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Effective elastic characteristics of honeycomb sandwich composite shells made of generally orthotropic materials
Composites: Part A 38 (2007) 1533–1546 www.elsevier.com/locate/compositesa
Effective elastic characteristics of honeycomb sandwich composite shells made of generally orthotropic materials Gobinda C. Saha, Alexander L. Kalamkarov *, Anastasis V. Georgiades Department of Mechanical Engineering, Dalhousie University, P.O. Box 1000, Halifax, NS, Canada B3J 2X4 Received 6 July 2006; accepted 3 January 2007
Abstract This paper works on the analytical development of the method of two-scale asymptotic homogenization. The technique is used to determine the effective elastic stiffnesses of hexagonal honeycomb-cored structural sandwich composite shells made of generally orthotropic materials. Orthotropy of the constituent materials leads to much more complex unit-cell problems and is considered in the present paper for the first time. At first, a 3D-to-2D general shell model based on a set of unit-cell problems is derived. This is followed by the exploitation of the model to the derivation of analytical estimate formulae; used to calculate the force and moment resultants present in the sandwich shell structure. The implication of the general shell model is further indicated by calculating similar design characteristics for a three-layered composite sandwich panel reinforced with hexagonal and triangular shaped cellular core made from the generally orthotropic material. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Material; B. Property; C. Analysis
1. Introduction The preponderance of uses for composite materials is in the form of plates and shells the optimum strength-toweight characteristics of which offer engineers attractive alternatives for different applications. A particular class of these structures, the structural sandwich shells and panels, find increasing use in lightweight construction, aerospace and automotive engineering, marine architecture and many other platforms. A typical structural sandwich element is a layered medium consisting of two high-density, highstrength face carriers bonded to a thick core made from a low-density material. Usually, the core is constructed from a foam material with a hexagonal or honeycomb configuration, while anisotropic laminae are commonly used as face carriers. The presence of the thick core means that the deformation characteristics of a sandwich shell are different
*
Corresponding author. Tel.: +1 902 494 6072; fax: +1 902 423 6711. E-mail address: [email protected] (A.L. Kalamkarov).
1359-835X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2007.01.002
from those of a classical laminated shell or a monolayer structure. Although the contribution from the core to the overall weight of the sandwich structure is minimal, the moment of inertia of the structure increases substantially by the separation of the face carriers from the core, thanks to the presence of the thick core. On the other hand, due to the limited flexural rigidity of the low-density core, the influences of the local phenomena, such as local instability, loadinduced thickness reduction, etc., on the global phenomena, such as stability, deformation, etc., have to be taken into account when designing with sandwich structures. Hence, micromechanical models dealing with sandwich shells should place proper emphasis on the importance of the core’s mechanical behavior. Classical analytical methods were derived for standard structures such as beams, plates and shells only under specific loading and boundary conditions. The development of enhanced models that can effectively determine the stresses, deformations and critical loads under general conditions is highly desirable. With the knowledge of the effective deformation, stress and stability performance of such structure, design parameters can then
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be optimized with respect to minimum weight and flexural and shear criteria of the sandwich structure. The incorporation of sandwich composite shells in new engineering applications will be facilitated if their properties are determined at the design stage. The micromechanical models that represent these shell structures will be considered as efficient provided that they are neither too complex to describe and use, nor too simple to reflect the ‘real’ properties and characteristics of the composite structure. One technique that has been employed in the modeling of composites with a periodic structure is the ‘asymptotic homogenization’ method. The mathematical framework of the method of asymptotic homogenization can be found in [1–3]. The premise of the method is based on the fact that many physical problems are characterized by some variables that exhibit a slow variation and some others that change rapidly. The coupling of the slow and fast variables renders the direct analytical and even numerical solution of the problem extremely difficult. Asymptotic homogenization however decouples the two sets of variables and solves them separately. In this paper, we intend to develop asymptotic homogenization model pertaining to sandwich composite thin shell made of generally orthotropic material. Duvaut [4] and Kalamkarov [5] have developed homogenized models of plates with periodical non-homogeneities in tangential coordinate(s). Their work led to a comprehensive definition of a single ‘periodicity cell’, wherein the elastic characteristics of the layer material and the shape of the upper and lower surfaces of the shell structure are assumed to be regularly periodic. Consequently, a unit-cell approach which, in fact, is based upon the notion of a regular periodic structure allows us to resort the method of averaging (or ‘homogenization’) in the design of thin composite shells. Furthermore, structural shells such as three-layered sandwich shells filled with a honeycomb structured core (or honeycomb shells) are neither one-dimensional nor twodimensional composites (such as classical laminated shells or fiber reinforced composites), nor are they three-dimensional composites (such as granulated materials). Periodicity is exhibited only in the two tangential coordinates introduced at the middle surface of the shell, with no such periodicity existing in the transverse coordinate. These features in terms of both period and shell thickness are commensurate in the present modeling and hence the force boundary conditions are specified at the upper and lower surfaces of the shell. The inherent features of the problem definition require a refined asymptotic analysis approach for a 3D thin shell problem which would allow the asymptotic transition from a 3D problem to the 2D shell problem. The technique, which is essentially a modified asymptotic approach developed by Caillerie [6], was used by Panasenko and Reztsov [7] to obtain a complete asymptotic expansion of a 3D problem in the theory of elasticity for a thin plate with a thickness equal to the characteristic dimension of the non-uniformities. Kohn and Vogelius [8] used a similar approach to analyze the 3D problem of a
homogeneous thin uniform plate with a rapidly oscillating thickness asymptotically. In this micromechanical modeling approach, the modified asymptotic homogenization technique is employed to study the elastic characteristics of non-uniform curved thin composite shells with regular structure and wavy surface. The exact formulation of the 3D shell problem associated with the elasticity problem is derived in Section 2. Befitting with the characteristic dimension of the non-uniformities, d, the derived 3D general shell model is amenable to a rigorous asymptotic analysis unifying an asymptotic 3D-to-2D process and a material homogeneous process. The general shell model in terms of a set of problems, called the ‘unit-cell’ problems, is dependent only on the nature of the local periodicity of the unit-cell and is independent of the global formulation of the composite shell structure. These unit-cell problems will generate the so-called effective elastic properties which replace the actual rapidly varying properties, and are in fact constant. Section 3 describes the homogenization procedure through which the effective elastic coefficients are actually obtained. In the second part of the study, the effectiveness of the general model is indicated by considering several applications with respect to structural non-uniform shells. Based on the solution of the local problems, explicit formulae are derived for elastic characteristics of hexagonal and hexagonal-triangular honeycomb sandwich shells of regular structure in Sections 4 and 5, respectively, made of generally orthotropic material. In Section 6, we continue our discussion through the use of numerical illustrations, based on the averaged property formulae obtained in Sections 4 and 5. A comparison is made between the two adopted geometries. Finally, Section 7 concludes the paper. It is noted that to the best of authors’ knowledge, this work represents the first attempt to model composite sandwich shells composed of entirely orthotropic constituents. As well, when an orthotropic material is rotated by an angle u about one its axes of orthotropy, the resulting stiffness tensor has the same format as a monoclinic material. The novel model developed in this work will consider orthotropic honeycomb core constituents which are not referenced with respect to their principal material directions, i.e., will exhibit monoclinic-like characteristics. 2. Problem formulation Consider an inhomogeneous shell of periodic structure, as illustrated in Fig. 1. The general shell structure is obtained by repeating a periodicity cell Xd, which is specified in the orthogonal dimensionless coordinate system (a1, a2, c) by the inequalities {dh1/2 < a1 < dh1/2, dh2/2 < a2 < dh2/2, c < c < c+}. The dimensionless small parameter d specifies the thickness of the shell, as well as the scale of the composite material inhomogeneity is assumed to be small when compared with the global dimensions of the solid. Here
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Fig. 1. A curvilinear reinforced composite layer with representative periodicity cell Xd.
h1, h2 are the ratios of the tangential dimensions of the shell to the thickness. The elasticity problem of this shell structure is characterized by means of the following boundary value problem: orij fi ¼ 0; oaj rij ¼ cijkl ekl ;
where 1 ouk oul ekl ¼ þ ; and 2 oal oak
ð1Þ
In Eq. (1), cijkl represents the fourth-order elasticity tensors of the materials involved and is periodic in a1 and a2 directions only with respective periods dh1 and dh2. Functions fi, pi, uk represent body forces, surface tractions and displacement fields, respectively, whereas nj is the unit normal vector pertaining to lateral wavy surfaces c= S ða1 ; a2 Þ of the shell and is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 oS oS oS =oa1 þ oS =oa2 þ 1 n ¼ ; ;1 oa1 oa2 ð2Þ We introduce the following notations: z = c/d, n = (n1, n2), a = (a1, a2). We also make the assumption that the elasticity coefficients cijkl(n, z) are piecewise smooth periodic functions for n1, n2 anywhere in the periodicity cell Xd: {n1, n2 2 (1/2, 1/2), z 2 (z, z+)}, exhibiting discontinuities of the first kind at a finite number of non-intersecting contact surfaces. Following the method of two-scale asymptotic expansions outlined in [9], we present the components of the displacement and stress vectors as follows: ð1Þ
ð2Þ
ui ða; n; z; dÞ ¼ ui ðaÞ þ dui ða; n; zÞ þ d2 ui ða; n; zÞ þ ð0Þ
z owðaÞ 2 lt 3 þ dU lt 1 elt þ d V 1 slt þ Oðd Þ A1 oa1 z owðaÞ 2 lt 3 þ dU lt u2 ¼ v2 ðaÞ d 2 elt þ d V 2 slt þ Oðd Þ A2 oa2 2 lt 3 u3 ¼ wðaÞ þ dU lt 3 elt þ d V 3 slt þ Oðd Þ u1 ¼ v1 ðaÞ d
rij nj ¼ pi
ð0 Þ
It has been demonstrated in [9] that for the principal terms in Eq. (3) and in the corresponding expansions for the components of the stress tensor over the small parameter d, the relationships that determine the local structure of the displacement and stress fields are
lt rij ¼ blt ij elt þ dbij slt
ð4Þ Here and in the sequel, summation is performed over identical indices, with the Latin indices taking on values of 1–3, while the Greek indices take on values of 1 and 2. A1(a), A2(a) are the coefficients of the first quadratic form of the shell middle surface (c = 0). A comparison of results in Eq. (4) with the classical general thin shell theory [10, page 439] results shows that functions v1(a), v2(a) are the displacements of a point in the directions of the tangent to the meridian and function w(a) is the displacement in the direction of the normal to middle surface of the shell element at that point. In the calculations we consider theses displacements to be very small and associate them with the displacement components z{w(a)/a1} and z{w(a)/a2} that account for rotation of the plane section around this middle surface. Strain functions e11 = e1, e22 = e2, e12 = e21 = e/2 represent the tensile and shearing strains in the middle surface; s11 = k1, s22 = k2, s12 = s21 = s characterize the flexural and torsional strains of the middle surface. In the last expression of Eq. (4), the following definitions are used: 1 oU lt oU lt k k þ cijlt and þ cijk3 hb onb oz 1 oV lt oV lt k ¼ cijkb þ cijk3 k þ zcijlt hb onb oz
blt ij ¼ cijkb
ð1 Þ
rij ða; n; z; dÞ ¼ rij ða; n; zÞ þ drij ða; n; zÞ ð2Þ
þ d2 uij ða; n; zÞ þ ð3Þ
blt ij
ð5Þ
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lt The functions U lt k and V k contained in relationships (4) and (5) depend on n1 = A1n1, n2 = A2n2 and z. With respect to n1 and n2, they represent the periodic solutions (with respective periods A1 and A2) of the following local problems on the periodicity cell Xd
Relationship (9) represents the equation of state for the averaged shell, while the coefficients of this relationship represent its effective elastic characteristics. On the basis of Eqs. (4)–(8), in arriving relationship (9), the following has been demonstrated:
1 o lt o lt b þ b ¼ 0 with hb onb ib oz i3 1 lt lt bib N b þ bi3 N 3 ¼0 hb z¼z
r lt hzr blt i3 i ¼ hz bi3 i ¼ 0
and 1 o lt o lt b þ bi3 ¼ 0 with hb onb ib oz 1 lt lt bib N b þ bi3 N 3 ¼0 hb z¼z
bk hblt bk i ¼ hblt i;
ð6Þ
where N i represents the component vectors normal to the surfaces z = z±. The problems expressed by Eq. (6), together with Eq. (5), respectively, are called the unit-cell local problems for the thin inhomogeneous shell structure illustrated in Fig. 1. With this, what we have achieved is that of the homogenized (because they involve only the derivatives with respect to rapid variables n and z) periodlt icity cell local functions U lt k ðn; zÞ and V k ðn; zÞ which are 1periodic in n1 and n2 and determine, in turn, the coefficients lt blt in Eq. (4). These local functions help us in ij and bij ð1Þ obtaining the first term in the stress function rij . It should be noted that, unlike the unit-cell problems of ‘classical’ homogenization schemes, those set by Eq. (6) depend on the boundary conditions z = z± rather than on periodicity in the z-direction. 3. Homogenization of unit-cell problems In the case of material discontinuity due to the presence of discontinued surfaces along the shell structure we add the following continuity conditions to the unit-cell problems in Eq. (6):
nb lt lt lt ½U k ¼ 0; b þ n3 bi3 ¼ 0 and hb ib
ð7Þ nb lt lt lt ½V k ¼ 0; b þ n3 bi3 ¼ 0 hb ib where ni is the component normal to the discontinuity surface. Local problems (5)–(7) have single solutions accurate lt to the constant terms blt ij and bij . This non-uniqueness is removed by introducing the operation of the averaging over the volume of the periodicity cell, which is defined on an arbitrary coefficient w by its integral as Z 1 wdn1 dn2 dz ð8Þ hwi ¼ jXd j Xd Averaging relationships in Eq. (4) by means of definition (8) over the volume Xd, we obtain (r = 0, 1): 2 r lt hzr rij i ¼ hzr blt ij ielt þ dhz bij islt þ Oðd Þ
ð9Þ
bk hzblt bk i ¼ hblt i;
bk hzblt bk i ¼ hzblt i
ð10Þ
Relationships (10) provide for the symmetry of the matrices comprised of the coefficients of the equations of state for the averaged shell. It is worth noting that, as demonstrated in [11], in the case of a uniform material and with constant shell thickness (z 2 (1/2, 1/2)) this average shell model reduces to the relationships taken from the theory of the elasticity of anisotropic shells, with the following formulas valid for the linear forces and the bending and torsional moments: N 1 ¼ dhr11 i; 2
M 1 ¼ d hzr11 i; N 12 ¼ dhr12 i;
N 2 ¼ dhr22 i; M 2 ¼ d2 hzr22 i;
ð11Þ
2
M 12 ¼ d hzr12 i
In summarizing, the way the above model is presented, the local problems (6) are completely determined by the structure of the unit-cell of the composite shell and are fully independent of the formulation of the global boundary value problem, set in Eq. (1). It follows that the solution of these local problems and, in particular, the effective elastic coefficients of the homogenized composite shell can be utilized in studying various types of boundary value problems associated with a given composite structure. In the rest of the paper, we shall dwell in some detail on certain applications of the general shell model to design the structural non-uniform shells of regular structure, these having been fabricated out of a uniform orthotropic material. 4. Sandwich shell with hexagonal honeycomb core To obtain the effective characteristics that appear in the elasticity relations involving problems from the theory of plates and shells, also present in the current solution of local problems in Eqs. (5) and (6), a method of approximation was proposed and verified by Kolpakov [12]. In that, the effective characteristics of small-cell skeletal constructions of periodic structure were studied. In combination with the above-mentioned general model of an averaged shell, this method makes it possible to derive explicit analytical estimate formulae and obtain effective characteristics for a large number of reinforced shells, such as those used in actual practice, with adequate accuracy. We examine the three layered shell reinforced with hexagonal honeycomb type sandwich filler with a periodicity cell consisting of ten individual elements (Fig. 2). As the governing equations of basic mechanics in relation with membrane and flexural rigidities can be simplified by assumptions typically fulfilled by sandwich structures,
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Fig. 2. A three-layered sandwich shell reinforced with hexagonal-honeycomb filler.
