Composite Structures 107 (2014) 620–628 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct A simple and accurate sandwich plate theory accounting for transverse normal strain and interfacial stress continuity ⇑ X. Wang, G. Shi Department of Mechanics, Tianjin University, Tianjin 300072, China article info abstract Article history: This paper presents a simple and accurate sandwich plate theory accounting for the transverse normal Available online 2 September 2013 strain and interfacial stress continuity. A refined cubic polynomial is used for the transverse shear func- tion and a linear function is adopted for the transverse normal strain. The Heaviside step function and Keywords: stress continuity coefficients are employed respectively to enforce the interfacial continuity of the in- Sandwich plate theory plane displacements and transverse stresses. By the enforcement of the traction conditions on the plate Transverse shear function surfaces, there are only five independent field variables in the present sandwich plate theory. The varia- Transverse normal strain tional consistent equilibrium equations and boundary conditions in terms of both displacements and Interfacial stress continuity stress resultants are derived by utilizing the variational principle. The analytical solutions of the bending Analytical solution analyses of sandwich plates with different aspect ratios and stiffness ratios are solved to demonstrate the accuracy of this new sandwich plate theory. The resulting analytical solutions of deflections, normal stresses and transverse shear stresses are compared with the 3D elasticity solutions, the numerical results and the results given by other sandwich plate theories. The comparison study shows that this equivalent single layer sandwich plate theory is not only simple, but also capable of achieving the accu- racy of layerwise sandwich plate theories. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction it is only applicable in the analysis of laminated plates with simple geometries and boundary conditions but not appropriate for the Sandwich plates are widely used in various engineering struc- large scale numerical analysis of laminated plates. Therefore, the tures such as aircrafts, spacecrafts, ships, automobiles, and infra- extensive application of sandwich plate structures has driven the structures, because of the low density and high specific stiffness. development of various sandwich plate theories since the middle The three-layered sandwich structures show the complicated ef- of last century. The accurate and efficient modeling of sandwich fects [1,2]. Firstly, sandwich plates are of high transverse shear plates presents many challenges because of their unique physical and normal deformability. Secondly, the zigzag form of the dis- and geometrical properties, especially when the core is much soft- placement field in the thickness directions (the displacement field er and the thickness of face panel is much thinner. There are a shows a discontinuous slope at each layer interface) and the inter- number of comprehensive reviews on the development and state laminar continuity of transverse shear and normal stresses should of the arts of sandwich plate theories [1–3,8–12]. The engineering be considered in refined models. The zigzag effects and interlami- sandwich plate theories are based on the displacement assumption 0 nar continuity were referred to as Cz -requirements in Ref. [3]. to derive the governing equations by using variational principles. Therefore, the key point in an accurate laminated plate theory is The displacement-based sandwich plate theories can be classified to ensure both the zigzag effects of displacement and interlaminar as: (1) the classical lamination theory (CLT); (2) the first-order continuity of displacement and stress along the plate thickness. shear deformation theories (FSDT) [13,14]; (3) the higher-order Carrera [2,4] also showed that the transverse normal stress plays shear deformation theories (HSDT) which include the plate theo- an important role in multilayered plates. ries with the zigzag effects [15–19] and the global–local higher-or- The continuum-based 3D elasticity theory presented by Pagano der plate theories [20,21]; and (4) the layerwise plate theories [5–7] is very accurate in the analysis of sandwich plates. However, [2,22]. In addition, the mixed theories based on Reissner’s mixed variational theorem show some advantages for accounting for the transverse normal stress effect [2]. ⇑ Corresponding author. The CLT is based on the Kirchhoff assumption, which cannot E-mail addresses: [email protected] (X. Wang), [email protected] (G. accurately describe the mechanical behavior of laminated plates. Shi). