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Review and assessment of various theories for modeling sandwich composites
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Composite Structures 84 (2008) 282–292 www.elsevier.com/locate/compstruct
Review and assessment of various theories for modeling sandwich composites Heng Hu b
a,b
, Salim Belouettar
a,*
, Michel Potier-Ferry b, El Mostafa Daya
b
a Centre de Recherche Public Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, Luxembourg LPMM, UMR CNRS 7554, I.S.G.M.P., Universite´ Paul Verlaine-Metz, Ile du Saulcy, F-57045 Metz Cedex 01, France
Available online 1 September 2007
Abstract In this paper, an attempt has been made to analyze and assess the various kinematics and theories used for the modeling of sandwich composites. Major classes of representative theories such as classical laminate theory (CLT), first-order shear deformation theory (FSDT) and high-order theories (HOTs) as well as Zig-Zag based theory models have been considered and a unified kinematic formulation is then proposed. Comparative studies with a finite element solution free of any kinematics assumptions have been conducted to address the applicability and the efficiency of previously considered models and theories. Qualitative and quantitative assessments of displacement, stress fields and modal parameters (natural frequency and loss factor) have been presented and discussed for several geometrical and mechanical sandwich beams configurations as well as a clear picture on suitable applications of each analyzed model. 2007 Elsevier Ltd. All rights reserved. Keywords: Sandwich beam; Viscoelastic layer; Damping; Zig-Zag; Inter-laminar continuity
1. Introduction Typically viscoelastic damped structures are sandwich composites in which a viscoelastic layer is sandwiched between two identical elastic ones. In this situation, the damping is introduced by the strong transverse shear in the core. It is due to the difference between in-plane displacement of the elastic faces and the low stiffness of the core. Concerning sandwich composites materials, their use in the past was mainly restricted to secondary non-critical components. In such cases, the main emphasis was to determine the overall global response of the sandwich component, for example cross-deflection, critical buckling loads and fundamental vibration frequencies and associated mode shapes. However, as this type of materials undergoes transition from secondary structural compo-
*
Corresponding author. Tel.: +352 54 55 80 530; fax: +352 42 59 91 333. E-mail address: [email protected] (S. Belouettar). 0263-8223/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.08.007
nents, the goals of analysis must be broadened to include highly accurate assessment. Numerical simulation of these structures requires, firstly an adequate kinematic model to obtain a reasonable computational cost, and secondly a proper account of the shear of the core. Many investigations have been devoted to the static and dynamic analyses of these structures and various theories and models have been proposed on the base of two-dimensional modeling of multi-layered plates and shells. An essential review of applied theories and models can be found in [1] by Carrera. Sandwich structures have been approached by referring to the classical models developed for the traditional onelayer structures. Application of the Kirchhoff approximations have led to the classical laminate theory, CLT. Later on, we took into account the transverse shear deformation effects by applying a Reissner–Mindlin model [2,3], formulating the first-order shear deformation theory, FSDT. However, according to the (3D) elasticity theory, the shear strains vary at least quadratically through the thickness and the transverse shear stress on the surfaces is equal to
H. Hu et al. / Composite Structures 84 (2008) 282–292
zero. The so-called shear correction factors were introduced to correct the discrepancy in shear forces of the first-order shear deformation theories (FSDTs) and 3D elasticity theory. In order to achieve a continuous through-the-thickness distribution of the transverse normal stress with zeros at the sandwich faces, higher-order shear deformation theories (HSDTs) were introduced. The HSDT based models are those in which the displacement is expanded up to higher powers, polynomial like the one proposed by Reddy [4] or sinus-expansion of the thickness coordinate like the one proposed by Touratier [5]. All these proposed theories differ mainly in the inclusion of effects of the shear deformation in their kinematics formulations. In higher-order models the equilibrium equations are obtained in a consistent manner using the principle of virtual work. Classical models and higher-order models (CLT, FSDT and HSDT) have been successfully and extensively applied to design multi-layered structural components. The discontinuity of physical and mechanical properties in the thickness direction makes inadequate those theories, which are originally developed for one-layered structures. Recently, the modeling of sandwich materials is seen to follow the same path of laminated composites, with sharing the same approaches. Comprehensive reviews and assessments of the modeling of multi-layered and sandwich composites can be found, e.g. in the recent survey papers by Noor et al. [6–8] and in the book by Reddy [4]. As focused in the papers [6–8] and in the book [9], the CLT and FSDT models as well as any higher-order smeared-laminate model, e.g. the HSDT model (see [4,10]), based on an overall approximation of the in-plane displacements across the thickness fail to yield accurate results when sandwich composites are (i) thick, (ii) the ratio of the transverse shear modulus to in-plane modulus is low, (iii) anisotropy is severe, and (iv) the ratios of longitudinal to transverse Young’s moduli are high. Several studies published in the literature [6–8] have shown that most of the higher models, while adding more effort in the analysis, do not result in higher accuracy than the first-order shear deformation model, used in conjunction with a post-processing approach based on 3D equations. In [11], Carrera and Ciuffreda uses a unified formulation to compare about 40 theories for multilayered, composites and sandwich plates which are loaded by transverse pressure with various inplane distributions (harmonic, constant, triangular and tent-like). These models, in most cases, describe only an approximation of the real structural behavior but are important for the physical understanding of the complicated behavior and the simplification of the calculations. However, it should be emphasized that the assumptions of these models are based on making them simple yet limit their validity, especially with modern sandwich panels, and in some cases even lead to erroneous results. The classical models are of limited value to analyse problems in which an accurate description of the transverse normal stress and related consequences is requested; it is concluded that such description
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would require the use of layer-wise or Zig-Zag models which describe piecewise continuous displacement fields. These Zig-Zag models are depicted here in two categories: 1. Interlaminar continuous shear stress zig-zag theories (IC-ZZTs). 2. Interlaminar discontinuous shear stress zig-zag theories (ID-ZZTs). In [1], Carrera gave a historical review of most of proposed Zig-Zag theories. He shown that major existing theories [12–16] need to include the interlaminar continuity of shear stress (ICSS) and therefore they are considered as ICZZT. On the contrary, ICSS condition has not been, in most cases, considered for the analysis and the modeling of the vibration problems of three-layer composites as in [17–20] and therefore they are identified as ID-ZZT. Although, all these proposed models are excellent, in authors’ opinions, there is still a need for numerical assessments to raise a clear picture on their applicability and accuracy. Looking at all these proposed theories, it is difficult to judge the efficacy of corresponding models. In this manuscript, an attempt will be made to compare and address the efficiency, the applicability and the limits of classical lamination theory (CLT), higher-order theory (HOT) and Zig-Zag theory based models. To achieve this, a comparative study, with a finite element based solution free of any kinematic assumptions, is then conducted as well as a qualitative and quantitative assessment of displacement, stress fields, modal parameters: natural frequency and loss factor. Static (Problem-1) and dynamic (Problem-2) situations have been considered and analyzed for various sandwich structures configurations. • Problem-1: three-point bending test analysis (displacement and stress fields) of a sandwich beam (Fig. 1). • Problem-2: free vibration analysis (natural frequency and loss factor) of simply supported viscoelastically damped sandwich beam (Fig. 2).
2. Formulation and kinematics Sandwich structures are considered to be plane and twodimensional with a viscoelastic core as in Fig. 3. Let x and z be longitudinal and the transverse coordinates. Hf is the thickness of the faces and Hc is the thickness of the core. The length of the beam and the total thickness are, respectively, L and Ht. The derivation of the general governing
F
Fig. 1. Sketch of Problem-1.
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equations is based on the following restrictive assumptions commons to many authors [14,17–21]: 1. All points on a normal to the beam axis have the same transverse displacement. 2. The displacement is continuous at the interfaces. 3. No shear stress upon the top and bottom surfaces.
8 > U ðx; tÞ z owðx;tÞ þ f ðzÞ þ H2c k 0 bðx; tÞ; > ox < 0 U ðx; z; tÞ ¼ U 0 ðx; tÞ z owðx;tÞ þ ½f ðzÞ þ k 0 zbðx; tÞ; ox > > : owðx;tÞ U 0 ðx; tÞ z ox þ f ðzÞ H2c k 0 bðx; tÞ;
Hc < z 6 H2t ; 2 H c 6 z 6 H2c ; 2 H t 6 z < H2 c ; 2
In addition, we assume that the sheet faces possess inplane and bending rigidities, the core and the faces are considered as homogeneous and isotropic and the faces are considered as 2D linear elastic continuum and its cross-section remain plane after deformation. Concerning sandwich composites modeling, two modeling approaches are commonly used namely, equivalent single-layer (ESL) and layer-wise (LW) theories. In ESL theories, the displacement components represent the weighted-average through the thickness of the sandwich panel. On the other hand, in LW theories, the displacement field is divided into three parts to reproduce piecewise continuous displacement in the thickness direction. The interlaminar equilibrium and displacement continuity is fulfilled by introducing a shear function f(z). The major drawback of LW theories lies in the fact that the number of unknowns is dependent on the number of layers. In correspondence to each interface, the transverse shear stress is discontinuous. Since the mechanical properties are layerdependent, the interlaminar equilibrium can be guaranteed if and only if, the strain components are discontinuous at the interfaces, i.e. if and only if the derivatives along the
Fig. 2. Sketch of Problem-2.
