u003e5.4.3 Multiple-scale Perturbation Analysis 125\u003c\/p\u003e \u003cp\u003e5.4.4 Analysis of Predicted Dispersion Shifts 127\u003c\/p\u003e \u003cp\u003e5.4.5 Numerical Simulation Validation Cases 129\u003c\/p\u003e \u003cp\u003eAnalysis Method 130\u003c\/p\u003e \u003cp\u003eOrthogonal and Oblique Interaction 131\u003c\/p\u003e \u003cp\u003e5.4.6 Application: Amplitude-tunable Focusing 133\u003c\/p\u003e \u003cp\u003eSummary 134\u003c\/p\u003e \u003cp\u003eAcknowledgements 135\u003c\/p\u003e \u003cp\u003eReferences 135\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Stability of Lattice Materials 139\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eFilippo Casadei, PaiWang and Katia Bertoldi\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction 139\u003c\/p\u003e \u003cp\u003e6.2 Geometry, Material, and Loading Conditions 140\u003c\/p\u003e \u003cp\u003e6.3 Stability of Finite-sized Specimens 141\u003c\/p\u003e \u003cp\u003e6.4 Stability of Infinite Periodic Specimens 142\u003c\/p\u003e \u003cp\u003e6.4.1 Microscopic Instability 142\u003c\/p\u003e \u003cp\u003e6.5 Post-buckling Analysis 145\u003c\/p\u003e \u003cp\u003e6.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves 146\u003c\/p\u003e \u003cp\u003e6.7 Conclusions 150\u003c\/p\u003e \u003cp\u003eReferences 151\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Impact and Blast Response of Lattice Materials 155\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eMatthew Smith,Wesley J. Cantwell and Zhongwei Guan\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 155\u003c\/p\u003e \u003cp\u003e7.2 Literature Review 155\u003c\/p\u003e \u003cp\u003e7.2.1 Dynamic Response of Cellular Structures 155\u003c\/p\u003e \u003cp\u003e7.2.2 Shock- and Blast-loading Responses of Cellular Structures 157\u003c\/p\u003e \u003cp\u003e7.2.3 Dynamic Indentation Performance of Cellular Structures 158\u003c\/p\u003e \u003cp\u003e7.3 Manufacturing Process 159\u003c\/p\u003e \u003cp\u003e7.3.1 The Selective Laser Melting Technique 159\u003c\/p\u003e \u003cp\u003e7.3.2 Sandwich Panel Manufacture 160\u003c\/p\u003e \u003cp\u003e7.4 Dynamic and Blast Loading of Lattice Materials 161\u003c\/p\u003e \u003cp\u003e7.4.1 ExperimentalMethod – Drop-hammer Impact Tests 161\u003c\/p\u003e \u003cp\u003e7.4.2 ExperimentalMethod – Blast Tests on Lattice Cubes 162\u003c\/p\u003e \u003cp\u003e7.4.3 ExperimentalMethod – Blast Tests on Composite-lattice Sandwich Structures 163\u003c\/p\u003e \u003cp\u003e7.5 Results and Discussion 165\u003c\/p\u003e \u003cp\u003e7.5.1 Drop-hammer Impact Tests 165\u003c\/p\u003e \u003cp\u003e7.5.2 Blast Tests on the Lattice Structures 167\u003c\/p\u003e \u003cp\u003e7.5.3 Blast Tests on the Sandwich Panels 170\u003c\/p\u003e \u003cp\u003eConcluding Remarks 173\u003c\/p\u003e \u003cp\u003eAcknowledgements 174\u003c\/p\u003e \u003cp\u003eReferences 174\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Pentamode Lattice Structures 179\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eAndrew N. Norris\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction 179\u003c\/p\u003e \u003cp\u003e8.2 Pentamode Materials 183\u003c\/p\u003e \u003cp\u003e8.2.1 General Properties 183\u003c\/p\u003e \u003cp\u003e8.2.2 Small Rigidity and Poisson’s Ratio of a PM 185\u003c\/p\u003e \u003cp\u003e8.2.3 Wave Motion in a PM 186\u003c\/p\u003e \u003cp\u003e8.3 Lattice Models for PM 187\u003c\/p\u003e \u003cp\u003e8.3.1 Effective PM Properties of 2D and 3D Lattices 187\u003c\/p\u003e \u003cp\u003e8.3.2 Transversely Isotropic PM Lattice 188\u003c\/p\u003e \u003cp\u003eEffective Moduli: 2D 190\u003c\/p\u003e \u003cp\u003e8.