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.08.033 X. Wang, G. Shi / Composite Structures 107 (2014) 620–628 621 The FSDTs for laminated plates are based on the shear flexible plate clamped boundaries of beams and plates than the cubic transverse theories proposed by Reissner [13] and Mindlin [14], in which a shear function used in Reddy’s plate theory [23]. Therefore, the cu- lamination scheme dependent shear coefficient is needed. The var- bic transverse shear function proposed by Shi [24] is used in this ious higher-order plate theories [15–19,23–26] are the most work to derive a new simple and accurate laminated composite widely used in engineering field. These higher-order theories do plate theory. But it should be noted that a modification for this not account for the interlaminar continuity conditions of displace- transverse shear function is needed in order to satisfy the continu- ments and transverse shear stresses; hence the resulting trans- ity conditions of both displacements and transverse shear stresses verse shear stresses are not accurate. The interlaminar continuity at the interlaminar surfaces of laminated plates. conditions can be accounted by the use of zigzag type kinematics The laminated plate is assumed to be built up of arbitrary num- across the laminated plate thickness. Carrera et al. recently showed ber, denoted by N, of orthotropic laminae. The global coordinates of that [2] the present equivalent single layer HSDTs, including the a typical laminated plate are depicted in Fig. 1. theories with the zigzag effects, are not able to predict accurate re- sults, particularly the transverse shear stresses. The global–local 2.1. The plate kinematics with the third-order shear deformation and higher-order plate theories [10,20,21] are capable of giving accu- satisfying interfacial shear stress continuity rate results, but more global field variables are utilized in this type of plate theories. For instance, 13 global unknowns are used in the The Heaviside step function introduced by He [29] to automat- global–local plate theories proposed by Wu and Chen [21]. While ically satisfy the displacement continuity at interfaces between dif- the layer-wise lamination theories [2,3,22] can both satisfy the ferent laminae of a sandwich plate can be used to modify the interlaminar continuity conditions, but the biggest shortcoming displacement field defined in Shi’s plate theory [24]. The displace- of this type of plate theories is that the unknowns used in the plate ment field considering the displacement interlaminar continuity of theories are dependent on the number of the laminae, which a laminated composite plate takes the form: results in the extensive computation of analysis. Therefore, the refined equivalent single layer HSDT with high computational effi- @w @w uðx; y; zÞ¼u0ðx; yÞz þ hxðzÞ þ /x ciency and accuracy is very desirable, particularly for the finite ele- @x @x ment modeling of sandwich plates although there are so many @w @w ð1a-cÞ vðx; y; zÞ¼v0ðx; yÞz þ hyðzÞ þ / HSDTs have been proposed. @y @y y The objective of this paper to present a new sandwich plate the- wðx; y; zÞ¼wðx; yÞ ory with five global field variables, three displacements and two rotations, but accounting for the transverse normal strain effect where hx(z) and hy(z) are the modified transverse shear functions and the interlaminar continuity conditions. The analytical solu- with the form: tions of the bending analyses of various sandwich plates with dif- XNÀ1 ferent aspect ratios and different face-to-core-stiffness ratios are 5 4z2 z 5z3 h ðzÞ¼ z 1 À þ kðiÞ À þðz À zðiÞÞHðz À zðiÞÞ ð2aÞ solved by using the present simple sandwich plate theory. And x 4 2 x 8 2 3h i¼1 6h the accuracy of the present sandwich plate theory is evaluated by the comparison of these analytical solutions with the 3D elastic- 5 4z2 XNÀ1 z 5z3 ity solutions and other results. The comparison study indicates that h ðzÞ¼ z 1 À þ kðiÞ À þðz À zðiÞÞHðz À zðiÞÞ ð2bÞ y 4 2 y 8 2 the sandwich plate theory presented in this paper is not only sim- 3h i¼1 6h ple, but also capable of achieving the accuracy of layerwise sand- In the equations above, N is the lamina number of the laminated wich plate theories. Since the sandwich plate theory possesses plate under consideration; the superscript i is the index of the inter- only five field variables and high accuracy, it can provide an accu- laminar interface; kðiÞ and kðiÞ are the continuity coefficients; z(i) rate and efficient theoretical model for the finite element analysis x y stands for the z coordinate at the ith interlaminar surface and of sandwich plates. H(z À z(i)) is the Heaviside step function defined as: ( 2. A laminated composite plate theory with third-order shear 1 for z P zðiÞ Hðz À zðiÞÞ¼ ð3Þ deformation and interlaminar shear stress continuity 0 for z < zðiÞ It was shown [27,28] that cubic transverse shear function pro- For a given laminated composite plate, the coefficients of the ðiÞ ðiÞ posed by Shi [24] can yield better predictions of both the trans- continuity kx and ky can be determined from the interlaminar verse shear stresses and the boundary layer effects at the continuity conditions of the transverse shear stresses.