Z
Elastic Viscoelastic Elastic
1 2 3
Fig. 3. Sketch of a Sandwich beam with a viscoelastic layer.
thickness coordinate of the displacement field are discontinuous too. Here, an attempt is made to propose a unified formulation of all considered models where major existing models could be represented only by choosing the appropriate mathematical form of the shear function. Hence, the following general form describes the displacement field:
X
ð1Þ
wðx; z; tÞ ¼ wðx; tÞ;
ð2Þ
where U(x, z, t) is the longitudinal displacement, U0(x, t) is the longitudinal displacement of the mid-plane of the core, w(x, z, t) is the transverse displacement and remains constant in the thickness direction, b(x, t) is the additional rotation of the normal to the mid-plane and f(z) is considered as a ‘shear function’. First-order and higher-order theories are represented. By choosing appropriate f(z) and k0 terms in the ‘shear function’ formulation, different well known models and kinematic refinements could be expressed: • Model-1: k0 = 0 and f(z) = 0,CLT based model. 4z2 • Model-2: k0 = 0 and f ðzÞ ¼ z 1 3H 2 , HSDT based on t Reddy kinematic’s. Ht • Model-3: k0 = 0 and f ðzÞ ¼ p sin Hpzt , HSDT based on Touratier kinematic’s. 4z2 • Model-4: k 0 ¼ GGcf 1 dfdzðzÞ H c and f ðzÞ ¼ z 1 3H 2 , z¼
2
t
IC-ZZT based on Reddy kinematic’s. • Model-5: k 0 ¼ GGcf 1 dfdzðzÞ H c and f ðzÞ ¼ Hpt sin Hpzt , ICz¼
2
ZZT based on Touratier kinematic’s. • Model-6: k0 5 0 and f(z) = 0, ID-ZZT based on Mindlin model in the core. • Model-7: k0 5 0 and f(z) = 0 and Kerwin’s hypotheses [22], ID-ZZT based on Rao’s model. Model-1 is a Kirchhoff–Love theory based model or CLT based model. In Model-1, deformations due to the transverse shear are neglected and therefore the deflection are underestimated and, consequently, natural frequencies are overestimated. Considering the importance of the transverse stress in the modeling of sandwich composites with soft cores, we state that Model-1 is unsuited for the core modeling. Nevertheless, it could be adequate for the faces modeling. Model-2 and Model-3 are HOT based models. They are mainly based on hypothesis of non-linear stress variation through the thickness. The third-order theory by Reddy [4] (Model-2) is based on same assumptions than classical
H. Hu et al. / Composite Structures 84 (2008) 282–292
and first-order theories, except that the assumption of straightness and normality of a transverse normal after deformation is relaxed by expanding the displacements as cubic functions of the thickness coordinate. Model-2 satisfies zero transverse shear stresses on the bounding planes and the equations of motion are derived from the principle of virtual displacements. As the CLT and FSDT, Reddy HOT based model does not satisfy the continuity conditions of transverse shear stresses at layer interfaces. In Model-3, Touratier [5] suggested trigonometric functions instead of polynomial developments of the transverse coordinate. The proposed theory recovers the classic thin plate and Reissner–Mindlin theories and satisfies zero transverse shear stress conditions on the top and bottom surface of plates and avoids shear correction factor. In Model-4 and Model-5, a parameter k0 is used to reproduce the interlaminar continuity of the shear stress (ICSS) at the interfaces. By choosing suitable parameter k0 and the shear function f(z), both displacement and transverse shear stress field are made piecewise continuous. Due to its physical and mechanical continuity at the interfaces, the ICSS condition is widely used for the modeling of laminate composites [1,14]. In Model-6, the core is modeled by using FSDT based model and the faces are modeled by using CLT based model. Thus, the shear stress is constant in the core and zero in the faces.Model-7 by Rao [18] uses same kinematic assumptions as in Model-6. In addition, Rao supposes that weak core layer deforms mainly through shear strain, and does not carry much axial force: the so-called Kerwin assumption [22]. Model-6 and Model-7 have been widely used for the modeling of vibration problems of viscoelastically damped sandwich structures [17,19,20]. Model-7 permits useful formulation of the vibration equations of viscoelastically damped sandwich structures. Indeed, considering Model-7, vibrations equations for these specific structures are generally obtained by using only shear and geometric parameters. 3. Basic equations The proposed two-dimensional theory is constructed accordingly to the following: material behavior is assigned, i.e. Hooke’s law is given and a geometrical relation, i.e. strain–displacement relation is assumed according to the assumption of small strain: 1 oui ouj ij ¼ þ ; ð3Þ 2 oxj oxi r ¼ 2l þ kI 3 TrðÞ; k¼
Em ; ð1 þ mÞð1 2mÞ
l¼
E ; 2ð1 þ mÞ ð4Þ
where I3 is the identity matrix, E and m are, respectively, Young’s modulus and Poisson’s ratio of the core or the faces.
285
As stated previously, the sandwich faces are elastic ones and their mechanical behavior is modeled by a real Young modulus, Ef. On the contrary, the viscoelastic behavior of the core is generally introduced by a convolution product [20]. This product becomes simple one when we deal with static or harmonic motion [17–20]. In this case, the viscoelastic behavior can be introduced by a complex modulus as: Ec ¼ Ec ð1 þ igc Þ where Ec is the delayed elasticity modulus and gc is the loss factor. 3.1. Balance of momentum The basic equilibrium equations are obtained by using the principal of virtual displacement as follows: P acc ðduÞ ¼ P int ðduÞ þ P ext ðduÞ;
ð5Þ
where Pacc(du) describes inertial terms, Pint(du) and Pext(du) are, respectively, internal and external virtual work. Notice that for Problem-1, the expressions of Pext(du) and Pacc(du) are: P acc ðduÞ ¼ 0;
L ; P ext ðduÞ ¼ F dw 2
ð6Þ ð7Þ
and that for Problem-2 Pext(du) and Pacc(du) are as follows: Z L o2 w ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ 2 dw dx; ð8Þ P acc ðduÞ ¼ ot 0 P ext ðduÞ ¼ 0:
ð9Þ
Assuming that the transverse normal stress is negligible (rzz = 0), the internal virtual work of sandwich beam for the two considered problems is expressed as: Z ðrxx dxx þ 2rxz dxz ÞdV t P int ðduÞ ¼ V
Z t H ck0 ¼ rxx zdw;xx þ f ðzÞ þ db;x 2 V1 Z þrxz f;z dbÞdV 1 ðrxx fzdw;xx þ ½f ðzÞ V2
þk 0 zdb;x þ rxz ðf;z þ k 0 Þdb dV 2
Z H ck0 rxx zdw;xx þ f ðzÞ db;x 2 V3 þrxz f;z dbÞdV 3 ; 2
ð10Þ
where w;xx ¼ ooxw2 ; b;x ¼ ob and f ;z ¼ of . ox oz In order to simplify and make more intuitive equation (10), additional definitions are introduced as Mt (Eq. (11)) which denotes the total bending moment to the middle face of sandwich beam, Q (Eq. (12)) which denotes the transverse shear force and the additional bending moment which is indicated by Ma (Eq. (13)) which denotes the axial bending Z Mt ¼ zrxx dS t ; ð11Þ St
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Q¼
Z
rxz f;z dS 1 þ
Z
S1
rxz ðf;z þ k 0 Þ dS 2 þ
Z
S2
ð12Þ
From Eq. (17), we derive two basic equations (Eq. (18)) that permit to describe Problem-1. Q M a;x ¼ 0; ð18Þ M t þ F2 x ¼ 0:
ð13Þ
It is to be noted that all boundary conditions of Problem-1 are included in the following equation: Z L 2 L L 2 2 ½M t dw;x 0 þ ½M t;x dw þ ½M a db0 þ M t;xx dw dx ¼ 0:
rxz f;z dS 3 ; S3
Z H ck0 Ma ¼ rxx f ðzÞ þ rxx ½f ðzÞ dS 1 þ 2 S1 S2
Z H ck0 rxx f ðzÞ þ k 0 z dS 2 þ dS 3 ; 2 S3 Z
where V1, V2, V3, S1, S2, S3 express the integration volumes and the integration areas of layer-1, layer-2 and layer-3 and where Vt and St are given by Vt = V1 + V2 + V3 and St = S1 + S2 + S3, respectively. Using the above defined equations, equilibrium equations of Problem-1 and Problem-2 are established and fully described by Eqs. (14) and (15), respectively Z L L ½M t dw;xx þ M a db;x þ Qdb dx ¼ F dw ; ð14Þ 2 0
x¼0
0
ð19Þ By inserting Eqs. (4) and 11, 12, 13 into Eq. (18), new differential equations are obtained for Problem-1: ( A0 w;xx A1 b;x þ F2 x ¼ 0; ð20Þ A1 w;xxx þ A2 b A3 b;xx ¼ 0; where A0, A1, A2 and A3 are constant and defined by the following general formulas:
R R R 8 A0 ¼ Ef S 1 z2 dS 1 þ Ec S 1 z2 dS 2 þ Ef S 3 z2 dS 3 ; > > > R R R > > < A1 ¼ Ef S z f ðzÞ þ H c2k0 dS 1 þ Ec S z½f ðzÞ þ k 0 zdS 2 þ Ef S z f ðzÞ H c2k0 dS 3 ; 1 2 3 R R R 2 > A2 ¼ Gf S 1 f;z2 dS 1 þ Gc S 2 ðf;z þ k 0 Þ dS 2 þ Gf S 3 f;z2 dS 3 ; > > > > 2 2 R R R : 2 A3 ¼ Ef S 1 f ðzÞ þ H c2k0 dS 1 þ Ec S 2 ½f ðzÞ þ k 0 z dS 2 þ Ef S 3 f ðzÞ H c2k0 dS 3 :
Z
L
½M t dw;xx þ M a db;x þ Qdb dx
0
¼
Z
L
ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ
0
o2 w dw dx: ot2
ð15Þ
3.2. Solution of Problem-1 In order to solve Problem-1, the definition domain of the equilibrium equation (14) is split into two parts ð½0; L2 and L2 ; L Þ. Since the considered problem is symmetrical, the equilibrium equation (14) takes the following integral form: Z
L 2
½M t dw;xx þ M a db;x þ Qdb dx ¼
0
F L dw : 2 2
ð16Þ
Integration by parts, Eq. (16) allows to obtain: L ½M t dw;x 02 þ ½M t;x dw
L
þ ½M a db02 þ
Z
Eq. (21) is valid for the analysis of all previously described models except Model-7 by Rao. In the later the axial forces are neglected in the core which lead to ignore the terms with Ec in expressions of A0, A1 and A3. Under hand the above-defined constants, Eq. (20) is solved analytically and therefore Problem-1 is completely solved by defining expressions of the deflection w and rotation b: bðxÞ ¼
A0 F 2 FL2 w;x þ x ; 4A1 A1 16A1
wðxÞ ¼
F 3 FL2 FA2 x þ xþ 21 12A0 16A0 2A0 A2
in which sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 A2 : n¼ A0 A3 A21
ð22Þ !
