<table><caption>**Tableau 5.14** Fréquences propres non-dimensionnelles (ω̄) d'une plaque sandwich carrée avec différentes séquences d'empilement dans les peaux</caption><thead><tr><th rowspan="2">Références</th><th rowspan="2">Modèles EF</th><th rowspan="2">Différentes types de séquences d'empilement</th><th colspan="4">Fréquences (Hz)</th></tr><tr><th>Mode 1</th><th>Mode 2</th><th>Mode 3</th><th>Mode 4</th></tr></thead><tbody><tr><td>Présent élément</td><td>QSFT52</td><td></td><td>14.582</td><td>27.494</td><td>28.110</td><td>36.619</td></tr><tr><td>Malekzadeh, K et al. [314]</td><td>FEM-3D-LW</td><td></td><td>14.740</td><td>26.830</td><td>27.530</td><td>35.600</td></tr><tr><td>Meunier et Shenoi [274]</td><td>Analytical-HSDT</td><td></td><td>15.280</td><td>28.690</td><td>30.010</td><td>38.860</td></tr><tr><td>Nayak et al. [277]</td><td>FEM-Q9-HSDT</td><td>[0/90/0/core/0/90/0]</td><td>15.040</td><td>28.100</td><td>29.200</td><td>37.760</td></tr><tr><td>Nayak et al. [277]</td><td>FEM-Q4-HSDT</td><td></td><td>15.340</td><td>30.180</td><td>31.960</td><td>40.940</td></tr><tr><td>Jam et al. [291]</td><td>Analytical-LW1</td><td></td><td>15.040</td><td>26.733</td><td>27.329</td><td>35.316</td></tr><tr><td>Jam et al. [291]</td><td>Analytical-LW2</td><td></td><td>15.061</td><td>26.753</td><td>27.343</td><td>35.330</td></tr><tr><td>Présent élément</td><td>QSFT52</td><td></td><td>15.534</td><td>28.066</td><td>28.066</td><td>36.619</td></tr><tr><td>Malekzadeh, K et al. [314]</td><td>FEM-3D-LW</td><td></td><td>15.810</td><td>27.230</td><td>27.230</td><td>36.260</td></tr><tr><td>Meunier et Shenoi [274]</td><td>Analytical-HSDT</td><td></td><td>16.380</td><td>29.650</td><td>29.650</td><td>40.000</td></tr><tr><td>Nayak et al. [277]</td><td>FEM-Q9-HSDT</td><td>[45/-45/45/core/-45/45/-45]</td><td>16.090</td><td>28.930</td><td>28.930</td><td>38.760</td></tr><tr><td>Nayak et al. [277]</td><td>FEM-Q4-HSDT</td><td></td><td>16.430</td><td>31.170</td><td>31.170</td><td>42.780</td></tr><tr><td>Jam et al. [291]</td><td>Analytical-LW1</td><td></td><td>15.786</td><td>27.316</td><td>27.316</td><td>36.216</td></tr><tr><td>Jam et al. [291]</td><td>Analytical-LW2</td><td></td><td>15.799</td><td>27.329</td><td>27.329</td><td>36.220</td></tr></tbody></table>
<table><caption>**Tableau 5.15** Fréquences propres non-dimensionnelles (ω̄), pour différents modes, d'une plaque carrée composite stratifiée biaise (90/0/90/0/90) avec deux types de conditions aux limites (SSSS et CCCC)</caption><thead><tr><th rowspan="2">Skew angle</th><th rowspan="2">Références</th><th rowspan="2">Modèles EF</th><th colspan="7">Fréquences (Hz)</th></tr><tr><th>Mode 1</th><th>Mode 2</th><th>Mode 3</th><th>Mode 4</th><th>Mode 5</th><th>Mode 6</th><th>Mode 7</th></tr></thead><tbody><tr><td colspan="10">Simplement appuyée (SSSS)</td></tr><tr><td rowspan="4">0°</td><td>Présent élément</td><td>QSFT52</td><td>1.6020</td><td>3.0936</td><td>3.9511</td><td>4.7734</td><td>5.3915</td><td>6.5139</td><td>6.6478</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>1.5699</td><td>3.0371</td><td>3.7324</td><td>4.5664</td><td>5.1469</td><td>6.0343</td><td>6.1820</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>1.5703</td><td>2.8917</td><td>3.8041</td><td>4.5314</td><td>4.7273</td><td>5.8888</td><td>6.2125</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>1.5699</td><td>3.0372</td><td>3.7325</td><td>4.5664</td><td>5.1475</td><td>6.0347</td><td>6.1825</td></tr><tr><td rowspan="3">15°</td><td>Présent élément</td><td>QSFT52</td><td>1.6556</td><td>3.1025</td><td>4.0721</td><td>4.7873</td><td>5.4713</td><td>6.4469</td><td>6.8284</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>1.6877</td><td>3.0458</td><td>4.0264</td><td>4.4818</td><td>5.1533</td><td>5.8277</td><td>6.4910</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>1.6874</td><td>3.1413</td><td>3.9600</td><td>4.6073</td><td>5.4606</td><td>6.0880</td><td>6.3125</td></tr><tr><td rowspan="4">30°</td><td>Présent élément</td><td>QSFT52</td><td>1.8572</td><td>3.1609</td><td>4.4985</td><td>4.8378</td><td>5.7640</td><td>6.4343</td><td>7.4741</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>2.0844</td><td>3.5127</td><td>4.6997</td><td>4.8855</td><td>6.2494</td><td>6.3118</td><td>7.2533</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>2.0840</td><td>3.4023</td><td>4.7176</td><td>4.7674</td><td>5.9852</td><td>6.1177</td><td>7.1966</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>2.0884</td><td>3.5147</td><td>4.7033</td><td>4.8864</td><td>6.2514</td><td>6.3149</td><td>7.2570</td></tr><tr><td rowspan="4">45°</td><td>Présent élément</td><td>QSFT52</td><td>2.3630</td><td>3.3989</td><td>4.9769</td><td>5.4698</td><td>6.5139</td><td>6.6802</td><td>8.4955</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>2.8825</td><td>4.2823</td><td>5.5868</td><td>6.1808</td><td>6.9022</td><td>7.8390</td><td>8.2114</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>2.8925</td><td>4.1906</td><td>5.4149</td><td>6.2868</td><td>6.6302</td><td>7.7433</td><td>7.8398</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>2.8932</td><td>4.2852</td><td>5.5886</td><td>6.1874</td><td>6.9035</td><td>7.8366</td><td>8.1971</td></tr></tbody></table>
<table><caption>**Tableau 5.15 Suite**</caption><thead><tr><th rowspan="2">Skew angle</th><th rowspan="2">Références</th><th rowspan="2">Modèle EF</th><th colspan="7">Fréquences (Hz)</th></tr><tr><th>Mode 1</th><th>Mode 2</th><th>Mode 3</th><th>Mode 4</th><th>Mode 5</th><th>Mode 6</th><th>Mode 7</th></tr></thead><tbody><tr><td colspan="10" style="background-color: #E0E0E0; font-weight: bold;">Encastrée (CCCC)</td></tr><tr><td rowspan="4">0°</td><td>Présent élément</td><td>QSFT52</td><td>2.4317</td><td>3.8296</td><td>4.3472</td><td>5.2745</td><td>5.9913</td><td>6.8568</td><td>7.0147</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>2.3820</td><td>3.7383</td><td>4.1122</td><td>5.0366</td><td>5.6127</td><td>6.2393</td><td>6.5591</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>2.3687</td><td>3.5399</td><td>4.2134</td><td>4.9852</td><td>5.1895</td><td>6.2801</td><td>6.4666</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>2.3820</td><td>3.7383</td><td>4.1122</td><td>5.0366</td><td>5.6133</td><td>6.2397</td><td>6.5595</td></tr><tr><td rowspan="3">15°</td><td>Présent élément</td><td>QSFT52</td><td>2.4610</td><td>3.8253</td><td>4.4573</td><td>5.2916</td><td>6.0549</td><td>6.9328</td><td>7.0249</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>2.4663</td><td>3.6255</td><td>4.4247</td><td>4.9395</td><td>5.5603</td><td>6.2086</td><td>7.2133</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>2.4750</td><td>3.7872</td><td>4.3418</td><td>5.0599</td><td>5.8762</td><td>6.4405</td><td>6.5178</td></tr><tr><td rowspan="4">30°</td><td>Présent élément</td><td>QSFT52</td><td>2.5812</td><td>3.8521</td><td>4.8330</td><td>5.3560</td><td>6.2860</td><td>6.9186</td><td>7.6454</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>2.7921</td><td>4.0566</td><td>5.0220</td><td>5.2906</td><td>6.5698</td><td>6.6042</td><td>7.4485</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>2.8001</td><td>3.9557</td><td>5.1079</td><td>5.1675</td><td>6.3624</td><td>6.4236</td><td>7.5067</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>2.7922</td><td>4.0568</td><td>5.0222</td><td>5.2909</td><td>6.5702</td><td>6.6047</td><td>7.4487</td></tr><tr><td rowspan="4">45°</td><td>Présent élément</td><td>QSFT52</td><td>2.9190</td><td>4.0216</td><td>5.5212</td><td>6.9025</td><td>7.1661</td><td>8.9485</td><td>8.9815</td></tr><tr><td>Wang [313]</td><td>Anal Solution</td><td>3.4738</td><td>4.7393</td><td>5.9554</td><td>6.3750</td><td>7.1961</td><td>8.0031</td><td>8.4468</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-TSDT</td><td>3.5215</td><td>4.7129</td><td>5.8789</td><td>6.5290</td><td>7.0275</td><td>7.9963</td><td>8.1687</td></tr><tr><td>Garg et al. [293]</td><td>FEM-Q9-FSDT</td><td>3.4739</td><td>4.7396</td><td>5.9553</td><td>6.3745</td><td>7.1920</td><td>7.9978</td><td>8.4213</td></tr></tbody></table>
# Conclusion générale & perspectives

