A family of higher-order single layer plate models meeting C 0 z -- requirements for arbitrary laminates
AA family of higher-order single layer plate models meeting C z − requirements for arbitrarylaminates A. Loredo a, ∗ , M. D’Ottavio b , P. Vidal b , O. Polit b a LEME, Université de Bourgogne–Franche-Comté, France b LEME, UPL, Univ. Paris Nanterre, France
Abstract
In the framework of displacement-based equivalent single layer (ESL) plate theories for laminates, this paper presents a generic and automaticmethod to extend a basis higher-order shear deformation theory (polynomial, trigonometric, hyperbolic. . . ) to a multilayer C z higher-order sheardeformation theory. The key idea is to enhance the description of the cross-sectional warping: the odd high-order C z function of the basis model isreplaced by one odd and one even high-order function and including the characteristic zig-zag behaviour by means of piecewise linear functions.In order to account for arbitrary lamination schemes, four such piecewise continuous functions are considered. The coe ffi cients of these fourwarping functions are determined in such a manner that the interlaminar continuity as well as the homogeneity conditions at the plate’s top andbottom surfaces are a priori exactly verified by the transverse shear stress field. These C z ESL models all have the same number of DOF as theoriginal basis HSDT. Numerical assessments are presented by referring to a strong-form Navier-type solution for laminates with arbitrary stackingsequences as well for a sandwich plate. In all practically relevant configurations for which laminated plate models are usually applied, the resultsobtained in terms of deflection, fundamental frequency and local stress response show that the proposed zig-zag models give better results thanthe basis models they are issued from.
Keywords:
Plate theory, Zig-Zag theory, Warping function, Laminates, Sandwich
1. Introduction
Among the numerous theories that have been developed for mul-tilayered plates, those belonging to the Equivalent-Single Layer (ESL)family are of practical interest due to their relatively small number ofunknowns, that is independent of the number of layers. Within thisESL class, the classical lamination theory (CLT) which has been pro-posed first, does not take into account the transverse shear behaviourand is, therefore, accurate only for thin plates, for which the transverseshear deformation can be neglected. First order shear-deformation the-ories (FSDT) have then been proposed to overcome this problem uponretaining a transverse shear deformation that is constant throughout theplate’s thickness. Its accuracy with regard to the plate’s gross response(transverse deflection, low vibration frequencies) results neverthelessdependent on shear correction factors. Higher-order shear deformationtheories (HSDT) have been subsequently proposed in order to avoidthe need of these problem-dependent shear correction factors. Thisis accomplished upon describing the through-thickness behaviour ofthe in-plane displacement field by means of functions of order greaterthan one, which thus introduces an enhanced description of the trans-verse shear deformation. The most well-known HSDT is the Vlasov–Levinson–Reddy’s third order theory (ToSDT). Polynomial functionsare not the unique way to enrich the kinematic field, a wide variety offunctions have been used, in particular trigonometric, hyperbolic, andexponential functions, as summarized in recent review papers [1, 2].In all the theories cited above, the transverse shear deformationis included into the kinematic field by means of functions of class C along the thickness direction z . This leads to continuous transverseshear strains and hence to a discontinuous transverse shear stress field, ∗ Corresponding author
Email address: [email protected] (A. Loredo) which violates equilibrium conditions within multilayered structures.Authors have thus proposed to use piecewise continuous, di ff erentiablefunctions of z , often referred to as zig-zag functions. These functionsare commonly constructed in such a manner that appropriate jumpsof their derivatives at the interfaces restore the transverse stress con-tinuity. Such Zig-Zag (ZZ) theories have appeared around the halfof the twentieth century with Ambartsumyan [3, 4], and Osternik andBarg [5], and have been continuously receiving attention until now.An earlier paper that belongs to this category is due to Lekhnitskii [6],but it was limited to the study of beams. The approach by Lekhnitskiihas been extended to plates 50 years later by Ren [7]. While the ap-proaches by Ambartsumyan and Lekhnitskii rely on the exact verifica-tion of the constitutive equation connecting the transverse shear stressand the kinematic fields, Murakami [8] formulated a ZZ theory by pos-tulating these two fields in an independent manner thanks to Reissner’smixed variational theorem (RMVT) dedicated to multilayered plates[9]. For more recent developments of RMVT-based ZZ theories, theinterested reader may refer to papers by Carrera [10], Demasi [11] andTessler [12, 13]. Murakami’s zig-zag function (MZZF) has been alsoextensively applied to classical displacement-based variable kinemat-ics approaches, see, e.g., [14, 15]. Within a comprehensive discussionabout ZZ theories, Groh and Weaver have recently proposed a mixedZZ theory based on Hellinger-Reissner’s principle [16].Among these several approaches, more details of which can befound in the review papers [17–19], only a subset of these ZZ-theoriesare able to satisfy the appropriate interlaminar continuity (IC) of both,the displacement and the transverse shear stress fields. These two re-quirements (ZZ and IC) have been summarized by the acronym C z –requirements [20]. In the following, we shall limit our attention tothose ZZ theories that satisfy exactly the C z − requirements. A moredetailed examination is next proposed of some pioneering works, withthe aim of establishing the background and highlighting the di ff erences Published in the
Composite Structures journal http://dx.doi.org/10.1016/j.compstruct.2019.111146 a r X i v : . [ phy s i c s . c l a ss - ph ] J un ith the family of models proposed in this paper. • Ambartsumyan’s approach is based on the early paper [3], inwhich it is assumed that the transverse shear stress vary along z according to a quadratic parabola with nil values at the outersurfaces. On page 20 of the later book [4], this assumption isformally expressed by the expressions τ xz = f ( z ) ϕ ( x , y ) and τ yz = f ( z ) ψ ( x , y ). It leads to 4 kinematic functions, but onlytwo of them are independent, based on the primitives of f ( z )and f ( z ). Conceived for orthotropic shells, the theory pre-sented in the 1958 paper is not yet a ZZ theory, but in the 1970book [4], page 75, an extension to symmetric multilayered or-thotropic plates is given, which exhibits 4 zig-zag kinematicfunctions issued from two modified functions f ( z ) and f ( z ).This extension is perhaps due to Osternik and Barg [5], whichis cited in the 1970 Ambartsumyan’s book (see also Carrera’sreview paper [17]). • In 1969, Whitney extends Ambrartsumyan’s theory to anisotropicplates, more specifically to general symmetric laminates andorthotropic non-symmetric laminates [21]. Following Ambart-sumyan’s approach, Whitney starts from an assumed transverseshear stress field and ends up with 4 kinematic functions thatare expressed as the superimposition of a polynomial third or-der ToSDT function and a zig-zag linear functions, see equation6 of [21]. • Sun and Whitney propose in 1973 a layerwise model, which isthe starting point for deriving an ESL model upon eliminatingthe parameters of the N − • The 1986 paper by Ren [7] proposes 4 kinematic functions from a priori given transverse shear stress functions and applies theresulting zig-zag model to cross-ply laminates. Four displace-ment unknowns are introduced to take into account the trans-verse shear behaviour, which yields a 7-parameter model whichis di ffi cult to compare with our 5-parameter models. • Cho and Parmerter formulate a ZZ theory for symmetric [24]and general [25] orthotropic composite plates. Starting pointare 2 kinematic functions which are the