the approximate analytical solution of the local problems (5) and (6) for a unit-cell of the relevant kind of geometry is found under the assumption that the thickness of each of the ten unit-cell elements is small in comparison with the other two dimensions, i.e., under the conditions t1 h1, t2 h2, h1, h2 H. The two face carriers are identical and symmetrically arranged with respect to the centroid of the sandwich structure. Both face carriers and hexagonal core of this sandwich composite structure is made of generally orthotropic material. For the sake of computational simplicity, we confine our attention to the case when both coefficients assigned to describe the quadratic form of the shell middle surface are made from unity, i.e., A1 = A2 = 1. For the case of structure shown in Fig. 2, the presence of rotated elements necessitates the use of coordinate systems which are at an angle with respect to the principal coordinate systems. Consequently, even though the structure is made from orthotropic material, these rotated elements will exhibit monoclinic-like behavior when they are not referred to with respect to their principal material coordinates. The appropriate elastic material tensor for monoclinic material is given below: 2
c11 6c 6 12 6 6 c13 6 6 0 6 6 4 0 c16
3
c12
c13
0
0
c16
c22
c23
0
0
c23 0
c33 0
0 c44
0 c45
0 c26
0 c36
c45 0
c55 0
c26 7 7 7 c36 7 7 0 7 7 7 0 5 c66
ð12Þ
For the same reason, after transformation of the coordinates from (a1, a2) to (n1, n2) according to the rules of transformation of coordinates pffiffiffiffiffi pffiffiffi h1 ¼ 3a; h2 ¼ 3a; t1 ¼ 2t= 30a ; pffiffiffi pffiffiffiffiffi ð13Þ t2 ¼ t= 3a ; sin u ¼ 3= 10 it is obvious from the periodicity unit-cell (see Fig. 3) that we first deal with a simpler type of unit-cell problem which consists only of a single element rotated at an arbitrary angle u with respect to the n2 axis. This rotated unit-cell is shown in Fig. 4. Once the solution for this rotated unit-cell is obtained, it is thus a matter of simple superposition which will allow us to compute the solution for the complete honeycomb core pertaining to appropriate angle u for participating elements X3 to X10. 4.1. Single-element rotated unit-cell We consider that the element is located along one of the coordinate axes, say g2. This alignment will essentially lead to the reduction of the order of the associated differential equations by making the problem independent of g2. The lt functions U lt k ðn; zÞ and V k ðn; zÞ will then depend on the variables g1 and z, and the local problems of the group lt blt (lt = 11, 22, 12) reduce to the determination ij and bij lt of the functions U lt 2 ðg1 ; zÞ, U 3 ðg1 ; zÞ, (lt = 11, 22) and 12 U 3 ðg1 ; zÞ from the following system of equations:
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a
b
c
Fig. 3. (a) Hexagonal core in 2-D; (b) a periodicity unit-cell; and (c) cross-section.
slt 22
slt 33
slt 23 slt 13 slt 12
c12 cos u c26 sin u oU lt 1 ¼ þ h1 h2 og1 c26 cos u c22 sin u oU lt oU lt 2 3 þ þ þ c23 h1 h2 og1 oz c13 cos u c36 sin u oU lt 1 ¼ þ h1 h2 og1 c36 cos u c23 sin u oU lt oU lt 2 3 þ þ þ c33 h1 h2 og1 oz oU lt oU lt c45 cos u c44 sin u oU lt 1 2 3 þ c44 þ ¼ c45 þ h1 h2 oz oz og1 oU lt oU lt c55 cos u c45 sin u oU lt 1 2 3 þ c45 þ ¼ c55 þ h1 h2 oz oz og1 c16 cos u c66 sin u oU lt 1 ¼ þ h1 h2 og1 c66 cos u c26 sin u oU lt oU lt 2 3 þ þ þ c36 h1 h2 og1 oz ð14Þ
Fig. 4. A rotated unit-cell comprising the honeycomb cellular core.
cos u oslt sin u oslt oslat i1 i2 þ þ i3 ¼ 0 h1 og1 h2 og1 oz lt c cos u c sin u oU 1 11 16 slt þ 11 ¼ h h2 og1 1 c16 cos u c12 sin u oU lt oU lt 2 3 þ þ þ c13 h1 h2 og1 oz
subject to the boundary conditions on outer surfaces (see Fig. 5) g1 = ±t1/2 and z = (0, H): 11 11 t11 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12 ¼ ðcos u=h1 Þc11 ðsin u=h2 Þc16 22 22 t22 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12
¼ ðcos u=h1 Þc12 ðsin u=h2 Þc26
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Fig. 5. Specification of boundary conditions on surfaces g1 = ±t1/2 and z = (0, H).
12 12 t12 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12
¼ ðcos u=h1 Þc16 ðsin u=h2 Þc66 11 11 t11 2 ¼ ðcos u=h1 Þs12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc16 ðsin u=h2 Þc12 22 22 t22 2 ¼ ðcos u=h1 Þs12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc26 ðsin u=h2 Þc22 t12 2
12 ¼ ðcos u=h1 Þs12 12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc66 ðsin u=h2 Þc26 t11 3
11 ¼ ðcos u=h1 Þs11 13 þ ðsin u=h2 Þs23 ¼ 0
22 22 t22 3 ¼ ðcos u=h1 Þs13 þ ðsin u=h2 Þs23 ¼ 0 12 12 t12 3 ¼ ðcos u=h1 Þs13 þ ðsin u=h2 Þs23 ¼ 0
ð15aÞ and
8 cos2 u sin2 u > c66 ðc213 c22 c11 c223 c212 c33 b22 > 11 ¼ > h21 h22 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > 22 > b22 ¼ cos44 u c66 ðc213 c22 c11 c223 c212 c33 > h1 1< ð16bÞ b22 ¼ þ c11 c22 c33 þ 2c12 c13 c23 Þ A> > > 22 22 22 >b ¼ b ¼ b ¼ 0 > 33 23 13 > > > > cos3 u sin u 22 > b12 ¼ h3 h c66 ðc213 c22 c11 c223 c212 c33 > > > 1 2 > : þ c11 c22 c33 þ 2c12 c13 c23 Þ 8 cos u sin3 u 12 2 2 2 > > > b11 ¼ h1 h32 c66 ðc13 c22 c11 c23 c12 c33 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > 12 cos3 u sin u > > b22 ¼ h3 h2 c66 ðc213 c22 c11 c223 c212 c33 1 1< b12 ¼ ð16cÞ þ c c 11 22 c33 þ 2c12 c13 c23 Þ > A> > 12 12 12 > b33 ¼ b23 ¼ b13 ¼ 0 > > > > > cos2 u sin2 u > b12 c66 ðc213 c22 c11 c223 c212 c33 > 12 ¼ > h21 h22 > > : þ c11 c22 c33 þ 2c12 c13 c23 Þ
22 11 2 12 12 t11 1 ¼ t 1 ¼ t 2 ¼ t 22 ¼ t 1 ¼ t 2 ¼ 0 11 t11 3 ¼ s33 ¼ c13 ;
22 t22 3 ¼ s33 ¼ c23 ;
12 t12 3 ¼ s33 ¼ c36
where
ð15bÞ A¼ The solution of the group blt ij local problems having been found, we then calculate the following functions: 8 sin4 u 11 2 2 2 > > > b11 ¼ h42 c66 ðc13 c22 c11 c23 c12 c33 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > cos2 u sin2 u 11 > > b22 ¼ h2 h2 c66 ðc213 c22 c11 c223 c212 c33 1 2 1< ð16aÞ b11 ¼ þ c c22 c33 þ 2c12 c13 c23 Þ 11 > A> > 11 11 11 > b33 ¼ b23 ¼ b13 ¼ 0 > > > > > cos u sin3 u > b11 c66 ðc213 c22 c11 c223 c212 c33 > 12 ¼ > h1 h32 > > : þ c11 c22 c33 þ 2c12 c13 c23 Þ
! cos12 u sin12 u þ c66 c11 c33 c213 4 4 h1 h2
ð16dÞ
A similar situation exists with the group blt ij (lt = 11, 22, 12) local problems, the two-dimensional forms of which are given as below: cos u oslt sin u oslt oslt i1 i2 þ þ i3 ¼ ci3lt h1 og1 h2 og1 oz lt c cos u c sin u oV 1 11 16 lt s11 ¼ þ h1 h2 og 1 c16 cos u c12 sin u oV lt oV lt 2 þ þ þ c13 3 h1 h2 og1 oz
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c12 cos u c26 sin u oV lt 1 þ h1 h2 og1 c26 cos u c22 sin u oV lt oV lt 2 þ þ þ c23 3 h1 h2 og1 oz lt c13 cos u c36 sin u oV 1 ¼ þ h1 h2 og1 c36 cos u c23 sin u oV lt oV lt 2 þ þ þ c33 3 h1 h2 og1 oz lt lt oV 1 oV 2 c45 cos u c44 sin u oV lt 3 þ c44 þ ¼ c45 þ h1 h2 oz oz og1 oV lt oV lt c55 cos u c45 sin u oV lt 1 2 3 þ c45 þ ¼ c55 þ h1 h2 oz oz og1 c16 cos u c66 sin u oV lt 1 ¼ þ h1 h2 og1 c66 cos u c26 sin u oV lt oV lt 2 þ þ þ c36 3 h1 h2 og1 oz
slt 22 ¼
slt 33
slt 23 slt 13 slt 12
ð17Þ with the appropriate conditions on surfaces g1 = ±t1/2 and z = (0, H): t11 s11 s11 1 ¼ ðcos u=h1 Þ 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc11 zðsin u=h2 Þc16 t22 1
¼ ðcos u=h1 Þs22 s22 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc12 zðsin u=h2 Þc26
