Electro- and Magneto-Mechanics of Soft Solids: Constitutive Modelling, Numerical Implementations, and Instabilities (CISM International Centre for Mechanical Sciences, 610)

This book examines the electro- and magneto-mechanics of soft composite materials and structures, and focuses on magnetorheological elastomers (MREs) and dielectric elastomer composites (DECs), which are composite materials that comprise ferromagnetic and high-dielectric/conducting filler nano- and micro-particles embedded in a soft polymeric matrix. This gives rise to a coupled magneto- and electro-mechanical response at the macroscopic (order of millimeters and larger) scale when they are subjected to magneto- electro-mechanical external stimuli. While such MRE and DEC materials and devices can become unstable at some critical electro-magneto-mechanical loading, their response may be well controlled in the post-instability regime. Moreover, recent advances on the complete electro-magneto-mechanical coupling are presented. All those aforementioned features motivate the operation of these devices in this unstable region to obtain controlled pattern formation, soft robotic motion and artificial muscles, controllable band-gap acoustic and electromagnetic properties, energy harvesting as well as actively controlled stiffness (for cell-growth). The book contains four individual chapters covering work on the fundamentals (O. Lopez-Pamies) and the modeling (M. Gei) of electroactive solids, the modeling of magnetoactive solids (K. Danas), and the analysis of elastic instabilities (Y. Fu). Preface Contents The Elastic Dielectric Response of Elastomers Filled with Liquid Inclusions: From Fundamentals to Governing Equations 1 Introduction 2 Initial Configuration and Kinematics 3 Conservation of Mass 4 Maxwell's Equations in the Presence of Material Interfaces 4.1 Bulk and Interface Charges, Electric Fields, and Electric Displacements 4.2 Gauss's Law 4.3 Faraday's Law 5 Balance of Momenta in the Presence of Material Interfaces 5.1 Bulk and Interface Electric and Mechanical Forces 5.2 Balance of Linear Momentum 5.3 Balance of Angular Momentum 6 Constitutive Behavior 6.1 Constitutive Behavior of the Bulk: The Solid Matrix and the Liquid Inclusions 6.2 Constitutive Behavior of the Solid/liquid Interfaces 7 Governing Equations 7.1 Boundary Conditions 7.2 The Choice of Independent Fields 7.3 The Strong Form of the Governing Equations 7.4 Residual Stresses References Modelling of Homogeneous and Composite Non-linear Electro-Elastic Elastomers 1 Introduction 2 Modelling of Electro-Elastic Materials 2.1 Electro-Elastic Constitutive Equations 2.2 Finite Electro-Elastic Actuation of Thin Films 3 Linearized Incremental Deformations 4 Global Bifurcations of Soft Dielectric Elastomers 4.1 Electro-Mechanical Instability 4.2 Diffuse-Mode Instabilities 5 Electro-Mechanics of Laminated Composites Under Plane-Strain Conditions 5.1 Constitutive Assumptions 5.2 Homogenized Solution Controlling the Voltage and Boundary Conditions 5.3 Macroscopic Performance 6 Introduction to Mechanical-to-Electrical Energy Conversion 6.1 Model of a DE Generator 6.2 The Load-Driven Harvesting Cycle and the Area of Admissible Configurations 6.3 Optimization of the Harvesting Cycle 7 Conclusions References A Unified Theoretical Modeling Framework for Soft and Hard Magnetorheological Elastomers 1 Introduction 2 Preliminary Definitions in Magneto-Elasto-Statics 2.1 Finite Strain Kinematics 2.2 Magnetostatics 3 Thermodynamics and General Variational Formulations 3.1 Scalar Potential-Based F-H Formulation 3.2 Vector Potential-Based F-B Formulation 4 Modeling of Isotropic Hard-MREs 4.1 Internal Variable for Magnetic Dissipation 4.2 General Properties of the Free Energy Density and the Dissipation Potential 4.3 The Isotropic Magneto-Mechanical Invariantspg for hh-MREs 4.4 Form of Energy Densities 4.5 The Mechanical Energy Density 4.6 The Magnetic and Coupled Energy Densities 4.7 The Dissipation Potential 4.8 Total Cauchy Stress in hh-MREs 5 Modeling of Isotropic Soft-MREs 5.1 F-H Expressions for ss-MREs 5.2 F-B Expressions for ss-MREs 5.3 Total Cauchy Stress in ss-MREs 6 Numerical Implementations for MREs 6.1 Time Discrete Variational Principle for F-H Formulation 6.2 Time Discrete Variational Principle for F-B Formulation 6.3 The Periodic Numerical Homogenization Problem 7 Results: Periodic RVE Simulations and Model Assessment 7.1 hh-MRE Models Versus FE Simulations 7.2 Magnetization Independent of Stretching in MREs 7.3 NdFeB-Based hh-MRE Versus CIP-Based ss-MRE Response 7.4 Energetic ss-MRE Models Versus hh-MRE Models with Zero Dissipation 8 Results: Numerical BVP Simulations 8.1 Generic Numerical BVP Setting 8.2 Treatment of Air 8.3 Magnetostriction and Magnetization Response of a Spherical ss-MRE Specimen 8.4 Uniformly Pre-magnetized hh-MRE Cantilever Beams 8.5 Non-uniformly Pre-magnetized, Functionally-Graded hh-MRE Cantilever Beams References Elastic Localizations 1 An Example of Bifurcation at Zero Wavenumber 2 Localized Bulging of an Inflated Hyperelastic Tube 2.1 Bifurcation Condition and Near-Critical Behaviour 2.2 Graphical Illustration of the Bifurcation Condition 2.3 Bulge Evolution—Fully Nonlinear Analysis 2.4 A 1D Gradient Model Under the Membrane Assumption 2.5 Evaluation of the Infinite Length Assumption 2.6 Tubes of Finite Wall Thickness 3 Necking of a Solid Cylinder Under Surface Tension 4 Axisymmetric Necking in a Circular Hyperelastic Plate Under Equibiaxial Stretching 4.1 Governing Equations and the Primary Solution 4.2 Linear Analysis 4.3 Weakly Nonlinear Analysis 4.4 Fully Nonlinear Regime 5 Conclusion References