superimposition of a cu-bic polynomial and a linear zig-zag function expressed in termsof the Heaviside function. The coe ffi cients are determined byenforcing the transverse stress continuity, which leads to thecoupling of the x − and y − directions in the kinematic field. Ittherefore appears that the theory is in fact based on 4 kinematicfunctions (see equation 4 of [25]). In his historical review,Carrera demonstrated the equivalence of Cho and Parmerter’smodel with Ambartsumyan’s model. • It is finally worth mentioning the ZZ theory fulfilling the C z –requirements that is based on trigonometric functions and de-veloped by Ossadzow and coworkers [26, 27]. The construc-tion of the kinematic zig-zag functions follows a similar pathas proposed by Cho and Parmerter, but the trigonometric cosand sin functions replace the quadratic and cubic terms of thepolynomial expansion, respectively.A conforming finite element based on a trigonometric ZZ theoryenhanced through a transverse normal strain has been developed forlaminated plates [28]. References [29, 30] extend the procedure of Choand Parmerter to a wider family including polynomial, trigonometric, exponential and hyperbolic functions. However, the authors only con-sider two kinematic functions, which reduces the applicability of theirmodels to cross-ply laminates.In [31, 32], corresponding polynomial and trigonometric C z zig-zag models have been constructed from basis polynomial and trigono-metric HSDT by using four functions, which shall be hereafter referredto as warping functions . While in these works the warping functions were obtained from transverse stress fields obtained from 3D solutionsor from equilibrium equations, the present paper presents a generalprocedure for extending a basis higher-order shear deformation the-ory (bHSDT) to a multilayer higher-order shear deformation theory(mHSDT) that meets the C z − requirements. This extension consists inthe construction of four C z warping functions starting from the nativefunctions that characterize the basis theory: it can be applied to anycouple of odd and even functions and has no limitation concerning thelamination scheme.The paper is organized as follows. Section 2 introduces the nota-tion and points out the properties that the four warping functions arerequired to fulfil. The extension of a bHSDT up to an mHSDT ful-filling all C z –requirements is described in Section 3. Three di ff erentbasis models are exemplarily considered, which pertain respectively tothe polynomial, trigonometric and hyperbolic type. It is also shownthat the warping functions are components of a second-order tensor,hence being covariant with rotations about the z − axis. Section 4 re-ports the numerical evaluations: the C z warping functions e ff ectivelyincrease the accuracy of the basis (non zig-zag) model and this en-hancement is quite insensitive with respect to the type of functionsused for the model. A discussion is proposed in Section 5 in order tosubstantiate the limitations of conventional ZZ models with respect toparticularly “constrained” configurations with very low number of lay-ers and length-to-thickness ratios: in these cases, accuracy may onlybe assured by resorting to warping functions that contain more layer-specific information, just as LayerWise models do. The main conclu-sions are finally summarised in Section 6.
2. Definitions and general properties
This paper deals with a generic method to extend a basis higher-order shear deformation theory (bHSDT) to a multilayer higher-ordershear deformation theory (mHSDT). This Section introduces the nota-tion employed for identifying the various plate theories along with thefundamental properties that the underlying approximating functionsare required to satisfy.
We consider a basis high-order shear deformation theory (bHSDT)for which the kinematic field can be written in the following generalform: u α ( z ) = u α − zw ,α + φ ( z ) γ − α u ( z ) = w (1a)(1b)where u α are the membrane displacements at z = w is the deflectionat z = γ − α are the transverse engineering strains at z = − , and φ ( z )is a C odd function. The choice of the 0 − coordinate is a conventionuseful to avoid undetermined shear strains if an interface lies at z = z = z = − h / z = h / φ ( z ) mustverify φ (0) = φ (cid:48) (0 − ) = φ (cid:48) (0) = φ (cid:48) ( ± h / = C property of φ ( z ), such theories o not have particular abilities to deal with multilayered plates. Indeed,the continuity of φ (cid:48) ( z ) induces discontinuities of the transverse shearstresses σ α at the interfaces. Table 1 summarizes the functions φ ( z ),along with the reference author and the type of the approximation, thatwill be extended to a ZZ model in Section 3. A multilayer theory is a plate theory which is dedicated to compos-ite plates upon fulfilling the C z –requirements. The generic expressionfor the kinematics of a multilayer HSDT (mHSDT) is of the form: u α ( z ) = u α − zw ,α + ϕ αβ ( z ) γ − β u ( z ) = w (2a)(2b)where ϕ αβ ( z ) are four piecewise C functions, sometimes called warp-ing functions , that are requested to fulfil specific properties, as it will bediscussed below. Among these properties, specific jump values needto be prescribed to their derivatives for enforcing continuity of trans-verse stresses at the interfaces. In order to construct an mHSDT that isapplicable to arbitrary laminates, it is important to consider four func-tions ϕ αβ ( z ), and hence to retain the coupling between γ − (resp. γ − )and u (resp. u ). In fact, models written with only two functions, i.e.,with ϕ ( z ) = ϕ ( z ) =
0, are only applicable to cross-ply laminates.Model Type φ ( z )Reddy[33] Polynomial z − z h Touratier[34] Trigonometric h π sin (cid:18) π zh (cid:19) Soldatos[35] Hyperbolic cosh (cid:16) k (cid:17) z − hk sinh (cid:16) k zh (cid:17) cosh (cid:16) k (cid:17) − Table 1:
Original bHSDT φ ( z ) functions. The parameter k of the hyperbolicfunction allows to generalise the original model with k = ϕ αβ ( z ) Formula (2a) shows that the piecewise functions ϕ αβ ( z ) must becontinuous at each interface to respect the continuity of the in-planedisplacements. Since u α denotes the membrane displacements at z = ϕ αβ must fulfil the following homogeneity condition ϕ αβ (0) = − / /
40] laminate, trigonometric mHSDT).The compatible strain field defined by generic mHSDT kinematicfield of Eq. (2) reads ε αβ ( z ) = ε αβ − z w ,αβ + ( ϕ αγ ( z ) γ − γ ,β + ϕ βγ ( z ) γ − γ ,α ) ε α ( z ) = ϕ (cid:48) αβ ( z ) γ − β ε ( z ) = ε α ( z ) must be defined in each layer, butthey also should be discontinuous at the interfaces for allowing thetransverse shear stresses σ α ( z ) to be continuous in order to fulfil theequilibrium condition. Indicating by ζ i the z -coordinate of the i thinterface, with i = , . . . N − ζ = − h / ζ N = h /
2, thefunctions ϕ αβ ( z ) are thus required to be piecewise C over the inter-vals ] ζ i − , ζ i [. Furthermore, since γ − β represents the engineering shear strain at z = − , the derivatives of the functions ϕ αβ ( z ) are required tofulfil the following homogeneity conditions : ϕ (cid:48) αβ (0 − ) = δ αβ (5)Figure 2 illustrates these conditions with the same practical exampleas before.Due to the continuity of the ϕ αβ ( z ), in-plane strains ε αβ ( z ) are con-tinuous. Following the classical plate approach, the normal stress σ ( z ) is set to 0, which leads to the use of reduced (in-plane) sti ff -nesses Q αβγδ ( z ) in place of the sti ff nesses C αβγδ ( z ). It is further recalledthat there is no physical reason for the in-plane stresses σ αβ ( z ) to becontinuous at interfaces between adjacent layers with dissimilar sti ff -ness coe ffi cients. − . − . . . − h / − h / h / h / ϕ ( z ) − · − ϕ ( z ) [ − / /
40] HomogeneityContinuityContinuity
Figure 1:
Illustration of the continuity and homogeneity conditions prescribedon the ϕ αβ ( z ) functions (a [ − / /
40] laminate, trigonometric mHSDT). Onlytwo of the four functions have been plotted.0 0 . − h / − h / h / h / ϕ ( z ) − . . ϕ ( z ) [ − / /
40] Homogeneity
Figure 2:
Illustration of the homogeneity conditions prescribed on the ϕ (cid:48) αβ ( z )functions (a [ − / /
40] laminate, trigonometric mHSDT). Only two of the fourfunctions have been plotted.