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D 192\u003c\/p\u003e \u003cp\u003e8.4.1 General Formulation with Rigidity 192\u003c\/p\u003e \u003cp\u003e8.4.2 Pentamode Limit 194\u003c\/p\u003e \u003cp\u003e8.4.3 Two-dimensional Results for Finite Rigidity 195\u003c\/p\u003e \u003cp\u003e8.5 Conclusion 195\u003c\/p\u003e \u003cp\u003eAcknowledgements 196\u003c\/p\u003e \u003cp\u003eReferences 196\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Modal Reduction of Lattice Material Models 199\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eDimitri Krattiger and Mahmoud I. Hussein\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction 199\u003c\/p\u003e \u003cp\u003e9.2 Plate Model 200\u003c\/p\u003e \u003cp\u003e9.2.1 Mindlin–Reissner Plate Finite Elements 200\u003c\/p\u003e \u003cp\u003e9.2.2 Bloch Boundary Conditions 202\u003c\/p\u003e \u003cp\u003e9.2.3 Example Model 203\u003c\/p\u003e \u003cp\u003e9.3 Reduced Bloch Mode Expansion 204\u003c\/p\u003e \u003cp\u003e9.3.1 RBME Formulation 204\u003c\/p\u003e \u003cp\u003e9.3.2 RBME Example 205\u003c\/p\u003e \u003cp\u003e9.3.3 RBME Additional Considerations 207\u003c\/p\u003e \u003cp\u003e9.4 Bloch Mode Synthesis 208\u003c\/p\u003e \u003cp\u003e9.4.1 BMS Formulation 208\u003c\/p\u003e \u003cp\u003e9.4.2 BMS Example 210\u003c\/p\u003e \u003cp\u003e9.4.3 BMS Additional Considerations 210\u003c\/p\u003e \u003cp\u003e9.5 Comparison of RBME and BMS 212\u003c\/p\u003e \u003cp\u003e9.5.1 Model Size 212\u003c\/p\u003e \u003cp\u003e9.5.2 Computational Efficiency 213\u003c\/p\u003e \u003cp\u003e9.5.3 Ease of Implementation 214\u003c\/p\u003e \u003cp\u003eReferences 214\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Topology Optimization of Lattice Materials 217\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eOsama R. Bilal and Mahmoud I. Hussein\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e10.1 Introduction 217\u003c\/p\u003e \u003cp\u003e10.2 Unit-cell Optimization 218\u003c\/p\u003e \u003cp\u003e10.2.1 Parametric, Shape, and Topology Optimization 218\u003c\/p\u003e \u003cp\u003e10.2.2 Selection of Studies from the Literature 218\u003c\/p\u003e \u003cp\u003e10.2.3 Design Search Space 219\u003c\/p\u003e \u003cp\u003e10.3 Plate-based Lattice Material Unit Cell 220\u003c\/p\u003e \u003cp\u003e10.3.1 Equation of Motion and FE Model 221\u003c\/p\u003e \u003cp\u003e10.3.2 Mathematical Formulation 222\u003c\/p\u003e \u003cp\u003e10.4 Genetic Algorithm 223\u003c\/p\u003e \u003cp\u003e10.4.1 Objective Function 223\u003c\/p\u003e \u003cp\u003e10.4.2 Fitness Function 224\u003c\/p\u003e \u003cp\u003e10.4.3 Selection 224\u003c\/p\u003e \u003cp\u003e10.4.4 Reproduction 224\u003c\/p\u003e \u003cp\u003e10.4.5 Initialization and Termination 225\u003c\/p\u003e \u003cp\u003e10.4.6 Implementation 225\u003c\/p\u003e \u003cp\u003e10.5 Appendix 226\u003c\/p\u003e \u003cp\u003eReferences 228\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Dynamics of Locally Resonant and Inertially Amplified Lattice Materials 233\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eCetin Yilmaz and Gregory M. Hulbert\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e11.1 Introduction 233\u003c\/p\u003e \u003cp\u003e11.2 Locally Resonant Lattice Materials 234\u003c\/p\u003e \u003cp\u003e11.2.1 1D Locally Resonant Lattices 234\u003c\/p\u003e \u003cp\u003e11.2.2 2D Locally Resonant Lattices 241\u003c\/p\u003e \u003cp\u003e11.2.3 3D Locally Resonant Lattices 243\u003c\/p\u003e \u003cp\u003e11.3 Inertially Amplified Lattice Materials 246\u003c\/p\u003e \u003cp\u003e11.3.1 1D Inertially Amplified Lattices 246\u003c\/p\u003e \u003cp\u003e11.3.2 2D Inertially Amplified Lattices 248\u003c\/p\u003e \u003cp\u003e11.3.3 3D Inertially Amplified Lattices 253\u003c\/p\u003e \u003cp\u003e11.4 Conclusions 255\u003c\/p\u003e \u003cp\u003eReferences 256\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Dynamics of Nanolattices: Polymer-Nanometal Lattices 259\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eCraig A. Steeves, Glenn D. Hibbard,Manan Arya, and Ante T. Lausic\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e12.1 Introduction 259\u003c\/p\u003e \u003cp\u003e12.2 Fabrication 259\u003c\/p\u003e \u003cp\u003e12.2.1 Case Study 262\u003c\/p\u003e \u003cp\u003e12.3 Lattice Dynamics 263\u003c\/p\u003e \u003cp\u003e12.3.1 Lattice Properties 264\u003c\/p\u003e \u003cp\u003eGeometries of 3D Lattices 264\u003c\/p\u003e \u003cp\u003eEffective Material Properties of Nanometal-coated Polymer Lattices 265\u003c\/p\u003e \u003cp\u003e12.3.2 Finite-elementModel 266\u003c\/p\u003e \u003cp\u003eDisplacement Field 266\u003c\/p\u003e \u003cp\u003eKinetic Energy 268\u003c\/p\u003e \u003cp\u003eStrain Potential Energy 269\u003c\/p\u003e \u003cp\u003eCollected Equation of Motion 270\u003c\/p\u003e \u003cp\u003e12.3.3 Floquet–Bloch Principles 271\u003c\/p\u003e \u003cp\u003eGeneralized Forces in Bloch Analysis 272\u003c\/p\u003e \u003cp\u003eReduced Equation of Motion 274\u003c\/p\u003e \u003cp\u003e12.3.4 Dispersion Curves for the Octet Lattice 275\u003c\/p\u003e \u003cp\u003e12.3.5 Lattice Tuning 277\u003c\/p\u003e \u003cp\u003eBandgap Placement 277\u003c\/p\u003e \u003cp\u003eLattice Optimization 277\u003c\/p\u003e \u003cp\u003e12.4 Conclusions 278\u003c\/p\u003e \u003cp\u003e12.