A Simple and Accurate Sandwich Plate Theory Accounting for Transverse Normal Strain and Interfacial Stress Continuity ⇑ X

Composite Structures 107 (2014) 620–628

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Composite Structures

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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 function 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 accuracy 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).

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 results, 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- XN1 ferent aspect ratios and different face-to-core-stiffness ratios are 5 4z2 z 5z3 h ðzÞ¼ z 1 þ kðiÞ þðz zðiÞÞHðz 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 XN1 z 5z3 ity solutions and other results. The comparison study indicates that h ðzÞ¼ z 1 þ kðiÞ þðz zðiÞÞHðz 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 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 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.

Fig. 1. The geometry and coordinates of a laminated composite plate with a constant thickness. 622 X. Wang, G. Shi / Composite Structures 107 (2014) 620–628

2.2. Strain–displacement relationships 2.5. Variational consistent equilibrium equations

Under small deformation assumption, the strains associated The strain energy P of a laminated plate made of orthotropic with the displacements defined in Eq. (1a-c) take the forms: laminae and with uniform thickness h as well as the domain X in the reference plane takes the form: @uðx; y; zÞ @vðx; y; zÞ Z Z e ðx; y; zÞ¼ ; e ðx; y; zÞ¼ ð4a; bÞ h=2 x @x y @y 1 P ¼ ½rxex þ ryey þ sxycxy þ sxzcxz þ syzcyzdzdX ð9Þ 2 X h=2 @uðx; y; zÞ @ ðx; y; zÞ v The work done by the distributed transverse load q(x,y) acting on cxyðx; y; zÞ¼ þ ð4cÞ @y @x the plate surface is of the form: Z @uðx; y; zÞ @wðx; yÞ Wðx; yÞ¼ qðx; yÞwdX ð10Þ c ðx; y; zÞ¼ þ ; xz @z @x X @vðx; y; zÞ @wðx; yÞ Then, for a plate with the transverse shear deformations and the exter- c ðx; y; zÞ¼ þ ð4d; eÞ yz @z @y nal load defined above, the minimum potential principle states as d½Pðx; yÞþWðx; yÞ ¼ 0 ð11Þ 2.3. Constitutive equations The governing equations in terms of displacements can be derived from Eq. (11) by making use of the strain–displacement relations gi- When each lamina is very thin and the transverse normal stress ven in Eqs. (4a-4e) and the constitutive equations defined in Eq. (7). r3 = rz is neglectable, the constitutive equations in the principal By setting the coefficients of du0, dv0, d/x, d/y, dw to zero separately, material coordinates of the lamina take the form: one obtains the following equilibrium equations in terms of the 8 9 2 38 9 displacements: <> r1 => Q 11 Q 12 0 <> e1 => 6 7 s Q 0 c 4 5 23 44 23 A u 2A u A u A A A > r2 > ¼ Q 12 Q 22 0 > e2 > and ¼ 11 0;11 þ 16 0;12 þ 66 0;22 þ 16v0;11 þð 12 þ 66Þv0;12 : ; : ; s13 0 Q 55 c13 s12 00Q 66 c12 þ A26v0;22 þðE11 B11Þw;111 þðE33 þ E34 2B66 þ E12 B12Þw;122

ð5Þ þðE16 þ 2E61 3B16Þw;112 þðE26 B26Þw;222 þ E11/1;11

þ 2E16/ þ E33/ þ E16/ þðE12 þ E34Þ/ Then the reduced stiffness components are can be written as in 1;12 1;22 2;11 2;12 E / 0 12 terms of the engineering constants as þ 26 2;22 ¼ ð Þ

E11 E22 A16u0;11 þðA12 þ A66Þu0;12 þ A26u0;22 þ A66v0;11 þ 2A26v0;12 Q 11 ¼ ; Q 22 ¼ ; Q 12 ¼ m21Q 11 1 m12m21 1 m12m21 ð6Þ þ A22v0;22 þðE61 B16Þw;111 þðE33 þ E34 2B66 þ E21 B12Þw;112

Q 66 ¼ G12; Q 44 ¼ G23; Q 55 ¼ G13; m21E11 ¼ m12E22 þðE62 þ 2E26 3B26Þw;122 þðE22 B22Þw;222 þ E61/1;11