The constitutive equation defines the transverse shear stresses interms of strains as follows: σ α ( z ) = C α β ( z ) ϕ (cid:48) βγ ( z ) γ − γ (6)For equilibrium reasons, these transverse stresses need to be continu-ous at the interfaces, and also to be null at z = ± h / C α γ ( ζ − i ) ϕ (cid:48) γβ ( ζ − i ) = C α γ ( ζ + i ) ϕ (cid:48) γβ ( ζ + i ) ( i ∈ { , N − } ) C α γ ( − h / ϕ (cid:48) γβ ( − h / = C α γ ( + h / ϕ (cid:48) γβ ( + h / = . − h / − h / h / h / ϕ ( z ) − . . ϕ ( z ) [ − / / Figure 3:
Illustration of the homogeneous bottom and top conditions, and thejump values prescribed on the ϕ (cid:48) αβ ( z ) functions (a [ − / /
40] laminate, trigono-metric mHSDT). Only two of the four functions have been plotted.
We can pre-multiply equations (7b) and (7c) with the compliance ten-sors S δ α ( − h /
2) and S δ α ( + h / ϕ (cid:48) δβ ( − h / = ϕ (cid:48) δβ ( + h / = ϕ (cid:48) αβ ( z ) functions. The following relationssummarize all the properties that the functions ϕ αβ ( z ) are required toverify: ϕ αβ (0) = ϕ αβ ( ζ − i ) = ϕ αβ ( ζ + i ) , ( i ∈ { , N − } ) ϕ (cid:48) αβ (0 − ) = δ αβ C α γ ( ζ − i ) ϕ (cid:48) γβ ( ζ − i ) = C α γ ( ζ + i ) ϕ (cid:48) γβ ( ζ + i ) ( i ∈ { , N − } ) ϕ (cid:48) αβ ( − h / = ϕ (cid:48) αβ ( + h / = + N − + + N − + + = N +
1) conditions forthe four functions ϕ αβ .
3. The extension process
This Section describes the procedure for extending a generic bHSDTto a corresponding mHSDT. The construction of the C z warping func-tions is described and their tensorial character highlighted. ϕ αβ ( z ) functions Given a composite plate consisting of N layers, the goal is to findfour ϕ αβ ( z ) functions that obey to all the properties summarized inEq. (9). Observing that the φ ( z ) function of the bHSDT is an odd func-tion, one possibility would be to use it directly and to merely find aneven γ ( z ) function with suitable properties, in order to build the four warping functions from the basis spanned by the following 2 N + φ ( z ) , γ ( z ) , Z i ( z ) , i ( z ) ) (10) Z i ( z ) and i ( z ) represent the restrictions on the interval ] ζ i − , ζ i [ of thelinear and the constant (unitary) functions, respectively. Instead of φ ( z ) and γ ( z ), the method proposed here uses more general and lessconstrained f ( z ) and g ( z ) functions, and hence is easier to use. The link that remains between the bHSDT and the corresponding mHSDT is thenature of the functions that will be used to form the basis (polynomial,trigonometric, hyperbolic. . . ). In any case, these high-order functionsare responsible for tailoring the transverse shear deformation, whilethe constant and linear elements introduce the characteristic zig-zagdistribution of the in-plane displacements.We need two functions of class C : an odd function f ( z ), and aneven function g ( z ) verifying g (cid:48) ( ± h / (cid:44) viz. : f ( z ) = − f ( − z ) g ( z ) = g ( − z ) g (cid:48) ( ± h / (cid:44) ϕ αβ ( z ) = a αβ f ( z ) + b αβ g ( z ) + c i αβ Z i ( z ) + d i αβ i ( z ) (12)where summation is implied over the dummy index i = , , . . . N .These four functions are defined with 4(2 + N ) = N +
1) constants.The expression for the derivatives of the four functions is ϕ (cid:48) αβ ( z ) = a αβ f (cid:48) ( z ) + b αβ g (cid:48) ( z ) + c i αβ i ( z ) (13)Just as the transverse shear strains, these four derivatives are not de-fined at the N − N +
1) equations: d i αβ = c i αβ ζ i + d i αβ = c i + αβ ζ i + d i + αβ ( i ∈ { , N − } ) a αβ f (cid:48) (0 − ) + b αβ g (cid:48) (0 − ) + c i αβ = δ αβ C i α γ (cid:16) a γβ f (cid:48) ( ζ i ) + b γβ g (cid:48) ( ζ i ) + c i γβ (cid:17) = C i + α γ (cid:16) a γβ f (cid:48) ( ζ i ) + b γβ g (cid:48) ( ζ i ) + c i + γβ (cid:17) ( i ∈ { , N − } ) a αβ f (cid:48) ( − h / + b αβ g (cid:48) ( − h / + c αβ = a αβ f (cid:48) ( + h / + b αβ g (cid:48) ( + h / + c N αβ = i corresponds to the number of the layer which contains the z = − coordinate. Since it seems di ffi cult to formulate a recursive pro-cess to determine all the coe ffi cients, the linear system (14) is solvedfor the 8( N +
1) unknown coe ffi cients a αβ , b αβ , c i αβ , d i αβ ( i = , . . . N ).Table 2 reports some functions f ( z ) and g ( z ) that can be chosento build an mHSDT model. While these functions allow to accommo-date the transverse shear behaviour inside each layer, the linear andconstant contributions are responsible for the ZZ behaviour, that is therespect of displacement and transverse stress continuities at the lay-ers’ interfaces. It should be noted that a “mixed” model can be con-structed by using functions of di ff erent nature, for example the hyper-bolic odd function sinh( kz / h ) can be considered in conjunction withthe even trigonometric function cos( π z / h ). Analytical expressions forthe warping functions for a single-layer plate are reported explicitly inAppendix. The construction of the linear system (14) can be automated be-cause its structure does not depend on the choice of the functions f ( z )and g ( z ). Indeed, only few values of these functions and of their deriva-tives, taken at specific z coordinates, have to be sent to the routine. Thesolution of the (8 N + × (8 N +
8) linear algebraic system can be carried ature f ( z ) g ( z ) NamePolynomial z z ToZZTrigonometric sin (cid:18) π zh (cid:19) cos (cid:18) π zh (cid:19) SiZZHyperbolic sinh (cid:18) k zh (cid:19) cosh (cid:18) k zh (cid:19)
HyZZ
Table 2:
