Structural continuum theories require a proper treatment of the kinematic, kinetic, and constitutive issues accounting for possible sources of non-local and non-classical continuum mechanics concepts and solving associated boundary value problems. In the case of structural mechanics, there is a wide range of theories, from higher gradient to truly nonlocal;however, there is a need for physical explanations and experimental observations that systematically corroborate these theories and provide physical interpretations of the parameters introduced in these theories. In this lecture, an overview of the author’s recent research with many colleagueson non-local elasticity and couple stress theories in developing the governing equations of functionally graded material beams and plates will be presented. In addition to Eringen’s non-local elasticity (1972), two different nonlinear gradient elasticity theories that account for (a) geometric non-linearity and (b) micro structure-dependent size effects are revisited to establish the connection between them. The first the oryis based on modified couple stress theory of Mindlin (1963)and the second one is based on Srinivasa Reddy gradient elasticity theory (2013). These two theories are used to derive the governing equations of beams and plates. In addition, the graph-based finite element framework (GraFEA) suitable for the study of damage in brittle materials will be discussed (see Reddy and Srinivasa (2015) and advanced by Khodabakhshi,Reddy, and Srinivasa (2016)).