## 1. Conclusion générale:

Dans cette étude, un nouveau modèle a été développé pour l'analyse du comportement des plaques composites multicouches et notamment les plaques sandwichs. Le modèle proposé est basé sur l'approche layerwise, mais en gardant tous les avantages de l'approche monocouche. Contrairement à tous les autres modèles layerwise présentés dans la littérature, le nombre de degrés de liberté dans ce modèle est indépendant du nombre de couches. Les deux peaux et l'âme sont modélisées individuellement en utilisant, respectivement, la théorie de déformation en cisaillement du premier ordre et celle du troisième ordre. Ces deux théories combinées, satisfont les exigences de continuités des déplacements interlaminaires ainsi que les conditions de compatibilité. Le nouveau modèle permet de calculer les contraintes de cisaillement transversal, d'une manière précise en utilisant les équations constitutives plutôt que les équations d'équilibre et sans recours à des facteurs de correction.

Sur la base de ce nouveau modèle, deux éléments finis, baptisés **RSFT52** et **QSFT52** (**Rectangular & Quadrilateral Sandwich First Third with 52-degrees of freedom**), ont été développés pour l'analyse statique et dynamique des plaques composites multicouches et des sandwichs. Ces éléments sont géométriquement simples et ne possèdent que quatre nœuds et treize degrés de liberté par nœud. Chaque nœud contient: deux degrés de liberté de rotation pour chaque peau, six degrés de liberté de rotation aussi pour l'âme, alors que les trois degrés de liberté de type translation (3ddl/nœud) sont communs aux trois couches. La formulation de ces deux éléments est basée sur le principe du travail virtuel.

La performance et l'efficacité des éléments développés RSFT52 et QSFT52 sont démontrés à travers une série d'applications en flexion et en vibration libre de plaques isotropes, composites multicouches et sandwichs symétrique / non symétrique, avec différents
cas de chargement, de géométrie et des conditions aux limites. Les résultats obtenus ont été comparés avec des solutions analytiques de références, et ceux obtenus par des modèles d'éléments finis basés sur différentes théories (HSDT, HZZT, LW, GLHSDT). Pour l'analyse statique, les exemples numériques montrent que les éléments finis développés sont capables de donner d'excellents résultats en particulier pour les contraintes de cisaillement transverse. Ces dernières ont été calculées directement en utilisant les équations constitutives plutôt que les équations d'équilibre. Ainsi, dans le cas de l'analyse dynamique, les résultats de l'élément développé QSFT52 sont en excellent accord avec ceux trouvés dans la littérature, où l'erreur n'atteint pas 0.004 % par comparaison avec la solution d'élasticité-3D.

Nous avons aussi montré l'influence de la distorsion géométrique sur la précision du résultat en étudiant le cas d'une plaque biaise (skew). En plus, on a montré à travers une étude paramétrique, l'effet de certains paramètres sur la variation du déplacement transversal ainsi que les fréquences naturelles d'une plaque sandwich, tels que le rapport d'épaisseur, les combinaisons de conditions aux limites, le rapport d'aspect, le degré d'anisotropie et d'orthotropie des sandwichs, etc.

On peut conclure que l'utilisation des éléments finis développé ainsi que la combinaison de la théorie de déformation en cisaillement du premier et celle du troisième ordre (FSDT, TSDT), pour modéliser respectivement, les deux peaux et l'âme, ont montré une bonne précision et une convergence rapide pour l'analyse les différents problèmes. Finalement, le potentiel des éléments finis développés peut être utilisé pour générer un certain nombre de nouveaux résultats, qui sont censés être utiles pour les futures recherches dans ce domaine.

## 2. Perspectives:

Les modèles d'éléments finis développés **RSFT52 et QSFT52** peuvent être considérés comme les plus simples parmi les éléments finis existants, quelle que soit l'approche ou le modèle utilisé. Les résultats obtenus sont fort encourageants et permettent d'augurer favorablement sur les possibilités réelles d'élargir ce modèle à l'étude d'autres phénomènes.

* Il serait intéressant d'utiliser l'élément **RSFT52** dans l'analyse non-linéaire et dans l'étude de comportement de flambement des plaques multicouches.
* Élaboration d'un nouveau modèle d'élément fini de type layerwise, nommé (QSFT36: Quadrilateral Sandwich First Third with 36 DOFs). Il serait pour l'analyse des structures isotropes, composites stratifiées et sandwichs en particulier. Ce nouveau modèle permettrait de réduire le nombre de degrés de liberté du modèle précédent (QSFT52); grâce à l'introduction des critères de continuités des contraintes de cisaillement transversal au niveau d'interface.

* Nous proposons d'étendre l'étude des phénomènes d'instabilités, i.e. le flambement global, le flambement local, le calcul thermique et thermo-mécanique des plaques sandwichs multicouches.

* Développer l'analyse non-linéaire matérielle, prenant en compte l'endommagement dans les structures sandwichs multicouches (possibilité de réaliser des tests expérimentaux afin de les comparer avec le modèle numérique).

* Étudier l'amortissement passif et actif des structures sandwichs comportant des couches viscoélastiques au niveau de l'âme, et des couches piézoélectriques au niveau des peaux, afin de réduire l'amplitude des vibrations et du bruit.

* L'intégration des éléments développés (QSFT52 et RSFT52) dans le code ABAQUS, en utilisant la routine utilisateur User-Element (UEL).

* Peut-on généraliser ce modèle aux problèmes des coques minces et épaisses? Cette question mérite également une investigation.
Références bibliographiques

[1] Reddy, J.N., Mechanics of laminated composite plates: theory and analysis. Vol. 1. 1997: CRC press Boca Raton.

[2] Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis. 2003: CRC press.

[3] Kollár, L.P. and G.S. Springer, Mechanics of composite structures. 2003: Cambridge university press.

[4] Vinson, J.R., The Behavior of Sandwich Structures of Isotropic and Composite Materials: Jack R. Vinson, 1999: CRC Press.

[5] Zenkert, D., An introduction to sandwich construction1997: Engineering Materials Advisory Services.

[6] Plantema, F.J., Sandwich construction. Vol. 966. 1966: Wiley New York.

[7] Jones, R.M., Mechanics of composite materials. Second Editions ed1998: CRC Press.

[8] Carlsson, L.A. and G.A. Kardomateas, Structural and failure mechanics of sandwich composites. Vol. 121. 2011: Springer.

[9] Kreja, I.,(2011), A literature review on computational models for laminated composite and sandwich panels, Central European Journal of Engineering, 1 (1) 59-80.

[10] Libove, C. and S. Batdorf, A general small-deflection theory for flat sandwich plates, 1948, DTIC Document.

[11] Reissner, E.,(1948), Finite Deflections of Sandwich Plates, Journal of the Aeronautical Sciences (Institute of the Aeronautical Sciences), 15 (7) 435-440.

[12] Reissner, E., Small bending and stretching of sandwich-type shells, 1950, National Advisory Committee for Aeronautics. Rpt 975.

[13] Hoff, N.J., Bending and buckling of rectangular sandwich plates.1950: National Advisory Committee for Aeronautics.

[14] Eringen, A.C. Bending and buckling of rectangular sandwich plates. in Proceedings of the First US National Congress of Applied Mechanics. 1951.
[15] Yen, K.T., S. Gunturkun, and F.V. Pohle, Deflections of a simply supported rectangular sandwich plate subjected to transverse loads1951: National Advisory Committee for Aeronautics.