Nature of the original bHSDT and corresponding couple of functions f ( z ) and g ( z ) used to build the mHSDT. out with a classical algorithm and provides the coe ffi cients defining thefour warping functions ϕ αβ ( z ). It may be noted that either numerical orsemi-analytical versions of the warping functions can be used. Numer-ical versions, which consist on a su ffi ciently dense table of values, aremore suitable for computing the numerous generalized sti ff ness andmass terms of the plate model within a numerical quadrature scheme. Once the ϕ αβ ( z ) functions are built, one can compute the corre-sponding stress functions ψ αβ ( z ). They do not bring new informa-tion to the models, as these stress functions are a direct consequenceof the warping functions , but they are useful to illustrate and under-stand the static response of the mHSDT. Let us replace in equation (6),the middle-plane transverse strains by the corresponding middle-planestresses: σ α ( z ) = C α β ( z ) ϕ (cid:48) βγ ( z ) γ − γ = C α β ( z ) ϕ (cid:48) βγ ( z ) S γ δ (0 − ) σ δ (15)Note that the 0 − exponent of γ − γ is not required to appear in the in-terlaminar continuous stress σ δ , but it is found in the S γ δ (0 − ) term.Eq. (15) permits to define the 4 stress functions of the model ψ αβ ( z ) = C α γ ( z ) ϕ (cid:48) γδ ( z ) S δ β (0 − ) (16)through which the transverse shear stresses are expressed as σ α ( z ) = ψ αβ ( z ) σ β (17) ϕ αβ ( z ) The tensorial character of the ϕ αβ ( z ) functions follows directlyfrom their definition, see Eq. (2). This tensorial character concernsonly the 2D ( x , y ) space. Also, the equations of the system (9) aretensor equations, i.e., their form is invariant with respect to rotationsabout the z axis. It implies that all the coe ffi cients a αβ , b αβ , c i αβ , d i αβ are second order tensors and must, therefore, obey to the formulas ofcoordinate transformation for second order tensors.The tensorial character of the warping functions implies that thefour functions ϕ αβ ( z ) of a laminate whose lamination sequence is s = [ θ /θ / . . . /θ N ] must be linked to the four functions ¯ ϕ αβ ( z ) of the lami-nate whose stacking sequence is ¯ s = s + θ = [( θ + θ ) / ( θ + θ ) / . . . / ( θ N + θ )]. In order to identify this relation, let us consider the ¯ s –laminate,in a Cartesian frame ( x , y , z ), and suppose it undergoes a pure sheardeformation of its middle plane ¯ γγγ . In this case, the kinematic fieldof Eq. (2a) can be written ¯ u α ( z ) = ¯ ϕ αβ ( z )¯ γ β or, in matrix notation,¯ u = ¯ ϕϕϕ ¯ γγγ . Consider the matrix of change of coordinates P θ from theCartesian frame ( x , y , z ) to a Cartesian frame ( x (cid:48) , y (cid:48) , z ), rotated from theprevious one by an angle θ about the z -axis. In the rotated frame, thisshear strain is γγγ = P θ ¯ γγγ and it “acts” on the s –laminate producing thein-plane kinematic field u . In the original frame, the ¯ s –laminate is thensubjected to the kinematic field ¯ u = P − θ u . Therefore, the following relation is established: ¯ ϕϕϕ = P − θ ϕϕϕ P θ . Since the transformation matrixis P θ = (cid:34) c s − s c (cid:35) with c = cos( θ ) and s = sin( θ ) (18)one can compute the warping functions for the ¯ s –laminate directlyfrom those for the s –laminate according to ¯ ϕ = ϕ c + ϕ s + ( ϕ + ϕ ) sc ¯ ϕ = − ( ϕ − ϕ ) sc + ϕ c − ϕ s ¯ ϕ = − ( ϕ − ϕ ) sc − ϕ s + ϕ c ¯ ϕ = ϕ s + ϕ c − ( ϕ + ϕ ) sc (19)Two examples of such transformations are given next for illustra-tion purposes. Figure 4 compares the native warping functions of a[45 / −
45] laminate against those obtained from a [0 /
90] laminate afterrotating them by an angle of 45 ◦ . The same comparison is proposed infigure 5 for the two laminates [ − / − /
25] and [ − / /
40] and witha rotation of − ◦ . − . . − h / h / ϕ ( z ) − . − . − . ϕ ( z ) − . − . − . − h / h / ϕ ( z ) − . − . . . ϕ ( z ) Native [0 /
90] Rotated [0 /
90] Native [45 / − Figure 4:
Tensorial behaviour of the warping functions : comparison betweenthe native warping functions of a [45 / −
45] laminate and those obtained by thecoordinate transformation from a [0 /
90] laminate.
4. Numerical results
A numerical evaluation is proposed in order to assess the accuracyof the basis models and their corresponding enhancement through ZZ warping functions with respect to the plate’s length-to-thickness ratio,number of layers, and stacking sequence. All bHSDT listed in Table 1are compared with their corresponding enhancements defined by thefunctions listed in Table 2. The factor k in the hyperbolic functions hasbeen set equal to 2 in the subsequent numerical investigations. Note . − . . . − h / − h / h / h / ϕ ( z ) − − · − ϕ ( z ) − · − − h / − h / h / h / ϕ ( z ) − . . ϕ ( z ) Native [ − / /
40] Rotated [ − / / − / − / Figure 5:
Tensorial behaviour of the warping functions : comparison betweenthe native polynomial [ − / − /
25] functions and those obtained by the coor-dinate transformation of the [ − / / finally that all considered models have the same number of DOF as thebHSDT, i.e., 5 DOF.In order to encompass a quite broad range of sti ff ness mismatchbetween adjacent layers, the study will investigate laminated as wellas sandwich plates with composite skins and a honeycomb core. Thematerial properties used for the composite and the honeycomb layersare reported in Table 3.The numerical assessment of the di ff erent models is carried out byreferring to an exact solution of the 2D di ff erential equations govern-ing the plate bending problem. Square, simply-supported plates areconsidered, for which we compute the fundamental eigenfrequencyas well as the static response under bi-sinusoidal transverse pressureloads of amplitude q / warping functions and stress functions of the bHSDT and mHSDT arecompared against those that have been extracted from the 3D solutionsfollowing the procedure detailed out in [31].The points at which quantities are output are defined as follows: A = ( a / , , A (cid:48) = ( a / , , h / B = (0 , a / , B (cid:48) = (0 , a / , h / C = ( a / , a / , C + = ( a / , a / , h /
2) (20)Deflections w , first natural frequencies ω and stresses are given ac- cording to following adimensionalisation w = E ref2 h q a w , ω = a h (cid:115) ρ ref E ref2 ωσ αβ = h q a σ αβ , σ α = hq a σ α (21)where for sandwich plates E ref2 and ρ ref are the values of the core ma-terial. It is important to specify that the transverse shear stress valuesreported in the tables and their distributions across the plate thicknessplotted in the figures are obtained from the equilibrium equations uponintegrating the in-plane stresses. E E E G G G ν α ν ρ Composite (c) 25 E c E c E c . E c . E c . E c .