t12 s12 s12 1 ¼ ðcos u=h1 Þ 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc16 zðsin u=h2 Þc66 t11 s11 s11 2 ¼ ðcos u=h1 Þ 12 þ ðsin u=h2 Þ 22 ¼ zðcos u=h1 Þc16 zðsin u=h2 Þc12 t22 2
¼
ðcos u=h1 Þs22 12
þ
ð18aÞ
ðsin u=h2 Þs22 22
where B ¼ h31 sin7 u c11 c33 c213 þ h32
cos6 u sin u c11 c33 c213 þ c12 c33 c13 c23 þ 2c33 c66
¼ zðcos u=h1 Þc26 zðsin u=h2 Þc22 t12 s12 s12 2 ¼ ðcos u=h1 Þ 12 þ ðsin u=h2 Þ 22 ¼ zðcos u=h1 Þc66 zðsin u=h2 Þc26 t11 3 t22 3
¼
ðcos u=h1 Þs11 13
¼
ðcos u=h1 Þs22 13
8 11 b ¼ h31 sin11 uðc213 c22 c11 c223 c212 c33 > > > 11 > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h1 h22 cos10 u sin uc66 c11 c33 c213 > > > > > 11 2 10 > > > b22 ¼ 2h1 h2 cos u sin uc66 ðc12 c33 c13 c23 Þ > < 11 11 11 z b33 ¼ b23 ¼ b13 ¼ 0 ð19aÞ b 11 ¼ B> > b11 ¼ h31 cosu sin10 uðc213 c22 c11 c223 12 > > > > > c212 c33 þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h31 cos u sin10 uc66 ðc11 c33 c213 > > > > > þ c12 c33 c13 c23 Þ > > : 2h21 h2 cos u sin10 uc66 ðc12 c33 c13 c23 Þ 8 22 b ¼ 2h21 h2 cosu sin10 uc66 ðc12 c33 c13 c23 Þ > > > 11 > 3 11 2 2 2 > b22 > 22 ¼ h2 cos uðc13 c22 c11 c23 c12 c33 > > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h21 h2 cos u sin10 uc66 c11 c33 c213 > > > 22 22 z < b22 33 ¼ b23 ¼ b13 ¼ 0 b 22 ¼ ð19bÞ > b22 ¼ h3 cos10 u sin uðc2 c22 c11 c2 B> 12 2 13 23 > > > > c212 c33 þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > > þ 2h32 cos10 u sin uc66 ðc11 c33 c213 > > > > > þ c12 c33 c13 c23 Þ > > : 2h1 h22 cos10 u sin uc66 ðc12 c33 c13 c23 Þ 8 12 b11 ¼ h32 cos11 uc66 c11 c33 c213 > > > > > > þ h21 h2 cos u sin10 uc66 ðc12 c33 c13 c23 Þ > > > > > þ 2h31 cos u sin10 uc66 c11 c33 c213 > > > > > > b12 ¼ h32 cos11 uc66 ðc12 c33 c13 c23 Þ z < 22 b 12 ¼ ð19cÞ þ h21 h2 cos u sin10 uc66 ðc11 c33 c213 Þ B> > 22 22 22 > > b ¼ b ¼ b ¼ 0 > 33 23 13 > > > 11 22 3 2 > b ¼ h sin uc > 66 c11 c33 c13 12 1 > > > > > þ h21 h2 cos10 u sin uc66 ðc12 c33 c13 c23 Þ > > : þ 2h32 cos10 u sin uc66 c13 c23 c213
þ
ðsin u=h2 Þs11 23
¼0
þ
ðsin u=h2 Þs22 23
¼0
þ h1 h22 cos6 u sin uðc12 c33 c13 c23 þ 2c33 c66 Þ
lt These blt ij and bij functions in Eqs. (16) and (19) will now be substituted to appropriate angle u for each of the elements X3 to X10 to obtain a solution for the entire honeycomb core. However, to calculate the effective elastic characteristics for the complete sandwich structure we also need to consider the top and bottom face carriers.
t12 s12 s12 3 ¼ ðcos u=h1 Þ 13 þ ðsin u=h2 Þ 23 ¼ 0 and t11 22 11 2 12 12 1 ¼ t 1 ¼ t 2 ¼ t 22 ¼ t 1 ¼ t 2 ¼ 0 t11 t22 s11 s22 3 ¼ 33 ¼ zc13 ; 3 ¼ 33 ¼ zc23 ;
t12 s12 3 ¼ 33 ¼ zc36 ð18bÞ
blt ij
ð19dÞ
local problems having The solution of the group been found, we then calculate the following functions:
4.2. Modeling of a unit-cell consisting of top/bottom face carrier Similar to the honeycomb core structure, the faces are made of generally orthotropic material. As such, we shall lt derive the U lt k ðn; zÞ and V k ðn; zÞ functions and hence
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
obtain the solution for the local problems of the group blt ij and blt ij (lt = 11, 22, 12) functions that appropriately represent the material orthotropic elastic behavior. We begin with the local unit-cell problems in Eq. (6) in a form: 1 oslt 1 oslt oslt i1 i2 þ þ i3 ¼ 0 h1 on1 h2 on2 oz 1 oU lt 1 oU lt oU lt lt 1 2 3 s11 ¼ c11 þ c12 þ c13 h1 h2 on1 on2 oz 1 oU lt 1 oU lt oU lt 1 2 3 slt c12 þ c22 þ c23 22 ¼ h1 h2 on1 on2 oz 1 oU lt 1 oU lt oU lt 1 2 3 slt ¼ c þ c þ c 13 23 33 33 h1 h2 on1 on2 oz 1 oU lt oU lt lt 3 2 s23 ¼ c44 þ h2 on2 oz lt lt 1 oU oU lt 3 1 s13 ¼ c55 þ h1 on1 oz 1 oU lt oU lt lt 1 2 s12 ¼ c66 þ h2 on2 on1
ð20Þ
s11 33 ¼ c13 ;
s22 33 ¼ c23 ;
The solution of the group blt ij local problems having been found, we then calculate the following functions: 8 c213 11 > > < b11 ¼ c11 c33 c23 c13 b11 ¼ b11 ð22aÞ > 22 ¼ c12 c33 > : 11 11 11 b33 ¼ b11 23 ¼ b13 ¼ b12 ¼ 0 8 22 b ¼ c12 c23c33c13 > > < 11 c223 ð22bÞ b22 ¼ b22 22 ¼ c22 c33 > > : 22 22 22 b ¼ b22 23 ¼ b13 ¼ b12 ¼ 0 ( 33 12 12 12 12 b12 11 ¼ b22 ¼ b33 ¼ b23 ¼ b13 ¼ 0 b12 ¼ ð22cÞ b12 12 ¼ c66 In a similar way we obtain the blt ij -type local problems from the governing unit-cell problem (6) and corresponding surface boundary conditions. The results are given below:
From Fig. 6, the above problem will be solved with the boundary conditions on surfaces {n1 = ±1/2, n2 = ±1/2 and z = ±t0/2}, and the surface boundary conditions must be satisfied on z = ±t0/2 and are lt slt 13 ¼ s23 ¼ 0;
1541
s12 33 ¼ 0
ð21Þ
1 oslt 1 oslt oslt i1 i2 þ þ i3 ¼ ci3lt h1 on1 h2 on2 oz lt 1 oV 1 1 oV lt oV lt 2 3 slt ¼ c þ c þ c 11 12 13 11 h1 h2 on1 on2 oz 1 oV lt 1 oV lt oV lt 1 2 3 slt ¼ c þ c þ c 12 22 23 22 h1 h2 on1 on2 oz 1 oV lt 1 oV lt oV lt slt c13 1 þ c23 2 þ c33 3 33 ¼ h1 h2 on1 on2 oz
Fig. 6. A local unit-cell of the sandwich structure in slow variables.
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G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
1 h2 1 lt s13 ¼ c55 h1 1 slt ¼ c 66 12 h2
slt 23 ¼ c44
oV lt oV lt 3 þ 2 on2 oz oV lt oV lt 3 þ 1 on1 oz oV lt oV lt 1 þ 2 on2 on1
4.3. Calculation of effective elastic coefficients of hexagonal honeycomb sandwich shell made of generally orthotropic material
The effective characteristics different from zero and contained in the equation of state (9), with consideration of relationships (10), we obtain
ð23Þ
with slt slt 13 ¼ 23 ¼ 0;
s11 33 ¼ zc13 ;
s22 33 ¼ zc23 ;
s12 33 ¼ 0
ð24Þ
blt ij
The solution of the group local problems having been found, we then calculate the following functions: 8 c213 11 > > > b11 ¼ c11 c33 < c23 c13 ð25aÞ b 11 ¼ z b11 22 ¼ c12 c33 > > > : 11 11 11 b33 ¼ b11 23 ¼ b13 ¼ b12 ¼ 0 8 22 b11 ¼ c12 c23c33c13 > > > < c223 b 22 ¼ z b22 ð25bÞ 22 ¼ c22 c33 > > > : 22 22 22 b33 ¼ b22 23 ¼ b13 ¼ b12 ¼ 0 ( 12 12 12 12 b11 ¼ b12 22 ¼ b33 ¼ b23 ¼ b13 ¼ 0 b 12 ¼ z ð25cÞ b12 12 ¼ c66 With this, we now have at our disposal all the information needed to calculate the effective elastic characteristics of the sandwich shell with hexagonal honeycomb core structure made of generally orthotropic material. In particular, the effective elastic coefficients will be determined in a straight-forward manner from Eqs. (16), (19), (22) and (25) by simple superposition from Eq. (8). The results are given in the following sub-section.
2E1 t0 E1 m21 Ht þ 1:1732 m12 a 1 m12 m21 2E2 t0 E2 Ht 22 hb22 i ¼ þ 0:5152 a 1 m12 m21 2m E t E1 m21 Ht 12 1 0 22 hb11 þ 0:3908 22 i ¼ hb11 i ¼ m12 a 1 m12 m21 E2 Ht 12 hb12 i ¼ 2G12 t0 þ 0:3908 a 2 2E t 3H 3Ht0 2 E1 m21 H 3 t 1 0 11 þ þ t0 þ 0:0976 hzb11 i ¼ m12 a 36ð1 m12 m21 Þ 4 2 2 2E t 3H 3Ht E2 H 3 t 2 0 0 2 þ þ t i ¼ þ 0:0429336 hzb22 22 0 a 36ð1 m12 m21 Þ 4 2 2 2m12 E1 t0 3H 3Ht0 2 22 þ þ t0 hzb11 22 i ¼ hzb11 i ¼ 36ð1 m12 m21 Þ 4 2
hb11 11 i ¼
E1 m21 H 3 t þ 0:03256 m12 a 2 2G12 t0 3H 3Ht0 2 E2 H 3 t hzb12 þ þ t i ¼ 12 0 þ 0:03256 a 36 4 2 ð26Þ It is imperative to note that in arriving the above set of effective elastic properties of the sandwich shell we needed to consider the appropriate cross-sectional areas and the inertia moments of the cross-sections of the each element (X1 to X10) participating in the sandwich structure, see [9]. Further, it should be noted that the effective
Fig. 7. A three-layered sandwich shell reinforced with hexagonal-triangular cellular core.