[16] Gerard, G., (1951), Note on Bending of Thick Sandwich Plates, Journal of the Aeronautical Sciences (Institute of the Aeronautical Sciences), 18 (6) 424-426.

[17] Ericksen, W.S., (1953, US Forest Products Lab, Madison WI, FPL-1828), The bending of a circular sandwich plate under normal load.

[18] Ericksen, W.S. and H. March, (1958, rapport FPL n°1583-B. ), Effects of shear deformation in the core of a flat rectangular sandwich panel: compressive buckling of sandwich panels having dissimilar facings of unequal thickness.

[19] Cheng, S., On the theory of bending of sandwich plates, 1961, DTIC Document: Proceedings of the 4 U.S. National Congress of A. M. p. 511-518.

[20] Lin, T.H. and L.T. Yokota, (1965), Deflections and bending moments of rectangular sandwich panels with clamped edges under combined biaxial compressions and pressure, AIAA Journal, 3 (6) 1162-1164.

[21] Liaw, B.D. and R.W. Little, (1967), Theory of bending multi-layer sandwich plates, AIAA Journal, 5 (2) 301-304.

[22] Azar, J.J., (1968), Bending theory for multilayer orthotropic sandwich plates, AIAA Journal, 6 (11) 2166-2169.

[23] Pagano, N., (1969), Exact solutions for composite laminates in cylindrical bending, Journal of composite materials, 3 (3) 398-411.

[24] Pagano, N., (1970), Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of composite materials, 4 (1) 20-34.

[25] Jones, A.T., (1970), Exact natural frequencies for cross-ply laminates, Journal of Composite materials, 4 (4) 476-491.

[26] Srinivas, S. and A. Rao, (1971), A three-dimensional solution for plates and laminates, Journal of the Franklin Institute, 291 (6) 469-481.

[27] Srinivas, S., C. Joga Rao, and A. Rao, (1970), An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates, Journal of Sound and Vibration, 12 (2) 187-199.

[28] Noor, A., J. Peters, and W. Burton, (1994), Three-Dimensional Solutions for Initially Stressed Structural Sandwiches, Journal of Engineering Mechanics, 120 (2) 284-303.

[29] Noor, A.K. and W.S. Burton, (1990), Three-dimensional solutions for antisymmetrically solutions for antisymmetrically laminated anisotropic plates, Journal of applied mechanics, 57 (1) 182-188.
[30] Kant, T. and K. Swaminathan,(2002), Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures, 56 (4) 329-344.

[31] Lascoup, B., et al.,(2010), Homogenization of the core layer in stitched sandwich structures, Composites Science and Technology, 70 (2) 350-355.

[32] Lewiński, T.,(1993), On recent developments in the homogenization theory of elastic plates and their application to optimal design: Part I, Structural optimization, 6 (1) 59-64.

[33] Rabczuk, T., et al.,(2004), Homogenization of sandwich structures, International Journal for Numerical Methods in Engineering, 61 (7) 1009-1027.

[34] Takano, N., M. Zako, and N. Kikuchi,(1995), Stress analysis of sandwich plate by the homogenization method, Materials science research international, 1 (2) 82-88.

[35] Reddy, J.N.,(1993), An evaluation of equivalent-single-layer and layerwise theories of composite laminates, Composite structures, 25 (1-4) 21-35.

[36] Carrera, E.,(2003), Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking, Archives of Computational Methods in Engineering, 10 (3) 215-296.

[37] Kirchhoff, G.,(1850), Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal für die reine und angewandte Mathematik, 40 51-88.

[38] Love, A.E.H., A treatise on the mathematical theory of elasticity1934: Cambridge University Press.

[39] Reissner, E. and Y. Stavsky,(1961), Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates, Journal of Applied Mechanics, 28 (3) 402-408.

[40] Librescu, L., Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Vol. 2. 1975, Noordhoff, Leyden, Netherlands

[41] Reddy, J.N.,(1989), On refined computational models of composite laminates, International Journal for Numerical Methods in Engineering, 27 (2) 361-382.

[42] Stavsky, Y.,(1965), On the theory of symmetrically heterogeneous plates having the same thickness variation of the elastic moduli, Topics in applied mechanics, 105.

[43] Rohwer, K., S. Friedrichs, and C. Wehmeyer,(2005), Analyzing laminated structures from fibre-reinforced composite material–an assessment, Technische Mechanik, 25 (1) 59-79.

[44] Ambarsumyan, S.,(1958), K tieorii izgiba anisotropnvch plasinok, Izv. Akad Nauk SSR, 5, 69-77.

[45] Dong S.B., P.K.S.e.T.R.L.,(1962), On the Theory of Laminated Anisotropic Shells and Plates, Journal of the Aerospace Sciences, 29 (8) 969-975.
[46] Stavsky, Y.,(1962), Bending and stretching of laminated aeolotropic plates, Transactions of the American Society of Civil Engineers, 127 (1) 1194-1219.

[47] Decolon, C., Analysis of composite structures, 2004: Butterworth-Heinemann.

[48] Pai, P.F.,(1995), A new look at shear correction factors and warping functions of anisotropic laminates, International Journal of Solids and Structures, 32 (16) 2295-2313.

[49] Yang, P.C., C.H. Norris, and Y. Stavsky,(1966), Elastic wave propagation in heterogeneous plates, International Journal of Solids and Structures, 2 (4) 665-684.

[50] Whitney, J.,(1969), The effect of transverse shear deformation on the bending of laminated plates, Journal of composite materials, 3 (3) 534-547.

[51] Reissner, E.,(1945), The effect of transverse shear deformation on the bending of elastic plates, J. appl. Mech, 12 (2) 69-77.

[52] Mindlin, R.,(1951), Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, J. of Appl. Mech., 18 31-38.

[53] Reissner, E.,(1985), Reflections on the Theory of Elastic Plates, Applied Mechanics Reviews, 38 (11) 1453-1464.

[54] Whitney, J. and N. Pagano,(1970), Shear deformation in heterogeneous anisotropic plates, Journal of Applied Mechanics, 37 (4) 1031-1036.

[55] Berthelot, J.-M., Matériaux composites: comportement mécanique et analyse des structures. Vol. 3. 1992: Masson Paris.

[56] Dong, S.B. and F.K.W. Tso,(1972), On a Laminated Orthotropic Shell Theory Including Transverse Shear Deformation, Journal of Applied Mechanics, 39 (4) 1091-1097.

[57] Barker, R.M. and J.C.W. Pryor,(1971), A finite-element analysis including transverse shear effects for applications to laminated plates, AIAA Journal, 9 (5) 912-917.

[58] Whitney, J.M.,(1972), Stress Analysis of Thick Laminated Composite and Sandwich Plates*, Journal of Composite materials, 6 (4) 426-440.

[59] Hu, H., et al.,(2008), Review and assessment of various theories for modeling sandwich composites, Composite Structures, 84 (3) 282-292.

[60] Vlachoutsis, S.,(1992), Shear correction factors for plates and shells, International Journal for Numerical Methods in Engineering, 33 (7) 1537-1552.

[61] Whitney, J.,(1973), Shear correction factors for orthotropic laminates under static load, Journal of applied mechanics, 40 302.

[62] Birman, V. and C.W. Bert,(2002), On the choice of shear correction factor in sandwich structures, Journal of Sandwich Structures and Materials, 4 (1) 83-95.
[63] Chatterjee, S.N. and S.V. Kulkarni,(1979), Shear Correction Factors for Laminated Plates, AIAA Journal, 17 (5) 498-499.

[64] Madabhusi-Raman, P. and J.F. Davalos,(1996), Static shear correction factor for laminated rectangular beams, Composites Part B: Engineering, 27 (3-4) 285-293.

[65] Laitinen, M., H. Lahtinen, and S.G. Sjölind,(1995), Transverse shear correction factors for laminates in cylindrical bending, Communications in numerical methods in engineering, 11 (1) 41-47.

[66] Carrera, E. and S. Brischetto,(2008), A Survey With Numerical Assessment of Classical and Refined Theories for the Analysis of Sandwich Plates, Applied Mechanics Reviews, 62 (1) 010803-010803.

[67] Reissner, E.,(1972), A consistent treatment of transverse shear deformations in laminated anisotropic plates, AIAA Journal, 10 (5) 716-718.

[68] Belarbi, M.T., Développement de nouveaux éléments finis basés sur le modèle en déformation. Application linéaire et non linéaire, 2000, Thèse de Doctorat d'état, Université de Constantine.

[69] Ayad, R., Eléments finis de plaque et coque en formulation mixte avec projection en cisaillement. Thèse de doctorat, Université de Technologie de Compiègne, France. 1993.

