Fundamentals of Continuum Mechanics: With Applications to Mechanical, Thermomechanical, and Smart Materials

Book Preface

CONTINUUM MECHANICS: THE NEW PEDAGOGY

Since my days as a graduate student at Berkeley in the early 1980s, the graduate  engineering mechanics curriculum has undergone three major changes: First, in previous years, the curriculum left time for courses in linear elasticity, finite elasticity, plasticity, viscoelasticity, inviscid fluid dynamics, and viscous fluid dynamics, followed by a unifying course in continuum mechanics. Today, with the onslaught of new materials and technological advances competing for space in the curriculum, the reality is that there is no longer room for this many traditionalmechanics courses. Second, back in the day, much of the teaching and learningwas accomplished through homework exercises worked by the students on their own; graded and annotated by the professor; and then returned to the students. Although the most effective, this approach is impractical given today’s time constraints on both the student and the professor. As such, there is a risk that important concepts may be overlooked if they are not illuminated through worked examples. Third, mathematics and applied mechanics have diverged, and this gap continues to widen. As Courant and Hilbert [1] mused in their treatise Methods of Mathematical Physics, and is still the case today: “Since the seventeenth century, physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinement and emphasized the postulational side of mathematics, and at times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and smaller rivulets and dry out.”

This textbook adjusts to each of these realities: First, the material is covered in themost time-efficient manner, that is, by first giving the unified situation (continuum mechanics), then applying it to special cases (finite elasticity, viscous fluid dynamics, and so on). Because these special cases are presented in a single textbook, the handoff between one subject and another is cleaner, and undue redundancy is avoided. Second, the majority of the problems in the textbook are presented as worked examples with full, detailed solutions. Each of these problems is designed to convey an important concept. Third, we place a strong emphasis on explicitly connecting the mathematics to the continuum physics. Indicial notation is jettisoned almost entirely in favor of the more compact and elegant direct notation, allowing us to be more fundamental in our treatment and cover much more material in a single book. Furthermore, we restrict our development of the mathematical parlance to only that which is required to rigorously present the physical concepts.

Continuum mechanics is presented here as a unifying course in the sense that it separates those concepts that are true for all materials (i.e., the fundamental laws) from those that vary from material to material (i.e., the constitutive equations). Our textbook is structured as follows: In Chapter 2, we discuss the mathematical  world in which continuum mechanics lives and acquaint the reader with direct notation. Once the reader is fluent in direct notation, which is the primary goal of Chapter 2, he or she will be able to think of physical quantities such as velocity and stress as elements of a vector space rather than just in terms of their components. This insight is powerful, allowing the reader to proceed further conceptually than is possible with component notation and enabling a more transparent interplay between the mathematics and the physics. Chapter 3 covers motion and deformation, which is merely a discussion of geometry. (My apologies to those researchers who have devoted their lives to the study of geometry.) Chapter 4 develops the fundamental laws (or first principles), valid for all materials. Chapter 5 introduces the notion of constitutive equations, which describe how different materials respond to loading and deformation.

A distinguishing feature of this book is the postulation of constitutive equations for various materials in their most general form, and their subsequent simplification using restrictions imposed by the second law of thermodynamics, invariance, conservation of angular momentum, and material symmetry. This is illustrated for elastic solids in Chapter 6 and viscous fluids in Chapter 7. Chapter 8 presents both traditional and modern approaches to modeling constraints such as incompressibility and explores the role of stability in constitutive modeling. Finally, Chapter 9 discusses coupled thermo-electro-magneto-mechanical behavior and illustrates the development of field theories for smart materials such as piezoelectrics.