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
1543
Fig. 8. A periodicity cell in coordinate systems (a1, a2, c) and (n1, n2, z).
properties of this structure can be tailored to meet the requirements of a particular application by changing some geometric parameters such as thickness of the face carriers, length and cross-sectional areas of the core elements or the aspect ratio of the shell, the relative height of the core material, the angular orientation of the comprising elements, etc. This will be demonstrated in the subsequent Section 5, followed by a discussion on the obtained results in Section 6. 5. Sandwich shell reinforced with hexagonal-and-triangular mixed core A further sophisticated sandwich core structure involves a combination of two frequently used cell forms – hexagonal and triangular. It is therefore interesting to obtain explicit formulas for all effective elastic characteristics of a three-layered shell filled with hexagonal and triangular
shaped cellular core (Fig. 7) and made of generally orthotropic material. A representative periodicity cell of the structure is given in Fig. 8. On the basis of the approximate analytic solution of local problems (5) and (6), obtained in analogy with the previous case, for all effective characteristics of the shell different from zero and contained in the equation of state (9) (with consideration of relationship (10)), we have 2E1 t0 E1 m21 Ht þ 16:6276 m12 a 1 m12 m21 2E2 t0 E2 Ht 22 hb22 i ¼ þ 2:4250 a 1 m12 m21 2m E t E1 m21 Ht 12 1 0 22 hb11 þ 5:5424 22 i ¼ hb11 i ¼ m12 a 1 m12 m21 E2 Ht 12 hb12 i ¼ 2G12 t0 þ 5:5424 a hb11 11 i ¼
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2 2E1 t0 3H 3Ht0 2 E1 m21 H 3 t þ þ t0 þ 1:3856 m12 a 36ð1 m12 m21 Þ 4 2 2 2E2 t0 3H 3Ht0 2 E2 H 3 t þ þ t i ¼ hzb22 þ 0:2022 0 22 a 36ð1 m12 m21 Þ 4 2 2 2m12 E1 t0 3H 3Ht0 2 22 þ þ t0 hzb11 22 i ¼ hzb11 i ¼ 36ð1 m12 m21 Þ 4 2
hzb11 11 i ¼
E1 m21 H 3 t þ 0:4620 m12 a 2 2G t 3H 3Ht0 2 E2 H 3 t 12 0 hzb12 þ þ t i ¼ þ 0:4620 12 0 a 36 4 2
Variant I (H = 8, t0 = t = 0.1): N 1 ¼ dð0:3545E1 Þe11 þ dð0:1115E1 Þe22 þ Oðd3 Þ N 2 ¼ dð0:1115E1 Þe11 þ dð0:3661E2 Þe22 þ Oðd3 Þ M 1 ¼ d3 ð1:0912E1 Þs11 þ d3 ð0:3548E1 Þs22 þ Oðd4 Þ M 2 ¼ d3 ð0:3548E1 Þs11 þ d3 ð1:1543E2 Þs22 þ Oðd4 Þ
ð28Þ
N 12 ¼ dð0:2G12 þ 0:1251E2 Þe12 þ Oðd3 Þ M 12 ¼ d3 ð0:2734G12 þ 0:6668E2 Þs12 þ Oðd4 Þ Variant II (H = 20, t0 = t = 0.5): N 1 ¼ dð2:9221E1 Þe11 þ dð0:9401E1 Þe22 þ Oðd3 Þ
ð27Þ
N 2 ¼ dð0:9401E1 Þe11 þ dð3:0667E2 Þe22 þ Oðd3 Þ M 1 ¼ d3 ð72:5740E1 Þs11 þ d3 ð23:9151E1 Þs22 þ Oðd4 Þ
6. Results and discussions
M 2 ¼ d3 ð23:9151E1 Þs11 þ d3 ð77:5025E2 Þs22 þ Oðd4 Þ
It is noted that first terms in formulas (26) and (27) represent the contribution of the stress carrying face layers, while the second terms represent the contribution made by the honeycomb filler. Formulas (26) and (27) for the effective rigidity moduli (elastic portion of the characteristics) are in good agreement with the familiar relationships from the structural anisotropic theory of reinforced plates (when A1 = A2 = 1). The formula for torsional rigidity hzb12 12 i is beyond the scope of this theory. For the purposes of demonstration, the results from the numerical calculation of effective elastic moduli, estimated in Eq. (26), in two sample variants (with E1 = 138.0 GPa, E2 = 9 GPa, G12 = 6.9 GPa, t12 = 0.3, t21 = 0.0196, for unidirectional AS/3501 graphite/epoxy laminae) are given in Table 1. From Table 1, by using Eq. (11), and in combination with Eq. (9), it is now possible to calculate the forces, moments and torsional effects present in the hexagonal honeycomb sandwich structure. The results from two different variants presented in Table 1 are given below:
ð29Þ
N 12 ¼ dðG12 þ 1:5632E2 Þe12 þ Oðd3 Þ M 12 ¼ d3 ð8:7569G12 þ 52:0690E2 Þs12 þ Oðd4 Þ The derived general shell model will now be used to tailor the effective properties of the sandwich structures in Figs. 2 and 7 to meet the requirements of any particular application by modifying some geometric parameters. To this end, Figs. 9–11 illustrate the variation of effective elastic properties of sandwich shells reinforced with hexagonal and hexagonal-triangular core with respect to the height of the core, H. In Fig. 9, the structural stiffness in hb11 11 i direction is significantly higher than the stiffness in hb11 22 i direction for both shells, as expected. It is observed that the hexagonal-triangular core structure provides higher stiffness to the overall sandwich configuration than its hexagonal honeycomb counterpart. This may be a fact that the former combines an increased number of structural elements, hence making it far stiffer, than the latter.
Table 1 Effective elastic characteristics of hexagonal honeycomb-cored sandwich composite shells (A1 = A2 = 1, a = 2.5) Effective elastic moduli
Variants
The insertion of face carriers
hb11 11 i=E1
The insertion of honeycomb core
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2012 1.0059
0.1533 1.9162
hb22 22 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2012 1.0059
0.1649 2.0608
hb11 22 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.0604 0.3018
0.0511 0.6383
hb12 12 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2000(G12/E2) (G12/E2)
0.1251 1.5632
hzb11 11 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2750 8.8087
0.8162 63.7653
hzb22 22 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2750 8.8087
0.8793 68.6938
hzb11 22 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.0825 2.6426
0.2723 21.2725
hzb12 12 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2734(G12/E2) 8.7569(G12/E2)
0.6668 52.0960
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
nal-triangular structure has contributed to making the structure stiffer than its hexagonal counterpart. It is demonstrated that the effective properties of the studied structures can be tailored to meet the requirements of a particular application by changing the geometric parameters such as thickness of the carrier faces, length and cross-sectional area of the cellular core, etc.
11
11
/E1 , /E1
30
/E1, hexagonaltriangular core
25 20
/E1, hexagonaltriangular core
15
/E1, hexagonal core /E1, hexagonal core
10 5
7. Conclusions
0 8
9
10
11
12
13
14
15
16
17
18
19
20
H 11 Fig. 9. Effective elastic characteristics (hb11 11 i, hb22 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
*11
>/E1 ,
*11
>/E1
1000
/E1, hexagonaltriangular core
800 600
/E1, hexagonal core
400
/E1, hexagonaltriangular core
200
/E1, hexagonal core
0 8
9
10
11
12
13
14 H
15
16
17
18
19
20
11 Fig. 10. Effective elastic characteristics (hzb11 11 i, hzb22 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
6000
/G12, hexagonaltriangular core
*12
>/G12
5000
1545
4000 3000
/G12, hexagonal core
2000 1000 0 12
13
14
15
16
17
18
19
20
H
Fig. 11. Effective elastic characteristics (hzb12 12 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
In Fig. 10, it is seen that the torsional stiffnesses (hzb11 11 i and hzb11 iÞ for the mentioned structures takes into a stee22 per shape than their respective extensional stiffnesses (hb11 11 i and hb11 iÞ (Fig. 9). Finally, Fig. 11 shows the variation of 22 hzb12 i vs. H for the two structures. It is obvious that the 12 presence of additional number of members in the hexago-
The paper develops a micromechanical model pertaining to a general composite sandwich shell. Consideration is given on the application of a modified two-scale asymptotic homogenization technique applied to a rigorously formulated 3D elastic problem for a thin curvilinear periodically-inhomogeneous composite layer made of generally orthotropic material. Orthotropy of the constituent materials leads to much more complex unit-cell problems and is considered in the present paper for the first time. Thanks to the presence of the dimensionless small parameter d representing the thickness of the shell, the original 3D problem then proves to be amenable to a rigorous asymptotic analysis unifying an asymptotic three-to-two dimensions process and a homogenization composite material-homogeneous material process. The explicit expressions called unit-cell problems that are used to determine the effective elastic characteristics of the shell structure are presented. In particular, it is shown that the effective stiffnesses of the homogenized shell generally depend on the local elastic constants of the material. This dependence is illustrated by application of the general model to a hexagonal honeycomb sandwich shell made of generally orthotropic material. The explicit formulas for a rotational member, which when substituted with appropriate angle of the individual element generates the complete expression for the structure, comprising the sandwich core is obtained. Theory is further illustrated by introducing a further complex problem involving a hexagonal-triangular mixed core in the sandwich shell. Through the use of numerical illustrations, it is shown that the effective elastic properties of the structure overwhelmingly depend on the geometric configuration of the stiffeners. It is noted that even though the analysis presented in the paper is applied to elastic materials, the model derived should be considered to hold equally well for structures exhibiting viscoelastic, piezoelectric or magnetostrictive characteristics, or are associated with some general transduction characteristics which can be used to induce strains and stresses in some coordinated fashion.
Acknowledgement The support by the Natural Sciences and Engineering Council of Canada (NSERC) is gratefully acknowledged.
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Appendix It is noted that in formulating the unit-cell problems for a rotated element of the sandwich honeycomb structure we have written the governing differential equations and associated surface boundary conditions in terms of the global coordinates. However, it makes practical sense to give the final closed-form expressions in Eqs. (26) and (27) in terms of the familiar material constants (Young’s moduli, shear moduli and Poisson’s ratios). To do that, we first need to relate the off-axis coefficients, cijkl, with the principal ðmÞ material coefficients, cijkl . This is achieved through the tensor transformation equation for a fourth-order tensor, i.e., cijkl ¼ aim ajn akp alq cðmÞ mnpq
ð30Þ
When expanded for an orthotropic material, Eq. (30) becomes, see [13]: ðmÞ ðmÞ ðmÞ c11 ¼ c11 cos4 u þ 2 c12 þ 2c66 cos2 u sin2 u ðmÞ
þ c22 sin4 u
ðmÞ ðmÞ ðmÞ ðmÞ c12 ¼ c12 cos4 u þ c11 þ c22 4c66 cos2 u sin2 u ðmÞ
þ c12 sin4 u ðmÞ
ðmÞ
c13 ¼ c13 cos2 u þ c23 sin2 u ðmÞ ðmÞ ðmÞ c16 ¼ c11 þ c12 2c66 cos3 u sin u ðmÞ ðmÞ ðmÞ þ 2c66 þ c12 2c22 cos u sin3 u ðmÞ ðmÞ ðmÞ c22 ¼ c22 cos4 u þ 2 c12 þ 2c66 cos2 u sin2 ua ðmÞ
þ c11 sin4 u ðmÞ
ðmÞ
c23 ¼ c23 cos2 u þ c13 sin2 u ðmÞ ðmÞ ðmÞ c26 ¼ c12 c22 þ 2c66 cos3 u sin u ðmÞ ðmÞ ðmÞ þ c11 c12 2c66 cos u sin3 u ðmÞ
c33 ¼ c33
ðmÞ ðmÞ c36 ¼ c13 c23 cos u sin u ðmÞ ðmÞ ðmÞ ðmÞ c66 ¼ c11 þ c22 2c12 2c66 cos2 u sin2 u ðmÞ þ c66 cos4 u þ sin4 u ð31Þ ðmÞ
Subsequently, the principal material coefficients, cijkl , are related to the material constants, E1, E2, G12, t12, etc., through well-known expressions, see for example [3]. References [1] Bensoussan A, Lions JL, Papanicolaou G. Asymptotic analysis for periodic structures. Amsterdam: North-Holland; 1978. [2] Cioranescu D, Donato P. An introduction to homogenization. Oxford lecture series in mathematics and its applications, vol. 17. London: Oxford University Press; 2000. [3] Kalamkarov AL. Composite and reinforced elements of construction. Chichester: Wiley; 1992. [4] Duvaut G. Analyse fontionnelle et me´chanique des milieux continues. Application a` l’e´tude des materiaux composites e´lastiques a structure pe´riodique-homoge´ne´isation. In: Proceedings of Theor and Appl Mech Prepr 14th IUTAM Congress. Delft, 1976. p. 119–32. [5] Kalamkarov AL. On the determination of the effective properties of network plates and shells with periodic structure. Izv Akad Nauk SSSR (Mekh Tv Tela) 1987(2):181–5. [6] Caillerie D. Thin elastic and periodic plates. Math Meth Appl Sci 1984;6:159–91. [7] Panasenko GP, Reztsov MV. Homogenization of the three-dimensional elasticity problem for a homogeneous plate. Dokl Akad Nauk SSSR 1987;294:1061–5. [8] Kohn RV, Vogelius M. A new model for thin plates with rapidly varying thickness. Int J Solids Struct 1984;20:333–50. [9] Kalamkarov AL, Saha GC, Georgiades AV. General micromechanical modeling of smart composite shells with application to smart honeycomb sandwich structures. Compos Struct 2007;79(1):18–33. [10] Timoshenko S. Theory of plates and shells. New York: McGrawHill Book Co., Inc; 1940. [11] Kalamkarov AL. The thermal conductivity of a twisted nonuniform anisotropic layer of periodic structure with wavy surfaces. JEP 1987;52(5). [12] Kolpakov AG. The determination of averaged characteristics for elastic skeletons. PMM 1985;49(6). [13] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton: CRC Press; 1997.
