Chun-Fu Chen

Analytical geometrical responses in large deflection of simply supported piezoelectric layered plate under initial tension

Abstract The analytical geometrical responses in large deflection of a simply supported and layered piezoelectric circular plate under initial tension due to lateral pressure are presented. The approach follows von Karman plate theory for large deflection with a consideration of a symmetrically laminated case including a piezoelectric layer. The related nonlinear governing equations are derived in a non‐dimensional form and are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel equation or a standard Bessel equation for the lateral slope. The associated analytical solutions are developed by imposing the simply supported edge conditions of the problem. For a 3‐layered nearly monolithic plate under a low pretension and a low applied voltage upon the piezoelectric layer, the results agree well with those obtained by using the classical plate theory for a single‐layered plate under pure mechanical loading, and thus the developed approach is validated. Typical 3‐layered piezoelectric plates are then implemented and the results show that, no apparent edge effect was found for the present problem. In additions, a piezoelectric effect appears to be present only up to a moderate initial tension. For a relatively high pretension, the tension effect tends to be dominant, resulting in nearly the same results for the geometrical responses, regardless of the magnitude of the applied voltage.

Piezoelectric effect on transition behavior in large deflection of layered-bossed plate under pretension

This study investigates the piezoelectric effect on the transition behavior of a layered plate bossed with piezoelectric patches in large deflection due to lateral load. The approach follows von Karmans large deflection theory but extends to a layered case including piezoelectric patches. The governing equations thus derived are simplified to a modified Bessel or Bessel equation for the lateral slope, depending on the relative magnitude of the piezoelectric load. The associated analytical solutions were developed by imposing the clamped-end boundary condition and the interface continuity between the piezoelectric patched boss and the annular plate. The solution involves with modified Bessel functions of both the first kind and the second kind, rather than only the first kind for a flat plate in literature. A further integration is conducted to derive the expression for the center deflection for various applied voltages across the piezoelectric patches to explore how the bossed plate transits from a plate mode to a membrane mode as the pretension proceeds, namely, the transition behavior. Typical silicon-based materials commonly used for a sensing or actuating device were considered for the patched plate. A parametric study was also conducted to study the effects of varying the geometry of the patched region and also the effect of piezoelectricity. Essentially, it is observed that piezoelectric effect appears to be present only in a low pretension condition. Yet, the higher the applied voltage, the more earlier in pre-stretching for the plate to transmit to the membrane mode is revealed.

Analytical study of large deflection of piezoelectric and elastically-bossed plate under pretension

Large deflection of a piezoelectric layered plate with an elastic boss due to lateral load is studied. von Karmans plate theory of large deflection is utilized and extended to a symmetrically-layered case including a piezoelectric layer. Governing equations thus derived are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel or Bessel equation for the lateral slope, depending on the relative magnitude of the piezoelectric load. The associated analytical solutions were developed by imposing the end support condition and the interface continuity between the center boss and the annular plate. Lateral deflection and curvature were further derived by the use of the corresponding recurrence relations. The approach was implemented with typical silicon-based materials used in miniaturized devices, for various geometrical sizes of the center boss and a wide range of pretension, as a parametric study. For a nearly monolithic plate with a narrow and shallow boss under a very low applied voltage, the results agree well with those available in literature, thus the developed approach is validated. For typically bossed plates, piezoelectric effect appears to be present only in a low pretension condition. In this case, the higher the applied voltage, the greater the normalized center deflection and lateral curvature are observed. The range of initial tension for the validity of plate behavior is lowered as well, as the applied voltage is raised.