x
sinhðnxÞ ; n sinhðnL Þ 2
ð23Þ
ð24Þ
L 2
It should be stated that A1, A2 and A3 are zeros in Model-1 and therefore the transverse displacement takes the following simplified form:
M t;xx dw dx ! Z L2 F L þ M t;x þ ðQ M a;x Þdb dx ¼ 0: dw 2 2 L 0 x¼ x¼0
ð21Þ
0
2
ð17Þ
wm1 ¼
F 3 FL2 x þ x: 12A0 16A0
ð25Þ
H. Hu et al. / Composite Structures 84 (2008) 282–292
By analyzing Eqs. (23) and (25), one can express the displacement solution, in the case of above-considered kinematics models, as an adding of the solution obtained by using Model-1 and a correction term " # FA21 sinhðnxÞ : wðxÞ ¼ wm1 þ 2 x ð26Þ n sinh nL 2A0 A2 2 3.3. Solution of Problem-2 To investigate the numerical accuracy of the analyzed models, natural frequencies and loss factors are investigated. Integrating by parts, Eq. (15) permits to derive basic equations of Problem-2: Z L L L L ðQ M a;x Þdb dx ½M t dw;x 0 ½M t;x dw0 þ ½M a db0 þ 0
Z L o2 w þ M t;xx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ 2 dw dx ¼ 0; ot 0 (
ð27Þ Q M a;x ¼ 0; 2
M t;xx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ ootw2 ¼ 0; L
L
L
½M t dw;x 0 ½M t;x dw0 þ ½M a db0 ¼ 0:
ð28Þ ð29Þ
While expressing Mt, Q and Ma according, respectively, to Eqs. (11)–(13), we obtain a system of differential equations in w and b: ( 2 A0 w;xxxx A1 b;xxx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ ootw2 ¼ 0; ð30Þ A1 w;xxx þ A2 b A3 b;xx ¼ 0: This differential equation appears as very complicated with no known general analytical close-form solution. In order to solve this equation, the amplitude of harmonic vibrations is assumed for Problem-2 as: wðx; tÞ ¼ A ejXt evx :
ð31Þ
By using the end conditions in Eq. (29), a sixth-order characteristic determinant which has to be zero for non-trivial solution is established and from which, the deflection w(x, t) can be expressed as: npx wðx; tÞ ¼ A ejXt sin ; ð32Þ L where n is the mode of vibration. Finally, substituting Eq. (32) into Eq. (30) yields the expression for the complex pulsation X as:
A21 j2 X2 ¼ X2m1 1 ; ð33Þ A0 ðA2 þ A3 j2 Þ where j = np/L, Xm1 is the complex pulsation obtained by using Model-1: X2m1 ¼
A0 j 4 : q1 S 1 þ q2 S 2 þ q3 S 3
ð34Þ
287
According to Eq. (33) or (34), we define the loss factor g and the natural pulsation x0 of the damped sandwich beam by using the following definitions: X2 ¼ x20 ð1 þ igÞ;
g ¼ ImðX2 Þ=ReðX2 Þ:
ð35Þ
4. Analytical and numerical assessments In order to assess models associated with the previously described kinematic assumptions, finite element calculations have been achieved for Problem-1 and Problem-2. A two-dimensional finite element analysis is carried out with a ‘2D-Q8’ element defined by eight nodes having two degrees of freedom at each node (2 in-plane translations). The presented results are obtained with in-house Matlab developed finite element (FE) code. The sandwich beam consists of two face sheets made of aluminum and a lightweight core whose mechanical and geometrical properties are given in Table 1. Three non-dimensional beam variables are used in this comparative study, namely, the ratio of core and skin Young moduli (Ec/Ef), the ratio of beam length to beam total thickness (L/Ht) and the ratio of core to skin thickness (Hc/Hf). Let us remark that under these considerations and by using material parameters as listed in Table 1, all sandwich possible configurations can be presented. For these purposes, five evaluations parameters are presented: 1. wmax(m): w(L/2,0); maximum deflection in Problem-1. 2. rcis(Pa): rxz(L/4,0); maximum shear stress in the section x = L/4 in Problem-1. 3. rnor(Pa): rxx(L/4, Ht/2); maximum normal stress in the section x = L/4 in Problem-1. 4. f1st(Hz): f0 = x0/2p; first natural frequency in Problem2. 5. g1st: g/gc; first loss factor ratio in Problem-2. The assessment of parameters wmax, rnor and f1st are global ones since they depend on the global rigidity of the structure. On the other hand, rcis and g1st mainly depend on the core properties since they describe shear stress and damping. The latter is introduced by the shear stress, which is mainly due to the difference between the longitudinal displacements of the elastic faces. Remark first that core-to-face stiffness ratio influences considerably the sandwich beam behavior and consequently it influences the choice of the adequate modeling theory. Tables 3 and 4 show the results obtained by using various types of kinematic modeling and with various core to face Young moduli (Ec/Ef) ratios. Table 1 Material parameters and dimensions Ef 6.9E10 Pa
qc
qf 3
968 kg/m
3
2770 kg/m
Ht
mc
mf
gc
0.01 m
0.3
0.3
0.3
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4.1. Mesh convergence study N x = 10
Prior to initiating the evaluation study, an analysis of mesh convergence is carried out to ensure the accuracy of the proposed finite element solution since it is considered in the present study as the reference. Thus, calculations were repeated three times, first time cutting back the typical element length by a factor of two and then by a factor of 2.5, see Fig. 4. Table 2 shows the numerical values, of wmax, rcis, rnor, obtained in each considered mesh configuration. The relative difference is almost zero in the first refinement and in the last refinement step.
N zf = 1 N cz = 1
4.2. Assessment of CLT and HSDT models N x = 20
Considering the global criteria, classical and high order theories based models are not sensitive to core-to-face stiffness ratio (Ec/Ef). Indeed, wmax, rnor and f1st remain nearly constant when the stiffness ratio decreases. Considering this, global rigidities are almost independent on core-to-face stiffness ratio. Moreover, these models overestimate the global rigidity of sandwich beams with an unacceptable margin. Regarding to stiff cores (Ec/Ef P 0.01), single layer theory yields acceptable results for wmax, rnor and f1st with maximum relative error of 5.8% and 1.9%, respectively, for wmax, and f1st (see Tables 3 and 4). In addition, global criteria based models yield the same rnor values as in the FE based solution. On the contrary, regarding to soft core (Ec/Ef < 0.01), global criteria and FE based solutions present big differences in terms of wmax, rnor and f1st, as for the core criteria, classical and high order based theory models give inaccurate results of rcis and g1st, since the transverse shear stress distribution is not simulated adequately by these models.
N zf = 1 N cz = 1
N x = 50 N zf = 3 N cz = 3
4.3. Post-processing for CLT and HSDT models The 3-D equilibrium equation is integrated to obtain transverse normal stress as follows: orxx orxz þ ¼ 0: ox oz
ð36Þ
In calculating transverse shear stresses in Eq. (36), both constitutive approach and equilibrium approach are used. Since the postprocess method does not guarantee the satisfaction of shear equilibrium equation of higher-order theory, constitutive approach shows same order of accuracy compared to that of equilibrium approach. Thus in the computation of transverse normal stress, it is sufficient to obtain transverse shear stresses in Eq. (36) by constitutive equation. The maximum transversal shear stress rcis is given as ! Z H c Z 0 2 rcis ¼ Ef z dz þ Ec z dz w;xxx H t 2
H c 2
Z Ef
H c 2
H t 2
f ðzÞ dz þ Ec
Z
0 H c 2
! f ðzÞ dz b;xx :
ð37Þ
Fig. 4. Three meshes of the half sandwich beam (Hc/Hf = 1, Ec/ Ef = 0.0001, L/Ht = 50).
Table 2 Convergence of the static test
wmax rcis rnorm
Mesh 1
Mesh 2
Mesh 3
2.2380E5 4.6390E+3 6.9149E+5
2.2416E5 4.6339E+3 6.9106E+5
2.2446E5 4.6367E+3 6.9172E+5
Notice that using a postprocess method yields very good results as long as Ec/Ef is greater than 0.001. On the contrary, the post-processing method is without any help for the dynamical analysis of sandwich composites with very soft core (Ec/Ef = 0.0001), since it cannot permit to obtain accurate loss factor. Tests with higher-order models (see for instance Table 5) have shown that unless the 3-D equilibrium equations are
H. Hu et al. / Composite Structures 84 (2008) 282–292
289
Table 3 Influence of Ec/Ef in static tests (F = bq and q = 100 N/m, Hc/Hf = 1, L/Ht = 50) CLT
HSDT
IC-ZZT
ID-ZZT
FEM
Model-1
Model-2
Model-3
Model-4
Model-5
Model-6
2D-Q8
Ec Ef
¼1
wmax rcis rnor
4.53E5 0.00E+0 3.75E+5
4.53E5 7.50E+3 3.75E+5
4.53E5 7.74E+3 3.75E+5
4.53E5 7.49E+3 3.75E+5
4.53E5 7.73E+3 3.75E+5
4.53E5 7.22E+3 3.75E+5
4.53E5 7.64E+3 3.75E+5
Ec Ef
¼ 101
wmax rcis rnor
4.69E5 0.00E+0 3.88E+5
4.70E5 1.56E+3 3.88E+5
4.70E5 1.70E+3 3.88E+5
4.72E5 6.98E+3 3.88E+5
4.72E5 7.01E+3 3.88E+5
4.72E5 6.95E+3 3.88E+5
4.72E5 7.00E+3 3.88E+5
Ec Ef
¼ 102
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.71E5 1.75E+2 3.89E+5
4.72E5 1.93E+2 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
Ec Ef
¼ 103
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.72E5 1.77E+1 3.89E+5
4.72E5 1.96E+1 3.89E+5
7.31E5 6.76E+3 3.98E+5
7.31E5 6.76E+3 3.98E+5
7.31E5 6.76E+3 3.98E+5
7.32E5 6.79E+3 3.98E+5
Ec Ef
¼ 104
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.72E5 1.77E+0 3.89E+5
4.72E5 1.96E+0 3.89E+5
2.23E4 4.63E+3 6.90E+5
2.23E4 4.63E+3 6.90E+5
2.23E4 4.63E+3 6.90E+5
2.24E4 4.64E+3 6.92E+5
Table 4 Influence of Ec/Ef in dynamic tests (Hc/Hf = 1, L/Ht = 50) CLT
HSDT
Model-1
Model-2
Model-3
Model-4
IC-ZZT Model-5
Model-6
ID-ZZT Model-7
2D-Q8
FEM
Ec Ef
¼1
f1st g1st
1.02E+2 3.70E2
1.02E+2 3.76E2
1.02E+2 3.76E2
1.02E+2 3.75E2
1.02E+2 3.75E2
1.02E+2 3.75E2
1.02E+2 4.82E4
1.02E+2 3.79E2
Ec Ef
¼ 101
f1st g1st
1.01E+2 3.84E3
1.00E+2 4.07E3
1.00E+2 4.07E3
1.00E+2 8.63E3
1.00E+2 8.63E3
1.00E+2 8.63E3
1.00E+2 4.80E3
1.00E+2 8.60E3
Ec Ef
¼ 102
f1st g1st
1.00E+2 3.84E4
1.00E+2 4.14E4
1.00E+2 4.20E4
9.80E+1 4.61E2
9.80E+1 4.61E2
9.81E+1 4.61E2
9.80E+1 4.57E2
9.81E+1 4.60E2
Ec Ef
¼ 103
f1st g1st
1.00E+2 3.84E5
1.00E+2 4.15E5
1.00E+2 4.21E5
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
Ec Ef
¼ 104
f1st g1st
1.00E+2 3.84E6
1.00E+2 4.15E6
1.00E+2 4.22E6
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
Table 5 Shear stress of CLT and HSDT models by post-processing (F = bq and q = 100 N/m, Hc/Hf = 1, L/Ht = 50)
Ec Ef Ec Ef Ec Ef Ec Ef Ec Ef
¼1 ¼ 101 ¼ 102 ¼ 103 ¼ 104
rcis rcis rcis rcis rcis
CLT
HSDT
Model-1
Model-2
Model-3
2D-Q8
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.64E+3 7.00E+3 6.93E+3 6.79E+3 4.64E+3
used in evaluating the thickness distribution of the transverse stresses, the resulting stresses are inaccurate. 4.4. Evaluation of IC-ZZT and ID-ZZT models Notice first, from Tables 3 and 4, that zig-zag based models are always more precise than CLT and HSDT theories based models, especially Model-4, Model-5 and Model-6, they present always very good results both for global criterion and for core criterion. For soft cores (Ec/
FEM
Ef 6 0.01), Model-7 give a good estimation compared to the FE based solutions with maximum relative error for f1st of 0.10% and of 0.70% for g1st. On the contrary, for stiff cores, where Kerwin’s assumptions are not satisfied, the previous concluding remarks are not valid and the relative errors are larger than those obtained by other zig-zag based models. For presentation convenience and simplicity, only Model-5 and Model-6 results are discussed. The transverse shear stress of the considered IC-ZZT and ID-ZZT models
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H. Hu et al. / Composite Structures 84 (2008) 282–292
−3
5 x 10
−3
5
Model−5 Model−6
4
3
2
2
Thickness (m)
Thickness (m)
Model−5 Model−6 (σxz corrected)
4
3
1 0 −1 −2
1 0 −1 −2
−3
−3
−4
−4
−5
x 10
−7000
−6000
−5000
−4000
−3000
−2000
−1000
−5
0
Transverse shear stress at the section x=L/4 (Ec/Ef=0.1)
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress at the section x=L/4 (Ec/Ef=0.1)
−3
5 4
x 10
−3
Model−5 Model−6
5 4
3
2
1
Thickness (m)
Thickness (m)
Model−5 Model−6 (σxz corrected)
3
2
0 −1 −2 −3
1 0 −1 −2 −3
−4 −5
x 10
−4
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress (Pa) at the section x=L/4 (Ec/Ef=0.0001)
−5
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress (Pa) at the section x=L/4 (Ec/Ef=0.0001)
Fig. 5. Comparison of shear stress in the thickness direction between Model-5 and Model-6.