Effective elastic characteristics of honeycomb sandwich composite shells made of generally orthotropic materials Gobinda C. Saha, Alexander L. Kalamkarov *, Anastasis V. Georgiades Department of Mechanical Engineering, Dalhousie University, P.O. Box 1000, Halifax, NS, Canada B3J 2X4 Received 6 July 2006; accepted 3 January 2007
Abstract This paper works on the analytical development of the method of two-scale asymptotic homogenization. The technique is used to determine the effective elastic stiffnesses of hexagonal honeycomb-cored structural sandwich composite shells made of generally orthotropic materials. Orthotropy of the constituent materials leads to much more complex unit-cell problems and is considered in the present paper for the first time. At first, a 3D-to-2D general shell model based on a set of unit-cell problems is derived. This is followed by the exploitation of the model to the derivation of analytical estimate formulae; used to calculate the force and moment resultants present in the sandwich shell structure. The implication of the general shell model is further indicated by calculating similar design characteristics for a three-layered composite sandwich panel reinforced with hexagonal and triangular shaped cellular core made from the generally orthotropic material. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Material; B. Property; C. Analysis
1. Introduction The preponderance of uses for composite materials is in the form of plates and shells the optimum strength-toweight characteristics of which offer engineers attractive alternatives for different applications. A particular class of these structures, the structural sandwich shells and panels, find increasing use in lightweight construction, aerospace and automotive engineering, marine architecture and many other platforms. A typical structural sandwich element is a layered medium consisting of two high-density, highstrength face carriers bonded to a thick core made from a low-density material. Usually, the core is constructed from a foam material with a hexagonal or honeycomb configuration, while anisotropic laminae are commonly used as face carriers. The presence of the thick core means that the deformation characteristics of a sandwich shell are different
*
Corresponding author. Tel.: +1 902 494 6072; fax: +1 902 423 6711. E-mail address: [email protected] (A.L. Kalamkarov).
1359-835X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2007.01.002
from those of a classical laminated shell or a monolayer structure. Although the contribution from the core to the overall weight of the sandwich structure is minimal, the moment of inertia of the structure increases substantially by the separation of the face carriers from the core, thanks to the presence of the thick core. On the other hand, due to the limited flexural rigidity of the low-density core, the influences of the local phenomena, such as local instability, loadinduced thickness reduction, etc., on the global phenomena, such as stability, deformation, etc., have to be taken into account when designing with sandwich structures. Hence, micromechanical models dealing with sandwich shells should place proper emphasis on the importance of the core’s mechanical behavior. Classical analytical methods were derived for standard structures such as beams, plates and shells only under specific loading and boundary conditions. The development of enhanced models that can effectively determine the stresses, deformations and critical loads under general conditions is highly desirable. With the knowledge of the effective deformation, stress and stability performance of such structure, design parameters can then
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be optimized with respect to minimum weight and flexural and shear criteria of the sandwich structure. The incorporation of sandwich composite shells in new engineering applications will be facilitated if their properties are determined at the design stage. The micromechanical models that represent these shell structures will be considered as efficient provided that they are neither too complex to describe and use, nor too simple to reflect the ‘real’ properties and characteristics of the composite structure. One technique that has been employed in the modeling of composites with a periodic structure is the ‘asymptotic homogenization’ method. The mathematical framework of the method of asymptotic homogenization can be found in [1–3]. The premise of the method is based on the fact that many physical problems are characterized by some variables that exhibit a slow variation and some others that change rapidly. The coupling of the slow and fast variables renders the direct analytical and even numerical solution of the problem extremely difficult. Asymptotic homogenization however decouples the two sets of variables and solves them separately. In this paper, we intend to develop asymptotic homogenization model pertaining to sandwich composite thin shell made of generally orthotropic material. Duvaut [4] and Kalamkarov [5] have developed homogenized models of plates with periodical non-homogeneities in tangential coordinate(s). Their work led to a comprehensive definition of a single ‘periodicity cell’, wherein the elastic characteristics of the layer material and the shape of the upper and lower surfaces of the shell structure are assumed to be regularly periodic. Consequently, a unit-cell approach which, in fact, is based upon the notion of a regular periodic structure allows us to resort the method of averaging (or ‘homogenization’) in the design of thin composite shells. Furthermore, structural shells such as three-layered sandwich shells filled with a honeycomb structured core (or honeycomb shells) are neither one-dimensional nor twodimensional composites (such as classical laminated shells or fiber reinforced composites), nor are they three-dimensional composites (such as granulated materials). Periodicity is exhibited only in the two tangential coordinates introduced at the middle surface of the shell, with no such periodicity existing in the transverse coordinate. These features in terms of both period and shell thickness are commensurate in the present modeling and hence the force boundary conditions are specified at the upper and lower surfaces of the shell. The inherent features of the problem definition require a refined asymptotic analysis approach for a 3D thin shell problem which would allow the asymptotic transition from a 3D problem to the 2D shell problem. The technique, which is essentially a modified asymptotic approach developed by Caillerie [6], was used by Panasenko and Reztsov [7] to obtain a complete asymptotic expansion of a 3D problem in the theory of elasticity for a thin plate with a thickness equal to the characteristic dimension of the non-uniformities. Kohn and Vogelius [8] used a similar approach to analyze the 3D problem of a
homogeneous thin uniform plate with a rapidly oscillating thickness asymptotically. In this micromechanical modeling approach, the modified asymptotic homogenization technique is employed to study the elastic characteristics of non-uniform curved thin composite shells with regular structure and wavy surface. The exact formulation of the 3D shell problem associated with the elasticity problem is derived in Section 2. Befitting with the characteristic dimension of the non-uniformities, d, the derived 3D general shell model is amenable to a rigorous asymptotic analysis unifying an asymptotic 3D-to-2D process and a material homogeneous process. The general shell model in terms of a set of problems, called the ‘unit-cell’ problems, is dependent only on the nature of the local periodicity of the unit-cell and is independent of the global formulation of the composite shell structure. These unit-cell problems will generate the so-called effective elastic properties which replace the actual rapidly varying properties, and are in fact constant. Section 3 describes the homogenization procedure through which the effective elastic coefficients are actually obtained. In the second part of the study, the effectiveness of the general model is indicated by considering several applications with respect to structural non-uniform shells. Based on the solution of the local problems, explicit formulae are derived for elastic characteristics of hexagonal and hexagonal-triangular honeycomb sandwich shells of regular structure in Sections 4 and 5, respectively, made of generally orthotropic material. In Section 6, we continue our discussion through the use of numerical illustrations, based on the averaged property formulae obtained in Sections 4 and 5. A comparison is made between the two adopted geometries. Finally, Section 7 concludes the paper. It is noted that to the best of authors’ knowledge, this work represents the first attempt to model composite sandwich shells composed of entirely orthotropic constituents. As well, when an orthotropic material is rotated by an angle u about one its axes of orthotropy, the resulting stiffness tensor has the same format as a monoclinic material. The novel model developed in this work will consider orthotropic honeycomb core constituents which are not referenced with respect to their principal material directions, i.e., will exhibit monoclinic-like characteristics. 2. Problem formulation Consider an inhomogeneous shell of periodic structure, as illustrated in Fig. 1. The general shell structure is obtained by repeating a periodicity cell Xd, which is specified in the orthogonal dimensionless coordinate system (a1, a2, c) by the inequalities {dh1/2 < a1 < dh1/2, dh2/2 < a2 < dh2/2, c < c < c+}. The dimensionless small parameter d specifies the thickness of the shell, as well as the scale of the composite material inhomogeneity is assumed to be small when compared with the global dimensions of the solid. Here
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Fig. 1. A curvilinear reinforced composite layer with representative periodicity cell Xd.
h1, h2 are the ratios of the tangential dimensions of the shell to the thickness. The elasticity problem of this shell structure is characterized by means of the following boundary value problem: orij fi ¼ 0; oaj rij ¼ cijkl ekl ;
where 1 ouk oul ekl ¼ þ ; and 2 oal oak
ð1Þ
In Eq. (1), cijkl represents the fourth-order elasticity tensors of the materials involved and is periodic in a1 and a2 directions only with respective periods dh1 and dh2. Functions fi, pi, uk represent body forces, surface tractions and displacement fields, respectively, whereas nj is the unit normal vector pertaining to lateral wavy surfaces c= S ða1 ; a2 Þ of the shell and is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 oS oS oS =oa1 þ oS =oa2 þ 1 n ¼ ; ;1 oa1 oa2 ð2Þ We introduce the following notations: z = c/d, n = (n1, n2), a = (a1, a2). We also make the assumption that the elasticity coefficients cijkl(n, z) are piecewise smooth periodic functions for n1, n2 anywhere in the periodicity cell Xd: {n1, n2 2 (1/2, 1/2), z 2 (z, z+)}, exhibiting discontinuities of the first kind at a finite number of non-intersecting contact surfaces. Following the method of two-scale asymptotic expansions outlined in [9], we present the components of the displacement and stress vectors as follows: ð1Þ
ð2Þ
ui ða; n; z; dÞ ¼ ui ðaÞ þ dui ða; n; zÞ þ d2 ui ða; n; zÞ þ ð0Þ
z owðaÞ 2 lt 3 þ dU lt 1 elt þ d V 1 slt þ Oðd Þ A1 oa1 z owðaÞ 2 lt 3 þ dU lt u2 ¼ v2 ðaÞ d 2 elt þ d V 2 slt þ Oðd Þ A2 oa2 2 lt 3 u3 ¼ wðaÞ þ dU lt 3 elt þ d V 3 slt þ Oðd Þ u1 ¼ v1 ðaÞ d
rij nj ¼ pi
ð0 Þ
It has been demonstrated in [9] that for the principal terms in Eq. (3) and in the corresponding expansions for the components of the stress tensor over the small parameter d, the relationships that determine the local structure of the displacement and stress fields are
lt rij ¼ blt ij elt þ dbij slt
ð4Þ Here and in the sequel, summation is performed over identical indices, with the Latin indices taking on values of 1–3, while the Greek indices take on values of 1 and 2. A1(a), A2(a) are the coefficients of the first quadratic form of the shell middle surface (c = 0). A comparison of results in Eq. (4) with the classical general thin shell theory [10, page 439] results shows that functions v1(a), v2(a) are the displacements of a point in the directions of the tangent to the meridian and function w(a) is the displacement in the direction of the normal to middle surface of the shell element at that point. In the calculations we consider theses displacements to be very small and associate them with the displacement components z{w(a)/a1} and z{w(a)/a2} that account for rotation of the plane section around this middle surface. Strain functions e11 = e1, e22 = e2, e12 = e21 = e/2 represent the tensile and shearing strains in the middle surface; s11 = k1, s22 = k2, s12 = s21 = s characterize the flexural and torsional strains of the middle surface. In the last expression of Eq. (4), the following definitions are used: 1 oU lt oU lt k k þ cijlt and þ cijk3 hb onb oz 1 oV lt oV lt k ¼ cijkb þ cijk3 k þ zcijlt hb onb oz
blt ij ¼ cijkb
ð1 Þ
rij ða; n; z; dÞ ¼ rij ða; n; zÞ þ drij ða; n; zÞ ð2Þ
þ d2 uij ða; n; zÞ þ ð3Þ
blt ij
ð5Þ
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lt The functions U lt k and V k contained in relationships (4) and (5) depend on n1 = A1n1, n2 = A2n2 and z. With respect to n1 and n2, they represent the periodic solutions (with respective periods A1 and A2) of the following local problems on the periodicity cell Xd
Relationship (9) represents the equation of state for the averaged shell, while the coefficients of this relationship represent its effective elastic characteristics. On the basis of Eqs. (4)–(8), in arriving relationship (9), the following has been demonstrated:
1 o lt o lt b þ b ¼ 0 with hb onb ib oz i3 1 lt lt bib N b þ bi3 N 3 ¼0 hb z¼z
r lt hzr blt i3 i ¼ hz bi3 i ¼ 0
and 1 o lt o lt b þ bi3 ¼ 0 with hb onb ib oz 1 lt lt bib N b þ bi3 N 3 ¼0 hb z¼z
bk hblt bk i ¼ hblt i;
ð6Þ
where N i represents the component vectors normal to the surfaces z = z±. The problems expressed by Eq. (6), together with Eq. (5), respectively, are called the unit-cell local problems for the thin inhomogeneous shell structure illustrated in Fig. 1. With this, what we have achieved is that of the homogenized (because they involve only the derivatives with respect to rapid variables n and z) periodlt icity cell local functions U lt k ðn; zÞ and V k ðn; zÞ which are 1periodic in n1 and n2 and determine, in turn, the coefficients lt blt in Eq. (4). These local functions help us in ij and bij ð1Þ obtaining the first term in the stress function rij . It should be noted that, unlike the unit-cell problems of ‘classical’ homogenization schemes, those set by Eq. (6) depend on the boundary conditions z = z± rather than on periodicity in the z-direction. 3. Homogenization of unit-cell problems In the case of material discontinuity due to the presence of discontinued surfaces along the shell structure we add the following continuity conditions to the unit-cell problems in Eq. (6):
nb lt lt lt ½U k ¼ 0; b þ n3 bi3 ¼ 0 and hb ib
ð7Þ nb lt lt lt ½V k ¼ 0; b þ n3 bi3 ¼ 0 hb ib where ni is the component normal to the discontinuity surface. Local problems (5)–(7) have single solutions accurate lt to the constant terms blt ij and bij . This non-uniqueness is removed by introducing the operation of the averaging over the volume of the periodicity cell, which is defined on an arbitrary coefficient w by its integral as Z 1 wdn1 dn2 dz ð8Þ hwi ¼ jXd j Xd Averaging relationships in Eq. (4) by means of definition (8) over the volume Xd, we obtain (r = 0, 1): 2 r lt hzr rij i ¼ hzr blt ij ielt þ dhz bij islt þ Oðd Þ
ð9Þ
bk hzblt bk i ¼ hblt i;
bk hzblt bk i ¼ hzblt i
ð10Þ
Relationships (10) provide for the symmetry of the matrices comprised of the coefficients of the equations of state for the averaged shell. It is worth noting that, as demonstrated in [11], in the case of a uniform material and with constant shell thickness (z 2 (1/2, 1/2)) this average shell model reduces to the relationships taken from the theory of the elasticity of anisotropic shells, with the following formulas valid for the linear forces and the bending and torsional moments: N 1 ¼ dhr11 i; 2
M 1 ¼ d hzr11 i; N 12 ¼ dhr12 i;
N 2 ¼ dhr22 i; M 2 ¼ d2 hzr22 i;
ð11Þ
2
M 12 ¼ d hzr12 i
In summarizing, the way the above model is presented, the local problems (6) are completely determined by the structure of the unit-cell of the composite shell and are fully independent of the formulation of the global boundary value problem, set in Eq. (1). It follows that the solution of these local problems and, in particular, the effective elastic coefficients of the homogenized composite shell can be utilized in studying various types of boundary value problems associated with a given composite structure. In the rest of the paper, we shall dwell in some detail on certain applications of the general shell model to design the structural non-uniform shells of regular structure, these having been fabricated out of a uniform orthotropic material. 4. Sandwich shell with hexagonal honeycomb core To obtain the effective characteristics that appear in the elasticity relations involving problems from the theory of plates and shells, also present in the current solution of local problems in Eqs. (5) and (6), a method of approximation was proposed and verified by Kolpakov [12]. In that, the effective characteristics of small-cell skeletal constructions of periodic structure were studied. In combination with the above-mentioned general model of an averaged shell, this method makes it possible to derive explicit analytical estimate formulae and obtain effective characteristics for a large number of reinforced shells, such as those used in actual practice, with adequate accuracy. We examine the three layered shell reinforced with hexagonal honeycomb type sandwich filler with a periodicity cell consisting of ten individual elements (Fig. 2). As the governing equations of basic mechanics in relation with membrane and flexural rigidities can be simplified by assumptions typically fulfilled by sandwich structures,
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Fig. 2. A three-layered sandwich shell reinforced with hexagonal-honeycomb filler.