þðE33 þ E21Þ/1;12 þ E62/1;22 þ E34/2;11 þ 2E26/2;12 in which Eij, mij and Gij (i,j = 1, 2) are, respectively, the equivalent Young’s modulus, Poisson ratio and shear modulus of the lamina þ E22/2;22 ¼ 0 ð13Þ in the principal material coordinates. When there is an orientation angle h between the principal material axis of a lamina and the ref- E11u0;11 þ 2E61u0;12 þ E33u0;22 þ E61v0;11 þðE33 þ E21Þv0;12 þ E62v0;22 erence coordinate x-axis, the stresses and strains of the lamina in þðH11 F11Þw;111 þðH33 þ H34 2F33 þ H12 F21Þw;122 the plate global coordinates have the following relationship þðH56 þ 2H55 3F61Þw;112 þðH65 F62Þw;222 þ H11/1;11 þ 2H55/1;12 8 9 2 38 9 "# þ H33/1;22 þ H56/2;11 þðH12 þ H34Þ/2;12 þ H65/2;22 A55ð/1 þ w;1Þ <> rx => Q 11 Q 12 Q 16 <> ex => 6 7 syz Q Q c r 4 5 e 44 45 yz A ð/ þ w Þ¼0 ð14Þ > y > ¼ Q 12 Q 22 Q 36 > y > and ¼ 45 2 ;2 : ; : ; sxz Q 45 Q 55 cxz cxy Q 16 Q 26 Q 66 cxy ð7Þ E16u0;11 þðE34 þ E12Þu0;12 þ E26u0;22 þ E34v0;11 þ 2E62v0;12 þ E22v0;22 þðH56 F16Þw;111 þðH44 þ H34 2F34 þ H12 F12Þw;112 where the Q is the off-axis stiffness matrix of the lamina deﬁned ij þðH þ 2H 3F Þw þðH F Þw þ H / that can be found from any textbook of mechanics of composite 65 66 26 ;122 22 22 ;222 56 1;11 materials. þðH12 þ H34Þ/1;12 þ H65/1;22 þ H44/2;11 þ 2H66/2;12 þ H22/2;22

A45ð/1 þ w;1ÞA44ð/2 þ w;2Þ¼0 ð15Þ 2.4. Interlaminar Transverse Shear Stress Continuity Conditions ðE11 B11Þu0;111 þðE33 þ E34 2B66 þ E12 B12Þu0;122 (i) The Heaviside step function H(z z ) in the modified trans- þðE16 þ 2E61 3B16Þu0;112 þðE26 B26Þu0;222 þðE61 B16Þv0;111 verse shear functions h(z) can ensure the continuity of displace- þðE33 þ E34 2B66 þ E21 B12Þv0;112 þðE62 þ 2E26 3B26Þv0;122 ment. However, the continuity of the transverse shear stresses at þðE22 B22Þv0;222 þðH11 F11Þ/1;111 þðH33 þ H34 2F33 the interfaces of laminae cannot be satisfied. Therefore, it is neces- þ H F Þ/ þðH þ 2H 3F Þ/ þðH F Þ/ sary to introduce the constraints defined in Eq. (8a,b) to enforce the 12 21 1;122 56 55 61 1;112 65 62 1;222 interlaminar shear stress continuity, which can be achieved by þðH56 F16Þ/2;111 þðH34 þ H44 2F34 þ H12 F12Þ/2;112 ðiÞ ðiÞ introducing continuity coefficients kx and ky which are deter- þðH65 þ 2H66 3F26Þ/2;122 þðH22 F22Þ/2;222 mined by enforcing the interlaminar continuity of transverse shear þðH11 2F11 þ D11Þw;1111 þ 2ðH55 þ H56 3F61 F16 þ 2D16Þw;1112 stresses defined below: þ 2ðH65 þ H66 3F26 F62 þ 2D26Þw;1222 þðH22 2F22 þ D22Þw;2222

ðkÞ ðiÞ ðkþ1Þ ðiÞ ðkÞ ðiÞ ðkþ1Þ ðiÞ þð2H12 2F12 2F21 þ 2D12 þ H33 þ 2H34 þ H44 4F33 4F34 sxz ðz ¼ z Þ¼sxz ðz ¼ z Þ syz ðz ¼ z Þ¼syz ðz ¼ z Þð8a; bÞ