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degrees\u003c\/p\u003e \u003cp\u003eof Freedom 279\u003c\/p\u003e \u003cp\u003eReferences 280\u003c\/p\u003e \u003cp\u003eIndex 283\u003c\/p\u003e   \u003cp\u003e \u003cem\u003eEditors \u003c\/em\u003e\u003cbr\u003e \u003cstrong\u003eA. Srikantha Phani,\u003c\/strong\u003e University of British Columbia, Canada\u003cbr\u003e \u003cstrong\u003eMahmoud I. Hussein,\u003c\/strong\u003e University of Colorado Boulder, USA \u003c\/p\u003e\u003cp\u003e      \u003c\/p\u003e\u003cp\u003e A lattice material is formed from a spatially periodic network of interconnected rods, beams, plates or other slender structures. The ability to tailor the unit-cell microstructure of a lattice material on multiple length scales is a way of attaining superior mechanical and vibroacoustic properties that are ordinarily not possible using conventional materials. This book focuses on the dynamic response of lattice materials, an area that has been greatly inspired by concepts from crystal physics. The methods and analyses covered directly apply to periodic materials in general, including phononic crystals and elastic metamaterials.  \u003c\/p\u003e\u003cp\u003e The text has been written by leading experts in the field, and covers  \u003c\/p\u003e\u003cul\u003e \u003cli\u003eelastostatics and elastodynamics\u003c\/li\u003e \u003cli\u003ethe effects of damping, nonlinearity, instabilities, and impact loads\u003c\/li\u003e \u003cli\u003eexotic dynamics such as pentamodes, local resonances, and inertial amplification\u003c\/li\u003e \u003cli\u003emodel reduction and optimization\u003c\/li\u003e \u003cli\u003enano-lattices\u003c\/li\u003e \u003c\/ul\u003e \u003cbr\u003e  \u003cp\u003e A systematic and unified synthesis of these topics is provided to help consolidate conceptual building blocks in this emerging field of research. \u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eKey features:\u003c\/strong\u003e \u003c\/p\u003e\u003cul\u003e   \u003cli\u003eComprehensive treatment of dynamics of lattice materials and periodic materials in general, including phononic crystals and elastic metamaterials\u003c\/li\u003e   \u003cli\u003eProvides an in-depth introduction to elastostatics and elastodynamics of lattice materials\u003c\/li\u003e   \u003cli\u003eCovers advanced topics such as damping, nonlinearity, instability, impact and nanoscale systems\u003c\/li\u003e   \u003cli\u003eIntroduces contemporary concepts including pentamodes, local resonances and inertial amplification\u003c\/li\u003e   \u003cli\u003eIncludes chapters on fast computation and design optimization tools\u003c\/li\u003e   \u003cli\u003eTopics are introduced using simple systems and generalized to more complex structures with a focus on dispersion characteristics\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003cem\u003eDynamics of Lattice Materials\u003c\/em\u003e is suitable for graduate students and research scientists with a background in dynamics, vibrations, mechanics of materials, and materials physics. It serves as a useful reference for researchers based in universities and practitioners in industrial research labs. It may also be used as a textbook for graduate courses on the mechanics of lattice materials.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989092745445,"sku":"NP9781118729595","price":167.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781118729595.jpg?v=1761782765","url":"https:\/\/k12savings.com\/products\/dynamics-of-lattice-materials-isbn-9781118729595","provider":"K12savings","version":"1.0","type":"link"}