are, respectively, presented in Fig. 5. Noting that for Ec/ Ef < 1, Model-5 and Model-6 present almost the same distribution of shear stress in the core. On the other hand, Model-5 and Model-6 present big discrepancies of the shear stress in the case of faces since Model-6 is based on Kirchhoff assumptions. However, this disparity is not of a real concern for sandwich structures modeling in general and for viscoelastically damped sandwich vibration modeling more precisely, since rigidity ratio is generally between 0.001 and 0.0001. Moreover, this disparity can be reduced with a precessing of ID-ZZT models, see Fig. 6, where the shear stress distribution in the faces is obtained by the equilibrium equation. 4.5. Other influences Tables 3 and 4 present the influence of rigidities ratios, where Hc/Hf and L/Ht remain constants. In Tables 6–8, we will show the influences of Hc/Hf and L/Ht. From Table 6, let us remark that the obtained results by using IC-ZZT and ID-ZZT models are always very close to FE based solutions. Except for very thick core (Hc/ Hf = 100), where one finds a larger error for loss factor with the Rao’s model (Model-7) than with two other ones. This is due, in fact, to the non-validity of Kerwin hypoth-
Fig. 6. Comparison of shear stress in the thickness direction between Model-5 and Model-6 when used in conjunction with a post-processing method in the faces.
eses since the core is very thick and the stress in the core cannot be reduced to shear. At the same time, Rao’s model underestimates the frequencies and the global stiffness of the sandwich beam. Notice from Tables 7 and 8 that for all L/Ht ratios, both IC-ZZT models and ID-ZZT models give acceptable results and that even for small L/Ht ratios, their efficiencies are also valid. Moreover, IC-ZZT and ID-ZZT models can be used in the study of high vibration modes, see Table 9. 5. Conclusion Beside the analysis of various in-the-literature models and theories, governing equations of viscoelastic damped beams have been derived by using the virtual work principle. A general kinematic model has then been introduced and used and the deformation as well as free vibration analyzes of sandwich beam have been performed. Global criteria, which depend on global rigidity as well as core criteria, which depend on rigidities ratio, have been introduced to evaluate the different proposed kinematics. Evaluation procedures have been performed and the results have been presented for static and dynamic problems.
H. Hu et al. / Composite Structures 84 (2008) 282–292
291
Table 6 Influence of Hc/Hf in dynamic tests (L/Ht = 20, Ec/Ef = 103) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
Hc Hf
¼ 101
f1st g1st
4.73E+2 2.72E1
4.73E+2 2.72E1
4.74E+2 2.73E1
4.74E+2 2.73E1
4.73E+2 2.72E1
Hc Hf
¼1
f1st g1st
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
Hc Hf
¼ 101
f1st g1st
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
Hc Hf
¼ 102
f1st g1st
2.00E+2 2.36E1
2.00E+2 2.36E1
2.00E+2 2.36E1
1.99E+2 2.20E1
2.00E+2 2.33E1
Model-4
Model-5
Model-6
Model-7
2D-Q8
Table 7 Influence of L/Ht in dynamic tests (Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
L Ht
¼ 10
f1st g1st
9.31E+2 4.09E1
9.31E+2 4.09E1
9.32E+2 4.10E1
9.31E+2 4.10E1
9.30E+2 4.10E1
L Ht
¼ 40
f1st g1st
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.89E1
L Ht
¼ 100
f1st g1st
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
Table 8 Influence of L/Ht in static tests (Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
L Ht
¼ 10
wmax rcis rnor
2.77E6 3.11E+3 1.99E+5
2.77E6 3.12E+3 1.99E+5
2.76E6 3.11E+3 1.99E+5
2.76E6 3.11E+3 1.99E+5
2.94E6 3.14E+3 1.72E+5
L Ht
¼ 40
wmax rcis rnor
4.42E5 6.58E+3 3.29E+5
4.42E5 6.58E+3 3.29E+5
4.41E5 6.58E+3 3.29E+5
4.41E5 6.58E+3 3.29E+5
4.43E5 6.58E+3 3.29E+5
L Ht
¼ 100
wmax rcis rnor
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.33E4 6.92E+3 7.79E+5
Table 9 Natural frequencies and loss factors(L/Ht = 50, Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
2nd mode
Frequency Loss factor
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
4th mode
Frequency Loss factor
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
8th mode
Frequency Loss factor
2.05E+3 2.34E1
2.05E+3 2.35E1
2.05E+3 2.36E1
2.05E+3 2.36E1
2.04E+3 2.35E1
It comes out from this assessment process that IC-ZZT and ID-ZZT models are more accurate than CLT and HSDT based models and that CLT and HSDT models always overestimate global rigidities and do not enable a
valid description of the distribution of the transverse shear stress. Nevertheless, CLT and HSDT models can be used for relatively rigid cores (Ec/Ef larger than 0.001), in conjunction with a post-processing method, to predict accu-
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rately the global stiffness, the core shear stress and the loss factor of the structure. For soft cores, a Zig-Zag model is necessary. Apart of the fact that Model-4 and Model-5 give the same results, they are often accurate and therefore they can be used for the modeling of all types of the three-layered structures. Compared to FE based solutions, Model-6 is more precise than Model-7. Considering this, we state that Model-7 is inadequate for modeling of sandwich beams with rigid core or with thick cores. On the other hand, Rao’s model is suitable for the modeling of sandwich composites with soft and thin cores. Acknowledgements This study has been supported by the European FP6 STREP project CASSEM: NMP3-CT-2005-013517. The authors would like also to acknowledge the financial support of the Luxembourgish Ministry of Research. References [1] Carrera E. Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 2003;56(3):287–308. [2] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945;12:69–76. [3] Mindlin RD. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J Appl Mech 1951;18:1031–6. [4] Reddy JN. A simple higher-order theory of laminated composite plate. J Appl Mech 1984;51:745–52. [5] Touratier M. An efficient standard plate theory. Int J Eng Sci 1991;29:901–16.
[6] Noor AK et al. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;41(1):1–12. [7] Noor AK et al. Assessment of computational models for multilayered anisotropic plates. Compos Struct 1990;14:233–65. [8] Noor AK et al. Computational models for sandwich panels and shells. Appl Mech Rev 1996;49(3):155–99. [9] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton (FL): CRC Press; 1997. [10] Jemielita G. On kinematical assumptions of refined theories of plates: a survey. J Appl Mech 1990;57(3):1088–91. [11] Carrera E, Ciuffreda A. A unified formulation to assess theories of multilayered plates for various bending problems. Compos Struct 2005;69:271–93. [12] Lekhnitskii SG. Strength calculation of composite beams. Vestn Inzhen Tekh 1935;9. [13] Ambartsumian SA. On a theory of bending of anisotropiv plates. Prikl Mat Mekh 1958;22(2):226–37. [14] Whitney JM. The effects of transverse shear deformation on the bending of laminated plates. J Compos Mater 1969;3:537–47. [15] Reissner E. On a certain mixed variational theory and a proposed application. Int J Numer Methods Eng 1984;20:1366–8. [16] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Eng State Art Rev 2003;10:215–97. [17] Mead DJ et al. The forced vibration of three-layer damped sandwich beam with arbitrary boundary conditions. J Sound Vib 1969;10:163–75. [18] Rao DK. Frequency and loss factor of sandwich beams under various boundary conditions. J Mech Eng Sci 1978;20(5):271–82. [19] He JF et al. Vibration analysis of viscoelastically damped sandwich plates. J Sound Vib 1992;152:107–23. [20] Daya EM et al. A shell element for viscoelastically damped sandwich structures. Rev Eur Elem Finis 2002;11(1):39–56. [21] Idlbi A et al. Comparison of various laminated plate theories. Compos Struct 1997;37:173–84. [22] Kerwin EM. Damping of flexural waves by a constrained visco-elastic layer. J Acoust Soc Am 1959;31:952–62.
