Abstract
In this paper, strain gradient theory is used in developing a mathematical model based on classical flexural Kirchhoff plate theory that can predict static response of rectangular micro-plates. The result of this new model is a sixth order differential equation. Order of differential terms in Galerkin weak form of the equation is reduced so that C2 hierarchical p-version finite elements with second order global smoothness can be used to solve the problem. With different boundary conditions, the computed deflection distribution of micro-plates is compared with those of the classical theory, in which length scale parameters are not present. A series of studies have revealed that when length scale parameters are considered, deflection of a rectangular plate decreases with increasing the length scale effect; in other words micro plates exhibit more rigidity than what is predicted by the classic model. Here, deflections are normalized with respect to results obtained from classical plate theory. Comparison of maximum deflection values obtained from the extended model for micro plates with those available from the classic plate model indicates that classical theory overestimates displacement values and the largest error is observed for square micro plates. The overestimation levels off for plates with aspect ratios greater than three.
Highlights
Effects of length scale parameter are studied for SSSS, CCCC, CCSS, CCF F and SSF F boundary conditions of rectangular micro-plate and relevant observations are discussed
Micro-plates with different aspect ratios for several boundary conditions of microplate are studied in detail
In order to account for non-local effects that are present in micro-structures, the relation between stress and strain fields can be generalized as follows [17, 20]: σ = f (σ0, ε, gn, η) where σ0 is the initial stress, ε is the strain variable, g is length scale parameter and η is strain gradient
Summary
Elastic constants in addition to the two classical Lame’s constants and contains only first-order strain gradients in the strain energy density expression. A review of the above-mentioned higher-order theories of elasticity can be found in the works of Tiersten and Bleustein [15], Vardoulakis and Sulem [16], Lakes [17]. Papargyri-Beskou and Beskos [18] investigated the static flexural response of rectangular micro-plates. Static deflection of a rectangular microplate, including one length scale parameter, is investigated by utilizing C2 hierarchical p-version finite-element setting. Effects of length scale parameter are studied for SSSS, CCCC, CCSS, CCF F and SSF F boundary conditions of rectangular micro-plate and relevant observations are discussed. Micro-plates with different aspect ratios for several boundary conditions of microplate are studied in detail
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