Continuum Mechanics

This chapter presents an introduction of the fundamentals of continuum mechanics. It starts with a revision of tensor analysis that discusses the definition of tensor and coordinate transformations. In the sequence, continuum motion is treated discussing the kinematics or the geometry of motion. Definitions of strain tensors are of concern. Material derivative and Reynolds transport theorem is also treated. Afterward, a discussion about stress is presented presenting the Cauchy principle. The definition of stress tensors is established presenting Cauchy and Piola-Kirchhoff tensors. Conservation principles are then analyzed: linear and angular momentum; mass; and energy. The principle of entropy is also treated. After these definitions, it is presented a summary of fundamental equations of mechanics, discussing the importance of constitutive equations. The generalized standard material approach is discussed as a framework to elaborate constitutive equations that respect the thermodynamical principles. As examples, it is discussed the elasticity, elastoplasticity, and also smart materials phenomena as piezoelectricity, pseudoelasticity, and shape memory effect.