[70] Lardeur, P., Développement et évaluation de deux nouveaux éléments finis de plaques et coques composites avec influence du cisaillement transversal, 1990, Thèse de Doctorat, University of Technology of Compiègne,France.

[71] Noor, A.K. and W.S. Burton,(1989), Assessment of Shear Deformation Theories for Multilayered Composite Plates, Applied Mechanics Reviews, 42 (1) 1-13.

[72] Bert, C.W.,(1973), Simplified analysis of static shear factors for beams of nonhomogeneous cross section, Journal of Composite materials, 7 (4) 525-529.

[73] Yu, Y.-Y.,(1959), Simple Thickness-shear Modes of Vibration in Infinite Sandwich Plates, (26) 679-681.

[74] Krishna Murty, A. and S. Vellaichamy,(1987), On higher order shear deformation theory of laminated composite panels, Composite Structures, 8 (4) 247-270.

[75] Lo, K., R. Christensen, and E. Wu,(1977), A high-order theory of plate deformation— part 2: laminated plates, Journal of Applied Mechanics, 44 (4) 669-676.

[76] Lo, K., R. Christensen, and E. Wu,(1977), A high-order theory of plate deformation— Part 1: Homogeneous plates, Journal of Applied Mechanics, 44 (4) 663-668.

[77] Nelson, R.B. and D.R. Lorch,(1974), A Refined Theory for Laminated Orthotropic Plates, Journal of Applied Mechanics, 41 (1) 177-183.

[78] Reddy, J.N.,(1984), A simple higher-order theory for laminated composite plates, Journal of applied mechanics, 51 (4) 745-752.
[79] Whitney, J. and C. Sun,(1973), A higher order theory for extensional motion of laminated composites, Journal of Sound and Vibration, 30 (1) 85-97.

[80] Bhimaraddi, A. and L.K. Stevens,(1984), A Higher Order Theory for Free Vibration of Orthotropic, Homogeneous, and Laminated Rectangular Plates, Journal of Applied Mechanics, 51 (1) 195-198.

[81] Murthy, M.,(1981), An improved transverse shear deformation theory for laminated antisotropic plates, NASA Technical, 1-37.

[82] Kant, T.,(1982), Numerical analysis of thick plates, Computer methods in applied mechanics and engineering, 31 (1) 1-18.

[83] Ren-Huai, L. and H. Ling-Hui,(1991), A simple theory for non-linear bending of laminated composite rectangular plates including higher-order effects, International journal of non-linear mechanics, 26 (5) 537-545.

[84] Touratier, M.,(1991), An efficient standard plate theory, International Journal of Engineering Science, 29 (8) 901-916.

[85] Hildebrand, F., E. Reissner, and G. Thomas, Notes on the foundations of the theory of small displacements of orthotropic shells1949: NACA Technical Note No. 1833, Washington, DC.

[86] Basset, A.B.,(1890), On the extension and flexure of cylindrical and spherical thin elastic shells, Philosophical Transactions of the Royal Society of London. A, 181 (6) 433-480.

[87] Nguyen-Van, H., Development and application of assumed strain smoothing finite element technique for composite plate/shell structures, Thèse de Doctorat, University of Southern Queensland, 2009, 295 pp.

[88] Whitney, J. and C.-T. Sun,(1974), A refined theory for laminated anisotropic, cylindrical shells, Journal of Applied Mechanics, 41 (2) 471-476.

[89] Kwon, Y. and J. Akin,(1987), Analysis of layered composite plates using a high-order deformation theory, Computers & structures, 27 (5) 619-623.

[90] Fares, M.,(1993), A refined two-dimensional theory of anisotropic plates, Mechanics Research Communications, 20 (4) 319-327.

[91] Tessler, A.,(1993), An improved plate theory of {1, 2}-order for thick composite laminates, International Journal of Solids and Structures, 30 (7) 981-1000.

[92] Sacco, E. and J. Reddy,(1992), On first-and second-order moderate rotation theories of laminated plates, International Journal for Numerical Methods in Engineering, 33 (1) 1-17.

[93] Reissner, E.,(1963), On the derivation of boundary conditions for plate theory, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 276 (1365) 178-186.
[94] Levinson, M.,(1980), An accurate, simple theory of the statics and dynamics of elastic plates, Mechanics Research Communications, 7 (6) 343-350.

[95] Reddy, J. and D. Robbins,(1994), Theories and computational models for composite laminates, Applied mechanics reviews, 47, 147.

[96] Bert, C.W.,(1984), A critical evaluation of new plate theories applied to laminated composites, Composite Structures, 2 (4) 329-347.

[97] Senthilnathan, N., et al.,(1987), Buckling of shear-deformable plates, AIAA Journal, 25 (9) 1268-1271.

[98] Lee, K., et al.,(1989), A simple higher-order non-linear shear deformation plate theory, International journal of non-linear mechanics, 24 (2) 127-137.

[99] Rohwer, K.,(1992), Application of higher order theories to the bending analysis of layered composite plates, International Journal of Solids and Structures, 29 (1) 105-119.

[100] Ferreira, A., C. Roque, and R. Jorge,(2006), Static and free vibration analysis of composite shells by radial basis functions, Engineering Analysis with Boundary Elements, 30 (9) 719-733.

[101] Krishna Murty, A.,(1987), Flexure of composite plates, Composite Structures, 7 (3) 161-177.

[102] Aydogdu, M.,(2009), A new shear deformation theory for laminated composite plates, Composite structures, 89 (1) 94-101.

[103] Levy, M.,(1877), Mémoire sur la théorie des plaques élastiques planes, Journal de mathématiques pures et appliquées, 219-306.

[104] Mantari, J., A. Oktem, and C. Guedes Soares,(2012), A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates, Computers & Structures, 94 45-53.

[105] Mantari, J.L., A.S. Oktem, and C. Guedes Soares,(2011), Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory, Composite structures, 94 (1) 37-49.

[106] Mantari, J., A. Oktem, and C. Guedes Soares,(2012), A new higher order shear deformation theory for sandwich and composite laminated plates, Composites Part B: Engineering, 43 (3) 1489-1499.

[107] Soldatos, K.P.,(1992), A general laminated plate theory accounting for continuity of displacements and transverse shear stresses at material interfaces, Composite Structures, 20 (4) 195-211.

[108] Stein, M.,(1986), Nonlinear theory for plates and shells including the effects of transverse shearing, AIAA Journal, 24 (9) 1537-1544.
[109] Sahoo, R. and B.N. Singh,(2013), A new inverse hyperbolic zigzag theory for the static analysis of laminated composite and sandwich plates, Composite structures, 105 (0) 385-397.

[110] Karama, M., K. Afaq, and S. Mistou,(2003), Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structures, 40 (6) 1525-1546.

[111] Ghugal, Y.M. and A.S. Sayyad,(2010), A static flexure of thick isotropic plates using trigonometric shear deformation theory, J Solid Mech, 2 (1) 79-90.

[112] Stein, M. and D.C. Jegley,(1987), Effects of transverse shearing on cylindrical bending, vibration, and buckling of laminated plates, AIAA Journal, 25 (1) 123-129.

[113] Nguyen, V.T., Modélisation globale et locale des structures multicouches par éléments finis de plaque. Thèse de Doctorat, Ecole des Ponts ParisTech., 2004, 216 pp.

[114] Idlbi, A., M. Karama, and M. Touratier,(1997), Comparison of various laminated plate theories, Composite Structures, 37 (2) 173-184.

[115] Soldatos, K.P.,(1992), A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica, 94 (3-4) 195-220.

[116] Soldatos, K. and T. Timarci,(1993), A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories, Composite Structures, 25 (1) 165-171.

[117] Aydogdu, M.,(2006), Comparison of various shear deformation theories for bending, buckling, and vibration of rectangular symmetric cross-ply plate with simply supported edges, Journal of Composite materials, 40 (23) 2143-2155.

[118] Karama, M., K. Afaq, and S. Mistou,(2009), A new theory for laminated composite plates, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials Design and Applications, 223 (2) 53-62.

[119] Nguyen, D.T., Benchmark d'un modèle layer wise de multicouches et implémentation du modèle dans Abaqus. Thèse de Doctorat, Université Paris-Est., 2012, 160 pp.

[120] Kaczkowski, Z.,(1968), Plates, Statical calculations, Arkady, Warsaw.

[121] Panc, V., Theories of elastic plates. Prague: Academia, 1975: Springer.

[122] Reissner, E.,(1975), On transverse bending of plates, including the effect of transverse shear deformation, International Journal of Solids and Structures, 11 (5) 569-573.

