Introduction to the Thermomechanics of Continua and Hyperbolic Systems

The primary aim of this book is to present a unified treatment of the thermomechanics of continua using the axiomatic approach typical of rational mechanics. While many books on continuum mechanics focus on specific types of continuous bodies, such as deformable solid bodies or fluids, this book adopts a general perspective. It presents the mathematical structure of balance laws and constitutive equations as a cohesive whole, with special attention given to the modern theory of constitutive equations. Notable principles such as the principle of material indifference and the contemporary interpretation of the principle of entropy are emphasized. This book will be beneficial not only to engineering students but also to students from other scientific disciplines where aspects of continuum mechanics are studied. It provides an opportunity to view traditionally distinct topics in a broader, interconnected context. To ensure self-consistency, the first part of the book addresses issues related to linear algebra, with a particular focus on linear operators within finite-dimensional vector spaces. The book then offers a detailed exploration of finite deformations of continua, followed by an overview of kinematics. It characterizes the various forces that can exist in a continuum, introduces the stress tensor, and presents the balance laws in both Eulerian and Lagrangian forms. Next, the modern theory of constitutive equations is defined, emphasizing the role of the general principles of material indifference and entropy as criteria for selecting physically acceptable classes of constitutive equations. The resulting field equations are specialized for various cases, including thermoelasticity, Eulerian fluids, Fourier-Navier‒Stokes fluids, and rigid heat conductors. In the final part of the book, partial differential equations in continuum mechanics are discussed, with particular attention given to hyperbolic systems. The method of characteristics is introduced in both linear and nonlinear cases, and the need to expand the class of solutions by introducing weak solutions is discussed, with shock waves being a significant case. As an illustrative example of a weak solution, the Riemann problem is presented for the fluid dynamic model of vehicular traffic, where cars are initially stopped at a red light and then start moving when the light turns green. Preface Acknowledgments Contents 1 Matrix Operators on Vectors 1.1 Matrix Operators and Cartesian Components 1.2 Identity Operator 1.3 Scalar Multiplication of a Matrix Operator 1.4 Sum of Two Operators 1.5 Product of Two Operators 1.6 Transpose Operator 1.7 Trace of an Operator 1.8 Determinant of an Operator 1.8.1 Expression of the Determinant for n=3 1.9 Inverse Operator 1.10 Cofactor Operator 1.11 Some Notable Identities of Matrix Operators 1.11.1 Some Notable Identities for n=3 1.12 Scalar Product Between Operators 1.13 Symmetric and Antisymmetric Operators 1.13.1 Dual Vector Associated with an Antisymmetric Operator 1.13.2 Symmetric and Antisymmetric Parts of an Operator 1.14 Deviatoric and Isotropic Parts of an Operator 1.15 Rotation Operator 1.16 Orthogonal Similarity Transformations 1.16.1 Principal Invariants of an Operator 1.17 Eigenvalues and Eigenvectors of an Operator 1.17.1 Eigenvalues and Invariants of Operator Powers 1.17.2 Eigenvalues and Eigenvectors for Symmetric Operators 1.17.3 Diagonalization of an Operator 1.17.4 Hamilton–Cayley Theorem 1.17.5 Relations Between Invariants and Derivatives of Principal Invariants in the Case n=3 1.18 Tensor Product 1.18.1 Semi-Cartesian Representation of an Operator 1.18.2 Eigenvalues