þ 4D66Þw;1122 A44ð/2;2 þ w;22ÞA55ð/1;1 þ w;11ÞA45ð/2;1 þ w;12Þ The superscript k of sðkÞ and sðkÞ in the equations above is the index xz yz A ð/ þ w Þþq ¼ 0 ð16Þ of the laminar number. 45 1;2 ;12 X. Wang, G. Shi / Composite Structures 107 (2014) 620–628 623 Z ( In the equations above, the extensional, coupling, bending and h=2 3 XN1 3 5z 5z ðmÞ z 5z H56; H65 ¼ ðQ 16; Q 26Þ þ k transverse shear rigidities Aij, Bij and Dij of the present laminated 2 x 2 h=2 4 3h 8 6h plate theory considering the interfacial continuity conditions are m¼1 ( deﬁned as follows: 3 XN1 3 ðmÞ ðmÞ 5z 5z ðmÞ z 5z Z Z Z þðz z ÞHðz z Þ þ ky h=2 h=2 h=2 4 2 8 2 2 3h m¼1 6h Aij ¼ Q ijdz; Bij ¼ Q ijzdz; Dij ¼ Q ijz dz i; j ¼ 1; 2; 6 h=2 h=2 h=2 () þðz zðmÞÞHðz zðmÞÞ dz Z 2 h=2 2 XN1 2 5 5z ðmÞ 1 5z ðmÞ ( A44 ¼ Q 44 þ k þ Hðz z Þ dz Z 2 y 2 h=2 3 XN1 3 h=2 4 h 8 2h 5z 5z z 5z m¼1 H ; H ; H ¼ ðQ ; Q ; Q Þ þ kðmÞ 22 44 66 22 66 26 4 2 y 8 2 ð17aÞ h=2 3h m¼1 6h 2 Z () þðz zðmÞÞHðz zðmÞÞ dz ð18cÞ h=2 5 5z2 XN1 1 5z2 A ¼ Q þ kðmÞ þ Hðz zðmÞÞ 45 45 4 2 x 8 2 The stress resultants, stress couples and generalized forces h=2 h m¼1 2h () [24,25] in the new laminated plate theory are deﬁned in the fol- 2 XN1 2 5 5z ðmÞ 1 5z lowing forms: þ k þ Hðz zðmÞÞ dz 4 2 y 8 2 Z h m¼1 2h h=2 ð17bÞ Nx; Ny; Nxy ¼ ðrx; ry; sxyÞdz; Mx; My; Mxy h=2 Z ()h=2 Z 2 h=2 2 XN1 2 ¼ ð ; ; Þzdz ð19aÞ 5 5z ðmÞ 1 5z ðmÞ rx ry sxy A ¼ Q þ k þ Hðz z Þ dz h=2 55 55 4 2 x 8 2 h=2 h m 1 2h ¼ Z h=2 e e 5z 5z2 Eij, Fij and Hij (i,j = 1,2,6) in Eqs. (12)–(16) are some entities associ- Mx; Mxy ¼ ðrx; sxyÞ 2 h=2 4 3h ated with the higher-order shear deformations, and they are of the ) XN1 forms z 5z3 þ kðmÞ þðz zðmÞÞHðz zðmÞÞ dz Z x 8 2 h=2 3 m¼1 6h 5z 5z Z ð19bÞ E11; E21; E61; E62; E33 ¼ ðQ 11; Q 12; Q 16; Q 26; Q 66Þ 2 h=2 2 h=2 4 3h e e 5z 5z ) My; Myx ¼ ðry; sxyÞ 2 XN1 4 z 5z3 h=2 3h ) þ kðmÞ þðz zðmÞÞHðz zðmÞÞ dz x 8 2 XN1 z 5z3 m¼1 6h þ kðmÞ þðz zðmÞÞHðz zðmÞÞ dz y 8 2 ð18aÞ m¼1 6h Z Z () h=2 XN1 h=2 5z 5z3 5 5z2 1 5z2 E ; E ; E ; E ; E ¼ ðQ ; Q ; Q ; Q ; Q Þ Q ¼ s þ kðmÞ þ Hðz zðmÞÞ dz 22 12 16 26 34 22 12 16 26 66 4 2 x xz 4 2 x 8 2 h=2 3h ) h=2 h m¼1 2h () XN1 z 5z3 Z ðmÞ ðmÞ ðmÞ h=2 5 5z2 XN1 1 5z2 þ ky 2 þðz z ÞHðz z Þ dz ðmÞ ðmÞ 8 Q ¼ syz þ k þ Hðz z Þ dz m¼1 6h y 4 2 y 8 2 h=2 h m¼1 2h Z h=2 5z2 5z4 ð19cÞ F11; F21; F61; F62; F33 ¼ ðQ 11; Q 12; Q 16; Q 26; Q 66Þ 2 h=2 4 3h The equilibrium equations of the present laminated plate theory ) can also be expressed in terms of the stress resultants, stress cou- XN1 z2 5z4 ðmÞ ðmÞ ðmÞ ples and generalized forces as þ kx þ zðz z ÞHðz z Þ dz 8 6h2 m¼1 @N @N @N @N x þ xy ¼ 0; xy þ y ¼ 0 ð20; 21Þ ð18bÞ @x @y @x @y Z h=2 2 4 e e e e 5z 5z @ @ @ @ F ; F ; F ; F ; F ¼ ðQ ; Q ; Q ; Q ; Q Þ Mx Mxy Myx My 22 12 16 26 34 22 12 16 26 66 2 þ Q x ¼ 0; þ Q y ¼ 0 ð22; 23Þ h=2 4 3h @x @y @x @y ) XN1 z2 5z4 "# kðmÞ z z zðmÞ H z zðmÞ dz 2 2 2 þ y 2 þ ð Þ ð Þ @ e @ e @ e e 8 6h ðM M Þþ ðM M Þþ ðM þ M 2M Þ m¼1 @x2 x x @y2 y y @x@y xy yx xy Z ( h=2 3 XN1 3 5z 5z ðmÞ z 5z @Q H ; H ; H ¼ ðQ ; Q ; Q Þ þ k @Q x y 11 33 55 11 66 16 4 2 x 8 2 q ¼ 0 ð24Þ h=2 3h m¼1 6h @x @y

