Asymptotic analysis of plates in static and dynamic strain gradient elasticity

Abstract

We study the steady-state and transient responses of a second-order elastic plate by implementing an asymptotic analysis of the three-dimensional equations with respect to two geometric characteristics seen as parameters: the thickness of the plate and an inner material length. Depending on their ratio, four different models arise. Conditions under which Reissner–Mindlin kinematics may appear are discussed while the influence of crystalline symmetries is studied. The transient situation is solved through Trotter’s theory of approximation of semi-groups of operators acting on variable spaces.

Keywords

• Approximation of semi-groups in the sense of Trotter
• Asymptotic analysis
• Strain gradient elasticity
• Plate models
• Transient problems
• $m$-dissipative operators
• Approximation of semi-groups in the sense of Trotter