the approximate analytical solution of the local problems (5) and (6) for a unit-cell of the relevant kind of geometry is found under the assumption that the thickness of each of the ten unit-cell elements is small in comparison with the other two dimensions, i.e., under the conditions t1 h1, t2 h2, h1, h2 H. The two face carriers are identical and symmetrically arranged with respect to the centroid of the sandwich structure. Both face carriers and hexagonal core of this sandwich composite structure is made of generally orthotropic material. For the sake of computational simplicity, we confine our attention to the case when both coefficients assigned to describe the quadratic form of the shell middle surface are made from unity, i.e., A1 = A2 = 1. For the case of structure shown in Fig. 2, the presence of rotated elements necessitates the use of coordinate systems which are at an angle with respect to the principal coordinate systems. Consequently, even though the structure is made from orthotropic material, these rotated elements will exhibit monoclinic-like behavior when they are not referred to with respect to their principal material coordinates. The appropriate elastic material tensor for monoclinic material is given below: 2
c11 6c 6 12 6 6 c13 6 6 0 6 6 4 0 c16
3
c12
c13
0
0
c16
c22
c23
0
0
c23 0
c33 0
0 c44
0 c45
0 c26
0 c36
c45 0
c55 0
c26 7 7 7 c36 7 7 0 7 7 7 0 5 c66
ð12Þ
For the same reason, after transformation of the coordinates from (a1, a2) to (n1, n2) according to the rules of transformation of coordinates pffiffiffiffiffi pffiffiffi h1 ¼ 3a; h2 ¼ 3a; t1 ¼ 2t= 30a ; pffiffiffi pffiffiffiffiffi ð13Þ t2 ¼ t= 3a ; sin u ¼ 3= 10 it is obvious from the periodicity unit-cell (see Fig. 3) that we first deal with a simpler type of unit-cell problem which consists only of a single element rotated at an arbitrary angle u with respect to the n2 axis. This rotated unit-cell is shown in Fig. 4. Once the solution for this rotated unit-cell is obtained, it is thus a matter of simple superposition which will allow us to compute the solution for the complete honeycomb core pertaining to appropriate angle u for participating elements X3 to X10. 4.1. Single-element rotated unit-cell We consider that the element is located along one of the coordinate axes, say g2. This alignment will essentially lead to the reduction of the order of the associated differential equations by making the problem independent of g2. The lt functions U lt k ðn; zÞ and V k ðn; zÞ will then depend on the variables g1 and z, and the local problems of the group lt blt (lt = 11, 22, 12) reduce to the determination ij and bij lt of the functions U lt 2 ðg1 ; zÞ, U 3 ðg1 ; zÞ, (lt = 11, 22) and 12 U 3 ðg1 ; zÞ from the following system of equations:
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a
b
c
Fig. 3. (a) Hexagonal core in 2-D; (b) a periodicity unit-cell; and (c) cross-section.
slt 22
slt 33
slt 23 slt 13 slt 12
c12 cos u c26 sin u oU lt 1 ¼ þ h1 h2 og1 c26 cos u c22 sin u oU lt oU lt 2 3 þ þ þ c23 h1 h2 og1 oz c13 cos u c36 sin u oU lt 1 ¼ þ h1 h2 og1 c36 cos u c23 sin u oU lt oU lt 2 3 þ þ þ c33 h1 h2 og1 oz oU lt oU lt c45 cos u c44 sin u oU lt 1 2 3 þ c44 þ ¼ c45 þ h1 h2 oz oz og1 oU lt oU lt c55 cos u c45 sin u oU lt 1 2 3 þ c45 þ ¼ c55 þ h1 h2 oz oz og1 c16 cos u c66 sin u oU lt 1 ¼ þ h1 h2 og1 c66 cos u c26 sin u oU lt oU lt 2 3 þ þ þ c36 h1 h2 og1 oz ð14Þ
Fig. 4. A rotated unit-cell comprising the honeycomb cellular core.
cos u oslt sin u oslt oslat i1 i2 þ þ i3 ¼ 0 h1 og1 h2 og1 oz lt c cos u c sin u oU 1 11 16 slt þ 11 ¼ h h2 og1 1 c16 cos u c12 sin u oU lt oU lt 2 3 þ þ þ c13 h1 h2 og1 oz
subject to the boundary conditions on outer surfaces (see Fig. 5) g1 = ±t1/2 and z = (0, H): 11 11 t11 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12 ¼ ðcos u=h1 Þc11 ðsin u=h2 Þc16 22 22 t22 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12
¼ ðcos u=h1 Þc12 ðsin u=h2 Þc26
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Fig. 5. Specification of boundary conditions on surfaces g1 = ±t1/2 and z = (0, H).
12 12 t12 1 ¼ ðcos u=h1 Þs11 þ ðsin u=h2 Þs12
¼ ðcos u=h1 Þc16 ðsin u=h2 Þc66 11 11 t11 2 ¼ ðcos u=h1 Þs12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc16 ðsin u=h2 Þc12 22 22 t22 2 ¼ ðcos u=h1 Þs12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc26 ðsin u=h2 Þc22 t12 2
12 ¼ ðcos u=h1 Þs12 12 þ ðsin u=h2 Þs22
¼ ðcos u=h1 Þc66 ðsin u=h2 Þc26 t11 3
11 ¼ ðcos u=h1 Þs11 13 þ ðsin u=h2 Þs23 ¼ 0
22 22 t22 3 ¼ ðcos u=h1 Þs13 þ ðsin u=h2 Þs23 ¼ 0 12 12 t12 3 ¼ ðcos u=h1 Þs13 þ ðsin u=h2 Þs23 ¼ 0
ð15aÞ and
8 cos2 u sin2 u > c66 ðc213 c22 c11 c223 c212 c33 b22 > 11 ¼ > h21 h22 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > 22 > b22 ¼ cos44 u c66 ðc213 c22 c11 c223 c212 c33 > h1 1< ð16bÞ b22 ¼ þ c11 c22 c33 þ 2c12 c13 c23 Þ A> > > 22 22 22 >b ¼ b ¼ b ¼ 0 > 33 23 13 > > > > cos3 u sin u 22 > b12 ¼ h3 h c66 ðc213 c22 c11 c223 c212 c33 > > > 1 2 > : þ c11 c22 c33 þ 2c12 c13 c23 Þ 8 cos u sin3 u 12 2 2 2 > > > b11 ¼ h1 h32 c66 ðc13 c22 c11 c23 c12 c33 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > 12 cos3 u sin u > > b22 ¼ h3 h2 c66 ðc213 c22 c11 c223 c212 c33 1 1< b12 ¼ ð16cÞ þ c c 11 22 c33 þ 2c12 c13 c23 Þ > A> > 12 12 12 > b33 ¼ b23 ¼ b13 ¼ 0 > > > > > cos2 u sin2 u > b12 c66 ðc213 c22 c11 c223 c212 c33 > 12 ¼ > h21 h22 > > : þ c11 c22 c33 þ 2c12 c13 c23 Þ
22 11 2 12 12 t11 1 ¼ t 1 ¼ t 2 ¼ t 22 ¼ t 1 ¼ t 2 ¼ 0 11 t11 3 ¼ s33 ¼ c13 ;
22 t22 3 ¼ s33 ¼ c23 ;
12 t12 3 ¼ s33 ¼ c36
where
ð15bÞ A¼ The solution of the group blt ij local problems having been found, we then calculate the following functions: 8 sin4 u 11 2 2 2 > > > b11 ¼ h42 c66 ðc13 c22 c11 c23 c12 c33 > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > cos2 u sin2 u 11 > > b22 ¼ h2 h2 c66 ðc213 c22 c11 c223 c212 c33 1 2 1< ð16aÞ b11 ¼ þ c c22 c33 þ 2c12 c13 c23 Þ 11 > A> > 11 11 11 > b33 ¼ b23 ¼ b13 ¼ 0 > > > > > cos u sin3 u > b11 c66 ðc213 c22 c11 c223 c212 c33 > 12 ¼ > h1 h32 > > : þ c11 c22 c33 þ 2c12 c13 c23 Þ
! cos12 u sin12 u þ c66 c11 c33 c213 4 4 h1 h2
ð16dÞ
A similar situation exists with the group blt ij (lt = 11, 22, 12) local problems, the two-dimensional forms of which are given as below: cos u oslt sin u oslt oslt i1 i2 þ þ i3 ¼ ci3lt h1 og1 h2 og1 oz lt c cos u c sin u oV 1 11 16 lt s11 ¼ þ h1 h2 og 1 c16 cos u c12 sin u oV lt oV lt 2 þ þ þ c13 3 h1 h2 og1 oz
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c12 cos u c26 sin u oV lt 1 þ h1 h2 og1 c26 cos u c22 sin u oV lt oV lt 2 þ þ þ c23 3 h1 h2 og1 oz lt c13 cos u c36 sin u oV 1 ¼ þ h1 h2 og1 c36 cos u c23 sin u oV lt oV lt 2 þ þ þ c33 3 h1 h2 og1 oz lt lt oV 1 oV 2 c45 cos u c44 sin u oV lt 3 þ c44 þ ¼ c45 þ h1 h2 oz oz og1 oV lt oV lt c55 cos u c45 sin u oV lt 1 2 3 þ c45 þ ¼ c55 þ h1 h2 oz oz og1 c16 cos u c66 sin u oV lt 1 ¼ þ h1 h2 og1 c66 cos u c26 sin u oV lt oV lt 2 þ þ þ c36 3 h1 h2 og1 oz
slt 22 ¼
slt 33
slt 23 slt 13 slt 12
ð17Þ with the appropriate conditions on surfaces g1 = ±t1/2 and z = (0, H): t11 s11 s11 1 ¼ ðcos u=h1 Þ 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc11 zðsin u=h2 Þc16 t22 1
¼ ðcos u=h1 Þs22 s22 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc12 zðsin u=h2 Þc26