Linear analytical study of large deflection of simply supported and elastically-bossed plate under pretension

A simplified linear problem of large deflection of a pre-tensioned and elastically bossed plate due to lateral load is studied. von Karmans plate theory of large deflection is employed and extended to a simply supported and symmetrically-layered case with an elastic boss. The governing equations reduce to a modified Bessel equation for the lateral slope, by neglecting the arising nonlinear terms. The analytical solutions were developed by imposing the end support condition and the interface continuity, leading to modified Bessel functions of both the first and the second kinds, in contrast to only the first kind for a flat plate. The recurrence relations of the modified Bessel functions were then employed to derive the lateral deflection and curvature. The central deflection of the bossed plate is further evaluated to study the transition behavior. The approach was implemented with typical silicon-based materials commonly used for a sensing device, for various radial sizes and thicknesses of the center boss and a wide range of pretension, as a parametric study. For a monolithic plate with a narrow and shallow boss under a slight pretension, the solutions agree well with those of classical plate approach under free pretension, thus the presented approach is validated. For typically bossed plates, no edge effect arises around the support end and the geometrical responses show that, the plate mode in low pretension may shift to a membrane mode in a comparatively high pretension. Compared to the available clamped-ended problem, the present study reveals a relatively lower pretension for the plate to change to a membrane mode. The center boss geometry shows an apparent influence upon the structural responses but is limited to a low pretension regime. In this case, a bigger radial size for the center boss may slightly extend the magnitude of pretension for the bossed plate to retain a plate behavior. For a pretension beyond a moderate threshold magnitude, the responses tend to be unified, implying that the pretension effect dominates those of the center boss.

Transition behavior in large deflection of elastically-bossed unsymmetrically layered plate under initial tension

The transition behavior of an elastically-bossed un-symmetrically layered plate in large deflection due to lateral load is studied. von Karman plate theory for large deflection is utilized and extended to the case of a pre-stressed un-symmetrically layered plate with an elastic boss. The governing equations are simplified to a modified Bessel equation for the lateral slope, by neglecting the arising nonlinear terms following a linear consideration. Analytical solution is developed by imposing the interface continuity and the clamped-end boundary condition, which is expressible via modified Bessel functions of the first and the second kinds, rather than only the first kind for a flat plate. A subsequent integration is performed to evaluate the deflection at the center of the plate for studying the transition behavior as a function of pretension. For a nearly monolithic plate with a shallow boss, the results agree well with those of a flat plate available in literature, thus validates the presented approach. For two-layered un-symmetric plates made of typical silicon based materials, various radial sizes and thickness for the center boss are implemented. The solutions show that, widening the center boss may appreciably extend the range of pretension for the plate mode and thus enlarge the pretension for transferring to a membrane behavior.

Transition behavior in large deflection of un-symmetrically layered piezo-electric plate under initial tension

Transition behavior in large deflection of un-symmetrically layered piezoelectric plate under pretension due to lateral load is studied. von Karman plate theory for large deflection is utilized and extended to the case of an un-symmetrically layered plate including a piezoelectric layer. The derived governing equations are simplified by neglecting the arising nonlinear terms for a linear consideration, yielding a Bessel or modified Bessel equation for the lateral slope. Analytical solutions for geometrical responses expressible in terms of Bessel and modified Bessel functions were developed, respectively, depending on the relative magnitude of piezoelectric effect. For a nearly monolithic plate under a very low applied voltage, the solutions correlate well with those for a single-layered case due to uniform lateral load available in literature and thus the present approach is checked. For a two-layered un-symmetric plate made of typical silicon-based materials, the transition behaviour is seen to be apparently influenced by the piezoelectric effect only in a low pretension condition.