Composite Structures 84 (2008) 282–292 www.elsevier.com/locate/compstruct
Review and assessment of various theories for modeling sandwich composites Heng Hu b
a,b
, Salim Belouettar
a,*
, Michel Potier-Ferry b, El Mostafa Daya
b
a Centre de Recherche Public Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, Luxembourg LPMM, UMR CNRS 7554, I.S.G.M.P., Universite´ Paul Verlaine-Metz, Ile du Saulcy, F-57045 Metz Cedex 01, France
Available online 1 September 2007
Abstract In this paper, an attempt has been made to analyze and assess the various kinematics and theories used for the modeling of sandwich composites. Major classes of representative theories such as classical laminate theory (CLT), first-order shear deformation theory (FSDT) and high-order theories (HOTs) as well as Zig-Zag based theory models have been considered and a unified kinematic formulation is then proposed. Comparative studies with a finite element solution free of any kinematics assumptions have been conducted to address the applicability and the efficiency of previously considered models and theories. Qualitative and quantitative assessments of displacement, stress fields and modal parameters (natural frequency and loss factor) have been presented and discussed for several geometrical and mechanical sandwich beams configurations as well as a clear picture on suitable applications of each analyzed model. 2007 Elsevier Ltd. All rights reserved. Keywords: Sandwich beam; Viscoelastic layer; Damping; Zig-Zag; Inter-laminar continuity
1. Introduction Typically viscoelastic damped structures are sandwich composites in which a viscoelastic layer is sandwiched between two identical elastic ones. In this situation, the damping is introduced by the strong transverse shear in the core. It is due to the difference between in-plane displacement of the elastic faces and the low stiffness of the core. Concerning sandwich composites materials, their use in the past was mainly restricted to secondary non-critical components. In such cases, the main emphasis was to determine the overall global response of the sandwich component, for example cross-deflection, critical buckling loads and fundamental vibration frequencies and associated mode shapes. However, as this type of materials undergoes transition from secondary structural compo-
*
Corresponding author. Tel.: +352 54 55 80 530; fax: +352 42 59 91 333. E-mail address: [email protected] (S. Belouettar). 0263-8223/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.08.007
nents, the goals of analysis must be broadened to include highly accurate assessment. Numerical simulation of these structures requires, firstly an adequate kinematic model to obtain a reasonable computational cost, and secondly a proper account of the shear of the core. Many investigations have been devoted to the static and dynamic analyses of these structures and various theories and models have been proposed on the base of two-dimensional modeling of multi-layered plates and shells. An essential review of applied theories and models can be found in [1] by Carrera. Sandwich structures have been approached by referring to the classical models developed for the traditional onelayer structures. Application of the Kirchhoff approximations have led to the classical laminate theory, CLT. Later on, we took into account the transverse shear deformation effects by applying a Reissner–Mindlin model [2,3], formulating the first-order shear deformation theory, FSDT. However, according to the (3D) elasticity theory, the shear strains vary at least quadratically through the thickness and the transverse shear stress on the surfaces is equal to
H. Hu et al. / Composite Structures 84 (2008) 282–292
zero. The so-called shear correction factors were introduced to correct the discrepancy in shear forces of the first-order shear deformation theories (FSDTs) and 3D elasticity theory. In order to achieve a continuous through-the-thickness distribution of the transverse normal stress with zeros at the sandwich faces, higher-order shear deformation theories (HSDTs) were introduced. The HSDT based models are those in which the displacement is expanded up to higher powers, polynomial like the one proposed by Reddy [4] or sinus-expansion of the thickness coordinate like the one proposed by Touratier [5]. All these proposed theories differ mainly in the inclusion of effects of the shear deformation in their kinematics formulations. In higher-order models the equilibrium equations are obtained in a consistent manner using the principle of virtual work. Classical models and higher-order models (CLT, FSDT and HSDT) have been successfully and extensively applied to design multi-layered structural components. The discontinuity of physical and mechanical properties in the thickness direction makes inadequate those theories, which are originally developed for one-layered structures. Recently, the modeling of sandwich materials is seen to follow the same path of laminated composites, with sharing the same approaches. Comprehensive reviews and assessments of the modeling of multi-layered and sandwich composites can be found, e.g. in the recent survey papers by Noor et al. [6–8] and in the book by Reddy [4]. As focused in the papers [6–8] and in the book [9], the CLT and FSDT models as well as any higher-order smeared-laminate model, e.g. the HSDT model (see [4,10]), based on an overall approximation of the in-plane displacements across the thickness fail to yield accurate results when sandwich composites are (i) thick, (ii) the ratio of the transverse shear modulus to in-plane modulus is low, (iii) anisotropy is severe, and (iv) the ratios of longitudinal to transverse Young’s moduli are high. Several studies published in the literature [6–8] have shown that most of the higher models, while adding more effort in the analysis, do not result in higher accuracy than the first-order shear deformation model, used in conjunction with a post-processing approach based on 3D equations. In [11], Carrera and Ciuffreda uses a unified formulation to compare about 40 theories for multilayered, composites and sandwich plates which are loaded by transverse pressure with various inplane distributions (harmonic, constant, triangular and tent-like). These models, in most cases, describe only an approximation of the real structural behavior but are important for the physical understanding of the complicated behavior and the simplification of the calculations. However, it should be emphasized that the assumptions of these models are based on making them simple yet limit their validity, especially with modern sandwich panels, and in some cases even lead to erroneous results. The classical models are of limited value to analyse problems in which an accurate description of the transverse normal stress and related consequences is requested; it is concluded that such description
283
would require the use of layer-wise or Zig-Zag models which describe piecewise continuous displacement fields. These Zig-Zag models are depicted here in two categories: 1. Interlaminar continuous shear stress zig-zag theories (IC-ZZTs). 2. Interlaminar discontinuous shear stress zig-zag theories (ID-ZZTs). In [1], Carrera gave a historical review of most of proposed Zig-Zag theories. He shown that major existing theories [12–16] need to include the interlaminar continuity of shear stress (ICSS) and therefore they are considered as ICZZT. On the contrary, ICSS condition has not been, in most cases, considered for the analysis and the modeling of the vibration problems of three-layer composites as in [17–20] and therefore they are identified as ID-ZZT. Although, all these proposed models are excellent, in authors’ opinions, there is still a need for numerical assessments to raise a clear picture on their applicability and accuracy. Looking at all these proposed theories, it is difficult to judge the efficacy of corresponding models. In this manuscript, an attempt will be made to compare and address the efficiency, the applicability and the limits of classical lamination theory (CLT), higher-order theory (HOT) and Zig-Zag theory based models. To achieve this, a comparative study, with a finite element based solution free of any kinematic assumptions, is then conducted as well as a qualitative and quantitative assessment of displacement, stress fields, modal parameters: natural frequency and loss factor. Static (Problem-1) and dynamic (Problem-2) situations have been considered and analyzed for various sandwich structures configurations. • Problem-1: three-point bending test analysis (displacement and stress fields) of a sandwich beam (Fig. 1). • Problem-2: free vibration analysis (natural frequency and loss factor) of simply supported viscoelastically damped sandwich beam (Fig. 2).
2. Formulation and kinematics Sandwich structures are considered to be plane and twodimensional with a viscoelastic core as in Fig. 3. Let x and z be longitudinal and the transverse coordinates. Hf is the thickness of the faces and Hc is the thickness of the core. The length of the beam and the total thickness are, respectively, L and Ht. The derivation of the general governing
F
Fig. 1. Sketch of Problem-1.
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H. Hu et al. / Composite Structures 84 (2008) 282–292
equations is based on the following restrictive assumptions commons to many authors [14,17–21]: 1. All points on a normal to the beam axis have the same transverse displacement. 2. The displacement is continuous at the interfaces. 3. No shear stress upon the top and bottom surfaces.
8 > U ðx; tÞ z owðx;tÞ þ f ðzÞ þ H2c k 0 bðx; tÞ; > ox < 0 U ðx; z; tÞ ¼ U 0 ðx; tÞ z owðx;tÞ þ ½f ðzÞ þ k 0 zbðx; tÞ; ox > > : owðx;tÞ U 0 ðx; tÞ z ox þ f ðzÞ H2c k 0 bðx; tÞ;
Hc < z 6 H2t ; 2 H c 6 z 6 H2c ; 2 H t 6 z < H2 c ; 2
In addition, we assume that the sheet faces possess inplane and bending rigidities, the core and the faces are considered as homogeneous and isotropic and the faces are considered as 2D linear elastic continuum and its cross-section remain plane after deformation. Concerning sandwich composites modeling, two modeling approaches are commonly used namely, equivalent single-layer (ESL) and layer-wise (LW) theories. In ESL theories, the displacement components represent the weighted-average through the thickness of the sandwich panel. On the other hand, in LW theories, the displacement field is divided into three parts to reproduce piecewise continuous displacement in the thickness direction. The interlaminar equilibrium and displacement continuity is fulfilled by introducing a shear function f(z). The major drawback of LW theories lies in the fact that the number of unknowns is dependent on the number of layers. In correspondence to each interface, the transverse shear stress is discontinuous. Since the mechanical properties are layerdependent, the interlaminar equilibrium can be guaranteed if and only if, the strain components are discontinuous at the interfaces, i.e. if and only if the derivatives along the
Fig. 2. Sketch of Problem-2.
Z
Elastic Viscoelastic Elastic
1 2 3
Fig. 3. Sketch of a Sandwich beam with a viscoelastic layer.
