Handout of Elasticity

Abstract

Context and Motivation The mechanical behavior of solid bodies under external loading constitutes one of the most fundamental and practically significant areas of engineering science. From the analysis of civil infrastructure to the design of aerospace structures, the ability to predict how materials deform, develop internal stresses, and ultimately reach their limits of safe operation is central to the work of structural and mechanical engineers. Elasticity theory provides the rigorous mathematical and physical framework that underpins all such analyses, offering tools to describe reversible deformation, stress distributions, and the recovery of shape upon removal of loads. The present course handout is designed for first-year Master's students in Civil Engineering, specializing in Structures, at Ziane Achour University of Djelfa. It responds to the need for a unified, self-contained, and pedagogically accessible treatment of elasticity theory at an advanced undergraduate and early graduate level. The course spans two theoretical pillars: the mathematical formalism necessary to express field equations in compact and invariant form, and the physical insight required to interpret solutions in terms of real engineering phenomena. Scope and Objectives The primary objective of this course is to equip students with both the analytical and conceptual tools needed to study the mechanical behavior of elastic bodies and structural elements. Students are expected to gain a deep understanding of the interrelated concepts of stress, strain, and constitutive laws, and to develop the capacity to formulate and solve elasticity problems in two and three dimensions. The course also prepares students for the study of advanced structural analysis techniques, in particular the finite element method, which is now indispensable in professional engineering practice. Beyond the acquisition of analytical skills, this course aims to cultivate a rational and critical approach to structural modeling. By working through the derivation of governing equations from first principles, students learn to distinguish between exact solutions and engineering approximations, to recognize the limitations of classical theories, and to apply simplifying assumptions judiciously in practical contexts. This capacity for theoretical rigor combined with engineering judgment is the hallmark of a well-trained structural engineer. Structure of the Course The course is organized into eight chapters, each addressing a distinct but interconnected aspect of elasticity theory. The first chapter introduces the foundational concepts of continuum mechanics, including the classification of elastic materials, Hooke's

Law in its elementary form, and the mathematical tools of index notation and tensor algebra. These tools are essential for the compact and invariant expression of field equations throughout the course. Chapters Two and Three are devoted to the theory of stress and strain states, respectively. In Chapter Two, the Cauchy stress tensor is introduced, including its physical interpretation, principal stresses, stress invariants, the differential equations of equilibrium, and Mohr's circle for graphical representation. Chapter Three treats the kinematics of deformation, covering the Green–Lagrange and linearized strain tensors, the small perturbation hypothesis, strain–displacement relations in multiple coordinate systems, principal strains, compatibility equations, and experimental measurement techniques. Chapter Four presents the constitutive laws governing the elastic response of materials, including the generalized Hooke's law in one, two, and three dimensions, the role of Lamé and engineering elastic constants, orthotropy and anisotropy, thermoelastic effects, and the strain energy concept. Chapter Five consolidates these developments into the general equations of linear elasticity, specifically the Navier–Cauchy displacement equations and the Beltrami–Michell stress compatibility equations, together with Saint-Venant's principle. The remaining three chapters shift the focus toward structural applications. Chapter Six develops the theory of plane elasticity, including plane stress and plane strain formulations and the Airy stress function approach for solving two-dimensional problems analytically. Chapter Seven addresses beam bending through both the Euler–Bernoulli and Timoshenko beam theories, comparing their assumptions, governing equations, and ranges of applicability. Chapter Eight introduces the study of thin plates using the Kirchhoff–Love model, covering plate kinematics, constitutive behavior, equilibrium equations, the governing biharmonic equation, and strain energy formulations. Prerequisites and Learning Outcomes This course assumes a solid background in mathematics, particularly in linear algebra, differential equations, and vector calculus, as well as a working knowledge of Strength of Materials at the undergraduate level. Upon successful completion of the course, students will be able to apply index notation and tensor calculus to formulate elasticity problems; analyze stress and strain states including principal values, invariants, and graphical representations; use constitutive laws to relate stress and strain for isotropic, orthotropic, and anisotropic materials; derive and apply the Navier–Cauchy and Beltrami–Michell equations; solve plane elasticity problems using Airy stress functions; analyze beam bending using Euler–Bernoulli and Timoshenko theories; and study the bending behavior of thin elastic plates. Together, these competencies form a comprehensive foundation for graduate-level work in structural mechanics, numerical methods, and advanced material design, positioning

students to contribute meaningfully to research and professional practice in civil and structural engineering.