t12 s12 s12 1 ¼ ðcos u=h1 Þ 11 þ ðsin u=h2 Þ 12 ¼ zðcos u=h1 Þc16 zðsin u=h2 Þc66 t11 s11 s11 2 ¼ ðcos u=h1 Þ 12 þ ðsin u=h2 Þ 22 ¼ zðcos u=h1 Þc16 zðsin u=h2 Þc12 t22 2
¼
ðcos u=h1 Þs22 12
þ
ð18aÞ
ðsin u=h2 Þs22 22
where B ¼ h31 sin7 u c11 c33 c213 þ h32
cos6 u sin u c11 c33 c213 þ c12 c33 c13 c23 þ 2c33 c66
¼ zðcos u=h1 Þc26 zðsin u=h2 Þc22 t12 s12 s12 2 ¼ ðcos u=h1 Þ 12 þ ðsin u=h2 Þ 22 ¼ zðcos u=h1 Þc66 zðsin u=h2 Þc26 t11 3 t22 3
¼
ðcos u=h1 Þs11 13
¼
ðcos u=h1 Þs22 13
8 11 b ¼ h31 sin11 uðc213 c22 c11 c223 c212 c33 > > > 11 > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h1 h22 cos10 u sin uc66 c11 c33 c213 > > > > > 11 2 10 > > > b22 ¼ 2h1 h2 cos u sin uc66 ðc12 c33 c13 c23 Þ > < 11 11 11 z b33 ¼ b23 ¼ b13 ¼ 0 ð19aÞ b 11 ¼ B> > b11 ¼ h31 cosu sin10 uðc213 c22 c11 c223 12 > > > > > c212 c33 þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h31 cos u sin10 uc66 ðc11 c33 c213 > > > > > þ c12 c33 c13 c23 Þ > > : 2h21 h2 cos u sin10 uc66 ðc12 c33 c13 c23 Þ 8 22 b ¼ 2h21 h2 cosu sin10 uc66 ðc12 c33 c13 c23 Þ > > > 11 > 3 11 2 2 2 > b22 > 22 ¼ h2 cos uðc13 c22 c11 c23 c12 c33 > > > > > þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > þ 2h21 h2 cos u sin10 uc66 c11 c33 c213 > > > 22 22 z < b22 33 ¼ b23 ¼ b13 ¼ 0 b 22 ¼ ð19bÞ > b22 ¼ h3 cos10 u sin uðc2 c22 c11 c2 B> 12 2 13 23 > > > > c212 c33 þ c11 c22 c33 þ 2c12 c13 c23 Þ > > > > > > þ 2h32 cos10 u sin uc66 ðc11 c33 c213 > > > > > þ c12 c33 c13 c23 Þ > > : 2h1 h22 cos10 u sin uc66 ðc12 c33 c13 c23 Þ 8 12 b11 ¼ h32 cos11 uc66 c11 c33 c213 > > > > > > þ h21 h2 cos u sin10 uc66 ðc12 c33 c13 c23 Þ > > > > > þ 2h31 cos u sin10 uc66 c11 c33 c213 > > > > > > b12 ¼ h32 cos11 uc66 ðc12 c33 c13 c23 Þ z < 22 b 12 ¼ ð19cÞ þ h21 h2 cos u sin10 uc66 ðc11 c33 c213 Þ B> > 22 22 22 > > b ¼ b ¼ b ¼ 0 > 33 23 13 > > > 11 22 3 2 > b ¼ h sin uc > 66 c11 c33 c13 12 1 > > > > > þ h21 h2 cos10 u sin uc66 ðc12 c33 c13 c23 Þ > > : þ 2h32 cos10 u sin uc66 c13 c23 c213
þ
ðsin u=h2 Þs11 23
¼0
þ
ðsin u=h2 Þs22 23
¼0
þ h1 h22 cos6 u sin uðc12 c33 c13 c23 þ 2c33 c66 Þ
lt These blt ij and bij functions in Eqs. (16) and (19) will now be substituted to appropriate angle u for each of the elements X3 to X10 to obtain a solution for the entire honeycomb core. However, to calculate the effective elastic characteristics for the complete sandwich structure we also need to consider the top and bottom face carriers.
t12 s12 s12 3 ¼ ðcos u=h1 Þ 13 þ ðsin u=h2 Þ 23 ¼ 0 and t11 22 11 2 12 12 1 ¼ t 1 ¼ t 2 ¼ t 22 ¼ t 1 ¼ t 2 ¼ 0 t11 t22 s11 s22 3 ¼ 33 ¼ zc13 ; 3 ¼ 33 ¼ zc23 ;
t12 s12 3 ¼ 33 ¼ zc36 ð18bÞ
blt ij
ð19dÞ
local problems having The solution of the group been found, we then calculate the following functions:
4.2. Modeling of a unit-cell consisting of top/bottom face carrier Similar to the honeycomb core structure, the faces are made of generally orthotropic material. As such, we shall lt derive the U lt k ðn; zÞ and V k ðn; zÞ functions and hence
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
obtain the solution for the local problems of the group blt ij and blt ij (lt = 11, 22, 12) functions that appropriately represent the material orthotropic elastic behavior. We begin with the local unit-cell problems in Eq. (6) in a form: 1 oslt 1 oslt oslt i1 i2 þ þ i3 ¼ 0 h1 on1 h2 on2 oz 1 oU lt 1 oU lt oU lt lt 1 2 3 s11 ¼ c11 þ c12 þ c13 h1 h2 on1 on2 oz 1 oU lt 1 oU lt oU lt 1 2 3 slt c12 þ c22 þ c23 22 ¼ h1 h2 on1 on2 oz 1 oU lt 1 oU lt oU lt 1 2 3 slt ¼ c þ c þ c 13 23 33 33 h1 h2 on1 on2 oz 1 oU lt oU lt lt 3 2 s23 ¼ c44 þ h2 on2 oz lt lt 1 oU oU lt 3 1 s13 ¼ c55 þ h1 on1 oz 1 oU lt oU lt lt 1 2 s12 ¼ c66 þ h2 on2 on1
ð20Þ
s11 33 ¼ c13 ;
s22 33 ¼ c23 ;
The solution of the group blt ij local problems having been found, we then calculate the following functions: 8 c213 11 > > < b11 ¼ c11 c33 c23 c13 b11 ¼ b11 ð22aÞ > 22 ¼ c12 c33 > : 11 11 11 b33 ¼ b11 23 ¼ b13 ¼ b12 ¼ 0 8 22 b ¼ c12 c23c33c13 > > < 11 c223 ð22bÞ b22 ¼ b22 22 ¼ c22 c33 > > : 22 22 22 b ¼ b22 23 ¼ b13 ¼ b12 ¼ 0 ( 33 12 12 12 12 b12 11 ¼ b22 ¼ b33 ¼ b23 ¼ b13 ¼ 0 b12 ¼ ð22cÞ b12 12 ¼ c66 In a similar way we obtain the blt ij -type local problems from the governing unit-cell problem (6) and corresponding surface boundary conditions. The results are given below:
From Fig. 6, the above problem will be solved with the boundary conditions on surfaces {n1 = ±1/2, n2 = ±1/2 and z = ±t0/2}, and the surface boundary conditions must be satisfied on z = ±t0/2 and are lt slt 13 ¼ s23 ¼ 0;
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s12 33 ¼ 0
ð21Þ
1 oslt 1 oslt oslt i1 i2 þ þ i3 ¼ ci3lt h1 on1 h2 on2 oz lt 1 oV 1 1 oV lt oV lt 2 3 slt ¼ c þ c þ c 11 12 13 11 h1 h2 on1 on2 oz 1 oV lt 1 oV lt oV lt 1 2 3 slt ¼ c þ c þ c 12 22 23 22 h1 h2 on1 on2 oz 1 oV lt 1 oV lt oV lt slt c13 1 þ c23 2 þ c33 3 33 ¼ h1 h2 on1 on2 oz
Fig. 6. A local unit-cell of the sandwich structure in slow variables.
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1 h2 1 lt s13 ¼ c55 h1 1 slt ¼ c 66 12 h2
slt 23 ¼ c44
oV lt oV lt 3 þ 2 on2 oz oV lt oV lt 3 þ 1 on1 oz oV lt oV lt 1 þ 2 on2 on1
4.3. Calculation of effective elastic coefficients of hexagonal honeycomb sandwich shell made of generally orthotropic material
The effective characteristics different from zero and contained in the equation of state (9), with consideration of relationships (10), we obtain
ð23Þ
with slt slt 13 ¼ 23 ¼ 0;
s11 33 ¼ zc13 ;
s22 33 ¼ zc23 ;
s12 33 ¼ 0
ð24Þ
blt ij
The solution of the group local problems having been found, we then calculate the following functions: 8 c213 11 > > > b11 ¼ c11 c33 < c23 c13 ð25aÞ b 11 ¼ z b11 22 ¼ c12 c33 > > > : 11 11 11 b33 ¼ b11 23 ¼ b13 ¼ b12 ¼ 0 8 22 b11 ¼ c12 c23c33c13 > > > < c223 b 22 ¼ z b22 ð25bÞ 22 ¼ c22 c33 > > > : 22 22 22 b33 ¼ b22 23 ¼ b13 ¼ b12 ¼ 0 ( 12 12 12 12 b11 ¼ b12 22 ¼ b33 ¼ b23 ¼ b13 ¼ 0 b 12 ¼ z ð25cÞ b12 12 ¼ c66 With this, we now have at our disposal all the information needed to calculate the effective elastic characteristics of the sandwich shell with hexagonal honeycomb core structure made of generally orthotropic material. In particular, the effective elastic coefficients will be determined in a straight-forward manner from Eqs. (16), (19), (22) and (25) by simple superposition from Eq. (8). The results are given in the following sub-section.
2E1 t0 E1 m21 Ht þ 1:1732 m12 a 1 m12 m21 2E2 t0 E2 Ht 22 hb22 i ¼ þ 0:5152 a 1 m12 m21 2m E t E1 m21 Ht 12 1 0 22 hb11 þ 0:3908 22 i ¼ hb11 i ¼ m12 a 1 m12 m21 E2 Ht 12 hb12 i ¼ 2G12 t0 þ 0:3908 a 2 2E t 3H 3Ht0 2 E1 m21 H 3 t 1 0 11 þ þ t0 þ 0:0976 hzb11 i ¼ m12 a 36ð1 m12 m21 Þ 4 2 2 2E t 3H 3Ht E2 H 3 t 2 0 0 2 þ þ t i ¼ þ 0:0429336 hzb22 22 0 a 36ð1 m12 m21 Þ 4 2 2 2m12 E1 t0 3H 3Ht0 2 22 þ þ t0 hzb11 22 i ¼ hzb11 i ¼ 36ð1 m12 m21 Þ 4 2
hb11 11 i ¼
E1 m21 H 3 t þ 0:03256 m12 a 2 2G12 t0 3H 3Ht0 2 E2 H 3 t hzb12 þ þ t i ¼ 12 0 þ 0:03256 a 36 4 2 ð26Þ It is imperative to note that in arriving the above set of effective elastic properties of the sandwich shell we needed to consider the appropriate cross-sectional areas and the inertia moments of the cross-sections of the each element (X1 to X10) participating in the sandwich structure, see [9]. Further, it should be noted that the effective
Fig. 7. A three-layered sandwich shell reinforced with hexagonal-triangular cellular core.