thickness coordinate of the displacement field are discontinuous too. Here, an attempt is made to propose a unified formulation of all considered models where major existing models could be represented only by choosing the appropriate mathematical form of the shear function. Hence, the following general form describes the displacement field:
X
ð1Þ
wðx; z; tÞ ¼ wðx; tÞ;
ð2Þ
where U(x, z, t) is the longitudinal displacement, U0(x, t) is the longitudinal displacement of the mid-plane of the core, w(x, z, t) is the transverse displacement and remains constant in the thickness direction, b(x, t) is the additional rotation of the normal to the mid-plane and f(z) is considered as a ‘shear function’. First-order and higher-order theories are represented. By choosing appropriate f(z) and k0 terms in the ‘shear function’ formulation, different well known models and kinematic refinements could be expressed: • Model-1: k0 = 0 and f(z) = 0,CLT based model. 4z2 • Model-2: k0 = 0 and f ðzÞ ¼ z 1 3H 2 , HSDT based on t Reddy kinematic’s. Ht • Model-3: k0 = 0 and f ðzÞ ¼ p sin Hpzt , HSDT based on Touratier kinematic’s. 4z2 • Model-4: k 0 ¼ GGcf 1 dfdzðzÞ H c and f ðzÞ ¼ z 1 3H 2 , z¼
2
t
IC-ZZT based on Reddy kinematic’s. • Model-5: k 0 ¼ GGcf 1 dfdzðzÞ H c and f ðzÞ ¼ Hpt sin Hpzt , ICz¼
2
ZZT based on Touratier kinematic’s. • Model-6: k0 5 0 and f(z) = 0, ID-ZZT based on Mindlin model in the core. • Model-7: k0 5 0 and f(z) = 0 and Kerwin’s hypotheses [22], ID-ZZT based on Rao’s model. Model-1 is a Kirchhoff–Love theory based model or CLT based model. In Model-1, deformations due to the transverse shear are neglected and therefore the deflection are underestimated and, consequently, natural frequencies are overestimated. Considering the importance of the transverse stress in the modeling of sandwich composites with soft cores, we state that Model-1 is unsuited for the core modeling. Nevertheless, it could be adequate for the faces modeling. Model-2 and Model-3 are HOT based models. They are mainly based on hypothesis of non-linear stress variation through the thickness. The third-order theory by Reddy [4] (Model-2) is based on same assumptions than classical
H. Hu et al. / Composite Structures 84 (2008) 282–292
and first-order theories, except that the assumption of straightness and normality of a transverse normal after deformation is relaxed by expanding the displacements as cubic functions of the thickness coordinate. Model-2 satisfies zero transverse shear stresses on the bounding planes and the equations of motion are derived from the principle of virtual displacements. As the CLT and FSDT, Reddy HOT based model does not satisfy the continuity conditions of transverse shear stresses at layer interfaces. In Model-3, Touratier [5] suggested trigonometric functions instead of polynomial developments of the transverse coordinate. The proposed theory recovers the classic thin plate and Reissner–Mindlin theories and satisfies zero transverse shear stress conditions on the top and bottom surface of plates and avoids shear correction factor. In Model-4 and Model-5, a parameter k0 is used to reproduce the interlaminar continuity of the shear stress (ICSS) at the interfaces. By choosing suitable parameter k0 and the shear function f(z), both displacement and transverse shear stress field are made piecewise continuous. Due to its physical and mechanical continuity at the interfaces, the ICSS condition is widely used for the modeling of laminate composites [1,14]. In Model-6, the core is modeled by using FSDT based model and the faces are modeled by using CLT based model. Thus, the shear stress is constant in the core and zero in the faces.Model-7 by Rao [18] uses same kinematic assumptions as in Model-6. In addition, Rao supposes that weak core layer deforms mainly through shear strain, and does not carry much axial force: the so-called Kerwin assumption [22]. Model-6 and Model-7 have been widely used for the modeling of vibration problems of viscoelastically damped sandwich structures [17,19,20]. Model-7 permits useful formulation of the vibration equations of viscoelastically damped sandwich structures. Indeed, considering Model-7, vibrations equations for these specific structures are generally obtained by using only shear and geometric parameters. 3. Basic equations The proposed two-dimensional theory is constructed accordingly to the following: material behavior is assigned, i.e. Hooke’s law is given and a geometrical relation, i.e. strain–displacement relation is assumed according to the assumption of small strain: 1 oui ouj ij ¼ þ ; ð3Þ 2 oxj oxi r ¼ 2l þ kI 3 TrðÞ; k¼
Em ; ð1 þ mÞð1 2mÞ
l¼
E ; 2ð1 þ mÞ ð4Þ
where I3 is the identity matrix, E and m are, respectively, Young’s modulus and Poisson’s ratio of the core or the faces.
285
As stated previously, the sandwich faces are elastic ones and their mechanical behavior is modeled by a real Young modulus, Ef. On the contrary, the viscoelastic behavior of the core is generally introduced by a convolution product [20]. This product becomes simple one when we deal with static or harmonic motion [17–20]. In this case, the viscoelastic behavior can be introduced by a complex modulus as: Ec ¼ Ec ð1 þ igc Þ where Ec is the delayed elasticity modulus and gc is the loss factor. 3.1. Balance of momentum The basic equilibrium equations are obtained by using the principal of virtual displacement as follows: P acc ðduÞ ¼ P int ðduÞ þ P ext ðduÞ;
ð5Þ
where Pacc(du) describes inertial terms, Pint(du) and Pext(du) are, respectively, internal and external virtual work. Notice that for Problem-1, the expressions of Pext(du) and Pacc(du) are: P acc ðduÞ ¼ 0;
L ; P ext ðduÞ ¼ F dw 2
ð6Þ ð7Þ
and that for Problem-2 Pext(du) and Pacc(du) are as follows: Z L o2 w ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ 2 dw dx; ð8Þ P acc ðduÞ ¼ ot 0 P ext ðduÞ ¼ 0:
ð9Þ
Assuming that the transverse normal stress is negligible (rzz = 0), the internal virtual work of sandwich beam for the two considered problems is expressed as: Z ðrxx dxx þ 2rxz dxz ÞdV t P int ðduÞ ¼ V
Z t H ck0 ¼ rxx zdw;xx þ f ðzÞ þ db;x 2 V1 Z þrxz f;z dbÞdV 1 ðrxx fzdw;xx þ ½f ðzÞ V2
þk 0 zdb;x þ rxz ðf;z þ k 0 Þdb dV 2
Z H ck0 rxx zdw;xx þ f ðzÞ db;x 2 V3 þrxz f;z dbÞdV 3 ; 2
ð10Þ
where w;xx ¼ ooxw2 ; b;x ¼ ob and f ;z ¼ of . ox oz In order to simplify and make more intuitive equation (10), additional definitions are introduced as Mt (Eq. (11)) which denotes the total bending moment to the middle face of sandwich beam, Q (Eq. (12)) which denotes the transverse shear force and the additional bending moment which is indicated by Ma (Eq. (13)) which denotes the axial bending Z Mt ¼ zrxx dS t ; ð11Þ St
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H. Hu et al. / Composite Structures 84 (2008) 282–292
Q¼
Z
rxz f;z dS 1 þ
Z
S1
rxz ðf;z þ k 0 Þ dS 2 þ
Z
S2
ð12Þ
From Eq. (17), we derive two basic equations (Eq. (18)) that permit to describe Problem-1. Q M a;x ¼ 0; ð18Þ M t þ F2 x ¼ 0:
ð13Þ
It is to be noted that all boundary conditions of Problem-1 are included in the following equation: Z L 2 L L 2 2 ½M t dw;x 0 þ ½M t;x dw þ ½M a db0 þ M t;xx dw dx ¼ 0:
rxz f;z dS 3 ; S3
Z H ck0 Ma ¼ rxx f ðzÞ þ rxx ½f ðzÞ dS 1 þ 2 S1 S2
Z H ck0 rxx f ðzÞ þ k 0 z dS 2 þ dS 3 ; 2 S3 Z
where V1, V2, V3, S1, S2, S3 express the integration volumes and the integration areas of layer-1, layer-2 and layer-3 and where Vt and St are given by Vt = V1 + V2 + V3 and St = S1 + S2 + S3, respectively. Using the above defined equations, equilibrium equations of Problem-1 and Problem-2 are established and fully described by Eqs. (14) and (15), respectively Z L L ½M t dw;xx þ M a db;x þ Qdb dx ¼ F dw ; ð14Þ 2 0
x¼0
0
ð19Þ By inserting Eqs. (4) and 11, 12, 13 into Eq. (18), new differential equations are obtained for Problem-1: ( A0 w;xx A1 b;x þ F2 x ¼ 0; ð20Þ A1 w;xxx þ A2 b A3 b;xx ¼ 0; where A0, A1, A2 and A3 are constant and defined by the following general formulas:
R R R 8 A0 ¼ Ef S 1 z2 dS 1 þ Ec S 1 z2 dS 2 þ Ef S 3 z2 dS 3 ; > > > R R R > > < A1 ¼ Ef S z f ðzÞ þ H c2k0 dS 1 þ Ec S z½f ðzÞ þ k 0 zdS 2 þ Ef S z f ðzÞ H c2k0 dS 3 ; 1 2 3 R R R 2 > A2 ¼ Gf S 1 f;z2 dS 1 þ Gc S 2 ðf;z þ k 0 Þ dS 2 þ Gf S 3 f;z2 dS 3 ; > > > > 2 2 R R R : 2 A3 ¼ Ef S 1 f ðzÞ þ H c2k0 dS 1 þ Ec S 2 ½f ðzÞ þ k 0 z dS 2 þ Ef S 3 f ðzÞ H c2k0 dS 3 :
Z
L
½M t dw;xx þ M a db;x þ Qdb dx
0
¼
Z
L
ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ
0
o2 w dw dx: ot2
ð15Þ
3.2. Solution of Problem-1 In order to solve Problem-1, the definition domain of the equilibrium equation (14) is split into two parts ð½0; L2 and L2 ; L Þ. Since the considered problem is symmetrical, the equilibrium equation (14) takes the following integral form: Z
L 2
½M t dw;xx þ M a db;x þ Qdb dx ¼
0
F L dw : 2 2
ð16Þ
Integration by parts, Eq. (16) allows to obtain: L ½M t dw;x 02 þ ½M t;x dw
L
þ ½M a db02 þ
Z
Eq. (21) is valid for the analysis of all previously described models except Model-7 by Rao. In the later the axial forces are neglected in the core which lead to ignore the terms with Ec in expressions of A0, A1 and A3. Under hand the above-defined constants, Eq. (20) is solved analytically and therefore Problem-1 is completely solved by defining expressions of the deflection w and rotation b: bðxÞ ¼
A0 F 2 FL2 w;x þ x ; 4A1 A1 16A1
wðxÞ ¼
F 3 FL2 FA2 x þ xþ 21 12A0 16A0 2A0 A2
in which sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 A2 : n¼ A0 A3 A21
ð22Þ !