2 þðz zðmÞÞHðz zðmÞÞ dz 2.6. Variational consistent boundary conditions Z ( h=2 5z 5z3 XN1 z 5z3 H ; H ¼ ðQ ; Q Þ þ kðmÞ The boundary conditions can also be derived from Eq. (11) by 12 34 12 66 4 2 x 8 2 h=2 3h m¼1 6h making use of the constitutive equations deﬁned in Eq. (7) and ( integrating the displacement gradients. The entities in the bound- 5z 5z3 XN1 z 5z3 þðz zðmÞÞHðz zðmÞÞ þ kðmÞ ary integral in Eq. (11) can be grouped in terms of du0, dv0, d/x, d/y, 4 2 y 8 2 @dw @dw 3h m¼1 6h dw, @x and @y . And by setting each expression on the boundary integrals to zero, one obtains that either one in the following seven ðmÞ ðmÞ þðz z ÞHðz z Þ dz pairs to be speciﬁed on the boundaries C of a laminated plate. 624 X. Wang, G. Shi / Composite Structures 107 (2014) 620–628 8 > u or N > n n > > us or Nns > e > / or M <> n n e / or M > s ns on C ð25Þ > e > worQ n > > @w e > @n or Mn Mn :> @w e @s or Mns Mns

@ @ @ @ @ @ where @n ¼ nx @x þ ny @y and @s ¼ny @x þ nx @y. And the following transformations of stress resultants and stress couples are used in the derivation of boundary integration in Eq. (25). un ¼ nxu0 þ ny 0; us ¼nyu0 þ nx 0; v v Fig. 2. The geometry and coordinates of a sandwich plate with a constant thickness. /n ¼ nx/x þ ny/y; /s ¼ny/x þ nx/y ð26Þ