[123] Ferreira, A., C. Roque, and R. Jorge,(2005), Analysis of composite plates by trigonometric shear deformation theory and multiquadrics, Computers & structures, 83 (27) 2225-2237.
[124] Mantari, J., A. Oktem, and C. Guedes Soares, (2011), Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory, Composite Structures, 94 (1) 37-49.

[125] Mantari, J., A. Oktem, and C. Guedes Soares, (2012), A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates, International Journal of Solids and Structures, 49 (1) 43-53.

[126] Mantari, J. and C. Guedes Soares, (2012), Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory, Composite Structures, 94 (8) 2640-2656.

[127] Grover, N., D. Maiti, and B. Singh, (2013), A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates, Composite Structures, 95 667-675.

[128] Sahoo, R. and B. Singh, (2013), A new shear deformation theory for the static analysis of laminated composite and sandwich plates, International Journal of Mechanical Sciences, 75 324-336.

[129] Carrera, E., (2002), Theories and finite elements for multilayered, anisotropic, composite plates and shells, Archives of Computational Methods in Engineering, 9 (2) 87-140.

[130] Saeedi, N., Une approche très efficace pour l'analyse du délaminage des plaques stratifiées infiniment longues. Thèse de Doctorat, Université Paris-Est., 2012, 223 pp.

[131] Kao, J.S. and R.J. Ross, (1968), Bending of multilayer sandwich beams, AIAA Journal, 6 (8) 1583-1585.

[132] Srinivas, S., (1973), A refined analysis of composite laminates, Journal of Sound and Vibration, 30 (4) 495-507.

[133] Sun, C.-T. and J. Whitney, (1973), Theories for the dynamic response of laminated plates, AIAA Journal, 11 (2) 178-183.

[134] Swift, G.W. and R.A. Heller, (1974), Layered beam analysis, Journal of the Engineering Mechanics Division, 100 (2) 267-282.

[135] Seide, P., (1980), An improved approximate theory for the bending of laminated plates. Chapter book.

[136] Reddy, J.N., (1987), A generalization of two-dimensional theories of laminated composite plates, Communications in Applied Numerical Methods, 3 (3) 173-180.

[137] Reddy, J.N., E.J. Barbero, and J.L. Teply, (1989), A plate bending element based on a generalized laminate plate theory, International Journal for Numerical Methods in Engineering, 28 (10) 2275-2292.
[138] Barbero, E.J., J.N. Reddy, and J. Teply,(1990), An accurate determination of stresses in thick laminates using a generalized plate theory, International Journal for Numerical Methods in Engineering, 29 (1) 1-14.

[139] Robbins, D. and J. Reddy,(1993), Modelling of thick composites using a layerwise laminate theory, International journal for numerical methods in engineering, 36 (4) 655-677.

[140] Cho, K., A. Striz, and C. Bert,(1989), Thermal stress analysis of laminate using higher-order theory in each layer, Journal of thermal stresses, 12 (3) 321-332.

[141] Cho, K., C. Bert, and A. Striz,(1991), Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory, Journal of Sound and Vibration, 145 (3) 429-442.

[142] Pagano, N.J. and S.R. Soni,(1983), Global-local laminate variational model, International Journal of Solids and Structures, 19 (3) 207-228.

[143] Ren, J.G.,(1986), A new theory of laminated plate, Composites Science and Technology, 26 (3) 225-239.

[144] Kassapoglou, C. and P.A. Lagace,(1986), An Efficient Method for the Calculation of Interlaminar Stresses in Composite Materials, Journal of Applied Mechanics, 53 (4) 744-750.

[145] Kassapoglou, C. and P.A. Lagace,(1987), Closed form solutions for the interlaminar stress field in angle-ply and cross-ply laminates, Journal of Composite materials, 21 (4) 292-308.

[146] Wan-Lee, Y.,(1994), Interlaminar stress analysis of composite laminates using a sublaminate/layer model, International Journal of Solids and Structures, 31 (11) 1549-1564.

[147] Yin, W.L.,(1994), Free-Edge Effects in Anisotropic Laminates Under Extension, Bending and Twisting, Part I: A Stress-Function-Based Variational Approach, Journal of Applied Mechanics, 61 (2) 410-415.

[148] Lee, K., et al.,(1990), An improved zig-zag model for the bending of laminated composite plates, Composite Structures, 15 (2) 137-148.

[149] Di Sciuva, M.,(1984), A refined transverse shear deformation theory for multilayered anisotropic plates, Atti. Accad. Sci. Torino, 118 279-295.

[150] Di Sciuva, M.,(1986), Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model, Journal of Sound and Vibration, 105 (3) 425-442.

[151] Murakami, H.,(1986), Laminated composite plate theory with improved in-plane responses, Journal of applied mechanics, 53 (3) 661-666.
[152] Cho, M. and R.R. Parmerter,(1992), An efficient higher-order plate theory for laminated composites, Composite Structures, 20 (2) 113-123.

[153] Cho, M. and R. Parmerter,(1993), Efficient higher order composite plate theory for general lamination configurations, AIAA journal, 31 (7) 1299-1306.

[154] Lu, X. and D. Liu,(1992), An Interlaminar Shear Stress Continuity Theory for Both Thin and Thick Composite Laminates, Journal of Applied Mechanics, 59 (3) 502-509.

[155] Di Sciuva, M.,(1987), An Improved Shear-Deformation Theory for Moderately Thick Multilayered Anisotropic Shells and Plates, Journal of Applied Mechanics, 54 (3) 589-596.

[156] Averill, R.C.,(1994), Static and dynamic response of moderately thick laminated beams with damage, Composites Engineering, 4 (4) 381-395.

[157] Cho, Y. and R. Averill,(2000), First-order zig-zag sublaminate plate theory and finite element model for laminated composite and sandwich panels, Composite Structures, 50 (1) 1-15.

[158] Chakrabarti, A. and A.H. Sheikh,(2005), Analysis of laminated sandwich plates based on interlaminar shear stress continuous plate theory, Journal of engineering mechanics, 131 (4) 377-384.

[159] He, L.-H.,(1994), A linear theory of laminated shells accounting for continuity of displacements and transverse shear stresses at layer interfaces, International Journal of Solids and Structures, 31 (5) 613-627.

[160] Khandelwal, R., A. Chakrabarti, and P. Bhargava,(2013), An efficient FE model based on combined theory for the analysis of soft core sandwich plate, Computational Mechanics, 51 (5) 673-697.

[161] Pandit, M., A.H. Sheikh, and B.N. Singh,(2010), Analysis of laminated sandwich plates based on an improved higher order zigzag theory, Journal of sandwich structures and materials, 12 (3) 307-326.

[162] Pandit, M.K., A.H. Sheikh, and B.N. Singh,(2008), An improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft core, Finite elements in analysis and design, 44 (9) 602-610.

[163] Topdar, P., A.H. Sheikh, and N. Dhang,(2003), Finite element analysis of composite and sandwich plates using a continuous inter-laminar shear stress model, Journal of sandwich structures and materials, 5 (3) 207-231.

[164] Icardi, U.,(2001), Higher-order zig-zag model for analysis of thick composite beams with inclusion of transverse normal stress and sublaminates approximations, Composites Part B: Engineering, 32 (4) 343-354.

[165] Kapuria, S., P. Dumir, and N. Jain,(2004), Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams, Composite Structures, 64 (3) 317-327.
[166] Sciuva, M.D.,(1992), Multilayered anisotropic plate models with continuous interlaminar stresses, Composite Structures, 22 (3) 149-167.

[167] Carrera, E.,(2003), Historical review of zig-zag theories for multilayered plates and shells, Applied mechanics reviews, 56 287-308.

[168] Duong, V.A., Développement en dynamique d'un élément fini multicouche avec interfaces imparfaites. Thèse de Doctorat, Ecole Nationale des Ponts et Chaussées, 2008, 196 pp.

[169] Lekhnitskii, S.,(1935), Strength calculation of composite beams, Vestnik inzhen i tekhnikov, 9.

[170] Ambartsumian, S.,(1958), On a general theory of anisotropic shells, Journal of Applied Mathematics and Mechanics, 22 (2) 305-319.

[171] Ambartsumian, S.A., Theory of anisotropic shells, 1961: State Publishing House for Physical and Mathematical Literature.

[172] Reissner, E.,(1984), On a certain mixed variational theorem and a proposed application, International Journal for Numerical Methods in Engineering, 20 (7) 1366-1368.

[173] Toledano, A. and H. Murakami,(1987), A Composite Plate Theory for Arbitrary Laminate Configurations, Journal of Applied Mechanics, 54 (1) 181-189.

[174] Toledano, A. and H. Murakami,(1987), A high-order laminated plate theory with improved in-plane responses, International Journal of Solids and Structures, 23 (1) 111-131.

