Mechanics of Microsystems Alberto Corigliano, Raffaele Ardito, Claudia Comi, Attilio Frangi, Aldo Ghisi and Stefano Mariani, Politecnico di Milano, Italy   A mechanical approach to microsystems, covering fundamental concepts including MEMS design, modelling and reliability   Mechanics of Microsystems takes a mechanical approach to microsystems and covers fundamental concepts including MEMS design, modelling and reliability. The book examines the mechanical behaviour of microsystems from a ‘design for reliability’ point of view and includes examples of applications in industry. Mechanics of Microsystems is divided into two main parts. The first part recalls basic knowledge related to the microsystems behaviour and offers an overview on microsystems and fundamental design and modelling tools from a mechanical point of view, together with many practical examples of real microsystems. The second part covers the mechanical characterization of materials at the micro-scale and considers the most important reliability issues (fracture, fatigue, stiction, damping phenomena, etc) which are fundamental to fabricate a real working device.     Key features: Provides an overview of MEMS, with special focus on mechanical-based Microsystems and reliability issues. Includes examples of applications in industry. Accompanied by a website hosting supplementary material.     The book provides essential reading for researchers and practitioners working with MEMS, as well as graduate students in mechanical, materials and electrical engineering.

Table of Contents:
Series Preface xiii Preface xv Acknowledgements xvii Notation xix About the Companion Websitexxiii 1 Introduction 1 1.1 Microsystems 1 1.2 Microsystems Fabrication 3 1.3 Mechanics in Microsystems 5 1.4 Book Contents 6 References 7 Part I Fundamentals 9 2 Fundamentals of Mechanics and Coupled Problems 11 2.1 Introduction 11 2.2 Kinematics and Dynamics of Material Points and Rigid Bodies 12 2.2.1 Basic Notions of Kinematics and Motion Composition 12 2.2.2 Basic Notions of Dynamics and Relative Dynamics 15 2.2.3 One-Degree-of-Freedom Oscillator 17 2.2.4 Rigid-Body Kinematics and Dynamics 22 2.3 Solid Mechanics 25 2.3.1 Linear Elastic Problem for Deformable Solids 26 2.3.2 Linear Elastic Problem for Beams 35 2.4 Fluid Mechanics 43 2.4.1 Navier–Stokes Equations 43 2.4.2 Fluid–Structure Interaction 48 2.5 Electrostatics and Electromechanics 49 2.5.1 Basic Notions of Electrostatics 49 2.5.2 Simple Electromechanical Problem 54 2.5.3 General Electromechanical Coupled Problem 58 2.6 Piezoelectric Materials in Microsystems 60 2.6.1 Piezoelectric Materials 60 2.6.2 Piezoelectric Modelling 62 2.7 Heat Conduction and Thermomechanics 64 2.7.1 Heat Problem 64 2.7.2 Thermomechanical Coupled Problem 67 References 70 3 Modelling of Linear and Nonlinear Mechanical Response 73 3.1 Introduction 73 3.2 Fundamental Principles 74 3.2.1 Principle of Virtual Power 74 3.2.2 Total Potential Energy Principle 74 3.2.3 Hamilton’s Principle 75 3.2.4 Specialization of the Principle of Virtual Powers to Beams 76 3.3 Approximation Techniques and Weighted Residuals Approach 76 3.4 Exact and Approximate Solutions for Dynamic Problems 79 3.4.1 Free Flexural Linear Vibrations of a Single-span Beam 79 3.4.2 Nonlinear Vibration of an Axially Loaded Beam 80 3.5 Example of Application: Bistable Elements 84 References 90 Part II Devices 91 4 Accelerometers 93 4.1 Introduction 93 4.2 Capacitive Accelerometers 94 4.2.1 In-Plane Sensing 94 4.2.2 Out-of-Plane Sensing 96 4.3 Resonant Accelerometers 98 4.3.1 Resonating Proof Mass 98 4.3.2 Resonating Elements Coupled to the Proof Mass 99 4.4 Examples 101 4.4.1 Three-Axis Capacitive Accelerometer 101 4.4.2 Out-of-Plane Resonant Accelerometer 104 4.4.3 In-Plane Resonant Accelerometer 105 4.5 Design Problems and Reliability Issues 107 References 107 5 Coriolis-Based Gyroscopes 109 5.1 Introduction 109 5.2 Basic Working Principle 109 5.2.1 Sensitivity of Coriolis Vibratory Gyroscopes 112 5.3 Lumped-Mass Gyroscopes 113 5.3.1 Symmetric and Decoupled Gyroscope 113 5.3.2 Tuning-Fork Gyroscope 114 5.3.3 Three-Axis Gyroscope 115 5.3.4 Gyroscopes with Resonant Sensing 115 5.4 Disc and Ring Gyroscopes 118 5.5 Design Problems and Reliability Issues 118 References 119 6 Resonators 121 6.1 Introduction 121 6.2 Electrostatically Actuated Resonators 123 6.3 Piezoelectric Resonators 125 6.4 Nonlinearity Issues 126 References 128 7 Micromirrors and Parametric Resonance 131 7.1 Introduction 131 7.2 Electrostatic Resonant Micromirror 132 7.2.1 Numerical Simulations with a Continuation Approach 136 7.2.2 Experimental Set-Up 140 References 145 8 Vibrating Lorentz Force Magnetometers 147 8.1 Introduction 147 8.2 Vibrating Lorentz Force Magnetometers 148 8.2.1 Classical Devices 148 8.2.2 Improved Design 151 8.2.3 