Continuum mechanics and thermodynamics. From fundamental concepts to governing equations.

*(English)*Zbl 1257.82002
Cambridge: Cambridge University Press (ISBN 978-1-107-00826-7/hbk; 978-1-139-20045-5/ebook). xxii, 350 p. (2012).

Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. They are essentially nonlinear theories dealing with the macroscopic response of material bodies to mechanical and thermal loading. The present book presents a fresh perspective on these important subjects, exploring their fundamentals and connecting them with micro- and nanoscopic theories. The book gives a self-contained treatment of topics directly related to nonliner materials modeling with an emphasis on the thermo-mechanical behavior of solid-state systems. It consists of two parts. Part I focuses on the basic concepts underlying continuum mechanics and thermodynamics, explaining abstract mathematical ideas and going to the response of real materials to mechanical and thermal loading. Part II discusses applications of the theory to solve actual physical problems.

After a short introduction, Part I of the work starts with scalars, vectors and tensors from the perspective of linear algebra. Chapter 2 deals with basic physical and mathematical concepts which have to be understood before one can discuss the mechanics of continuum bodies. The kinematics of deformation of bodies is covered in Chapter 3. The study of kinematics is concerned exclusively with the abstract motion of bodies, making no consideration of the forces that cause such a motion. Thus, the reader learns how to describe all possible deformations a body can undergo. Having laid out the geometry of deformation, next, the authors turn to the laws of nature to determine how a body will respond to applied loadings. Chapter 4 deals with this question from a pure mechanical perspective taking into account only mechanical loading, i.e., Chapter 4 concentrates on the principles of classical mechanics, the conservation of mass and the balance of linear and angular momentum. The extension of these laws to continuous media leads to Cauchy’s stress principle realizing that there is no inherent difference between external forces acting on the physical surfaces of a body and internal forces acting across virtual surfaces within the body.

In reality, a material is not just subjected to mechanical loading causing stresses and strains in a body. It also experiences thermal loading which may create an internally varying temperature field. Thus, Chapter 5 considers the response of bodies to thermal loading and explains the laws of thermodynamics. The equations relating the response of a special material (e.g., steel or butter) to the loading applied to it are called constitutive relations. Such relations are discussed in Chapter 6. They are generally nonlinear expressions which are found either empirically through experimentation or more recently using multiscale modeling approaches as described in the textbook [Modeling materials. Continuum, atomistic and multiscale techniques. Cambridge: Cambridge University Press (2011; Zbl 1235.80001)] by the first two authors. The authors concentrate on the topic how continuum mechanics can place constraints on the allowable forms of these expressions taking into account the second law of thermodynamics, the principle of material frame-indifference, or the symmetry properties of the material.

Combining the introduced conservation laws with the constitutive expressions, it is possible to write down a system of coupled nonlinear partial differential equations that fully characterizes a thermo-mechanical system. Together with appropriate boundary or initial conditions a well-defined initial or boundary-value problem may be constructed. How this is done is described in Chapter 7. Here, a special emphasis is placed on purely mechanical static problems, which may be treated as a variational problem, called the “principle of minimum potential energy” (PMPE). Part I of the book ends with the discussion of the stability of mechanical systems.

Having read Part I of the work, the reader should be able to write down a complete description of any problem in continuum mechanics, but it will be almost always impossible for her to obtain a closed-form analytical solution of the problem. To solve the problem, indeed, three different courses of action exist, which are described in Part II of the book.

First, in certain cases, it is indeed possible to find closed-form solutions. Moreover, some of these solutions apply to all materials (in a given class) regardless of the form of the constitutive relations. In addition to their academic interest, these solutions have important practical implications for the design of experiments that measure the nonlinear constitutive relations for materials. The known universal solutions are presented in Chapter 8. The second option for solving a continuum problem is to adopt a numerical approach. The nowadays most popular numerical approach is the finite element method described in Chapter 9. The authors focus on static boundary-value problems and approach the problem from the perspective of the PMPE. The third option for solving continuum problems is to simplify the equations by linearizing the kinematics and/or the constitutive expressions. This method is discussed in Chapter 10. Chapter 11 contains information on further reading to expand the understanding of the topics covered in the book.