[175] Carrera, E.,(1995), A class of two-dimensional theories for anisotropic multilayered plates analysis, Atti della accademia delle scienze di Torino. Classe di scienze fisiche matematiche e naturali, 19, 1-39.

[176] Di Sciuva, M. and M. Gherlone,(2003), A global/local third-order Hermitian displacement field with damaged interfaces and transverse extensibility: analytical formulation, Composite Structures, 59 (4) 419-431.

[177] Ha, K.,(1990), Finite element analysis of sandwich plates: an overview, Computers & Structures, 37 (4) 397-403.

[178] Kant, T. and K. Swaminathan,(2000), Estimation of transverse/interlaminar stresses in laminated composites–a selective review and survey of current developments, Composite structures, 49 (1) 65-75.

[179] Khandan, R., et al.,(2012), The development of laminated composite plate theories: a review, Journal of Materials Science, 47 (16) 5901-5910.

[180] Liu, D. and X. Li,(1996), An overall view of laminate theories based on displacement hypothesis, Journal of Composite materials, 30 (14) 1539-1561.
[181] Noor, A.K., W.S. Burton, and C.W. Bert,(1996), Computational models for sandwich panels and shells, Applied mechanics reviews, 49 155.

[182] Zhang, Y. and C. Yang,(2009), Recent developments in finite element analysis for laminated composite plates, Composite Structures, 88 (1) 147-157.

[183] Altenbach, H.,(1998), Theories for laminated and sandwich plates, Mechanics of composite materials, 34 (3) 243-252.

[184] Ghugal, Y. and R. Shimpi,(2002), A review of refined shear deformation theories of isotropic and anisotropic laminated plates, Journal of Reinforced Plastics and Composites, 21 (9) 775-813.

[185] Wanji, C. and W. Zhen,(2008), A selective review on recent development of displacement-based laminated plate theories, Recent patents on mechanical engineering, 1 (1) 29-44.

[186] Chen, W. and P. Jia,(2012), Interlaminar stresses analysis and the limit state function approximating methods for composite structure reliability assessment: A selected review and some perspectives, Journal of Composite materials, .

[187] Li, X. and D. Liu,(1995), A laminate theory based on global–local superposition, Communications in numerical methods in engineering, 11 (8) 633-641.

[188] Shariyat, M.,(2010), A generalized global–local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads, International Journal of Mechanical Sciences, 52 (3) 495-514.

[189] Wu, Z., R. Chen, and W. Chen,(2005), Refined laminated composite plate element based on global–local higher-order shear deformation theory, Composite Structures, 70 (2) 135-152.

[190] Wu, Z., et al.,(2012), A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations, Finite Elements in Analysis and Design, 48 (1) 1346-1357.

[191] Zhen, W. and C. Wanji,(2010), A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates, Computational Mechanics, 45 (5) 387-400.

[192] Zhen, W. and C. Wanji,(2010), A C⁰-type higher-order theory for bending analysis of laminated composite and sandwich plates, Composite Structures, 92 (3) 653-661.

[193] Zhen, W. and C. Wanji,(2007), A study of global–local higher-order theories for laminated composite plates, Composite Structures, 79 (1) 44-54.

[194] Chen, R.-g. and W.-j. Chen,(2000), Higher-order shear deformation theory and triangular plate element based on global-local superposition, Acta Materiae Compositae Sinica. (China), 17 96-102.
[195] Sedira, L., Contribution à la Modélisation de Composites 2D/3D à l'Aide d'Éléments Finis Spéciaux. Thèse de Doctorat, Université de Biskra, 2013, 195 pp.

[196] Li, X. and D. Liu,(1997), Generalized laminate theories based on double superposition hypothesis, International Journal for Numerical Methods in Engineering, 40 (7) 1197-1212.

[197] Khalili, S., M. Shariyat, and I. Rajabi,(2014), A finite element based global-local theory for static analysis of rectangular sandwich and laminated composite plates, Composite structures, 107, 177-189.

[198] Xiaohui, R., C. Wanji, and W. Zhen,(2012), A C0-type zig-zag theory and finite element for laminated composite and sandwich plates with general configurations, Archive of Applied Mechanics, 82 (3) 391-406.

[199] Zhen, W. and C. Wanji,(2006), Free vibration of laminated composite and sandwich plates using global-local higher-order theory, Journal of Sound and Vibration, 298 (1) 333-349.

[200] Zhen, W. and C. Wanji,(2007), Buckling analysis of angle-ply composite and sandwich plates by combination of geometric stiffness matrix, Computational Mechanics, 39 (6) 839-848.

[201] Tessler, A., M. DiSciuva, and M. Gherlone,(2009), Refined zigzag theory for laminated composite and sandwich plates.

[202] Oh, J., et al.,(2008), A finite element formulation based on an enhanced first order shear deformation theory for composite and sandwich structures, Journal of Mechanical Science and Technology, 22 (5) 871-878.

[203] Xiaohui, R., C. Wanji, and W. Zhen,(2011), A new zig-zag theory and C0 plate bending element for composite and sandwich plates, Archive of Applied Mechanics, 81 (2) 185-197.

[204] Nemeth, M.P.,(2012), Cubic zig-zag enrichment of the classical Kirchhoff kinematics for laminated and sandwich plates.

[205] Sedira, L., et al.,(2012), An enhanced discrete Mindlin finite element model using a zigzag function, European Journal of Computational Mechanics/Revue Européenne de Mécanique Numérique, 21 (1-2) 122-140.

[206] Natarajan, S. and G. Manickam,(2012), Bending and vibration of functionally graded material sandwich plates using an accurate theory, Finite Elements in Analysis and Design, 57 32-42.

[207] Abel, J. and E. Popov,(1968), Static and dynamic finite element analysis of sandwich structures, Proceedings of the Second Conference of Matrix Methods in Structural Mechanics, 68 213-245,.
[208] Monforton, G. and L. Schmit. Finite element analysis of sandwich plates and cylindrical shells with laminated faces. in Proceedings of the Second Conference of Matrix Methods in Structural Mechanics, AFFSL-TR-68-150. 1968.

[209] Ahmed, K.,(1971), Static and dynamic analysis of sandwich structures by the method of finite elements, Journal of Sound and Vibration, 18 (1) 75-91.

[210] Khatua, T. and Y. Cheung,(1973), Bending and vibration of multilayer sandwich beams and plates, International journal for numerical methods in engineering, 6 (1) 11-24.

[211] Khatua, T.P. and Y.K. Cheung,(1972), Triangular element for multilayer sandwich plates, Journal of the Engineering Mechanics Division, 98 (5) 1225-1238.

[212] Cook, R.D.,(1972), Two hybrid elements for analysis of thick, thin and sandwich plates, International journal for numerical methods in engineering, 5 (2) 277-288.

[213] Cook, R.D.,(1975), Finite element buckling analysis of homogeneous and sandwich plates, International Journal for Numerical Methods in Engineering, 9 (1) 39-50.

[214] Fazio, P.P. and K.H. Ha,(1974), Sandwich plate structure analysis by finite element, Journal of the Structural Division, 100 (6) 1243-1262.

[215] Sheikh, A.H., S. Haldar, and D. Sengupta,(2002), A high precision shear deformable element for the analysis of laminated composite plates of different shapes, Composite Structures, 55 (3) 329-336.

[216] Tafla, A., Eléments finis mixtes-hybrides naturels sans facteurs correctifs du cisaillement pour les plaques et les coques composites multicouches. Thèse de Doctorat, University of Reims Champagne Ardenne, 2007, 132 pp.

[217] Sakami, S., Modélisation numérique des structures composites multicouches à l'aide d'une approche discrète au sens de Mindlin. Le modèle DDM (Displacement Discrete Mindlin). Thèse de Doctorat, University of Reims Champagne Ardenne, 2008, 188 pp

[218] Katili, I., Formulation et évaluation de nouveaux éléments finis pour l'analyse linéaire des plaques et coques de forme quelconque. Thèse de doctorat, Université de Technologie de Compiègne, France, 1993.

[219] Kant, T., D. Owen, and O. Zienkiewicz,(1982), A refined higher-order C plate bending element, Computers & structures, 15 (2) 177-183.

[220] Engblom, J.J. and O.O. Ochoa,(1985), Through-the-thickness stress predictions for laminated plates of advanced composite materials, International journal for numerical methods in engineering, 21 (10) 1759-1776.

[221] Pandya, B. and T. Kant,(1988), Flexural analysis of laminated composites using refined higher-order C° plate bending elements, Computer Methods in Applied Mechanics and Engineering, 66 (2) 173-198.
[222] Pandya, B. and T. Kant, (1988), A refined higher-order generally orthotropic C⁰ plate bending element, Computers & structures, 28 (2) 119-133.

