This monograph presents a comprehensive and rigorous new framework for the theoretical description and modelling of enriched continua. In other words, continua that exhibit more complex behaviour than their conventional counterparts and, in particular, multicomponent systems.
It employs gradient theories, exhibiting multiple transition layers described by phase fields. As a point of departure, we account for multiple continuum kinematic processes, including motion and various phase fields. These gradient theories arise by considering various kinematic processes which are tightly linked to the level of the arbitrariness of the Euler–Cauchy cuts. The surface defining the Euler–Cauchy cut may lose its smoothness along a curve, and the curve may also lose its smoothness at a point. Additionally, we postulate the principle of virtual power on surfaces. Then, the first and second laws of thermodynamics with the power balance provide suitable and consistent choices for the constitutive equations. Finally, the complementary balances, namely the balances on surfaces, are tailored to coincide with different parts of the boundaries of the body. These surface balances are then called environmental surface balances and aid in determining suitable and consistent boundary conditions. Ultimately, the environmental surface power balance is relaxed to yield an environmental surface imbalance of powers, rendering a more general type of boundary condition.
A detailed introduction sets the scene for the mathematical chapters that follow, ensuring that graduate students and newcomers can profit from the material presented.
Author(s): Luis Espath
Publisher: Springer
Year: 2023
Language: English
Pages: 160
City: Cham
Preface
Acknowledgments
Contents
Part I Point of Departure and Machinery
1 Introduction
1 Fields, Hyperfields & Couple-Fields
2 The Principle of Virtual Powers' Formalism
2 Integro-Differential Machinery
1 Differential Relations on a part
2 Differential Relations on surfaces
3 Differential Relations on curves
4 Integro-Differential Relations on surfaces
5 Integro-Differential Relations on curves
6 Transport Relations and Mass Balance
Part II The Principle of Virtual Powers and Complementary Balances
3 Power Balance, Fields, and Hyperfields
1 The Principle of Virtual Powers on bar P in B
2 Inertia Components
4 Complementary Balances, Jump Conditions, and Couple-Fields
1 Frame Indifference
2 Partwise Balances on P in B
3 Action-Reaction Principle
4 Balances on S in P
5 Jump Conditions
5.1 Jump Conditions Arising from Surface Balances
6 The Principle of Virtual Powers on bar S in bar P
Part III Thermodynamics and Constitutive Relations
5 Thermodynamics
1 First and Second Laws of Thermodynamics for a Spatial Part Pt
2 Partwise and Pointwise Free-Energy Imbalances for Isothermal Processes for a Spatial Part Pt
3 First and Second Laws of Thermodynamics for a Control Volume bar P
4 Partwise Free-Energy Imbalances for Isothermal Processes for a Control Volume bar P
5 Derivatives, Inner Products, and Consequences in Constrained Spaces
5.1 Derivatives
5.2 Inner Products
6 Representation Theorem for Isotropic Tensors
6.1 Decomposition of Tensors
6.2 Fourth-Order Isotropic Tensor
6.3 Sixth-Order Isotropic Tensor
6.4 Eighth-Order Isotropic Tensor
6 Coupling
1 Nonconserved Coupling with an Incompressible Fluid
2 Conserved Coupling with an Incompressible Fluid
Part IV Environmental Conditions and Boundary Conditions
7 Environmental Surface Balances and Imbalances
1 Surface Balances
2 Surface Imbalances
3 Uncoupled Conditions
8 Boundary Conditions
1 Natural Boundary Conditions
2 Essential Boundary Conditions
3 Mixed Boundary Conditions
Part V A Special Theory
9 Bulk-Surface Dynamics
1 Introduction
1.1 Synopsis of Purely Variational Models
1.2 Synopsis of this Work
2 Virtual Power Principle
3 Conserved Species
4 Free-Energy Imbalance
5 Additional Constitutive Response Functions
5.1 Further Connections: Boundary Conditions
5.2 Specialized Equations
6 Decay Relations
6.1 Decay Relations for Mixed Boundary Conditions
Part VI Conclusion
10 Summary and Specializations
1 Final Remarks
Appendix Bibliography
