Two multilayered plate models with transverse shear warping functions issued from three dimensional elasticity equations

Loredo, Alexandre; Castel, Alexis

doi:10.1016/j.compstruct.2014.07.001

Cited by 10 publications

(18 citation statements)

References 40 publications

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“…A Navier-type strong-form solution is found for all considered laminates, where the simply-support conditions of arbitrary, non cross-ply laminates are opportunely relaxed as discussed in [36]. The warping functions and stress functions of the bHSDT and mHSDT are compared against those that have been extracted from the 3D solutions following the procedure detailed out in [31].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In [31,32], corresponding polynomial and trigonometric C 0 z zigzag models have been constructed from basis polynomial and trigonometric HSDT by using four functions, which shall be hereafter referred to as warping functions. While in these works the warping functions were obtained from transverse stress fields obtained from 3D solutions or from equilibrium equations, the present paper presents a general procedure for extending a basis higher-order shear deformation theory (bHSDT) to a multilayer higher-order shear deformation theory (mHSDT) that meets the C 0 z −requirements.…”

Section: Introductionmentioning

confidence: 99%

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A family of higher-order single layer plate models meeting Cz0-requirements for arbitrary laminates

Loredo

D′Ottavio

Vidal

et al. 2019

Composite Structures

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In the framework of displacement-based equivalent single layer (ESL) plate theories for laminates, this paper presents a generic and automatic method to extend a basis higher-order shear deformation theory (polynomial, trigonometric, hyperbolic. . . ) to a multilayer C 0 z higher-order shear deformation theory. The key idea is to enhance the description of the cross-sectional warping: the odd high-order C 1 z function of the basis model is replaced by one odd and one even high-order function and including the characteristic zig-zag behaviour by means of piecewise linear functions. In order to account for arbitrary lamination schemes, four such piecewise continuous functions are considered. The coefficients of these four warping functions are determined in such a manner that the interlaminar continuity as well as the homogeneity conditions at the plate's top and bottom surfaces are a priori exactly verified by the transverse shear stress field. These C 0 z ESL models all have the same number of DOF as the original basis HSDT. Numerical assessments are presented by referring to a strong-form Navier-type solution for laminates with arbitrary stacking sequences as well for a sandwich plate. In all practically relevant configurations for which laminated plate models are usually applied, the results obtained in terms of deflection, fundamental frequency and local stress response show that the proposed zig-zag models give better results than the basis models they are issued from.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A family of higher-order single layer plate models meeting Cz0-requirements for arbitrary laminates

Loredo

D′Ottavio

Vidal

et al. 2019

Composite Structures

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“…In reference [39], authors present a plate model with four warping functions (WFs) ϕ αβ (z) which embed the transverse shear behaviour into the displacement field. These functions are issued, for each lamination sequence, from 3D elasticity solutions.…”

Section: Introductionmentioning

confidence: 99%

“…An important conclusion of this paper is that, as all "without-normal-stretching" models use reduced stiffnesses according to the generalized plane stress assumption, the comparison of these models with an exact 3D solution has no sense for very low length-to thickness ratios like 2 or 4. The results provided by these models can however be compared to an exact solution for virtual laminates which have been highly stiffened in the z direction, see the paper [39] for more details. The corollary is that, to compare models with 3D solutions at very low length-to-thickness ratios, models must integrate the normal stretching behaviour.…”

Section: Introductionmentioning

confidence: 99%

“…It presents an interesting mechanism to adapt the slopes of the shear and normal WFs, with a choice depending on the lengthto-thickness ratio, which illustrate the need to such adaptation. In the present paper, as in the corresponding previous work [39], we use WFs that depends on length-to-thickness ratios because they are issued from 3D elasticity solutions. In the same reference [44], there is no coupling between the x and y directions in the kinematic.…”

Section: Introductionmentioning

confidence: 99%

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A multilayered plate theory with transverse shear and normal warping functions

Loredo

2016

Composite Structures

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A multilayered plate theory taking into account transverse shear and normal stretching is presented. The theory is based on a seven-unknowns kinematic field with five warping functions. Four warping functions are related to the transverse shear, the fifth to the normal stretching. The warping functions are issued from exact three-dimensional solutions. They are related to the variations of transverse shear and normal stresses computed at specific points for a simply supported bending problem. Reddy, Cho-Parmerter and (a modified version of) Beakou-Touratier theories have been retained for comparisons. Extended versions of these theories, able to manage the normal stretching, are also considered. These theories, which use the same kinematic field with different warping functions, are compared to analytical solutions for the bending of simply supported plates. Various plates are considered, with special focus on low length-to-thickness ratios: an isotropic plate, two homogeneous orthotropic plates with ply orientation of 0 and 5 degrees, a [0/c/0] sandwich panel and a [−45/0/45/90] s composite plate. Results show that models are more accurate if their kinematic fields (i) depend on all material properties (not only the transverse shear stiffnesses) (ii) depend on the length-to-thickness ratios (iii) present a coupling between the x and y directions.

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Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach

Meyer-Coors

Kienzler

Schneider

2021

Continuum Mech. Thermodyn.

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Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

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Two multilayered plate models with transverse shear warping functions issued from three dimensional elasticity equations

Cited by 10 publications

References 40 publications

A family of higher-order single layer plate models meeting Cz0-requirements for arbitrary laminates

A family of higher-order single layer plate models meeting Cz0-requirements for arbitrary laminates

A multilayered plate theory with transverse shear and normal warping functions

Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach

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