N ¼ N n2 þ 2N n n þ N n2; n x x xy x y y y In order to take account of the transverse normal strain effect 2 2 and to satisfy the interfacial transverse stress continuity, a layerw- Nns ¼Nxnxny þ Nxy nx ny þ Nynxny ð27Þ ised quadratic transverse displacement in terms of the thickness coordinate defined in the equation bellow can be used to replace M ¼ M n2 þ 2M n n þ M n2; n x x xy x y y y constant deflection through the plate thickness given in Eq. (1c). 2 2 Mns ¼Mxnxny þ Mxy nx ny þ Mynxny ð28Þ ðkÞ ðkÞ ðkÞ 2 w ðx; y; zÞ¼w0ðx; yÞþA ðx; yÞz þ B ðx; yÞz ð36Þ e e e e e where A(k)(x,y) and B(k)(x,y) are the function of x and y only, and the M ¼ M n2 þ M n n þ M n n þ M n2; ð29Þ n x x xy x y yx x y y y superscript k is the layer index of a sandwich plate. It follows from Eq. (36) that the transverse normal strain in e e 1 e 1 e e M ¼M n n þ M n2 n2 þ M n2 n2 þ M n n thekth-layer is of the form ns x x y 2 xy x y 2 yx x y y x y ðkÞ ðkÞ ðkÞ ð30Þ ez ðx; y; zÞ¼A ðx; yÞþ2B ðx; yÞz ð37Þ When the transverse normal strain effect is taken into account, the e e e 2 e 2 e Mns ¼M xnxny Mxyny þ Myxnx þ Mynxny ð31Þ constitutive equation defined in Eq. (5) should be replaced by the following one: Q n ¼ Q xnx þ Q yny ð32Þ 8 9 2 38 9 > r > C C C 000> e > > 1 > 11 12 13 > 1 > The system of 14th-order differential equations in Eqs. (12)–(16) to- > > 6 7> > > r2 > 6 C12 C22 C23 0007> e2 > gether with the seven pairs of boundary conditions on each plate <> => 6 7<> => r 6 C C C 0007 e edge in Eq. (25) compose of the complete set of the differential gov- 3 6 13 23 33 7 3 > > ¼ 6 7> > ð38Þ erning equations for the present higher-order shear deformation > s23 > 6 000C44 007> c23 > > > 6 7> > > > 4 0000C 0 5> > theory of laminated plates. :> s13 ;> 55 :> c13 ;> The following are some typical boundary conditions of lami- s12 00000C66 c12 nated plates. Firstly, the seven boundary conditions on a simply supported edge of a plate are of the form in which

1m23m32 m12 þm13m32 m13 þm12m23 m23 þm21m13 @w e C11 ¼ ; C12 ¼ ; C13 ¼ ; C23 ¼ ð39aÞ u ¼ w ¼ / ¼ ¼ 0; N ¼ M ¼ M ¼ 0 ð33Þ E E D E E D E E D E E D s s @s n n n 2 3 2 3 2 3 1 3 1m13m31 1m12m21 C22 ¼ ; C33 ¼ ; C44 ¼ G23; C55 ¼ G13; C66 ¼ G12 ð39bÞ Secondly, the boundary conditions on a clamped edge of a plate are E1E3D E1E2D 1m m m m m m 2m m m @w @w D ¼ 12 21 23 32 13 31 12 23 31 ð39cÞ u ¼ u ¼ w ¼ / ¼ / ¼ ¼ ¼ 0 ð34Þ E1E2E3 n s n s @n @s The coefficient functions A(k)(x,y) and B(k)(x,y) in Eq. (36) can be And thirdly, the boundary conditions on an edge of a plate with determined by the traction conditions on the plate surfaces and the specified stresses are interfacial continuity of the transverse normal stress in the follow- e e ing expressions: Nn ¼ Nn; Nns ¼ Nns; Mn ¼ Mn; M n ¼ Mn; e e e e e e 1 h 3 h M ¼ M ; Q ¼ Q ; ðM M Þ¼ðM M Þð35Þ rð Þ x; y; ¼ 0; rð Þ x; y; ¼ pðx; yÞð40Þ ns ns n n ns ns ns ns z 2 z 2 The values with ‘‘’’ in the above expressions denote the values gi- ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ð2Þ ð3Þ ð2Þ rz ðx; y; z Þ¼rz ðx; y; z Þ; rz ðx; y; z Þ¼rz ðx; y; z Þð41Þ ven by the specified stresses on the stress edges. wð1Þðx; y; zð1ÞÞ¼wð2Þðx; y; zð1ÞÞ; wð2Þðx; y; zð2ÞÞ¼wð3Þðx; y; zð2ÞÞð42Þ