Further Improvements 155 8.3 Topology or Geometry Optimization 156 References 159 9 Mechanical Energy Harvesters 161 9.1 Introduction 161 9.2 Inertial Energy Harvesters 162 9.2.1 Classification of Resonant Energy Harvesters 162 9.2.2 Mechanical Model of a Simple Piezoelectric Harvester 165 9.3 Frequency Upconversion and Bistability 174 9.4 Fluid–Structure Interaction Energy Harvesters 176 9.4.1 Synopsis of Aeroelastic Phenomena 177 9.4.2 Energy Harvesting through Vortex-Induced Vibration 179 9.4.3 Energy Harvesting through Flutter Instability 180 References 181 10 Micropumps 185 10.1 Introduction 185 10.2 Modelling Issues for Diaphragm Micropumps 186 10.3 Modelling of Electrostatic Actuator 188 10.3.1 Simplified Electromechanical Model 188 10.3.2 Reliability Issues 192 10.4 Multiphysics Model of an Electrostatic Micropump 196 10.5 Piezoelectric Micropumps 198 10.5.1 Modelling of the Actuator 198 10.5.2 Complete Multiphysics Model 201 References 202 Part III Reliability and Dissipative Phenomena 205 11 Mechanical Characterization at the Microscale 207 11.1 Introduction 207 11.2 Mechanical Characterization of Polysilicon as a Structural Material for Microsystems 209 11.2.1 Polysilicon as a Structural Material for Microsystems 209 11.2.2 Testing Methodologies 210 11.2.3 Quasi-Static Testing 211 11.2.4 High-Frequency Testing 214 11.3 Weibull Approach 215 11.4 On-Chip Testing Methodology for Experimental Determination of Elastic Stiffness and Nominal Strength 219 11.4.1 On-Chip Bending Test through a Comb-Finger Rotational Electrostatic Actuator 220 11.4.2 On-Chip Bending Test through a Parallel-Plate Electrostatic Actuator 225 11.4.3 On-Chip Tensile Test through an Electrothermomechanical Actuator 229 11.4.4 On-Chip Test for Thick Polysilicon Films 233 References 240 12 Fracture and Fatigue in Microsystems 245 12.1 Introduction 245 12.2 Fracture Mechanics: An Overview 245 12.3 MEMS Failure Modes due to Cracking 249 12.3.1 Cracking and Delamination at Package Level 249 12.3.2 Cracking at Silicon Film Level 250 12.4 Fatigue in Microsystems 256 12.4.1 An Introduction to Fatigue in Mechanics 256 12.4.2 Polysilicon Fatigue 259 12.4.3 Fatigue in Metals at the Microscale 261 12.4.4 Fatigue Testing at the Microscale 263 References 266 13 Accidental Drop Impact 271 13.1 Introduction 271 13.2 Single-Degree-of-Freedom Response to Drops 272 13.3 Estimation of the Acceleration Peak Induced by an Accidental Drop 276 13.4 A Multiscale Approach to Drop Impact Events 277 13.4.1 Macroscale Level 277 13.4.2 Mesoscale Level 279 13.4.3 Microscale Level 279 13.5 Results: Drop-Induced Failure of Inertial MEMS 280 References 287 14 Fabrication-Induced Residual Stresses and Relevant Failures 291 14.1 Main Sources of Residual Stresses in Microsystems 291 14.2 The Stoney Formula and its Modifications 292 14.3 Experimental Methods for the Evaluation of Residual Stresses 299 14.4 Delamination, Buckling and Cracks in Thin Films due to Residual Stresses 304 References 310 15 Damping in Microsystems 313 15.1 Introduction 313 15.2 Gas Damping in the Continuum Regime with Slip Boundary Conditions 314 15.2.1 Experimental Validation at Ambient Pressure 317 15.2.2 Effects of Decreasing Working Pressure 318 15.3 Gas Damping in the Rarefied Regime 320 15.3.1 Evaluation of Damping at Low Pressure using Kinetic Models 321 15.3.2 Linearization of the BGK Model 323 15.3.3 Numerical Implementation 324 15.3.4 Application to MEMS 325 15.4 Gas Damping in the Free-Molecule Regime 328 15.4.1 Boundary Integral Equation Approach 328 15.4.2 Experimental Validations 330 15.5 Solid Damping: Thermoelasticity 335 15.6 Solid Damping: Anchor Losses 338 15.6.1 Analytical Estimation of Dissipation 339 15.6.2 Numerical Estimation of Anchor Losses 342 15.7 Solid Damping: Additional unknown Sources – Surface Losses 346 15.7.1 Solid Damping: Deviations from Thermoelasticity 346 15.7.2 Solid Damping: Losses in Piezoresonators 346 References 348 16 Surface Interactions 351 16.1 Introduction 351 16.2 Spontaneous Adhesion or Stiction 352 16.3 Adhesion Sources 353 16.3.1 Capillary Attraction 353 16.3.2 Van der Waals Interactions 356 16.3.3 Casimir Forces 358 16.3.4 Hydrogen Bonds 359 16.3.5 Electrostatic Forces 360 16.4 Experimental Characterization 361 16.4.1 Experiments by Mastrangelo and Hsu 361 16.4.2 Experiments by the Sandia Group 362 16.4.3 Experiments by the Virginia Group 365 16.4.4 Peel Experiments 367 16.4.5 Pull-in Experiments 368 16.4.6 Tests for Sidewall Adhesion 372 16.5 Modelling and Simulation 374 16.5.1 Lennard-Jones Potential 374 16.5.2 Tribological Models: Hertz, JKR, DMT 375 16.5.3 Computation of Adhesion Energy 377 16.6 Recent Advances 380 16.6.1 Finite Element Analysis of Adhesion between Rough Surfaces 380 16.6.2 Accelerated Numerical Techniques 383 References 387 Index 393