Together with its companion book [loc. cit.], the new work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, chemistry and engineering. A solutions manual is available at http://www.cambridge.org/9781107008267, along with a link to the authors’ website which provides a variety of supplementary material for both this book and [loc. cit.]. (Text composed based on the “Introduction” of the book.)

After a short introduction, Part I of the work starts with scalars, vectors and tensors from the perspective of linear algebra. Chapter 2 deals with basic physical and mathematical concepts which have to be understood before one can discuss the mechanics of continuum bodies. The kinematics of deformation of bodies is covered in Chapter 3. The study of kinematics is concerned exclusively with the abstract motion of bodies, making no consideration of the forces that cause such a motion. Thus, the reader learns how to describe all possible deformations a body can undergo. Having laid out the geometry of deformation, next, the authors turn to the laws of nature to determine how a body will respond to applied loadings. Chapter 4 deals with this question from a pure mechanical perspective taking into account only mechanical loading, i.e., Chapter 4 concentrates on the principles of classical mechanics, the conservation of mass and the balance of linear and angular momentum. The extension of these laws to continuous media leads to Cauchy’s stress principle realizing that there is no inherent difference between external forces acting on the physical surfaces of a body and internal forces acting across virtual surfaces within the body.

In reality, a material is not just subjected to mechanical loading causing stresses and strains in a body. It also experiences thermal loading which may create an internally varying temperature field. Thus, Chapter 5 considers the response of bodies to thermal loading and explains the laws of thermodynamics. The equations relating the response of a special material (e.g., steel or butter) to the loading applied to it are called constitutive relations. Such relations are discussed in Chapter 6. They are generally nonlinear expressions which are found either empirically through experimentation or more recently using multiscale modeling approaches as described in the textbook [Modeling materials. Continuum, atomistic and multiscale techniques. Cambridge: Cambridge University Press (2011; Zbl 1235.80001)] by the first two authors. The authors concentrate on the topic how continuum mechanics can place constraints on the allowable forms of these expressions taking into account the second law of thermodynamics, the principle of material frame-indifference, or the symmetry properties of the material.

Combining the introduced conservation laws with the constitutive expressions, it is possible to write down a system of coupled nonlinear partial differential equations that fully characterizes a thermo-mechanical system. Together with appropriate boundary or initial conditions a well-defined initial or boundary-value problem may be constructed. How this is done is described in Chapter 7. Here, a special emphasis is placed on purely mechanical static problems, which may be treated as a variational problem, called the “principle of minimum potential energy” (PMPE). Part I of the book ends with the discussion of the stability of mechanical systems.

Having read Part I of the work, the reader should be able to write down a complete description of any problem in continuum mechanics, but it will be almost always impossible for her to obtain a closed-form analytical solution of the problem. To solve the problem, indeed, three different courses of action exist, which are described in Part II of the book.

First, in certain cases, it is indeed possible to find closed-form solutions. Moreover, some of these solutions apply to all materials (in a given class) regardless of the form of the constitutive relations. In addition to their academic interest, these solutions have important practical implications for the design of experiments that measure the nonlinear constitutive relations for materials. The known universal solutions are presented in Chapter 8. The second option for solving a continuum problem is to adopt a numerical approach. The nowadays most popular numerical approach is the finite element method described in Chapter 9. The authors focus on static boundary-value problems and approach the problem from the perspective of the PMPE. The third option for solving continuum problems is to simplify the equations by linearizing the kinematics and/or the constitutive expressions. This method is discussed in Chapter 10. Chapter 11 contains information on further reading to expand the understanding of the topics covered in the book.

Together with its companion book [loc. cit.], the new work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, chemistry and engineering. A solutions manual is available at http://www.cambridge.org/9781107008267, along with a link to the authors’ website which provides a variety of supplementary material for both this book and [loc. cit.]. (Text composed based on the “Introduction” of the book.)

Reviewer: Claudia-Veronika Meister (Darmstadt)

##### MSC:

82-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics |

80A05 | Foundations of thermodynamics and heat transfer |