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
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Fig. 8. A periodicity cell in coordinate systems (a1, a2, c) and (n1, n2, z).
properties of this structure can be tailored to meet the requirements of a particular application by changing some geometric parameters such as thickness of the face carriers, length and cross-sectional areas of the core elements or the aspect ratio of the shell, the relative height of the core material, the angular orientation of the comprising elements, etc. This will be demonstrated in the subsequent Section 5, followed by a discussion on the obtained results in Section 6. 5. Sandwich shell reinforced with hexagonal-and-triangular mixed core A further sophisticated sandwich core structure involves a combination of two frequently used cell forms – hexagonal and triangular. It is therefore interesting to obtain explicit formulas for all effective elastic characteristics of a three-layered shell filled with hexagonal and triangular
shaped cellular core (Fig. 7) and made of generally orthotropic material. A representative periodicity cell of the structure is given in Fig. 8. On the basis of the approximate analytic solution of local problems (5) and (6), obtained in analogy with the previous case, for all effective characteristics of the shell different from zero and contained in the equation of state (9) (with consideration of relationship (10)), we have 2E1 t0 E1 m21 Ht þ 16:6276 m12 a 1 m12 m21 2E2 t0 E2 Ht 22 hb22 i ¼ þ 2:4250 a 1 m12 m21 2m E t E1 m21 Ht 12 1 0 22 hb11 þ 5:5424 22 i ¼ hb11 i ¼ m12 a 1 m12 m21 E2 Ht 12 hb12 i ¼ 2G12 t0 þ 5:5424 a hb11 11 i ¼
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2 2E1 t0 3H 3Ht0 2 E1 m21 H 3 t þ þ t0 þ 1:3856 m12 a 36ð1 m12 m21 Þ 4 2 2 2E2 t0 3H 3Ht0 2 E2 H 3 t þ þ t i ¼ hzb22 þ 0:2022 0 22 a 36ð1 m12 m21 Þ 4 2 2 2m12 E1 t0 3H 3Ht0 2 22 þ þ t0 hzb11 22 i ¼ hzb11 i ¼ 36ð1 m12 m21 Þ 4 2
hzb11 11 i ¼
E1 m21 H 3 t þ 0:4620 m12 a 2 2G t 3H 3Ht0 2 E2 H 3 t 12 0 hzb12 þ þ t i ¼ þ 0:4620 12 0 a 36 4 2
Variant I (H = 8, t0 = t = 0.1): N 1 ¼ dð0:3545E1 Þe11 þ dð0:1115E1 Þe22 þ Oðd3 Þ N 2 ¼ dð0:1115E1 Þe11 þ dð0:3661E2 Þe22 þ Oðd3 Þ M 1 ¼ d3 ð1:0912E1 Þs11 þ d3 ð0:3548E1 Þs22 þ Oðd4 Þ M 2 ¼ d3 ð0:3548E1 Þs11 þ d3 ð1:1543E2 Þs22 þ Oðd4 Þ
ð28Þ
N 12 ¼ dð0:2G12 þ 0:1251E2 Þe12 þ Oðd3 Þ M 12 ¼ d3 ð0:2734G12 þ 0:6668E2 Þs12 þ Oðd4 Þ Variant II (H = 20, t0 = t = 0.5): N 1 ¼ dð2:9221E1 Þe11 þ dð0:9401E1 Þe22 þ Oðd3 Þ
ð27Þ
N 2 ¼ dð0:9401E1 Þe11 þ dð3:0667E2 Þe22 þ Oðd3 Þ M 1 ¼ d3 ð72:5740E1 Þs11 þ d3 ð23:9151E1 Þs22 þ Oðd4 Þ
6. Results and discussions
M 2 ¼ d3 ð23:9151E1 Þs11 þ d3 ð77:5025E2 Þs22 þ Oðd4 Þ
It is noted that first terms in formulas (26) and (27) represent the contribution of the stress carrying face layers, while the second terms represent the contribution made by the honeycomb filler. Formulas (26) and (27) for the effective rigidity moduli (elastic portion of the characteristics) are in good agreement with the familiar relationships from the structural anisotropic theory of reinforced plates (when A1 = A2 = 1). The formula for torsional rigidity hzb12 12 i is beyond the scope of this theory. For the purposes of demonstration, the results from the numerical calculation of effective elastic moduli, estimated in Eq. (26), in two sample variants (with E1 = 138.0 GPa, E2 = 9 GPa, G12 = 6.9 GPa, t12 = 0.3, t21 = 0.0196, for unidirectional AS/3501 graphite/epoxy laminae) are given in Table 1. From Table 1, by using Eq. (11), and in combination with Eq. (9), it is now possible to calculate the forces, moments and torsional effects present in the hexagonal honeycomb sandwich structure. The results from two different variants presented in Table 1 are given below:
ð29Þ
N 12 ¼ dðG12 þ 1:5632E2 Þe12 þ Oðd3 Þ M 12 ¼ d3 ð8:7569G12 þ 52:0690E2 Þs12 þ Oðd4 Þ The derived general shell model will now be used to tailor the effective properties of the sandwich structures in Figs. 2 and 7 to meet the requirements of any particular application by modifying some geometric parameters. To this end, Figs. 9–11 illustrate the variation of effective elastic properties of sandwich shells reinforced with hexagonal and hexagonal-triangular core with respect to the height of the core, H. In Fig. 9, the structural stiffness in hb11 11 i direction is significantly higher than the stiffness in hb11 22 i direction for both shells, as expected. It is observed that the hexagonal-triangular core structure provides higher stiffness to the overall sandwich configuration than its hexagonal honeycomb counterpart. This may be a fact that the former combines an increased number of structural elements, hence making it far stiffer, than the latter.
Table 1 Effective elastic characteristics of hexagonal honeycomb-cored sandwich composite shells (A1 = A2 = 1, a = 2.5) Effective elastic moduli
Variants
The insertion of face carriers
hb11 11 i=E1
The insertion of honeycomb core
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2012 1.0059
0.1533 1.9162
hb22 22 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2012 1.0059
0.1649 2.0608
hb11 22 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.0604 0.3018
0.0511 0.6383
hb12 12 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2000(G12/E2) (G12/E2)
0.1251 1.5632
hzb11 11 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2750 8.8087
0.8162 63.7653
hzb22 22 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2750 8.8087
0.8793 68.6938
hzb11 22 i=E1
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.0825 2.6426
0.2723 21.2725
hzb12 12 i=E2
H = 8, t0 = t = 0.1 H = 20, t0 = t = 0.5
0.2734(G12/E2) 8.7569(G12/E2)
0.6668 52.0960
G.C. Saha et al. / Composites: Part A 38 (2007) 1533–1546
nal-triangular structure has contributed to making the structure stiffer than its hexagonal counterpart. It is demonstrated that the effective properties of the studied structures can be tailored to meet the requirements of a particular application by changing the geometric parameters such as thickness of the carrier faces, length and cross-sectional area of the cellular core, etc.
11
11
30
25 20
15
10 5
7. Conclusions
0 8
9
10
11
12
13
14
15
16
17
18
19
20
H 11 Fig. 9. Effective elastic characteristics (hb11 11 i, hb22 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
*11
>/E1 ,
*11
>/E1
1000
800 600
400
200
0 8
9
10
11
12
13
14 H
15
16
17
18
19
20
11 Fig. 10. Effective elastic characteristics (hzb11 11 i, hzb22 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
6000
*12
>/G12
5000
1545
4000 3000
2000 1000 0 12
13
14
15
16
17
18
19
20
H
Fig. 11. Effective elastic characteristics (hzb12 12 iÞ of sandwich shells reinforced by a regular system of hexagonal or hexagonal-triangular core.
In Fig. 10, it is seen that the torsional stiffnesses (hzb11 11 i and hzb11 iÞ for the mentioned structures takes into a stee22 per shape than their respective extensional stiffnesses (hb11 11 i and hb11 iÞ (Fig. 9). Finally, Fig. 11 shows the variation of 22 hzb12 i vs. H for the two structures. It is obvious that the 12 presence of additional number of members in the hexago-
The paper develops a micromechanical model pertaining to a general composite sandwich shell. Consideration is given on the application of a modified two-scale asymptotic homogenization technique applied to a rigorously formulated 3D elastic problem for a thin curvilinear periodically-inhomogeneous composite layer made of generally orthotropic material. Orthotropy of the constituent materials leads to much more complex unit-cell problems and is considered in the present paper for the first time. Thanks to the presence of the dimensionless small parameter d representing the thickness of the shell, the original 3D problem then proves to be amenable to a rigorous asymptotic analysis unifying an asymptotic three-to-two dimensions process and a homogenization composite material-homogeneous material process. The explicit expressions called unit-cell problems that are used to determine the effective elastic characteristics of the shell structure are presented. In particular, it is shown that the effective stiffnesses of the homogenized shell generally depend on the local elastic constants of the material. This dependence is illustrated by application of the general model to a hexagonal honeycomb sandwich shell made of generally orthotropic material. The explicit formulas for a rotational member, which when substituted with appropriate angle of the individual element generates the complete expression for the structure, comprising the sandwich core is obtained. Theory is further illustrated by introducing a further complex problem involving a hexagonal-triangular mixed core in the sandwich shell. Through the use of numerical illustrations, it is shown that the effective elastic properties of the structure overwhelmingly depend on the geometric configuration of the stiffeners. It is noted that even though the analysis presented in the paper is applied to elastic materials, the model derived should be considered to hold equally well for structures exhibiting viscoelastic, piezoelectric or magnetostrictive characteristics, or are associated with some general transduction characteristics which can be used to induce strains and stresses in some coordinated fashion.
Acknowledgement The support by the Natural Sciences and Engineering Council of Canada (NSERC) is gratefully acknowledged.
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Appendix It is noted that in formulating the unit-cell problems for a rotated element of the sandwich honeycomb structure we have written the governing differential equations and associated surface boundary conditions in terms of the global coordinates. However, it makes practical sense to give the final closed-form expressions in Eqs. (26) and (27) in terms of the familiar material constants (Young’s moduli, shear moduli and Poisson’s ratios). To do that, we first need to relate the off-axis coefficients, cijkl, with the principal ðmÞ material coefficients, cijkl . This is achieved through the tensor transformation equation for a fourth-order tensor, i.e., cijkl ¼ aim ajn akp alq cðmÞ mnpq
ð30Þ
When expanded for an orthotropic material, Eq. (30) becomes, see [13]: ðmÞ ðmÞ ðmÞ c11 ¼ c11 cos4 u þ 2 c12 þ 2c66 cos2 u sin2 u ðmÞ
þ c22 sin4 u
ðmÞ ðmÞ ðmÞ ðmÞ c12 ¼ c12 cos4 u þ c11 þ c22 4c66 cos2 u sin2 u ðmÞ
þ c12 sin4 u ðmÞ
ðmÞ
c13 ¼ c13 cos2 u þ c23 sin2 u ðmÞ ðmÞ ðmÞ c16 ¼ c11 þ c12 2c66 cos3 u sin u ðmÞ ðmÞ ðmÞ þ 2c66 þ c12 2c22 cos u sin3 u ðmÞ ðmÞ ðmÞ c22 ¼ c22 cos4 u þ 2 c12 þ 2c66 cos2 u sin2 ua ðmÞ
þ c11 sin4 u ðmÞ
ðmÞ
c23 ¼ c23 cos2 u þ c13 sin2 u ðmÞ ðmÞ ðmÞ c26 ¼ c12 c22 þ 2c66 cos3 u sin u ðmÞ ðmÞ ðmÞ þ c11 c12 2c66 cos u sin3 u ðmÞ
c33 ¼ c33
ðmÞ ðmÞ c36 ¼ c13 c23 cos u sin u ðmÞ ðmÞ ðmÞ ðmÞ c66 ¼ c11 þ c22 2c12 2c66 cos2 u sin2 u ðmÞ þ c66 cos4 u þ sin4 u ð31Þ ðmÞ
Subsequently, the principal material coefficients, cijkl , are related to the material constants, E1, E2, G12, t12, etc., through well-known expressions, see for example [3]. References [1] Bensoussan A, Lions JL, Papanicolaou G. Asymptotic analysis for periodic structures. Amsterdam: North-Holland; 1978. [2] Cioranescu D, Donato P. An introduction to homogenization. Oxford lecture series in mathematics and its applications, vol. 17. London: Oxford University Press; 2000. [3] Kalamkarov AL. Composite and reinforced elements of construction. Chichester: Wiley; 1992. [4] Duvaut G. Analyse fontionnelle et me´chanique des milieux continues. Application a` l’e´tude des materiaux composites e´lastiques a structure pe´riodique-homoge´ne´isation. In: Proceedings of Theor and Appl Mech Prepr 14th IUTAM Congress. Delft, 1976. p. 119–32. [5] Kalamkarov AL. On the determination of the effective properties of network plates and shells with periodic structure. Izv Akad Nauk SSSR (Mekh Tv Tela) 1987(2):181–5. [6] Caillerie D. Thin elastic and periodic plates. Math Meth Appl Sci 1984;6:159–91. [7] Panasenko GP, Reztsov MV. Homogenization of the three-dimensional elasticity problem for a homogeneous plate. Dokl Akad Nauk SSSR 1987;294:1061–5. [8] Kohn RV, Vogelius M. A new model for thin plates with rapidly varying thickness. Int J Solids Struct 1984;20:333–50. [9] Kalamkarov AL, Saha GC, Georgiades AV. General micromechanical modeling of smart composite shells with application to smart honeycomb sandwich structures. Compos Struct 2007;79(1):18–33. [10] Timoshenko S. Theory of plates and shells. New York: McGrawHill Book Co., Inc; 1940. [11] Kalamkarov AL. The thermal conductivity of a twisted nonuniform anisotropic layer of periodic structure with wavy surfaces. JEP 1987;52(5). [12] Kolpakov AG. The determination of averaged characteristics for elastic skeletons. PMM 1985;49(6). [13] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton: CRC Press; 1997.