3. A {3,2} sandwich plate theory accounting for transverse Pai and Palazotto [30] pointed out that the thickness change of a normal strain effect and interfacial stress continuity plate has no significant influence on the transverse shear strains. Consequently the contributions of A(k)(x,y) and B(k)(x,y) to the A typical rectangular sandwich plate is depicted in Fig. 2.In transverse shear strains defined in Eqs. (4e, 4d) can be neglected. addition to the interlaminar continuity involving in a laminated Similarly, the Poisson’s effect from the deformations in the x- composite plate, the transverse normal strain is the most impor- and y-directions on the transverse normal stress can be neglected. tant issue that should be taken into account in a refined sandwich Then the kth-layer transverse normal stress can be written as the plate theory. following simplified form: X. Wang, G. Shi / Composite Structures 107 (2014) 620–628 625

ðkÞ ðkÞ Eq. (25) by making use of the constitutive equations defined in rz ðx; y; zÞ¼C33 ez ð43Þ Eq. (38). Therefore, the coefficient A(k)(x,y) and B(k)(x,y) in the layer-wised transverse displacement accounting for the transverse normal 4. Accuracy evaluation of the new sandwich plate theory strain and interfacial stress continuity defined in Eq. (36) take the considering the transverse normal strain and interfacial following forms: transverse stress continuity 2 2 24C33c þ 15C33f C33c þ 12C33f AðtopÞðx; yÞ¼ pðx; yÞð44Þ In this section, the accuracy of the sandwich plate theory pre- 2 2 9C33f 2C33c þ 5C33f C33c þ C33f sented in the previous section will be demonstrated by its applica- tions to the bending analyses of the sandwich plates with different 2 2 aspect ratios and face to core stiffness ratios. The Navier approach 5 2C33c 3C33f C33c þ 3C33f BðtopÞðx; yÞ¼ pðx; yÞð45Þ is employed to derive the analytical solutions of the bending of 2 2 sandwich plates. The analytical solutions of both the deflections 9C33f 2C33c þ 5C33f C33c þ C33f h and stresses given by the present sandwich theory are compared the 3D elasticity solutions, the numerical solutions obtained from 2ð2C þ 3C Þ AðcoreÞðx; yÞ¼ 33c 33f pðx; yÞð46Þ ABAQUS and the analytical solutions given by some other sand- 2 2 32C33c þ 5C33f C33c þ C33f wich plate theories reported in the literature. The notation of {3,1} is used to represent the present laminated plate theory and 5ðC þ 3C Þ {3,2} is used to denote the present sandwich plate theory account- BðcoreÞðx; yÞ¼ 33c 33f pðx; yÞð47Þ 2 2 ing for the transverse normal strain in this section. 62C33c þ 5C33f C33c þ C33f h Example 1. A simply supported square sandwich plate under 10C AðbottomÞðx; yÞ¼ 33c pðx; yÞð48Þ sinusoidally distributed pressure 2 2 The 3D continuum based elasticity theory was used by Pagano 32C33c þ 5C33f C33c þ C33f [5–7] to derive the 3D analytical solutions of laminated composite plates, and these 3D elasticity solutions have been used as the 10C BðbottomÞðx; yÞ¼ 33c pðx; yÞð49Þ exact solutions of the laminated plates by researchers. Both the 2 2 32C33c þ 5C33f C33c þ C33f h deflection and stresses of a simply supported square sandwich plate subjected to sinusoidally distributed pressure were pre- When the transverse normal strain effect is included, the strain en- sented in Ref. [6]. Therefore, the accuracy of the present laminated ergy P of a sandwich plate with the domain X in the reference plate theories can be evaluated by the analytical solutions of the plane and a uniform thickness h takes the form: same example given by the present laminated plate theory against Z Z 1 h=2 the 3D elasticity solutions given by Pagano [6]. P ¼ ½rxex þ ryey þ sxycxy þ sxzcxz þ syzcyz þ rzezdzdX The mechanical properties of the face and core of the sandwich 2 X h=2 under consideration are tabulated in Table 1. The thickness of each ð50Þ face sheet is hf = h/10, and thickness of the core is hc =4h/5. In this The work done by the distributed transverse load q(x,y) acting on example, the length to thickness ratios a/h ranged from 2 to 50 are the plate surface is of the form: considered. Z h The normal stresses at both the top surface and bottom surface Wðx; yÞ¼ qðx; yÞwx; y; dX ð51Þ of the top face skin as well as transverse shear stresses of the 2 X sandwich plates given by the present laminated plate theory and The governing equations and boundary conditions corresponding to the 3D solutions in Ref. [6] are listed in Table 2. The analytical Eq. (50), respectively, have the similar form as Eqs. (12)–(16) and solutions given by a third-order layerwise sandwich plate theory,