[223] Pandya, B. and T. Kant, (1988), Higher-order shear deformable theories for flexure of sandwich plates—finite element evaluations, International Journal of Solids and Structures, 24 (12) 1267-1286.

[224] Pandya, B. and T. Kant, (1987), A consistent refined theory for flexure of a symmetric laminate, Mechanics Research Communications, 14 (2) 107-113.

[225] Kant, T. and J. Kommineni, (1992), C0 Finite element geometrically non-linear analysis of fibre reinforced composite and sandwich laminates based on a higher-order theory, Computers & Structures, 45 (3) 511-520.

[226] Manjunatha, B. and T. Kant, (1992), A comparison of 9 and 16 node quadrilateral elements based on higher-order laminate theories for estimation of transverse stresses, Journal of reinforced plastics and composites, 11 (9) 968-1002.

[227] Manjunatha, B. and T. Kant, (1993), On evaluation of transverse stresses in layered symmetric composite and sandwich laminates under flexure, Engineering computations, 10 (6) 499-518.

[228] Polit, O. and M. Touratier, (1997), A new laminated triangular finite element assuring interface continuity for displacements and stresses, Composite Structures, 38 (1) 37-44.

[229] Polit, O. and M. Touratier, (2000), High-order triangular sandwich plate finite element for linear and non-linear analyses, Computer Methods in Applied Mechanics and Engineering, 185 (2) 305-324.

[230] Polit, O. and M. Touratier, (2002), A multilayered/sandwich triangular finite element applied to linear and non-linear analyses, Composite Structures, 58 (1) 121-128.

[231] Nayak, A., S.J. Moy, and R. Shenoi, (2003), Quadrilateral finite elements for multilayer sandwich plates, The Journal of Strain Analysis for Engineering Design, 38 (5) 377-392.

[232] Rezaiee-Pajand, M., F. Shahabian, and F. Tavakoli, (2012), A new higher-order triangular plate bending element for the analysis of laminated composite and sandwich plates, Structural Engineering and Mechanics, 43 (2) 253-271.

[233] Sheikh, A.H. and A. Chakrabarti, (2003), A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates, Finite elements in analysis and design, 39 (9) 883-903.

[234] Dvorkin, E.N. and K.-J. Bathe, (1984), A continuum mechanics based four-node shell element for general non-linear analysis, Engineering computations, 1 (1) 77-88.

[235] Huang, H. and E. Hinton, (1984), A nine node Lagrangian Mindlin plate element with enhanced shear interpolation, Engineering computations, 1 (4) 369-379.
[236] Tu, T.M., L.N. Thach, and T.H. Quoc, (2010), Finite element modeling for bending and vibration analysis of laminated and sandwich composite plates based on higher-order theory, Computational Materials Science, 49 (4) S390-S394.

[237] Pandya, B. and T. Kant, (1988), Finite element analysis of laminated composite plates using a higher-order displacement model, Composites Science and Technology, 32 (2) 137-155.

[238] Goswami, S. and W. Becker, (2013), A New Rectangular Finite Element Formulation Based on Higher Order Displacement Theory for Thick and Thin Composite and Sandwich Plates, World, 3 194-201.

[239] Lee, S.J. and H.R. Kim, (2013), FE analysis of laminated composite plates using a higher order shear deformation theory with assumed strains, Latin American Journal of Solids and Structures, 10 (3) 523-547.

[240] Spilker, R., (1982), Hybrid-stress eight-node elements for thin and thick multilayer laminated plates, International journal for numerical methods in engineering, 18 (6) 801-828.

[241] Fazio, P., K. Gowri, and K. Ha, (1987), Rectangular hybrid elements for the analysis of sandwich plate structures, Canadian Journal of Civil Engineering, 14 (4) 455-460.

[242] Wu, C.-P. and C.-C. Lin, (1993), Analysis of sandwich plates using a mixed finite element, Composite Structures, 25 (1) 397-405.

[243] Lee, L. and Y. Fan, (1996), Bending and vibration analysis of composite sandwich plates, Computers & Structures, 60 (1) 103-112.

[244] Oskooei, S. and J. Hansen, (2000), Higher-order finite element for sandwich plates, AIAA journal, 38 (3) 525-533.

[245] Ramtekkar, G., Y. Desai, and A. Shah, (2002), Mixed finite-element model for thick composite laminated plates, Mechanics of Advanced Materials and Structures, 9 (2) 133-156.

[246] Ramtekkar, G., Y. Desai, and A. Shah, (2003), Application of a three-dimensional mixed finite element model to the flexure of sandwich plate, Computers & Structures, 81 (22) 2183-2198.

[247] Demiray, S., W. Becker, and J. Hohe, (2004), A triangular v. Kármán type finite element for sandwich plates with transversely compressible core, Computer methods in applied mechanics and engineering, 193 (23) 2239-2260.

[248] Demiray, S., W. Becker, and J. Hohe, A Triangular Finite Element for Sandwich Plates Accounting for Transverse Core Compressibility, Sandwich Structures 7: Advancing with Sandwich Structures and Materials, 2005, Springer. p. 301-310.

[249] Linke, M., W. Wohlers, and H.-G. Reimerdes, (2007), Finite element for the static and stability analysis of sandwich plates, Journal of sandwich structures and materials, 9 (2) 123-142.
[250] Ramesh, S.S., Wang, CM, Reddy, JN, Ang, KK, (2009), A higher-order plate element for accurate prediction of interlaminar stresses in laminated composite plates, Composite Structures, 91 (3) 337-357.

[251] Ćetković, M. and D. Vuksanović,(2009), Bending, free vibrations and buckling of laminated composite and sandwich plates using a layerwise displacement model, Composite structures, 88 (2) 219-227.

[252] Darilmaz, K.,(2012), Analysis of sandwich plates: A three-dimensional assumed stress hybrid finite element, Journal of sandwich structures and materials, 14 (4) 487-501.

[253] Darilmaz, K.,(2005), A hybrid 8-node hexahedral element for static and free vibration analysis, Structural Engineering and Mechanics, 21 (5) 571-590.

[254] Di Sciuva, M.,(1993), A general quadrilateral multilayered plate element with continuous interlaminar stresses, Computers & structures, 47 (1) 91-105.

[255] Di Sciuva, M.,(1995), A third-order triangular multilayered plate finite element with continuous interlaminar stresses, International Journal for Numerical Methods in Engineering, 38 (1) 1-26.

[256] Carrera, E.,(1996), C0 Reissner-Mindlin multilayered plate elements including zig-zag and interlaminar stress continuity, International Journal for Numerical Methods in Engineering, 39 (11) 1797-1820.

[257] Carrera, E.,(1998), A refined multilayered finite-element model applied to linear and non-linear analysis of sandwich plates, Composites Science and Technology, 58 (10) 1553-1569.

[258] Averill, R. and Y.C. Yip,(1996), Development of simple, robust finite elements based on refined theories for thick laminated beams, Computers & structures, 59 (3) 529-546.

[259] Cho, Y.-B. and R.C. Averill,(1997), An improved theory and finite-element model for laminated composite and sandwich beams using first-order zig-zag sublaminate approximations, Composite Structures, 37 (3) 281-298.

[260] Chakrabarti, A. and A.H. Sheikh,(2004), A new triangular element to model interlaminar shear stress continuous plate theory, International journal for numerical methods in engineering, 60 (7) 1237-1257.

[261] Singh, S.K., Chakrabarti, A., Bera, P. and Sony, J. S. D., (2011), An efficient C0 FE model for the analysis of composites and sandwich laminates with general layup, Latin American Journal of Solids and Structures, 8 (2) 197-212.

[262] Chalak, H.D., Chakrabarti, A., Iqbal, M. A. and Sheikh, H. A., (2012), An improved C0 FE model for the analysis of laminated sandwich plate with soft core, Finite Elem. Anal. Des., 56 20-31.
[263] Rao, G.V., J. Venkataramana, and I. Raju, (1974), A high precision triangular plate bending element for the analysis of thick plates, Nuclear Engineering and Design, 30 (3) 408-412.

[264] Yuan, F.G. and R.E. Miller, (1989), A cubic triangular finite element for flat plates with shear, International journal for numerical methods in engineering, 28 (1) 109-126.

[265] Kant, T., Gupta, Avani B., Pendhari, S. S., Desai, Yogesh M., (2008), Elasticity solution for cross-ply composite and sandwich laminates, Composite Structures, 83 (1) 13-24.

[266] Kulkarni, S. and S. Kapuria, (2007), A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates, Computational Mechanics, 39 (3) 237-246.