x
sinhðnxÞ ; n sinhðnL Þ 2
ð23Þ
ð24Þ
L 2
It should be stated that A1, A2 and A3 are zeros in Model-1 and therefore the transverse displacement takes the following simplified form:
M t;xx dw dx ! Z L2 F L þ M t;x þ ðQ M a;x Þdb dx ¼ 0: dw 2 2 L 0 x¼ x¼0
ð21Þ
0
2
ð17Þ
wm1 ¼
F 3 FL2 x þ x: 12A0 16A0
ð25Þ
H. Hu et al. / Composite Structures 84 (2008) 282–292
By analyzing Eqs. (23) and (25), one can express the displacement solution, in the case of above-considered kinematics models, as an adding of the solution obtained by using Model-1 and a correction term " # FA21 sinhðnxÞ : wðxÞ ¼ wm1 þ 2 x ð26Þ n sinh nL 2A0 A2 2 3.3. Solution of Problem-2 To investigate the numerical accuracy of the analyzed models, natural frequencies and loss factors are investigated. Integrating by parts, Eq. (15) permits to derive basic equations of Problem-2: Z L L L L ðQ M a;x Þdb dx ½M t dw;x 0 ½M t;x dw0 þ ½M a db0 þ 0
Z L o2 w þ M t;xx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ 2 dw dx ¼ 0; ot 0 (
ð27Þ Q M a;x ¼ 0; 2
M t;xx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ ootw2 ¼ 0; L
L
L
½M t dw;x 0 ½M t;x dw0 þ ½M a db0 ¼ 0:
ð28Þ ð29Þ
While expressing Mt, Q and Ma according, respectively, to Eqs. (11)–(13), we obtain a system of differential equations in w and b: ( 2 A0 w;xxxx A1 b;xxx þ ðq1 S 1 þ q2 S 2 þ q3 S 3 Þ ootw2 ¼ 0; ð30Þ A1 w;xxx þ A2 b A3 b;xx ¼ 0: This differential equation appears as very complicated with no known general analytical close-form solution. In order to solve this equation, the amplitude of harmonic vibrations is assumed for Problem-2 as: wðx; tÞ ¼ A ejXt evx :
ð31Þ
By using the end conditions in Eq. (29), a sixth-order characteristic determinant which has to be zero for non-trivial solution is established and from which, the deflection w(x, t) can be expressed as: npx wðx; tÞ ¼ A ejXt sin ; ð32Þ L where n is the mode of vibration. Finally, substituting Eq. (32) into Eq. (30) yields the expression for the complex pulsation X as:
A21 j2 X2 ¼ X2m1 1 ; ð33Þ A0 ðA2 þ A3 j2 Þ where j = np/L, Xm1 is the complex pulsation obtained by using Model-1: X2m1 ¼
A0 j 4 : q1 S 1 þ q2 S 2 þ q3 S 3
ð34Þ
287
According to Eq. (33) or (34), we define the loss factor g and the natural pulsation x0 of the damped sandwich beam by using the following definitions: X2 ¼ x20 ð1 þ igÞ;
g ¼ ImðX2 Þ=ReðX2 Þ:
ð35Þ
4. Analytical and numerical assessments In order to assess models associated with the previously described kinematic assumptions, finite element calculations have been achieved for Problem-1 and Problem-2. A two-dimensional finite element analysis is carried out with a ‘2D-Q8’ element defined by eight nodes having two degrees of freedom at each node (2 in-plane translations). The presented results are obtained with in-house Matlab developed finite element (FE) code. The sandwich beam consists of two face sheets made of aluminum and a lightweight core whose mechanical and geometrical properties are given in Table 1. Three non-dimensional beam variables are used in this comparative study, namely, the ratio of core and skin Young moduli (Ec/Ef), the ratio of beam length to beam total thickness (L/Ht) and the ratio of core to skin thickness (Hc/Hf). Let us remark that under these considerations and by using material parameters as listed in Table 1, all sandwich possible configurations can be presented. For these purposes, five evaluations parameters are presented: 1. wmax(m): w(L/2,0); maximum deflection in Problem-1. 2. rcis(Pa): rxz(L/4,0); maximum shear stress in the section x = L/4 in Problem-1. 3. rnor(Pa): rxx(L/4, Ht/2); maximum normal stress in the section x = L/4 in Problem-1. 4. f1st(Hz): f0 = x0/2p; first natural frequency in Problem2. 5. g1st: g/gc; first loss factor ratio in Problem-2. The assessment of parameters wmax, rnor and f1st are global ones since they depend on the global rigidity of the structure. On the other hand, rcis and g1st mainly depend on the core properties since they describe shear stress and damping. The latter is introduced by the shear stress, which is mainly due to the difference between the longitudinal displacements of the elastic faces. Remark first that core-to-face stiffness ratio influences considerably the sandwich beam behavior and consequently it influences the choice of the adequate modeling theory. Tables 3 and 4 show the results obtained by using various types of kinematic modeling and with various core to face Young moduli (Ec/Ef) ratios. Table 1 Material parameters and dimensions Ef 6.9E10 Pa
qc
qf 3
968 kg/m
3
2770 kg/m
Ht
mc
mf
gc
0.01 m
0.3
0.3
0.3
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H. Hu et al. / Composite Structures 84 (2008) 282–292
4.1. Mesh convergence study N x = 10
Prior to initiating the evaluation study, an analysis of mesh convergence is carried out to ensure the accuracy of the proposed finite element solution since it is considered in the present study as the reference. Thus, calculations were repeated three times, first time cutting back the typical element length by a factor of two and then by a factor of 2.5, see Fig. 4. Table 2 shows the numerical values, of wmax, rcis, rnor, obtained in each considered mesh configuration. The relative difference is almost zero in the first refinement and in the last refinement step.
N zf = 1 N cz = 1
4.2. Assessment of CLT and HSDT models N x = 20
Considering the global criteria, classical and high order theories based models are not sensitive to core-to-face stiffness ratio (Ec/Ef). Indeed, wmax, rnor and f1st remain nearly constant when the stiffness ratio decreases. Considering this, global rigidities are almost independent on core-to-face stiffness ratio. Moreover, these models overestimate the global rigidity of sandwich beams with an unacceptable margin. Regarding to stiff cores (Ec/Ef P 0.01), single layer theory yields acceptable results for wmax, rnor and f1st with maximum relative error of 5.8% and 1.9%, respectively, for wmax, and f1st (see Tables 3 and 4). In addition, global criteria based models yield the same rnor values as in the FE based solution. On the contrary, regarding to soft core (Ec/Ef < 0.01), global criteria and FE based solutions present big differences in terms of wmax, rnor and f1st, as for the core criteria, classical and high order based theory models give inaccurate results of rcis and g1st, since the transverse shear stress distribution is not simulated adequately by these models.
N zf = 1 N cz = 1
N x = 50 N zf = 3 N cz = 3
4.3. Post-processing for CLT and HSDT models The 3-D equilibrium equation is integrated to obtain transverse normal stress as follows: orxx orxz þ ¼ 0: ox oz
ð36Þ
In calculating transverse shear stresses in Eq. (36), both constitutive approach and equilibrium approach are used. Since the postprocess method does not guarantee the satisfaction of shear equilibrium equation of higher-order theory, constitutive approach shows same order of accuracy compared to that of equilibrium approach. Thus in the computation of transverse normal stress, it is sufficient to obtain transverse shear stresses in Eq. (36) by constitutive equation. The maximum transversal shear stress rcis is given as ! Z H c Z 0 2 rcis ¼ Ef z dz þ Ec z dz w;xxx H t 2
H c 2
Z Ef
H c 2
H t 2
f ðzÞ dz þ Ec
Z
0 H c 2
! f ðzÞ dz b;xx :
ð37Þ
Fig. 4. Three meshes of the half sandwich beam (Hc/Hf = 1, Ec/ Ef = 0.0001, L/Ht = 50).
Table 2 Convergence of the static test
wmax rcis rnorm
Mesh 1
Mesh 2
Mesh 3
2.2380E5 4.6390E+3 6.9149E+5
2.2416E5 4.6339E+3 6.9106E+5
2.2446E5 4.6367E+3 6.9172E+5
Notice that using a postprocess method yields very good results as long as Ec/Ef is greater than 0.001. On the contrary, the post-processing method is without any help for the dynamical analysis of sandwich composites with very soft core (Ec/Ef = 0.0001), since it cannot permit to obtain accurate loss factor. Tests with higher-order models (see for instance Table 5) have shown that unless the 3-D equilibrium equations are
H. Hu et al. / Composite Structures 84 (2008) 282–292
289
Table 3 Influence of Ec/Ef in static tests (F = bq and q = 100 N/m, Hc/Hf = 1, L/Ht = 50) CLT
HSDT
IC-ZZT
ID-ZZT
FEM
Model-1
Model-2
Model-3
Model-4
Model-5
Model-6
2D-Q8
Ec Ef
¼1
wmax rcis rnor
4.53E5 0.00E+0 3.75E+5
4.53E5 7.50E+3 3.75E+5
4.53E5 7.74E+3 3.75E+5
4.53E5 7.49E+3 3.75E+5
4.53E5 7.73E+3 3.75E+5
4.53E5 7.22E+3 3.75E+5
4.53E5 7.64E+3 3.75E+5
Ec Ef
¼ 101
wmax rcis rnor
4.69E5 0.00E+0 3.88E+5
4.70E5 1.56E+3 3.88E+5
4.70E5 1.70E+3 3.88E+5
4.72E5 6.98E+3 3.88E+5
4.72E5 7.01E+3 3.88E+5
4.72E5 6.95E+3 3.88E+5
4.72E5 7.00E+3 3.88E+5
Ec Ef
¼ 102
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.71E5 1.75E+2 3.89E+5
4.72E5 1.93E+2 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
4.99E5 6.93E+3 3.89E+5
Ec Ef
¼ 103
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.72E5 1.77E+1 3.89E+5
4.72E5 1.96E+1 3.89E+5
7.31E5 6.76E+3 3.98E+5
7.31E5 6.76E+3 3.98E+5
7.31E5 6.76E+3 3.98E+5
7.32E5 6.79E+3 3.98E+5
Ec Ef
¼ 104
wmax rcis rnor
4.70E5 0.00E+0 3.89E+5
4.72E5 1.77E+0 3.89E+5
4.72E5 1.96E+0 3.89E+5
2.23E4 4.63E+3 6.90E+5
2.23E4 4.63E+3 6.90E+5
2.23E4 4.63E+3 6.90E+5
2.24E4 4.64E+3 6.92E+5
Table 4 Influence of Ec/Ef in dynamic tests (Hc/Hf = 1, L/Ht = 50) CLT
HSDT
Model-1
Model-2
Model-3
Model-4
IC-ZZT Model-5
Model-6
ID-ZZT Model-7
2D-Q8
FEM
Ec Ef
¼1
f1st g1st
1.02E+2 3.70E2
1.02E+2 3.76E2
1.02E+2 3.76E2
1.02E+2 3.75E2
1.02E+2 3.75E2
1.02E+2 3.75E2
1.02E+2 4.82E4
1.02E+2 3.79E2
Ec Ef
¼ 101
f1st g1st
1.01E+2 3.84E3
1.00E+2 4.07E3
1.00E+2 4.07E3
1.00E+2 8.63E3
1.00E+2 8.63E3
1.00E+2 8.63E3
1.00E+2 4.80E3
1.00E+2 8.60E3
Ec Ef
¼ 102
f1st g1st
1.00E+2 3.84E4
1.00E+2 4.14E4
1.00E+2 4.20E4
9.80E+1 4.61E2
9.80E+1 4.61E2
9.81E+1 4.61E2
9.80E+1 4.57E2
9.81E+1 4.60E2
Ec Ef
¼ 103
f1st g1st
1.00E+2 3.84E5
1.00E+2 4.15E5
1.00E+2 4.21E5
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
8.26E+1 3.04E1
Ec Ef
¼ 104
f1st g1st
1.00E+2 3.84E6
1.00E+2 4.15E6
1.00E+2 4.22E6
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
4.67E+1 5.41E1
Table 5 Shear stress of CLT and HSDT models by post-processing (F = bq and q = 100 N/m, Hc/Hf = 1, L/Ht = 50)
Ec Ef Ec Ef Ec Ef Ec Ef Ec Ef
¼1 ¼ 101 ¼ 102 ¼ 103 ¼ 104
rcis rcis rcis rcis rcis
CLT
HSDT
Model-1
Model-2
Model-3
2D-Q8
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.50E+3 6.98E+3 6.93E+3 6.92E+3 6.92E+3
7.64E+3 7.00E+3 6.93E+3 6.79E+3 4.64E+3
used in evaluating the thickness distribution of the transverse stresses, the resulting stresses are inaccurate. 4.4. Evaluation of IC-ZZT and ID-ZZT models Notice first, from Tables 3 and 4, that zig-zag based models are always more precise than CLT and HSDT theories based models, especially Model-4, Model-5 and Model-6, they present always very good results both for global criterion and for core criterion. For soft cores (Ec/
FEM
Ef 6 0.01), Model-7 give a good estimation compared to the FE based solutions with maximum relative error for f1st of 0.10% and of 0.70% for g1st. On the contrary, for stiff cores, where Kerwin’s assumptions are not satisfied, the previous concluding remarks are not valid and the relative errors are larger than those obtained by other zig-zag based models. For presentation convenience and simplicity, only Model-5 and Model-6 results are discussed. The transverse shear stress of the considered IC-ZZT and ID-ZZT models
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H. Hu et al. / Composite Structures 84 (2008) 282–292
−3
5 x 10
−3
5
Model−5 Model−6
4
3
2
2
Thickness (m)
Thickness (m)
Model−5 Model−6 (σxz corrected)
4
3
1 0 −1 −2
1 0 −1 −2
−3
−3
−4
−4
−5
x 10
−7000
−6000
−5000
−4000
−3000
−2000
−1000
−5
0
Transverse shear stress at the section x=L/4 (Ec/Ef=0.1)
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress at the section x=L/4 (Ec/Ef=0.1)
−3
5 4
x 10
−3
Model−5 Model−6
5 4
3
2
1
Thickness (m)
Thickness (m)
Model−5 Model−6 (σxz corrected)
3
2
0 −1 −2 −3
1 0 −1 −2 −3
−4 −5
x 10
−4
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress (Pa) at the section x=L/4 (Ec/Ef=0.0001)
−5
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
Transverse shear stress (Pa) at the section x=L/4 (Ec/Ef=0.0001)
Fig. 5. Comparison of shear stress in the thickness direction between Model-5 and Model-6.