Table 1 The face and core mechanical properties of Example 1.

E1(/GPa) E2(/GPa) E3(/GPa) G12(/GPa) G13(/GPa) G23(/GPa) m12 m13 m23 Face 172.375 6.895 6.895 3.4475 3.4475 1.379 0.25 0.25 0.25 Core 0.2758 0.2758 3.4475 0.11032 0.4137 0.4137 0.25 0.25 0.25

Table 2 Nondimensional normal stresses and transverse shear stressesa of Example 1.

a/h Theories a b h a b 2h a b h b a rx 2 ; 2 ; 2 rx 2 ; 2 ; 5 ry 2 ; 2 ; 2 sxz 0; 2 ; 0 syz 2 ; 0; 0 2 Pagano [6] 3.2780 2.2200 0.4517 0.1850 0.1399 LD3 [31] 3.2426 – – 0.1785 – Present {3,1} 3.1253 2.1297 0.4099 0.1828 0.1363 10 Pagano [6] 1.1550 0.6280 0.1104 0.3002 0.0527 LD3 [31] 1.1324 – – 0.2802 – Present {3,1} 1.1547 0.6270 0.1092 0.3002 0.0523 50 Pagano [6] 1.0990 0.8670 0.0569 0.3230 0.0306 LD3 [31] 1.0785 – – 0.3018 – Present {3,1} 1.0991 0.8665 0.0569 0.3231 0.0305

a The deﬁnition of nondimensional deﬂections and stresses can be found in Ref. [6]. 626 X. Wang, G. Shi / Composite Structures 107 (2014) 620–628

denoted by LD3, are also listed in Table 2 for comparison. The sandwich plate is Ef3/Ec3 = 2, so the core is relatively hard in the results in Table 2 show the both normal and transverse shear and thickness direction. The squared sandwich plate with softer and stresses of the sandwich plates with aspect ratios from 2 to 50 harder cores are considered in this example to verify the accuracy predicted by the new laminated plate theory agree very well with and reliability of the new {3,1} laminated composite plate theory the 3D elasticity solutions. It can also be seen from Table 2 that the and the {3,2} sandwich plate theory accounting for the transverse present laminated plate theory yields more accurate stress pre- normal strain. The mechanical properties of the face and core are dictions than the third-order layerwise sandwich plate theory LD3 shown in Table 3. The thickness of each face is hf = h/10, and the presented in Ref. [30]. Therefore, the present laminated plate thickness of core is hc =4h/5. The length to thickness ratio of the theory can be used directly to the analysis of sandwich plates. square sandwich plates are a/h = 10. The characteristic of the predicted stress distributions, espe- Example 2. Simply supported square sandwich plates with cially transverse shear stresses, across the plate thickness is a very softer and harder cores important measure for the accuracy of a laminated plate theory. It was shown [2] that the accuracy of the predicted results by a Figs. 3 and 4, respectively, display the predicted distributions of the sandwich plate depends strongly on the stiffness ratio of the face to normal stress and transverse shear stress across the thickness of the core. In the previous example the stiffness ratio of the sandwich plate with softer core and harder core. The results in

Table 3 The mechanical properties of sandwich plates with softer and harder core.