25 0 . ρ c Honeycomb (h) E h E c /
25 12 . E h . E h . E h . E h .
02 0 . ρ c / Table 3:
Material properties ( α = , [0 / n laminates The models are tested for the antisymmetric cross-ply laminates[0 / n , where di ff erent numbers of layers are considered with n = , , , , ,
10. In table 4, non-dimensional deflection, transverse shearstresses and fundamental frequency are given for the Sin and the SiZZmodels, and compared to the exact solution. The length-to-thicknessratio is set to a / h =
10. Very similar results are obtained with poly-nomial and hyperbolic bHSDT / mHSDT models and are omitted fromTable 4 for the sake of clarity. The results clearly shows the accu-racy improvement introduced by the C z warping functions , in particu-lar for the deflection and the fundamental frequency: the enhancementon these two quantities appears to decrease as the number of layersincreases, although for n =
10 it is still larger than 5% and 3%, respec-tively.Only two warping functions are required for a cross-ply lami-nate because the cross-coupling functions are identically nil, ϕ ( z ) = ϕ ( z ) =
0. The functions ϕ ( z ) and ϕ ( z ) of the polynomial bHSDT(ToSDT) and of the polynomial, trigonometric and hyperbolic mHSDTare compared in Figures 6 and 9 for the n = n = ff erences between the 3 mHSDT are seento be negligible, and the curves for the trigonometric and hyperbolicbHSDT have been omitted for the sake of clarity because they are prac-tically coincident with those of the ToSDT.As far as the impact of warping functions on the local stress re-sponse is concerned, the values in Table 4 for the transverse shearstresses at the selected points do not allow to well appreciate it, buttheir through-the-thickness distributions obtained with the extendedmHSDT model are closer to the exact solution in comparison to thebHSDT models. This can be seen by comparing the two stress func-tions ψ ( z ) and ψ ( z ) depicted in Figures 7 and 10 for the cases n = n =
4, respectively. On the other hand, Figures 8 and 11 reportthe transverse shear stress distributions computed for the two configu-rations n = n =
4, respectively, upon integrating the equilibriumequations starting from the in-plane stresses. This post-processing pro-cedure is seen to annihilate all di ff erences between the bHSDT and themHSDT, thus providing distributions that very accurately recover theexact 3D solution.All considered models for the cross-ply laminate [0 / are as-sessed in Table 5 with respect to the length-to-thickness ratio a / h . It isobvious that the improvement of the mHSDT over the bHSDT is more mportant for thick plates than for thin plates, it decreases from morethan 10% for a / h = a / h = ff ect of the transverse shear increasesas a / h diminishes.It is worthwhile to make some comments about an expected sym-metry for the warping functions of the considered antisymmetric cross-ply laminates. Indeed, one should expect the ϕ ( z ) functions to beequal to the corresponding ϕ ( − z ) functions, but figures 6 and 9 showthat this is not the case. This is due to the fact that the z = − coordinatehas been chosen for prescribing the ϕ (cid:48) (0 − ) = ϕ (cid:48) (0 − ) = ϕ ( − z ) by ϕ (cid:48) (0 + ) yields in fact exactly thefunction ϕ ( z ). Note that, since there are no such constraints on the stress functions , the symmetry ψ ( z ) = ψ ( − z ) is immediately appar-ent in Figures 7 and 10. − . − . . . − h / − h / h / h / ϕ ( z ) − . . ϕ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 6:
Warping functions of the [0 / square plate with a / h =
10 for eachconsidered model.0 0 . − h / − h / h / h / ψ ( z ) . ψ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 7:
Transverse shear stress functions of the [0 / square plate with a / h =
10 for each considered model. [30 / − n laminates The [30 / − n laminate family is next considered, for which allfour warping functions are required due to the o ff -axis orientation an-gles. For the moderately thick plate characterized by a / h =
10 with n =
4, figure 12 compares the warping functions ϕ αβ ( z ) of the ToSDT − h / − h / h / h / σ ( B ) σ ( A ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 8:
Post-processed transverse shear stresses of the [0 / square platewith a / h =
10 for each considered model. − . − . . . − h / − h / h / h / ϕ ( z ) − . . ϕ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 9:
Warping functions of the [0 / square plate with a / h =
10 for eachconsidered model. and of the 3 mHSDT with those obtained from the 3D solution; thecorresponding stress functions ψ αβ ( z ) are plotted in figure 13. The dif-ferences between the three mHSDT are again negligible. Even if thediscrepancy with respect to the exact stress functions may be relevant,very accurate transverse shear stresses are obtained from the integra-tion of the equilibrium equations, as shown in figure 14.The numerical values for deflection, stresses and fundamental fre-quency are reported in tables 6 and 7. The results in table 6 refer tothe trigonometric models Sin / SiZZ for a fixed length-to-thickness ra-tio a / h =
10 and for di ff erent numbers of layers n = , , , , , σ ( B (cid:48) ) and σ ( A (cid:48) ) for n = a / h and the fixed n = / n laminates: exception madefor the 2-layer n = ff erence is more significantif the a / h ratio is low. The improvement is more systematic for theglobal response than for the local stress response, for which a certaindependency is observed with respect to the number n of the stackingsequence. . − h / − h / h / h / ψ ( z ) . ψ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 10:
Transverse shear stress functions of the [0 / square plate with a / h =
10 for each considered model.0 1 2 − h / − h / h / h / σ ( B ) σ ( A ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 11:
Post-processed transverse shear stresses of the [0 / square platewith a / h =
10 for each considered model.
In order to give evidence of the generality of the proposed methodfor constructing an mHSDT, a laminate with a stacking sequence of themost general nature is next considered. For this example, the arbitrarystacking sequence [ − / / / − / / / −
60] has been taken. Thefour warping functions of the polynomial, trigonometric and hyper-bolic mHSDT are plotted in figure 15 along with the warping functions of the polynomial ToSDT and of the 3D solution. The corresponding stress functions are plotted in figure 16. The distribution across thethickness at points A and B of the transverse shear stresses obtainedfrom the equilibrium equations are illustrated in figure 17. As in theprevious examples, all curves are in good agreement.Numerical results for the global and local stress response are re-ported in table 8. A substantial improvement is evident of the pre-dictions provided by the mHSDT over those provided by the bHSDTfor laminates with low length-to-thickness ratios, i.e., when the trans-verse shear behavior plays a certain role. The improvement is hereclearly visible not only for the global response (deflection and funda-mental frequency), but also for the local bending and transverse shearstresses. − . − . . . − h / − h / h / h / ϕ ( z ) − . − . . ϕ ( z ) − . − h / − h / h / h / ϕ ( z ) − . − . . . ϕ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 12:
The four warping functions for the [30 / − laminate with a / h =
10 0 0 . − h / − h / h / h / ψ ( z ) − . . ψ ( z ) − . . − h / − h / h / h / ψ ( z ) . ψ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 13:
Transverse shear stress functions for the [30 / − laminate with a / h = − h / − h / h / h / σ ( B ) − . . σ ( B ) − − h / − h / h / h / σ ( A ) . . σ ( A ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 14:
Transverse shear stresses from equilibrium equations for the[30 / − laminate with a / h = − . . − h / − h / − h / − h / h / h / h / h / ϕ ( z ) − . . ϕ ( z ) − . . − h / − h / − h / − h / h / h / h / h / ϕ ( z ) − . − . . . ϕ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 15:
Warping functions of the [ − / / / − / / / −
60] laminatewith a / h =
10 for each considered model Seq. Model w ( C ) % σ ( B ) % σ ( C + ) % ω %[0 /
90] Sin 1 . − .
