Stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. However, in contrast to the tremendous development of computational methods for large-scale problems, which have been proposed and implemented extensively in recent years, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains.
Based on the idea that every large-scale problem starts as a small-scale data problem, this book combines fundamental aspects of finite (large-strain) elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials. The problems treated in this book are drawn from the analytical continuum mechanics literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right but they are also meant to inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex physical systems.
Author(s): L. Angela Mihai
Series: Interdisciplinary Applied Mathematics, 55
Publisher: Springer
Year: 2022
Language: English
Pages: 282
City: Cham
Preface
Acknowledgements
Contents
1 Introduction
2 Finite Elasticity as Prior Information
2.1 Finite Elastic Deformations
2.1.1 Hyperelastic Materials
2.1.2 Strains
2.1.3 Stresses
2.1.4 Adscititious Inequalities
2.2 Nonlinear Elastic Moduli in Isotropic Elasticity
2.2.1 Incremental Elastic Moduli
2.2.2 Stretch Moduli
2.2.3 Poisson Functions
2.2.4 Bulk Moduli
2.2.5 Shear Moduli
2.2.6 Universal Relations
2.2.7 Torsion Moduli
2.2.8 Poynting Moduli
2.3 Nonlinear Elastic Moduli in Anisotropic Elasticity
2.4 Quasi-Equilibrated Motion
3 Are Elastic Materials Like Gambling Machines?
3.1 Stochastic Hyperelastic Models
3.1.1 Model Calibration
3.1.2 Multiple-Term Models
3.1.3 One-Term Models
3.2 Stochastic Modelling of Rubber-Like Materials
3.2.1 Hypothesis Testing
3.2.2 Parameter Estimation
3.2.3 Bayesian Model Selection
4 Elastic Instabilities
4.1 Necking
4.2 Equilibria of an Elastic Cube Under Equitriaxial Surface Loads
4.2.1 The Rivlin Cube
4.2.2 The Stochastic neo-Hookean Cube
4.2.3 The Stochastic Mooney–Rivlin Cube
4.3 Cavitation of Spheres Under Radially Symmetric Tensile Loads
4.4 Inflation and Perversion of Fibre-Reinforced Tubes
4.4.1 Stochastic Anisotropic Hyperelastic Tubes
4.4.2 Inflation of Stochastic Anisotropic Tubes
4.4.3 Perversion of Stochastic Anisotropic Tubes
5 Oscillatory Motions
5.1 Cavitation and Radial Quasi-equilibrated Motion of Homogeneous Spheres
5.1.1 Oscillatory Motion of Stochastic Neo-Hookean Spheres Under Dead-Load Traction
5.1.2 Static Deformation of Stochastic Neo-Hookean Spheres Under Dead-Load Traction
5.1.3 Non-oscillatory Motion of Stochastic Neo-Hookean Spheres Under Impulse Traction
5.1.4 Static Deformation of Stochastic Neo-Hookean Spheres Under Impulse Traction
5.2 Cavitation and Radial Quasi-equilibrated Motion of Concentric Homogeneous Spheres
5.2.1 Oscillatory Motion of Spheres with Two Stochastic Neo-Hookean Phases Under Dead-Load Traction
5.2.2 Static Deformation of Spheres with Two Stochastic Neo-Hookean Phases Under Dead-Load Traction
5.2.3 Non-oscillatory Motion of Spheres with Two Stochastic Neo-Hookean Phases Under Impulse Traction
5.2.4 Static Deformation of Spheres with Two Stochastic Neo-Hookean Phases Under Impulse Traction
5.3 Cavitation and Radial Quasi-equilibrated Motion of Inhomogeneous Spheres
5.3.1 Oscillatory Motion of Stochastic Radially Inhomogeneous Spheres Under Dead-Load Traction
5.3.2 Static Deformation of Stochastic Radially Inhomogeneous Spheres Under Dead-Load Traction
5.3.3 Alternative Modelling of Radially InhomogeneousSpheres
5.3.4 Non-oscillatory Motion of Stochastic Radially Inhomogeneous Spheres Under Impulse Traction
5.3.5 Static Deformation of Stochastic Radially Inhomogeneous Spheres Under Impulse Traction
5.4 Oscillatory Motion of Circular Cylindrical Tubes
5.4.1 Tubes of Stochastic Mooney–Rivlin Material Under Impulse Traction
5.4.2 Tubes with Two Stochastic Neo-Hookean Phases Under Impulse Traction
5.4.3 Stochastic Radially Inhomogeneous Tubes Under Impulse Traction
5.5 Generalised Shear Motion of a Stochastic Cuboid
5.5.1 Shear Oscillations of a Cuboid of Stochastic Neo-Hookean Material
6 Liquid Crystal Elastomers
6.1 Uniaxial Elastic Models
6.2 Stresses in Liquid Crystal Elastomers
6.2.1 The Case of Free Nematic Director
6.2.2 The Case of Frozen Nematic Director
6.3 A Continuum Elastic–Nematic Model
6.4 Shear Striping Instability
6.5 Inflation of a Nematic Spherical Shell
6.6 Necking of a Nematic Elastomer
6.7 Auxeticity and Biaxiality
7 Conclusion
A Notation
B Fundamental Concepts
B.1 Tensors and Tensor Fields
B.2 Polar Systems of Coordinates
B.2.1 Cylindrical Coordinate System
B.2.2 Spherical Coordinate System
B.3 Random Variables and Random Fields
B.4 Normal Distribution as Limiting Case of the Gamma Distribution
B.5 Linear Combination of Independent Gamma Distributions
B.6 Maximum Entropy Probability
Bibliography
Index