are, respectively, presented in Fig. 5. Noting that for Ec/ Ef < 1, Model-5 and Model-6 present almost the same distribution of shear stress in the core. On the other hand, Model-5 and Model-6 present big discrepancies of the shear stress in the case of faces since Model-6 is based on Kirchhoff assumptions. However, this disparity is not of a real concern for sandwich structures modeling in general and for viscoelastically damped sandwich vibration modeling more precisely, since rigidity ratio is generally between 0.001 and 0.0001. Moreover, this disparity can be reduced with a precessing of ID-ZZT models, see Fig. 6, where the shear stress distribution in the faces is obtained by the equilibrium equation. 4.5. Other influences Tables 3 and 4 present the influence of rigidities ratios, where Hc/Hf and L/Ht remain constants. In Tables 6–8, we will show the influences of Hc/Hf and L/Ht. From Table 6, let us remark that the obtained results by using IC-ZZT and ID-ZZT models are always very close to FE based solutions. Except for very thick core (Hc/ Hf = 100), where one finds a larger error for loss factor with the Rao’s model (Model-7) than with two other ones. This is due, in fact, to the non-validity of Kerwin hypoth-
Fig. 6. Comparison of shear stress in the thickness direction between Model-5 and Model-6 when used in conjunction with a post-processing method in the faces.
eses since the core is very thick and the stress in the core cannot be reduced to shear. At the same time, Rao’s model underestimates the frequencies and the global stiffness of the sandwich beam. Notice from Tables 7 and 8 that for all L/Ht ratios, both IC-ZZT models and ID-ZZT models give acceptable results and that even for small L/Ht ratios, their efficiencies are also valid. Moreover, IC-ZZT and ID-ZZT models can be used in the study of high vibration modes, see Table 9. 5. Conclusion Beside the analysis of various in-the-literature models and theories, governing equations of viscoelastic damped beams have been derived by using the virtual work principle. A general kinematic model has then been introduced and used and the deformation as well as free vibration analyzes of sandwich beam have been performed. Global criteria, which depend on global rigidity as well as core criteria, which depend on rigidities ratio, have been introduced to evaluate the different proposed kinematics. Evaluation procedures have been performed and the results have been presented for static and dynamic problems.
H. Hu et al. / Composite Structures 84 (2008) 282–292
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Table 6 Influence of Hc/Hf in dynamic tests (L/Ht = 20, Ec/Ef = 103) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
Hc Hf
¼ 101
f1st g1st
4.73E+2 2.72E1
4.73E+2 2.72E1
4.74E+2 2.73E1
4.74E+2 2.73E1
4.73E+2 2.72E1
Hc Hf
¼1
f1st g1st
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
3.33E+2 5.57E1
Hc Hf
¼ 101
f1st g1st
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
3.06E+2 6.66E1
Hc Hf
¼ 102
f1st g1st
2.00E+2 2.36E1
2.00E+2 2.36E1
2.00E+2 2.36E1
1.99E+2 2.20E1
2.00E+2 2.33E1
Model-4
Model-5
Model-6
Model-7
2D-Q8
Table 7 Influence of L/Ht in dynamic tests (Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
L Ht
¼ 10
f1st g1st
9.31E+2 4.09E1
9.31E+2 4.09E1
9.32E+2 4.10E1
9.31E+2 4.10E1
9.30E+2 4.10E1
L Ht
¼ 40
f1st g1st
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.90E1
1.19E+2 3.89E1
L Ht
¼ 100
f1st g1st
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
2.37E+1 1.06E1
Table 8 Influence of L/Ht in static tests (Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
L Ht
¼ 10
wmax rcis rnor
2.77E6 3.11E+3 1.99E+5
2.77E6 3.12E+3 1.99E+5
2.76E6 3.11E+3 1.99E+5
2.76E6 3.11E+3 1.99E+5
2.94E6 3.14E+3 1.72E+5
L Ht
¼ 40
wmax rcis rnor
4.42E5 6.58E+3 3.29E+5
4.42E5 6.58E+3 3.29E+5
4.41E5 6.58E+3 3.29E+5
4.41E5 6.58E+3 3.29E+5
4.43E5 6.58E+3 3.29E+5
L Ht
¼ 100
wmax rcis rnor
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.32E4 6.92E+3 7.79E+5
4.33E4 6.92E+3 7.79E+5
Table 9 Natural frequencies and loss factors(L/Ht = 50, Ec/Ef = 103, Hc/Hf = 1) IC-ZZT
ID-ZZT
FEM
Model-4
Model-5
Model-6
Model-7
2D-Q8
2nd mode
Frequency Loss factor
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
2.42E+2 5.33E1
4th mode
Frequency Loss factor
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
6.60E+2 4.85E1
8th mode
Frequency Loss factor
2.05E+3 2.34E1
2.05E+3 2.35E1
2.05E+3 2.36E1
2.05E+3 2.36E1
2.04E+3 2.35E1
It comes out from this assessment process that IC-ZZT and ID-ZZT models are more accurate than CLT and HSDT based models and that CLT and HSDT models always overestimate global rigidities and do not enable a
valid description of the distribution of the transverse shear stress. Nevertheless, CLT and HSDT models can be used for relatively rigid cores (Ec/Ef larger than 0.001), in conjunction with a post-processing method, to predict accu-
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rately the global stiffness, the core shear stress and the loss factor of the structure. For soft cores, a Zig-Zag model is necessary. Apart of the fact that Model-4 and Model-5 give the same results, they are often accurate and therefore they can be used for the modeling of all types of the three-layered structures. Compared to FE based solutions, Model-6 is more precise than Model-7. Considering this, we state that Model-7 is inadequate for modeling of sandwich beams with rigid core or with thick cores. On the other hand, Rao’s model is suitable for the modeling of sandwich composites with soft and thin cores. Acknowledgements This study has been supported by the European FP6 STREP project CASSEM: NMP3-CT-2005-013517. The authors would like also to acknowledge the financial support of the Luxembourgish Ministry of Research. References [1] Carrera E. Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 2003;56(3):287–308. [2] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945;12:69–76. [3] Mindlin RD. Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates. J Appl Mech 1951;18:1031–6. [4] Reddy JN. A simple higher-order theory of laminated composite plate. J Appl Mech 1984;51:745–52. [5] Touratier M. An efficient standard plate theory. Int J Eng Sci 1991;29:901–16.
[6] Noor AK et al. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;41(1):1–12. [7] Noor AK et al. Assessment of computational models for multilayered anisotropic plates. Compos Struct 1990;14:233–65. [8] Noor AK et al. Computational models for sandwich panels and shells. Appl Mech Rev 1996;49(3):155–99. [9] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton (FL): CRC Press; 1997. [10] Jemielita G. On kinematical assumptions of refined theories of plates: a survey. J Appl Mech 1990;57(3):1088–91. [11] Carrera E, Ciuffreda A. A unified formulation to assess theories of multilayered plates for various bending problems. Compos Struct 2005;69:271–93. [12] Lekhnitskii SG. Strength calculation of composite beams. Vestn Inzhen Tekh 1935;9. [13] Ambartsumian SA. On a theory of bending of anisotropiv plates. Prikl Mat Mekh 1958;22(2):226–37. [14] Whitney JM. The effects of transverse shear deformation on the bending of laminated plates. J Compos Mater 1969;3:537–47. [15] Reissner E. On a certain mixed variational theory and a proposed application. Int J Numer Methods Eng 1984;20:1366–8. [16] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Eng State Art Rev 2003;10:215–97. [17] Mead DJ et al. The forced vibration of three-layer damped sandwich beam with arbitrary boundary conditions. J Sound Vib 1969;10:163–75. [18] Rao DK. Frequency and loss factor of sandwich beams under various boundary conditions. J Mech Eng Sci 1978;20(5):271–82. [19] He JF et al. Vibration analysis of viscoelastically damped sandwich plates. J Sound Vib 1992;152:107–23. [20] Daya EM et al. A shell element for viscoelastically damped sandwich structures. Rev Eur Elem Finis 2002;11(1):39–56. [21] Idlbi A et al. Comparison of various laminated plate theories. Compos Struct 1997;37:173–84. [22] Kerwin EM. Damping of flexural waves by a constrained visco-elastic layer. J Acoust Soc Am 1959;31:952–62.