17 1 . − .
84 7 . + .
41 8 . + . . − .
87 1 . − .
45 7 . + .
62 9 . + . . . . . / Sin 0 . − .
86 2 . + .
37 5 . − .
43 11 . + . . − .
49 2 . + .
33 5 . + .
21 11 . + . . . . . / Sin 0 . − .
91 2 . − .
87 5 . + .
64 12 . + . . − .
16 2 . − .
03 5 . + .
70 11 . + . . . . . / Sin 0 . − .
05 2 . + .
36 5 . + .
12 12 . + . . − .
02 2 . − .
36 5 . + .
84 12 . + . . . . . / Sin 0 . − .
61 2 . − .
35 5 . + .
17 12 . + . . + .
06 2 . − .
67 5 . + .
84 12 . + . . . . . / Sin 0 . − .
99 2 . + .
06 5 . + .
71 12 . + . . + .
19 2 . − .
41 5 . + .
59 12 . + . . . . . Table 4:
Comparison between the Sin and the SiZZ models for the [0 / n with n = , , , , ,
10 ( a / h = / h Model w ( C ) % σ ( B ) % σ ( C + ) % ω %4 ToSDT 1 . − . . + . . − .
63 7 . + . . − .
49 2 . + .
30 7 . + .
69 7 . + . . − . . + . . − .
02 7 . + . . − .
05 2 . − .
81 7 . + .
36 7 . + . . − . . + . . − .
22 7 . + . . − .
75 2 . + .
72 7 . + .
39 7 . + . . . . . . − .
95 2 . + .
46 5 . − .
81 12 . + . . − .
97 2 . + .
52 5 . + .
97 11 . + . . − .
86 2 . + .
37 5 . − .
43 11 . + . . − .
49 2 . + .
33 5 . + .
21 11 . + . . − .
99 2 . + .
49 5 . − .
95 12 . + . . − .
18 2 . + .
59 5 . + .
87 11 . + . . . . . . − .
16 2 . + .
03 4 . − .
02 14 . + . . − .
02 2 . + .
01 4 . + .
01 14 . + . . − .
16 2 . + .
02 4 . − .
02 14 . + . . − .
01 2 . + .
00 4 . + .
01 14 . + . . − .
16 2 . + .
03 4 . − .
02 14 . + . . − .
02 2 . + .
01 4 . + .
01 14 . + . . . . . Table 5:
Comparison between the di ff erent models for the square [0 / platewith a varying length to thickness ratio. A symmetric sandwich plate [0 / / / − / c ] s is finally consid-ered, which consists of a honeycomb core and two laminated faces.The core has a thickness 32 h /
40 and each composite layer in the facelaminates has the thickness h /
40. The material properties are thoselisted in table 3. In this case, di ff erent interfaces are present, i.e., be-tween plies of di ff erent orientations as well as between faces and core.The results for all considered bHSDT and corresponding mHSDT arereported in table 9. The improvement introduced by the mHSDT isclearly appreciable for all output variables and is more important forlow length-to-thickness ratios (short wavelength). Figure 18 illustratesthe warping functions and the corresponding stress functions are com-pared in figure 19. As in the previous case studies involving laminatedplates, the transverse shear stresses across the sandwich section ob-tained from the equilibrium equations are very accurate even for thebHSDT model, as shown in figure 20. eq. Model w ( C ) % σ ( B ) % σ ( A ) % σ ( B (cid:48) ) % σ ( A (cid:48) ) % σ ( C + ) % σ ( C + ) % ω %[30 / −
30] Sin 0 . − .
86 1 . − .
85 0 . − . − . + . − . + .
44 4 . + .
73 1 . + .
13 10 . + . . − .
24 1 . − .
27 0 . − . − . + . − . + .
41 4 . + .
97 1 . − .
58 10 . + . . . . − . − . . . . / − Sin 0 . − . . + .
39 2 . + . − . + . − . + .
75 2 . − .
49 1 . − .
23 14 . + . . − .
60 3 . + .
66 2 . + . − . + . − . + .
84 2 . − .
03 1 . − .
63 13 . + . . . . − . − . . . . / − Sin 0 . − .
14 2 . − .
76 1 . − . − . + . − . + . . − .
67 1 . − .
22 14 . + . . − .
82 2 . − .
89 1 . − . − . + . − . + . . + .
29 1 . − .
04 14 . + . . . . − . − . . . . / − Sin 0 . − .
89 3 . + .
43 1 . + . − . − . . + .
59 1 . − .
80 14 . + . . − .
46 3 . − .
52 1 . − . − . − . . + .
85 1 . + .
66 14 . + . . . . − . . . . . / − Sin 0 . − .
25 2 . − .
75 1 . − . − . + . − . + . . + .
00 1 . − .
29 14 . + . . − .
27 2 . − .
15 1 . − . − . + . − . + . . + .
04 1 . + .
95 14 . + . . . . − . − . . . . / − Sin 0 . − .
34 2 . + .
00 1 . − . − . + . − . + . . + .
90 1 . − .
15 15 . + . . + .
03 2 . − .
62 1 . − . − . + . − . + .
69 2 . + .
95 1 . + .
04 14 . + . . . . − . − . . . . Table 6:
Comparison between the Sin and the SiZZ models for the [30 / − n with n = , , , , ,
10 ( a / h = / h Model w ( C ) % σ ( B ) % σ ( A ) % σ ( B (cid:48) ) % σ ( A (cid:48) ) % σ ( C + ) % σ ( C + ) % ω %4 ToSDT 1 . − . . + . . + . − . + . − . + . . − .
91 1 . − . . + . . − .
49 2 . − .
57 1 . − . − . + . − . + . . + .
04 1 . − .
56 7 . + . . − . . + . . + . − . + . − . + . . − .
78 1 . − . . + . . − .
16 2 . − .
14 1 . − . − . + . − . + .
21 4 . + .
07 1 . + .
58 7 . + . . − . . + . . + . − . + . − . + . . − .
70 1 . − . . + . . − .
71 2 . + .
02 1 . − . − . + . − . + . . + .
24 1 . − .
39 7 . + . . . . − . − . . . . . − . . + .
53 2 . + . − . + . − . + .
89 2 . − .
10 1 . − .
83 14 . + . . − .
26 3 . + .
93 2 . + . − . + . − . + .
41 2 . − .
78 1 . − .
41 13 . + . . − . . + .
39 2 . + . − . + . − . + .
75 2 . − .
49 1 . − .
23 14 . + . . − .
60 3 . + .
66 2 . + . − . + . − . + .
84 2 . − .
03 1 . − .
63 13 . + . . − . . + .
58 2 . + . − . + . − . + .
94 2 . − .
32 1 . − .
04 14 . + . . − .
55 3 . + .
03 2 . + . − . + . − . + .
62 2 . − .
06 1 . − .