[267] Meftah, K., Modélisation numérique des solides par éléments finis volumiques basés sur le concept SFR (Space Fiber Rotation). Thèse de Doctorat, Université Mohamed Khider-Biskra, 2013.

[268] Belinha, J. and L.M.J.S. Dinis, (2006), Analysis of plates and laminates using the element-free Galerkin method, Computers & structures, 84 (22-23) 1547-1559.

[269] Ghosh, A. and S. Dey, (1992), A simple finite element for the analysis of laminated plates, Computers & Structures, 44 (3) 585-596.

[270] Xiao, J., et al., (2008), Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method, Composites Part B: Engineering, 39 (2) 414-427.

[271] Pagano, N.J. and H.J. Hatfield, (1972), Elastic Behavior of Multilayered Bidirectional Composites, AIAA Journal, 10 (7) 931-933.

[272] Folie, G., (1970), Bending of clamped orthotropic sandwich plates, Journal of the Engineering Mechanics Division, 96 (3) 243-265.

[273] Kanematsu, H.H., Y. Hirano, and H. Iyama, (1988), Bending and vibration of CFRP-faced rectangular sandwich plates, Composite Structures, 10 (2) 145-163.

[274] Meunier, M. and R. Shenoi, (1999), Free vibration analysis of composite sandwich plates, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 213 (7) 715-727.

[275] Tati, A., Analyse du flambage des plaques stratifiées en matériaux composites par la méthode des éléments finis. Thèse de Doctorat, Université Mohamed Khider de Biskra, 2007, 144 pp.

[276] Ergatoudis, I., B.M. Irons, and O.C. Zienkiewicz, (1968), Curved, isoparametric, “quadrilateral” elements for finite element analysis, International Journal of Solids and Structures, 4 (1) 31-42.
[277] Nayak, A., S. Moy, and R. Shenoi, (2002), Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory, Composites Part B: Engineering, 33 (7) 505-519.

[278] Lee, S., (2004), Free vibration analysis of plates by using a four-node finite element formulated with assumed natural transverse shear strain, Journal of sound and vibration, 278 (3) 657-684.

[279] Kabir, H.R.H., (1995), A shear-locking free robust isoparametric three-node triangular finite element for moderately-thick and thin arbitrarily laminated plates, Computers & Structures, 57 (4) 589-597.

[280] Chalak, H.D., et al., (2014), C0 FE model based on HOZT for the analysis of laminated soft core skew sandwich plates: Bending and vibration, Applied Mathematical Modelling, 38 (4) 1211-1223.

[281] Dey, P., A. Sheikh, and D. Sengupta, (2014), A new element for the analysis of composite plates, Finite Elements in Analysis and Design, 82 62-71.

[282] Yu, Y.-Y., (1960), Flexural vibrations of elastic sandwich plates, Journal of Aerospace Sciences, (27) 272-282.

[283] Yu, Y.-Y., (1960), Simplified Vibration Analysis of Elastic Sandwich Plates, Journal of the Aerospace Sciences, 27 (12) 894-900.

[284] Srinivas, S. and A.K. Rao, (1970), Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, International Journal of Solids and Structures, 6 (11) 1463-1481.

[285] Chan, H.C. and Y.K. Cheung, (1972), Static and dynamic analysis of multi-layered sandwich plates, International Journal of Mechanical Sciences, 14 (6) 399-406.

[286] Noor, A.K., (1973), Free vibrations of multilayered composite plates, AIAA Journal, 11 (7) 1038-1039.

[287] Ibrahim, I.M., M.N.F. Rizk, and A. Farah, (1981), Dynamic analysis of unbalanced anisotropic sandwich plates, Journal of the Engineering Mechanics Division, 107 (2) 405-418.

[288] Reddy, J.N. and T. Kuppusamy, (1984), Natural vibrations of laminated anisotropic plates, Journal of sound and vibration, 94 (1) 63-69.

[289] Wang, C., et al., (2000), Free vibration of skew sandwich plates with laminated facings, Journal of sound and vibration, 235 (2) 317-340.

[290] Kant, T. and K. Swaminathan, (2001), Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory, Composite Structures, 53 (1) 73-85.
[291] Jam, J.E., B. Eftari, and S.H. Taghavian, (2010), A new improved high-order theory for analysis of free vibration of sandwich panels, Polymer Composites, 31 (12) 2042-2048.

[292] Khare, R.K., A.K. Garg, and T. Kant, (2005), Free vibration of sandwich laminates with two higher-order shear deformable facet shell element models, Journal of Sandwich Structures and Materials, 7 (3) 221-244.

[293] Garg, A.K., R.K. Khare, and T. Kant, (2006), Free vibration of skew fiber-reinforced composite and sandwich laminates using a shear deformable finite element model, Journal of Sandwich Structures and Materials, 8 (1) 33-53.

[294] Nabarrete, A., S.F. M. De Almeida, and J.S. Hansen, (2003), Sandwich-plate vibration analysis: three-layer quasi-three-dimensional finite element model, AIAA Journal, 41 (8) 1547-1555.

[295] Roque, C., A. Ferreira, and R. Jorge, (2006), Free vibration analysis of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions, Journal of Sandwich Structures and Materials, 8 (6) 497-515.

[296] Zhen, W., C. Wanji, and R. Xiaohui, (2010), An accurate higher-order theory and CO finite element for free vibration analysis of laminated composite and sandwich plates, Composite Structures, 92 (6) 1299-1307.

[297] Elmalich, D. and O. Rabinovitch, (2012), A high-order finite element for dynamic analysis of soft-core sandwich plates, Journal of Sandwich Structures and Materials, 14 (5) 525-555.

[298] Chakrabarti, A. and A.H. Sheikh, (2004), Vibration of laminate-faced sandwich plate by a new refined element, Journal of Aerospace Engineering, 17 (3) 123-134.

[299] Chakrabarti, A. and A. Sheikh. A new triangular element based on a layer wise zigzag theory for the analysis of composite panels having sandwich construction. in Proceedings of the International Conference on Ship and Ocean Technology. 2002.

[300] Kulkarni, S. and S. Kapuria, (2008), Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory, Computational Mechanics, 42 (6) 803-824.

[301] Batoz, J.L. and M.B. Tahar, (1982), Evaluation of a new quadrilateral thin plate bending element, International Journal for Numerical Methods in Engineering, 18 (11) 1655-1677.

[302] Jeyachandrabose, C., J. Kirkhope, and L. Meekisho, (1987), An improved discrete Kirchhoff quadrilateral thin-plate bending element, International Journal for Numerical Methods in Engineering, 24 (3) 635-654.

[303] Chalak, H.D., et al., (2013), Free vibration analysis of laminated soft core sandwich plates, Journal of Vibration and Acoustics, 135 (1) 011013.
[304] Sahoo, R. and B. Singh, (2014), A new trigonometric zigzag theory for buckling and free vibration analysis of laminated composite and sandwich plates, Composite Structures, 117 316-332.

[305] Ferreira, A.J., MATLAB codes for finite element analysis: solids and structures. Vol. 157. 2008: Springer.

[306] Marjanović, M. and D. Vuksanović, (2014), Layerwise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations, Composite Structures, 108 9-20.

[307] Vuksanović, D., (2000), Linear analysis of laminated composite plates using single layer higher-order discrete models, Composite Structures, 48 (1-3) 205-211.

[308] Owen, D.R.J. and Z.H. Li, (1987), A refined analysis of laminated plates by finite element displacement methods—II. Vibration and stability, Computers & Structures, 26 (6) 915-923.

[309] Khandelwal, R.P., A. Chakrabarti, and P. Bhargava, (2013), Vibration and buckling analysis of laminated sandwich plate having soft core, International Journal of Structural Stability and Dynamics, 13 (08).

[310] Rao, M., et al., (2004), Natural Vibrations of Laminated and Sandwich Plates, Journal of Engineering Mechanics, 130 (11) 1268-1278.

[311] Kant, T. and K. Swaminathan, (2001), free vibration of isotropic, orthotropic, and multilayer plates based on higher order refined theories, Journal of sound and vibration, 241 (2) 319-327.

[312] Rao, M.K. and Y.M. Desai, (2004), Analytical solutions for vibrations of laminated and sandwich plates using mixed theory, Composite Structures, 63 (3-4) 361-373.

[313] Wang, S., (1997), Free vibration analysis of skew fibre-reinforced composite laminates based on first-order shear deformation plate theory, Computers & Structures, 63 (3) 525-538.

[314] Malekzadeh, K. and A. Sayyidmousavi, (2009), Free vibration analysis of sandwich plates with a uniformly distributed attached mass, flexible core and different boundary conditions, Journal of Sandwich Structures and Materials, 12(6) 709-732.