and Eigenvectors of a Tensor Product for n=3 1.19 Definite Sign Operators 1.19.1 Sylvester's Criterion 1.19.2 Square Root Operator of a Definite Positive Operator 1.20 Polar Theorem 2 Deformation of a Continuum 2.1 Configuration of a Continuum 2.2 Deformation Gradient Operator 2.3 Deformation Operators 2.4 Inverse Deformation Operator 2.5 Linear Dilatation Coefficient 2.6 Shear 2.7 Surface Dilatation Coefficient 2.8 Volume Dilatation Coefficient 2.9 Incompressible Bodies 2.10 Unimodular Deformation 2.11 Homogeneous Deformation 2.12 Small Deformations 3 Kinematics of a Continuum 3.1 Velocity and Acceleration 3.2 Velocity Gradient Operator 4 Forces on a Continuum and Stress Tensor 4.1 Forces on a Continuum 4.2 Cauchy's Theorem and Stress Tensor 5 Balance Laws in Continuum Mechanics 5.1 Conservation of Mass Law 5.1.1 Lagrangian Formulation 5.1.2 Eulerian Formulation 5.2 Cardinal Equations 5.2.1 Boundary Conditions 5.3 Principle of Virtual Work 5.4 General Balance Laws 5.4.1 Transport Theorem 5.4.2 Energy Balance Law 5.4.3 Thermomechanical Balance Laws in Eulerian Form 5.4.4 Galilean Invariance 5.4.5 Lagrangian Formulation of Balance Laws 5.4.6 Balance Law of Momentum in Lagrangian Form and the First Piola–Kirchhoff Stress Tensor 5.4.7 Boundary Conditions in Lagrangian Variables 5.4.8 Balance Laws of Energy in Lagrangian Variables 5.5 Physical Interpretation of the First Piola–Kirchhoff Tensor 5.5.1 Second Piola–Kirchhoff Tensor 5.5.2 Power of Internal Forces in Terms of Piola–Kirchhoff Tensors 6 Constitutive Equations 6.1 General Principles for Constitutive Laws 6.1.1 Principle of Material Indifference 6.1.2 The Entropy Principle 7 Elasticity and Thermoelasticity 7.1 Elastic Bodies 7.1.1 Consequences of the Principle of Material Indifference in the Elastic Case 7.2 Thermoelastic Bodies 7.2.1 Principle of Material Indifference in Thermoelasticity 7.2.2 Field Equations of Thermoelasticity 7.2.3 Consequences of the Entropy Principle in Thermoelasticity 7.2.4 Isotropic Materials 7.3 Principle of Dissipation in Elasticity 7.3.1 Compressible Mooney–Rivlin Potential 7.3.2 Incompressible Hyperelastic Solids 7.3.3 Nonlinear One-Dimensional Elasticity 7.4 Linear Elasticity 7.4.1 Linear Isotropic Elasticity Equations 7.4.1.1 Boundary Conditions 7.4.1.2 Plane Waves and Propagation Velocity 7.4.1.3 Wave Equation 8 Fluids 8.1 Ideal Fluids and Euler's Equations 8.1.1 Boundary Conditions for Ideal Fluids 8.1.2 Work of Internal Forces in an Ideal Fluid 8.2 Fourier–Navier–Stokes Dissipative Fluids 8.3 Entropy Principle for a Fluid 8.4 Some Special Cases of Fluids 8.4.1 Perfect Gases 8.4.1.1 Static Solution of a Gas in the Presence of Gravity 8.4.2 Incompressible Navier–Stokes Fluids 8.4.3 Compressible Euler Fluids 8.4.3.1 Linearized Equations and Speed of Sound 8.4.4 Incompressible Euler Ideal Fluids and Bernoulli's Theorem 8.5 Fluid Equations in Lagrangian Variables 9 Rigid Heat Conductor 9.1 Heat Equation 9.2 Cattaneo Equation 10 Hyperbolic Systems 10.1 Classification 10.2 Examples of Hyperbolic Systems 10.2.1 Euler Equations 10.2.2 Cattaneo's Equation 10.2.3 Thermoelasticity Equations 10.3 Wave Equation and Method of Characteristics 10.3.1 Homogeneous Linear Systems 10.4 A Nonlinear Example: The Burgers' Equation 11 Weak Solutions and Shock Waves 11.1 Shock Waves and Weak Solutions 11.2 Shock Waves in Eulerian Fluid 11.2.1 Entropy Growth and Admissible Shocks 11.2.2 Characteristic Shocks 11.3 Riemann Problem and Nonuniqueness of Weak Solutions 11.4 Lax Conditions and Entropy Growth 11.5 Traffic Flow 11.5.1 The Traffic Light Problem 11.6 Riemann Problem for a General System 12 Beyond Classical Thermomechanics 12.1 Outlook References Author Index Analitical Index