Plane Wave Solutions and Modal Analysis in Higher Order Shear and Normal Deformable Plate Theories

Batra, R.C.; Vidoli, Stefano; Vestroni, Fabrizio

doi:10.1006/jsvi.2002.5029

Cited by 67 publications

(28 citation statements)

References 17 publications

(37 reference statements)

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“…Batra et al 21 have analyzed plane traveling waves and frequencies of free vibrations of a simply supported plate. Frequencies computed with the fth-order plate theory are found to match extremely well with those computed from the three-dimensional elasticity theory.…”

Section: Discussionmentioning

confidence: 99%

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Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

2002

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A three-dimensional mixed variational principle is used to derive a Kth-order two-dimensional linear theory for an anisotropic homogeneous piezoelectric (PZT) plate. The mechanical displacements, the electric potential, the inplane components of the stress tensor, and the in-plane components of the electric displacement are expressed as a nite series of order K in the thickness coordinate by taking Legendre polynomials as the basis functions. However, the transverse shear stress, the transverse normal stress, and the transverse electric displacement are expressed as a nite series of order (K + 2) in the thickness coordinate. The formulation accounts for the double forces without moments that may change the thickness of the plate. Results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1) normal tractions and 2) tangential tractions. Results are also computed for the bending of a cantilever thick PZT beam loaded by 1) a uniformly distributed charge density on the top and the bottom surfaces and 2) equal and opposite normal tractions distributed uniformly only on a part of the beam. The seventh-order plate theory captures well the boundary-layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two. Also, through-the-thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three-dimensional elasticity equations. The governing partial differential equations are second order, so that Lagrange basis functions can be used to solve the problem by the nite element method.

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Section: Discussionmentioning

confidence: 99%

“…Lee et al 17 use the principle of virtual work to derive boundary condtions [Eqs. (21) 1 of Ref. 17] on the top and the bottom surfaces of the plate.…”

Section: Introduction Bmentioning

confidence: 99%

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

2002

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“…As observed by other authors, such as Batra et al [28,29], enforcing the equilibrium equations in the variational statement is a powerful technique for predicting accurate 3D stress fields in multilayered structures in a variationally consistent manner. However, the present contracted HR functional results in a structural model with fewer degree of freedom than the generalised model by Batra et al as the in-plane and transverse shear stresses are based on the same degrees of freedom and fewer displacement Lagrange multipliers are needed.…”

Section: Governing Equations From the Hellingerreissner Mixed-variatimentioning

confidence: 74%

“…Nevertheless, we would like to suggest two possible ways of incorporating the effects of transverse normal deformation if these effects are deemed to be significant. The first is to use the generalised approach presented by Batra and co-workers [28,29] of assuming independent Taylor or Legendre series expansions for the six stress and three displacement fields. The drawback of this approach is that the number of variables increases significantly.…”

Section: Discussionmentioning

confidence: 99%

“…Batra and Vidoli [28] and Batra et al [29] used the HR principle to develop a higher-order theory for studying vibrations and plane waves in piezoelectric and anisotropic plates, accounting for both transverse shear and transverse normal deformations with all functional unknowns expanded in the thickness direction using orthonormal Legendre polynomials. The researchers showed that the major advantage of the HR principle is that by enforcing stresses to satisfy the natural boundary conditions at the top and bottom surfaces, and by deriving transverse stresses from the plate equations directly, the stress fields are closer to 3D elasticity solutions than a purely displacement-based model that relies on the constitutive equations to compute the stress fields.…”

Section: Mixed-variational Theoriesmentioning

confidence: 99%

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A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Groh

2016

Composite Structures

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Current consensus suggests a trade-off for predicting accurate 3D stress fields in multilayered plates. Relatively expensive layerwise models facilitate accurate stress predictions from underlying models assumptions, whereas more efficient equivalent single-layer theories often rely on variationally inconsistent transverse stresses from recovery steps. In contrast, we show that variationally consistent 3D stress fields are predicted from an equivalent single-layer model using a contracted form of the Hellinger-Reissner functional. Notably, this procedure facilitates computationally efficient analysis of 3D heterogeneous plates, i.e. laminates comprising layers with material properties that differ by multiple orders of magnitude and that can also vary continuously in-plane. The model includes effects of higher-order transverse shearing and zig-zag deformations by expressing the in-plane stress field as a Taylor series expansion of global and local higher-order stress resultants. Equilibrated transverse stress assumptions are derived from Cauchy's 3D equilibrium equations, which identically satisfy the interfacial and surface traction equilibrium conditions when applied within a variational statement that enforces the 3D equilibrium equations as constraints, hence the Hellinger-Reissner mixed-variational statement. By using inherently equilibrated stress fields as model assumptions, it is shown that only the classical membrane and bending equilibrium equations have to be enforced explicitly. As a result, a contracted Hellinger-Reissner functional emerges that requires fewer displacement Lagrange multipliers than when independent stress fields are assumed. The governing equations are derived in a generalised framework such that the order of the model, and hence its accuracy, increases automatically when implemented in a computer code. By refining the order of the stress assumptions, the model is adaptable to plate-like structures ranging from thin engineering laminates to highly heterogeneous, thick laminates comprising straight-fibre or tow-steered variable-stiffness laminates, foam, honeycomb or other compliant layers.

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Wave Propagation in Orthotropic Elastic Shells: Theoretical and Numerical Modeling

Tié

Aubry

2006

Computational Methods in Engineering &Amp; Science

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Plane Wave Solutions and Modal Analysis in Higher Order Shear and Normal Deformable Plate Theories

Cited by 67 publications

References 17 publications

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Wave Propagation in Orthotropic Elastic Shells: Theoretical and Numerical Modeling

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