A first course on solid mechanics

Preface Preface

In engineering schools it has become quite customary to provide an introductory course to Solid mechanics, better known in Italy as Scienza delle costruzioni, comprising two fairly distinct themes: Continuum mechanics and Statics. The first theme studies deformable bodies while the second one, in engineering and architecture schools, embraces a series of topics that go well beyond the equilibrium of rigid systems to which the word “Statics” refers. Covering, even only at an introductory level, both topics within a six-month course surfing the vast number of texts and web resources already available on the two themes, however, would give rise to a labyrinthine path for the student. With respect to Continuum mechanics the student would tackle textbooks which, for the chosen language, are largely devoted to algebra and tensor calculus basics or, for the part regarding Statics, with texts that use a completely different language and methodologies that can be placed in the field of Mechanics of structures.

The previous considerations led me to elaborate a text, certainly still in a experimental phase, which, with a language as uniform as possible, is intended to be a support for a six-month course on two topics that from the didactic point of view, and not only, have two traditions distinct enough.

Another central aspect of the proposal is the chosen medium, i.e. a web interface available online. The web interface allows using different tools - traditional text, short video lessons, animations, references to web resources - and various computational resources. The latter ones can be used for the study of the theory and for carrying out exercises.

With respect to the latter aspect, this text proposes the use of MATLAB® as symbolic and numerical processing tool. This allows the student to concentrate only on the formulation of the models and of the required mathematical operations whose actual execution can be entrusted to MATLAB®. In my opinion, the teaching of Solid Mechanics can avoid to repeat topics to be considered learned in previous courses, and must stimulate the use of IT tools which are now the basic skills of any engineer in terms of computational language to be known and automatic calculation tools to be used in the daily work.

Where does the need to know a computational language and to use automatic calculation tools come from? Probably this requirement is not at all a peculiarity of the Mechanics of solids since it is now at the basis of all the disciplines that can be placed in the field of applied mathematics. But let's limit our gaze to what interests us most, namely how an engineer does his job. Fifty years ago a fresh engineering graduate was very good at using a slide rule or a manual mechanical analog calculator. But now thanks to the digital revolution, which began in the 80s with the advent of the first personal computers and still in progress, there is the possibility of holding in one hand a computing power that in 1969 was used by NASA to send the man on the moon. To complete a drastically changed scenario, the widespread diffusion of the Internet over the last decade has done the rest by opening the door to the “internet of things”.

Returning to MATLAB®, an acronym for MATrix LABoratory, the software was born ¹ expressly as a tool for the automatic manipulation of matrices but over the years it has been enriched with such and many features that it is more correct to define it as follows: “a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks®” (source Wikipedia). Therefore, being a subject in itself very vast, no information on its use is provided in this text. This information is available, as indicated in using MATLAB®, on the MATLAB® supplier website, a source that should be considered the most reliable and authoritative. The applications and exercises provided in MATLAB® in the text therefore serve to acquire a computational language and to learn how to use an automatic calculation tool, certainly not the only one available nowadays, suitable for an uptodate approach to the study of Solid mechanics.

The material proposed here is composed of the following chapters: the first three deal with continuum mechanics and the following two are dedicated to rigid systems.

• The first chapter deals with the kinematics of deformable bodies by defining the basic tools for describing the geometric transformation of bodies regardless of the causes that produced it. The topic is used as a “gym” to get familiar ourselves with all the mathematical tools necessary for solid mechanics. However, a systematic description, which very often becomes a list, of the necessary notions of algebra and tensor calculus is avoided. But the individual topics of algebra or calculus are recalled by keeping as much contact as possible with the kinematics subject for which are required. As delivering medium I chose short video lessons that are more accessible, in a preliminary instance, than the descriptions provided in the reference texts of Continuum Mechanics or Mathematics to which the student should always draw for further information.
• In the second chapter the notion of continuity hypothesis at the basis of continuum mechanics is clarified to subsequently move on to the treatment of bodies under the action of external agents. By this way we arrive at the definition of the Cauchy stress tensor and the related equilibrium equations. The link between the static description of the bodies and the kinematic description presented in the first chapter is also established through the principle of virtual work.
• In the third chapter it is verified that the equations presented in the previous chapters are not enough to solve the problem of a deformable body, however constrained, subjected to assigned actions. Additional experimental information must be added to the model, defining a relationship between stress and deformation in a generic point of the body. To this end, Hooke's law is used as a reference model. Its generalization to pluri-axial states allows general form of the elastic constituive law. After defining the general formulation of the elastic problem, the reduction of the 3D model to the simpler and more manageable one-dimensional model for the analysisi of 2D straight beams is also discussed.
• In the fourth chapter, kinematics presented in first chapter is applied to rigid systems allowing to experiment the kinematic description of bodies in a simplified context. The main types of constraints applicable to a body are discussed and a systematic procedure to be used for the kinematic analysis of rigid body systems is identified.
• The fifth chapter is devoted to the discussion of rigid systems from a static point of view. The relative equations of equilibrium, the notion of constraint reaction and free body diagram allow to identify a systematic procedure for the static analysis of rigid body systems.

To conclude, it is observed that the order given to the chapters reflects the need to group them with respect to the two themes continuum mechanics and rigid systems, themes which are not exactly distinct but rather accessory to each other's understanding. An alternative reading order could be 1, 4, 5, 2 and 3. This order, assuming the first chapter as a prerequisite, allows to study the basic ideas with respect to the simplest rigid systems, chapters 4 and 5, and then complete continuum mechanics till to the formulation of the elastic problem, chapters 2 and 3.