71 13 . + . . . . − . − . . . . . − .
23 3 . + .
04 2 . + . − . + . − . + .
08 2 . − .
07 0 . − .
09 17 . + . . − .
05 3 . + .
01 2 . + . − . + . − . + .
04 2 . − .
02 0 . − .
04 17 . + . . − .
23 3 . + .
04 2 . + . − . + . − . + .
08 2 . − .
06 0 . − .
08 17 . + . . − .
04 3 . + .
01 2 . + . − . + . − . + .
03 2 . − .
01 0 . − .
03 17 . + . . − .
23 3 . + .
04 2 . + . − . + . − . + .
08 2 . − .
07 0 . − .
09 17 . + . . − .
06 3 . + .
01 2 . + . − . + . − . + .
04 2 . − .
02 0 . − .
04 17 . + . . . . − . − . . . . Table 7:
Comparison between the di ff erent models for the square [30 / − plate with a varying length to thickness ratioa / h Model w ( C ) % σ ( B ) % σ ( A ) % σ ( B ) % σ ( A ) % σ ( C + ) % σ ( C + ) % ω %4 ToSDT 1 . − . . − .
81 2 . + .
95 0 . +
121 0 . +
137 1 . − . . − .
03 8 . + . . − .
35 2 . − .
34 1 . − .
42 0 . + .
77 0 . + .
54 1 . − .
61 3 . − .
73 7 . + . . − . . − .
71 1 . + .
67 0 . + . . +
109 1 . − . . − .
66 8 . + . . − .
48 2 . − .
22 1 . − .
96 0 . + .
98 0 . + .
35 1 . − .
83 3 . − .
52 7 . + . . − . . − .
50 2 . + .
40 0 . +
133 0 . +
146 1 . − . . − .
55 8 . + . . − .
34 2 . − .
02 1 . − .
22 0 . + .
15 0 . + .
20 1 . − .
97 3 . − .
80 7 . + . . . . . . . . . . − .
91 2 . − .
43 2 . + .
53 0 . + . . + .
56 0 . − .
21 2 . − .
19 14 . + . . − .
49 2 . + .
11 2 . − .
15 0 . + . . + .
80 0 . − .
58 2 . − .
32 13 . + . . − .
96 2 . − .
61 2 . + .
23 0 . + .
58 0 . − .
62 0 . − .
62 2 . − .
71 14 . + . . − .
47 2 . − .
05 2 . − .
29 0 . + .
42 0 . − .
25 0 . − .
30 2 . − .
21 13 . + . . − .
92 2 . − .
37 2 . + .
63 0 . + . . + . . − .
42 2 . − .
36 14 . + . . − .
53 2 . + .
17 2 . − .
11 0 . + . . + .
11 0 . − .
71 2 . − .
37 13 . + . . . . . . . . . . − .
16 2 . − .
00 2 . + .
00 0 . + .
17 0 . − .
32 0 . − .
06 2 . − .
03 17 . + . . − .
01 2 . + .
00 2 . − .
00 0 . + .
18 0 . − .
02 0 . − .
04 2 . − .
02 17 . + . . − .
16 2 . − .
01 2 . − .
00 0 . − .
03 0 . − .
76 0 . − .
06 2 . − .
02 17 . + . . − .
01 2 . + .
00 2 . − .
00 0 . + .
11 0 . − .
19 0 . − .
03 2 . − .
02 17 . + . . − .
16 2 . − .
00 2 . + .
00 0 . + .
23 0 . − .
17 0 . − .
07 2 . − .
03 17 . + . . − .
01 2 . + .
00 2 . − .
00 0 . + .
20 0 . + .
04 0 . − .
04 2 . − .
02 17 . + . . . . . . . . . Table 8:
Comparison between the di ff erent models for the square [ − / / / − / / / −
60] plate with a varying length to thickness ratio / h Model w ( C ) % σ ( B ) % σ ( A ) % σ ( B ) % σ ( A ) % σ ( C + ) % σ ( C + ) % ω %4 ToSDT 0 . − .
30 1 . − .
10 1 . + . − . + . − . + .
65 11 . − .
33 0 . − . . + . . + .
03 1 . − .
20 1 . − . − . + . − . + .
93 12 . + .
63 0 . − .
82 10 . + . . − .
65 1 . − .
93 1 . + . − . + . − . + .
96 11 . − .
06 0 . − . . + . . − .
28 1 . − .
22 1 . − . − . + . − . + .
82 12 . + .
79 0 . − .
77 10 . + . . − .
30 1 . − .
17 1 . + . − . + . − . + .
96 11 . − .
83 0 . − . . + . . + .
11 1 . − .
19 1 . − . − . + . − . + .
98 12 . + .
58 0 . − .
84 10 . + . . . . − . − . .
154 0 . . . − .
19 1 . − .
49 1 . + . − . + . − . − .
24 8 . − .
60 0 . − .
96 21 . + . . − .
22 1 . − .
02 1 . − . − . + . − . − .
48 8 . + .
05 0 . − .
72 20 . + . . − .
35 1 . − .
39 1 . + . − . + . − . − .
11 8 . − .
11 0 . − .
82 21 . + . . − .
40 1 . − .
02 1 . − . − . + . − . − .
51 8 . + .
10 0 . − .
71 20 . + . . − .
21 1 . − .
53 1 . + . − . + . − . − .
28 8 . − .
78 0 . − .
02 21 . + . . − .
17 1 . − .
03 1 . − . − . + . − . − .
47 8 . + .
03 0 . − .
73 20 . + . . . . − . − . . . . . − .
12 1 . − .
01 1 . + . − . + . − . − .
03 7 . − .
04 0 . − .
06 30 . + . . − .
01 1 . − .
00 1 . − . − . + . − . − .
01 7 . + .
00 0 . − .
03 30 . + . . − .
13 1 . − .
01 1 . + . − . + . − . − .
03 7 . − .
03 0 . − .
05 30 . + . . − .
01 1 . − .
00 1 . − . − . + . − . − .
01 7 . + .
00 0 . − .
03 30 . + . . − .
12 1 . − .
01 1 . + . − . + . − . − .
03 7 . − .
04 0 . − .
06 30 . + . . − .
01 1 . − .
00 1 . − . − . + . − . − .
01 7 . + .
00 0 . − .