Nonlinear geometrical responses in large deflection of un-symmetrically layered piezo-electric plate under initial tension

The nonlinear geometrical responses in large deflection of an un-symmetrically piezo-electric layered plate under initial tension are studied. von Karman plate theory for large deflection is utilized and extended to an un-symmetrically layered plate including a piezoelectric layer. The nonlinear governing equations are derived, first, in a non-dimensional form in terms of lateral slope and radial force resultant. These equations are solved u sin g a numerical finite difference method with the aid of the clamped-ended boundary conditions of the problem and an iteration procedure, by taking the associated linear analytical solution of lateral slope as the initial guess. For an early monolithic plate under a very low applied voltage, the results agree well with available solutions for a single-layered case due to uniform lateral load in literature and thus the present approach is validated. For a two-layered un-symmetric plate made of typical silicon-based materials, the results show that piezoelectric effect seems to be apparent only up to a moderate initial tension and a moderate lateral pressure. Under this circumstance, the higher the applied voltage, the greater the central deflection; and hence the plate may transit to a membrane in a relatively low pretension condition. For a relatively high pretension or a severe lateral load, however, the piezoelectric effect becomes insignificant. Moreover, the effects of initial tension and lateral load may merge to become dominant, yielding nearly the same responses, regardless of the magnitude of the applied voltage.

Nonlinear mechanical sensitivity in large deflection of elastically-bossed sensor plate under initial tension

The nonlinear mechanical sensitivity in large deflection of an elastically bossed-layered plate under initial tension due to lateral load is studied. The approach follows von-Karman plate theory for large deflection. The nonlinear governing equations were developed, for both the center boss and the annular layered plate, in terms of lateral slope and radial force resultant. These equations were solved numerically using a finite difference method with the aid of an iteration procedure, by taking the simplified linear analytical solution of lateral slope as the initial guess for the nonlinear problem. Upon solving for the geometrical responses, the laminate constitutive law was utilized and the outermost radial stress (mechanical sensitivity) was further evaluated. For a nearly monolithic plate with a thin boss, the obtained solutions correlate very well with those available in literature for a single-layered flat plate, thus validates the presented approach. For typically-layered annular plates combined with monolithic center boss, the nonlinear behavior is found to arise normally as the lateral pressure or the intial tension turns to be large. Varying the central boss size may also have a sensible influence upon the behavior of the bossed layered plate. Specifically, mechanical sensitivity is found to increase as the rigidity, i. e., the thickness ratio between the central boss and that of the annular plate is incresed, but the magnitude decreases as the initial tension is raised.

Analytical Linear study of Large Deflection of Simply Supported Layered-Plate under Initial Tension

The problem of large deflection of a simply supported layered plate under initial tension is studied. The approach is based on von Karman plate theory for large deflection in deriving the nonlinear governing equations for the lateral slope and radial force resultant, followed by a non-dimensional scheme. To have a preliminary insight, however, only the linear problem is considered by neglecting the arising nonlinear terms, yielding a modified Bessel equation for the lateral slope. This equation is solved analytically by considering the boundary conditions of simply supported ends along the edge. The related geometrical responses are then obtained by utilizing the re-occurrence relationships between the modified Bessel functions. Emphasis is placed upon the investigation of the effects of initial tension and deviation in layer moduli upon the transition behavior between a plate and a membrane as well as the deviation in the edge zone behavior near the boundary of the plate, between a simply supported case and a clamped one, for typical micro-plates made of common silicon-based materials.

Nonlinear analysis of large deflection of bossed layered-plate under initial tension

The nonlinear problem of large deflection of a bossed and laterally loaded circular layered plate with clamped end under initial tension is studied. The approach follows Von Karman plate theory for large deflection for an isotropic symmetrically layered case simulating a typical micro-pressure sensing device. The thus derived nonlinear governing equations for lateral slope and radial force resultant were solved using a numerical finite difference method incorporated with the boundary conditions along the central boss and the clamped edge. A three-layered symmetric plate with nearly monolithic material properties and a narrowed boss was considered first and the results correlate well with available solutions for an isotropic single-layered problem. Typical bossed and layered plates are then implemented and the results show that, changing the boss width may sensibly affect the structural behavior of the layered sensing plate, in general, in a comparatively low initial tension condition. As the initial tension becomes relatively large, on the other hand, the effect of pretension appears to be dominant, rendering nearly the same results for the structural responses regardless of the boss size and the deviation in the layer moduli.