A general exact elastic shell solution for bending analysis of functionally graded structures

Torre

2018

Applied Sciences

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This work deals with the study of the convergence ratio of the exponential matrix method used in the 3D static analysis of functionally graded structures subjected to harmonic loads. The equilibrium equations are written in mixed orthogonal curvilinear coordinates. This feature allows plates, cylinders, spherical and cylindrical shells to be studied with the same and unique formulation. After a reduction to first order differential equations, the obtained system is solved through the thickness by means of the exponential matrix method. The coefficients of these equations are not constant because the mechanical properties of the considered functionally graded materials change through the thickness. Moreover, the curvature terms introduce a further dependence of the coefficients from the thickness coordinate. The use of several mathematical layers allows for evaluating both the material properties and the curvature terms at certain points through the thickness direction. The M number of mathematical layers to be introduced is here studied in combination with the N expansion order of the exponential matrix. These values are influenced by several parameters. Therefore, different geometries, thickness ratios, lamination sequences, functionally graded material laws and half-wave numbers are considered.

“…One physical layer (N L = 1) divided into M global mathematical layers. 3D reference solution in Brischetto [23]. NaN means Not a Number.…”

Section: Resultsmentioning

confidence: 99%

Section: D Exact Static Analysis Of Fgm Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effects of Order of Expansion for the Exponential Matrix and Number of Mathematical Layers in the Exact 3D Static Analysis of Functionally Graded Plates and Shells

Torre

2018

Applied Sciences

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“…The author proposed in [29] the 3D exact static analysis of multilayered composite and sandwich plates and shells when transverse normal loads were positioned at the external surfaces. This model was extended to the static analysis of functionally graded plates and shells in [30]. Paper [31] modified the boundary loading conditions in order to introduce combinations of transverse normal and transverse shear loads at the external surfaces for the cases analyzed in [29].…”

Section: Introductionmentioning

confidence: 99%

A 3D layer-wise model for the correct imposition of transverse shear/normal load conditions in FGM shells

International Journal of Mechanical Sciences

2018

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A general elasticity 3D layer-wise shell model is proposed for the static investigation plates and shells including functionally graded material (FGM) layers. A closed-form solution is used considering simply-supported sides and harmonic forms for displacements and loads. The partial differential equations obtained from the 3D equilibrium relations developed in curvilinear orthogonal coordinates are solved using the exponential matrix methodology. These equations have constant coefficients because an opportune number of mathematical layers has been introduced in order to calculate the parametric coefficients including the radii of curvature for the shells and the elastic coefficients that are variable through the thickness direction in the case of functionally graded materials. The main aim of the present work is to fill the gap found in the literature where the 3D elasticity theories always give solutions for FGM plates or shells in the only case of a transverse normal load positioned at the top or at the bottom surfaces. The present work proposes an exhaustive static analysis where the load boundary conditions have been appropriately rewritten in order to allow the use of several transverse normal and transverse shear loads separately or simultaneously positioned at top and/or bottom surfaces. One-layered and sandwich FGM plates, cylinders, cylindrical shells and spherical shells are analyzed changing the material laws and properties, the applied loads and the thickness ratios. The importance of the zigzag features, the interlaminar continuity in terms of compatibility and equilibrium requirements, the boundary load requirements, the considerations about the symmetry, the thickness ratio effect and the three-dimensional behavior have been opportunely discussed. Advantages connected with the use of FGM layers have also been analyzed. These new 3D exact results will allow the validation of recent advanced 2D shell models in the literature for the static investigation of FGM structures subjected to different load conditions. The proposed 3D model is general for several geometries (plates and shells) and materials (classical ones, composites and FGMs) and it allows a unique 3D exact solution for a large variety of structures.

“…All of the details and the missing steps to reach the final solution may be found in the first author's past works [22,33].…”

Section: Use Of the Exponential Matrix And Mathematical Layers In Thementioning

confidence: 99%

Convergence Investigation for the Exponential Matrix and Mathematical Layers in the Static Analysis of Multilayered Composite Structures

Torre

2017

J. Compos. Sci.

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Abstract:The exact three-dimensional analysis of a large group of geometries is accomplished here using the same formulation written in orthogonal mixed curvilinear coordinates. This solution is valid for plates, cylindrical shells, cylinders and spherical shells. It does not need specialized equations for each proposed geometry. It makes use of a formulation that is valid for spherical shells and automatically degenerates in the simpler geometries. Second order differential equations are reduced of an order redoubling the number of variables, and then they are solved via the exponential matrix method. Coefficients of equations vary through the thickness when shells are considered. M mathematical layers must be introduced into each physical layer to approximate the curvature. The correlation between M and the order of expansion N for the exponential matrix is analyzed in this paper in order to find their opportune combined values to obtain the exact results. As their effects may depend on different parameters, several geometries, lamination sequences, thickness ratios and imposed half-wave numbers are taken into consideration.