End effects for a generalized biharmonic equation with applications to functionally graded materials

Abstract

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a < r < b, 0 < theta < alpha, where (r, theta) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile

Phragmén-Lindelöf Alternative Results for a Class of Thermoelastic Plate

“…Usually, these decays are exponential with the spatial distance from the finite end to the infinity. In order to have some understandings about the study of the Saint-Venant Principle, one could refer to the papers [4][5][6][7][8][9]. In recent years, the studies of Saint-Venant principle for hyperbolic or quasihyperbolic equations are abundant.…”

Structural Stability for a Coupled System of Wave Plate Type

“…Lin's study [19] was improved by Song [21,22] who studied the time-dependent Stokes flow. For more articles on the spatial behavior, see [23][24][25][26][27]. For the viscoelasticity equations, there are some recent contributions [28][29][30][31].…”

Spatial Behavior of a Coupled System of Wave-Plate Type

“…Roughly speaking, these results assert that the solution of the problem decays exponentially with distance from the boundary. For a more complete view for the spatial behavior, one could refer to [4][5][6][7][8].…”