Fundamentals of the Mechanics of Solids

This book emerged from an “experiment in didactics” that has been under development at the University of Florence (Italy) since the academic year 2006/2007: teaching the mechanics of solids to undergraduate students in mechanical engineering in a deductive way from a few first principles, including essential elements of the description of finite-strain behavior and paying attention to the role of invariance properties under changes in observers. We do not claim originality for this program and are also conscious of the advantages of other approaches: ours is just a description of the origin of a choice dictated by personal taste and history. It has been also motivated by the consciousness that a merely descriptive style can risk reducing the treatment to a list of special examples or formulas with unexplained origin, while an inductive approach could give, even indirectly, prominence to reasoning by analogy, blurring in some way the logical structure of the theory. Since its inception, about 170 have taken the 84-hour course each year. All students had previous training in analysis and geometry, including basic elements of linear algebra. They were also trained in rational mechanics of mass points and rigid bodies, which is taught in a course of the same length. Their mathematical background had been enlarged during the course by requisite notions from tensor algebra and analysis. We collect the pertinent material in an appendix, where we clarify further the notation adopted in the text. One of us (PMM) developed the “experiment” varying the course every year according to student response and the changes in his perception. The other (LG) began later to transfer appropriate portions of the spirit of the course to analogous courses in other fields of engineering.There are several introductory textbooks on the mechanics of solids. Some of them follow a strictly deductive program rather than being primarily descriptive or having an inductive approach. We have written this book by following our personal taste, with the goal of organizing the subject matter in a way that prepares the readers for further study, being conscious that the development of mechanics could require even modifications of the first principles. Beyond the technical aspects, our conviction is that the subject must be presented in a critical way, without givingthe reader the impression that it has been constructed as an immutable structure crystallized once and for all. In fact, it seems to us that a merely dogmatic approach to mechanics does not contribute to the possibility of deeply investigating the foundational aspects of the subject. In contrast, the attention to foundational aspects is the primary tool for constructing new models, even new theories: families of interconnected models. The interest for the analysis of the theoretical foundations is not a mere interest for the formal structure of the theories; rather, it has to be stimulated in the students in university courses, even though perhaps only a few of them will be involved in research activities after the completion of their education. To us, even those who will work as professional engineers can take meaningful advantage of this type of program so that they might eventually have the flexibility to learn and (perhaps) manage new models and techniques, those that might be developed to satisfy future technological needs or, above all, for giving us a better knowledge of nature. Moreover, an attitude that favors the comprehension and analysis of the foundational aspects of mechanical theories encourages one to search for the physical meaning of every formal step we do, on the basis of our analytical, geometric, and/or computational skills. In this spirit, we begin with the definition of bodies and deformation, recovering the kinematics of the rigid ones as a special case. In this way, we establish a link with the basic courses in rational mechanics of mass points and rigid bodies, showing how the subject matter we present is a natural continuation of the previous topics. We distinguish between the space in which we select the reference point for a body and the one in which we record shapes that we consider deformed. The second space is what we consider the physical one, the first being just a “room” used for comparing lengths, areas, volumes, with their prototypical counterparts that we declare to be undeformed. This unusual distinction allows us to clarify some statements concerning changes in observers and related invariance properties. We distinguish also between material and spatial metrics, each defined in the pertinent space. Then finite-strain measures emerge from the comparison between one metric and the pullback of the other in the space where we decide to compare the two. Small-strain deformation tensors arise from the linearization process. This is the topic of Chapter 1. Chapter 2 deals with the definition of observers and a class of their possible changes, those determined by rotating and translating frames (i.e., coordinate systems) in the ambient physical space.We call these changes in observers classical. We suggest options for them, all pertaining to the way in which we alter frames in space, indeed, irrespectively of the type of body considered; in fact, the class of changes in observers is not to be confused with the class of admissible motions for a body, although the two classes intersect. In Chapter 3, we tackle the representation of bulk and contact actions in terms of the power they develop. We write just the external power on a generic part of the body and require its invariance under classes of isometric changes in observers. The integral balances of forces and couples emerge as a result. Then they are used to derive the action–reaction principle, the existence of the stress tensor, the balance equation in Eulerian and Lagrangian descriptions, the expression of the internal (or inner) power in both representations. The approach follows the spirit of a 1963 proposal by Walter Noll. Chapter 4 deals with constitutive issues. We discuss the way of restricting a priori the set of possible constitutive structures on the basis of the second law of thermodynamics—here presented as a mechanical dissipation inequality—and on requirements of objectivity. Our attention is essentially focused on nonlinear and linearized elasticity. We discuss also the notion of material isomorphism. Incidentally, when we foresee changes in observers in the reference (material) space, the requirement that the observers record the same material forces the change in observer itself to preserve the volume, according to the definition of material diffeomorphism, irrespectively of the type of body under scrutiny. Such classes of changes in observers become crucial in the description of material mutations, a topic not treated here, since it goes beyond the scope of this book. In Chapter 5, we discuss variational principles in linearized elasticity. Among them, the Hellinger–Prange–Reissner and Hu–Washizu principles are additional to the material constituting the course mentioned repeatedly above. The chapter includes also Kirchhoff’s uniqueness theorem, and the Navier and Beltrami– Donati–Michell equations. The latter equations are essential tools for the analyses developed in the subsequent chapter. We end the chapter with some remarks on two-dimensional equilibrium problems. Chapter 6 deals with the de Saint-Venant problem: the statics of a linear elastic slender cylinder, free of weight, loaded just on its bases. There are two ways of discussing such a problem: in terms of displacements or stresses. We follow the second approach and are indebted to the 1984 treatise in Italian on the matter by Riccardo Baldacci.2 The chapter ends with a proof of the basic Toupin’s theorem on the de Saint-Venant principle. Chapter 7 includes a description of some yield criteria and a discussion of their role in the representation of the material behavior. There are several criteria, introduced for various reasons, not all of the same importance. Our choice is to include in this book just the classical ones, and nothing more. In one aspect, Chapter 8 is separate from the program followed in the course mentioned above. The chapter includes director-based models of rods, a term used here in a broad sense for rods themselves, beams, shafts, columns, etc. Their description is a revisitation in terms of invariance of the external power under changes in observers—the view followed for the three-dimensional continuum—of a 1985 proposal by Juan Carlos Simo. In the chapter, we include both the finitestrain and linearized treatments; the course that we taught involved just the latter one. Chapter 9 is an overview of some bifurcation phenomena. Attention is essentially focused on the Euler rod. This book can be used variously for a course in the mechanics of solids, with the instructor selecting some parts and neglecting others. Ours is just a proposal.