03 30 . + . . . . − . − . . . . Table 9:
Comparison between the di ff erent models for the square [0 / / / − / c ] s sandwich panel with a varying length to thickness ratio
5. Discussion: limitations of ZZ theories
The presented results show that the mHSDT, constructed in sucha manner that all C z − requirements are a priori satisfied, can providevery accurate solutions in terms of deflection, transverse stresses andfundamental frequency for a very large class of composite plates. Inparticular, mHSDT improve the results of bHSDT for moderately thickplates that consist of a quite large number of layers.As a matter of fact, for configurations with a very low number oflayers and strong anisotropy, the constructed mHSDT appear to be notbetter than the corresponding bHSDT. This is true in general for anyZZ theory and is due to the ESL nature of these theories. In order topoint out the limitations of these ZZ theories, we shall next considerthe warping functions obtained from the exact 3D solution [31, 32].While such warping functions can be defined for general laminatesonly in a numerical manner, an analytical expression can be found forthe elementary case of an orthotropic single-layer plate. Exact solu-tions obtained, e.g., by a state-space approach are expressed in termsof hyperbolic functions. Therefore, the following two new hyperbolic warping functions are proposed: ϕ ( z ) = a − (cid:32) a z − a sinh (cid:18) a zh (cid:19)(cid:33) ϕ ( z ) = a − (cid:32) a z − a sinh (cid:18) a zh (cid:19)(cid:33) (22)where the coe ffi cients a , a , a , a have the expressions a = cosh (cid:18) a (cid:19) ; a = π hl x (cid:114) Q G a = cosh (cid:18) a (cid:19) ; a = π hl y (cid:114) Q G (23)and where l x and l y denote the lengths of the plate edges along the x and y directions, respectively.It is emphasised that the resulting model, referred to as HySpemodel, is not a member of the hyperbolic mHSDT described in Sec-tion 3. Two major points should be remarked: first, the HySpe modelemploys two odd functions instead of only one odd function (see equa-tion (A.5)); second, the material properties of the layer as well as thelength-to-thickness ratio l / h appear in the argument of the hyperbolic function. This HySpe model is next compared against the correspond-ing “conventional” ZZ model HyZZ, whose analytical expression isreported in Appendix, equation (A.5) with k =
2, in order to point outthe limitations of the latter model.Figure 21 compares these two hyperbolic HSDT against the exact3D solution in terms of transverse shear stress functions for di ff erentvalues of the elastic moduli: for this, the shear flexibilities 1 / G , / G and 1 / G of table 3 are multiplied by the same factor b . From fig-ure 21 it is apparent that the HyZZ is independent from this coe ffi cientand that it provides a satisfying approximation to the 3D solution onlyfor b =
1. As the shear flexibility increases, the error of the “conven-tional” HyZZ model increases while the stress functions of the HySpemodel are capable of well reproducing the 3D solution.The influence of the length-to-thickness ratio on the accuracy ofZZ models is shown in figure 22, where the transverse shear stressfunctions obtained by the HySpe model, the HyZZ ( k =
2) and the 3Dsolution are compared for di ff erent values of the l x / h = l y / h = a / h ratio. The results show that HySpe model has a slight discrepancywith the 3D solution only for the extremely thick case a / h =
2. Notethat this discrepancy comes from the plane stress assumption that hasallowed to derive the analytical expressions in equation (22). Further-more, the HyZZ model recovers the correct solution only if the length-to-thickness ratio is su ffi ciently large, say a / h ≥ warp-ing functions should depend on the material properties in a far morecomplex manner than through the linear coe ffi cients a αβ and b αβ asidentified by equation (12). More specifically, the warping functions should depend on the ratio between the longitudinal and the transverseshear moduli: ZZ models whose warping functions depend only onthe transverse shear moduli will always su ff er a certain limitation withrespect to highly constrained configurations such as laminated plateswith low number of layers. On the other hand, ‘conventional” ZZ mod-els su ff er a certain inaccuracy in case of thick laminates unless thelength-to-thickness ratio is explicitly taken into account in the warp-ing functions . . − h / − h / − h / − h / h / h / h / h / ψ ( z ) − . − . − . . ψ ( z ) − . − h / − h / − h / − h / h / h / h / h / ψ ( z ) . ψ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 16:
Transverse shear stress functions of the[ − / / / − / / / −
60] laminate with a / h =
10 for each consid-ered model
6. Conclusion
In this paper, a general method has been presented to extend ba-sis higher order shear deformation theories (bHSDT) to their multi-layer counterpart, denoted mHSDT, which a priori and exactly meetall C z − requirements along with the homogeneous shear stress condi-tions at the plate’s top and bottom surfaces, without introducing anyadditional DOF. The method is purely displacement-based as no as-sumption is introduced about the transverse shear stress behaviour.The extension process constructs zig-zag models from a couple of oneeven and one odd high-order functions of z of arbitrary nature. Asexamples, warping functions based on polynomial, trigonometric andhyperbolic functions have been explicitly addressed by referring to thecouples of functions (cid:16) z , z (cid:17) , (cid:16) sin( π zh ) , cos( π zh ) (cid:17) , (cid:16) sinh( k zh ) , cosh( k zh ) (cid:17) ,respectively. For general stacking sequences of N layers, four warp-ing functions are to be determined by solving a linear system of size8 N +
8. Furthermore, the tensorial character of the warping functions has been pointed out, which implies that warping functions of a stack-ing sequence s + θ , defined upon a rotation θ about the z − axis of astacking sequence s , must correspond to those of the stacking sequence s through the well known tensorial transformation.The considered ESL ZZ mHSDT models have been comparedagainst the bHSDT for a large variety of composite plates, includingcross-ply, angle-ply, arbitrary laminates and sandwich plates. Exactsolutions are obtained based on a previously developed method, forthe static bending and the fundamental frequency responses of simply-supported plates. The results indicate that mHSDT substantially in-crease the accuracy of the bHSDT in most practically relevant applica-tions. No appreciable di ff erence is found between models formulated − h / − h / − h / − h / h / h / h / h / σ ( B ) − . σ ( B ) − − . . − h / − h / − h / − h / h / h / h / h / σ ( A ) σ ( A ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 17:
Post-processed transverse shear stresses of the[ − / / / − / / / −
60] laminate with a / h =
10 for each consid-ered model in terms of polynomial, trigonometric or hyperbolic functions. A finaldiscussion has been proposed in order to point out the limitations ofZZ models with respect to “constrained” configurations characterisedby a low number of layers (say, N ≤
4) and low length-to-thicknessratio: in order to recover with good accuracy the exact solution, theseconfigurations require warping functions that take into account in anexplicit manner also the in-plane sti ff nesses of each individual layer.This contrasts with the “conventional” ZZ approaches, in which onlythe transverse shear moduli are considered. While an explicit represen-tation of the mechanical and geometrical properties of each individuallayer is inherent to LayerWise models, the possibility of formulatingmore refined ZZ theories capable of overcoming the limitations of cur-rently available ESL models will be an object of further studies. Appendix A. Analytical expressions for one layer
For a plate consisting of one layer, the system of equations (14)reduces to: d αβ = a αβ f (cid:48) (0 − ) + b αβ g (cid:48) (0 − ) + c αβ = δ αβ a αβ f (cid:48) ( − h / + b αβ g (cid:48) ( − h / + c αβ = a αβ f (cid:48) ( + h / + b αβ g (cid:48) ( + h / + c αβ = δ αβ is Kronecker’s delta. Since f (cid:48) ( − h / = f (cid:48) ( h /
2) and g (cid:48) ( − h / = − g (cid:48) ( h / b αβ = a αβ f (cid:48) ( h / + c αβ = . − . . . − h / − h / h / h / ϕ ( z ) − · − ϕ ( z ) − − · − − h / − h / h / h / ϕ ( z ) − . − . . . ϕ ( z ) ToSDT ToZZ SiZZ HyZZ Exact
Figure 18:
Warping functions of the square [0 / / / − / c ] s sandwich panelwith a / h =
10 for each considered model found to be expressed in terms of only one odd function f ( z ) as ϕ αβ ( z ) = δ αβ f (cid:48) (0 − ) − f (cid:48) ( h /
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