Basic Continuum Mechanics

The purpose of this chapter is to provide an introduction into basic continuum mechanical concepts, including stress and strain tensor and equilibrium conditions. Using the concepts introduced in this chapter, we will derive simple solutions to elastic beam bending problems to demonstrate how problems can be addressed in the realm of continuum mechanics. The objective of solving such elasticity problems is to provide a link between applied forces, boundary conditions, and the stresses, strains, and displacements inside the material. Such information is critical for instance to find out if a designed structure can operate safely, away from its failure condition. There are several additional applications and significant components of elasticity problems in many other fields, as will be discussed later in this chapter. Topics covered in this chapter include definition of elasticity, elastic response, energetic vs. entropic elasticity, Young’s modulus, stress and strain tensor, as well as the mechanics of a beam. The continuum mechanics concepts introduced in this chapter will be used later on, when a direct comparison between atomistic and continuum methods is carried out.