E1(/GPa) E2(/GPa) E3(/GPa) G12(/GPa) G13(/GPa) G23(/GPa) m12 m13 m23 Face 96 96 96 35.294 35.294 35.294 0.36 0.36 0.36 Softer core 0.00276 0.00276 0.0048 0.000684 0.00102 0.00102 0.5 0.207 0.207 Harder core 27.6 27.6 48 6.84 10.2 10.2 0.5 0.207 0.207

Fig. 3. The distributions of normal stress in the x-direction across the thickness of sandwich plates.

Fig. 4. The distributions of transverse shear stress sxz across the thickness of sandwich plates. X. Wang, G. Shi / Composite Structures 107 (2014) 620–628 627

Figs. 3 and 4 indicate that the results given by the present {3,2} equivalent single layer sandwich plate theories cannot only predict sandwich plate theory are more close to the 3D numerical much better results than other ESLDMs, but also are capable of solutions obtained from ABAQUS when the core is softer especially delivering accurate results as the layerwise plate theories. at the surface with the applied pressure, while the results given by the present {3,1} laminated plate theory performs better when the 5. Conclusion core is harder. Therefore, it can be concluded that the transverse normal strain effect in a sandwich plate plays a more important Based on the Heaviside step function and the variational princi- role when the face to core stiffness ratio Ef3/Ec3 of the sandwich ple, a simple higher-order {3,2} sandwich plate theory accounting plates is large. for both the transverse shear deformations and transverse normal strain is proposed in this paper. The higher-order {3,2} displace- Example 3. Accuracy comparison of the new laminated plate ment field in the present sandwich plate theory satisfies the inter- theory and sandwich plate theory with the results of different facial stress continuity conditions and the traction free conditions types of sandwich plate theories on the plate surfaces. The number of unknowns in the present re- Recently Carrera and Brischetto [2] presented a very compre- fined sandwich plate theory is independent of the number of lam- hensive survey and accuracy assessment of various refined sand- inae in a laminated plate, and it has the same five unknowns as the wich plate theories. They also proposed new benchmarks for the first-order shear deformation plate theories for single-layered accuracy evaluations of refined sandwich plate theories in addition plates. The analytical solutions for both the displacements and to the benchmarks proposed in Ref. [30] which was used in the stresses of the sandwich plates with different aspect ratios Example 1. Benchmark 1 proposed in Ref. [2] is considered in this and stiffness ratios are derived. The comparison of the present ana- example to compare the present {3,1} and {3,2} sandwich plate lytical solutions with the 3D elasticity solutions and the results gi- theories with other refined sandwich plate theories. In this ven by other refined sandwich plate theories indicates that the benchmark, the face sheet is made of Al 2024 and core is made present sandwich plate theory is not only simple, but also capable of Nomex. The material and geometrical properties of Benchmark 1 of achieving the accuracy of layerwise sandwich plate theories. The present results also indicate that the transverse normal strain ef- are listed in Table 4, in which the stiffness ratios are E1f/E1c = 7.3E6 fect in a sandwich plate plays a more important role when the face and E3f/E3c = 962.4. The normalized deflection and transverse shear stresses of the to core stiffness ratio Ef3/Ec3 of the sandwich plates is large. benchmark given by different models are tabulated in Table 5.In The present sandwich plate theory can be used as an accurate the table, ED3 denoted the third-order Equivalent Single Layer Dis- and efficient theoretical model for the finite element analysis of placement-based Model (ESLDM) in which the transverse normal composite sandwich plates, since only five field variables are employed in this new sandwich plate theory. This paper focuses only strain is taken into account; ED4d is the fourth-order ESLDM dis- carding transverse normal strain; EDZ3 is the third-order ESLDM on the theoretical derivation and the analytical solutions of the including zigzag effects; EM3 is the third-order equivalent single new sandwich plate theory accounting for the interfacial stress layer mixed plate theory; and LD3 is the layerwise displacement- continuity and transverse normal strain effect. The finite element based lamination theory. It can be seen from the results in Table 5 formulation based on present refined sandwich plate theory will that among the various sandwich plate models employed in Ref. be reported in a following-up paper. [2] only LD3 could yield reliable results when the sandwich plates are thick. But the layerwise plate theories have a severe shortcom- Acknowledgements ing that is the unknowns used in this type of lamination theories are dependent on the number of the laminae, which results in The financial support provided by the grants of NSFC–91230113 the extensive computation of analysis. Nevertheless, the present is thankfully acknowledged.

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