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Dynamic wrinkling in sandwich beams
Composites: Part B 35 (2004) 665–672 www.elsevier.com/locate/compositesb
Dynamic wrinkling in sandwich beams Victor Birman* Engineering Education Center, University of Missouri-Rolla, 8001 Natural Bridge Road, St Louis, MO 63121, USA Received 2 June 2003; accepted 5 August 2003 Available online 9 April 2004
Abstract Facings of sandwich structures employed in typical applications are often subject to parametric periodic loading. Such loading can cause local dynamic instability of the facings, i.e. large-amplitude small wavelength lateral vibrations. This phenomenon, called in the paper dynamic wrinkling, may result in fatigue damage or immediate failure. The problem of dynamic wrinkling of the facings is analyzed in the present paper for sandwich beams and for large aspect ratio wide panels that vibrate forming a cylindrical surface. The solution is obtained for the case of a relatively thick or compliant core where the Winkler elastic foundation model of the core is applicable. In addition, the problem is formulated as an extension of the Plantema core model that may be preferable for thinner and stiffer cores. In addition, a new simplified elasticity model is introduced in the paper that is based on the assumption that both facings experience simultaneous and interactive dynamic wrinkling instability. Numerical results shown for the elastic foundation model include the criterion for the onset of dynamic wrinkling and the critical value of the damping coefficient of the facing that is sufficient to prevent such wrinkling. As follows from these results, dynamic wrinkling is unlikely in most engineering applications, except for the case in which the maximum stresses in the facing approach the static wrinkling value. q 2004 Elsevier Ltd. All rights reserved. Keywords: Sandwich beams and panels
1. Introduction Although the problem of wrinkling of the facings of sandwich structures was first addressed by Gough et al. [1], it is still far from being resolved. Numerous theories of wrinkling under static loads have been published, including the Winkler-type elastic foundation model [1], the models of Hoff and Mautner [2], Plantema [3], Goodier and Neou [4], and Allen [5] and others. Recent studies of various aspects of static wrinkling include the work of Fagerberg [6], Vonach and Rammerstorfer [7,8], Kardomateas [9], Birman and Bert [10]. A comprehensive review of extensive literature on static wrinkling is outside the scope of the present paper. In addition to wrinkling under static loads, the facings of sandwich structures can experience wrinkling under dynamic in-plane (parametric) periodic-in-time stresses. In marine sandwich structures such parametric dynamic stresses can be caused by vibration of the supporting stiffeners (kinematic excitation) generated by vibrating * Tel.: þ1-3145-165436; fax: þ1-3145-165434. E-mail address: [email protected] (V. Birman). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.08.012
machinery supported by these stiffeners. Propeller-induced vibrations in the stern section of a vessel represent another potential source of dynamic parametric loads. In the case where wrinkling is caused by parametric dynamic loads, it is characterized by large-amplitude local vibrations of the facing. Such vibrations may cause immediate failure or fatigue damage of the facing or the facing – core interface. The present paper illustrates the solution of the problem of dynamic wrinkling of sandwich beams or wide panels that vibrate forming a cylindrical surface using three different theories. The first theory is a dynamic version of the approach suggested by Gough et al. [1], who represented the effect of the core by the Winkler elastic foundation. The second method is a modification of the Plantema approach [3], accounting for dynamic effects. Finally, ‘the simplified theory of elasticity model’ enables us to fully account for the interaction of both facings that experience a simultaneous and interrelated wrinkling. Numerical examples presented in the paper are based on the Winkler model of the core that was shown conservative and suitable for the analysis of sandwich structures with a relatively thick and compliant core [10].
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2. Analysis The study of dynamic wrinkling is based on the theory of dynamic stability discussed in numerous references; in particular, the formulae for the regions of dynamic instability and for the critical damping coefficient employed below are obtained by modifying the corresponding equations in the monograph of Bolotin [11]. Note that these are approximate equations and the simplifications utilized to derive them are not discussed here for brevity (the reader is referred to the above-mentioned book of Bolotin for details). The accuracy of approximations used in this paper is justified in applications where the amplitude of the dynamic in-plane load represents a fraction of the static wrinkling load. 2.1. Analysis of the facing wrinkling instability based on the Winkler elastic foundation model of the core This model is acceptable if the core is relatively light. For example, in the case of finite aspect ratio sandwich panels, the model provided conservative estimates of the wrinkling stress for balsa and Divinylcell cores of grades H-30, H-45, H-100 and H-160 in numerous sandwich configurations [10]. If the facings wrinkle simultaneously as a result of an applied in-plane dynamic load forming a symmetric pattern relative to the middle plane, the modulus of the foundation provided by the core is given by k ¼ 2Ec =hc where Ec and hc are the modulus of the core in the z-direction and its thickness, respectively. 2.1.1. Boundary frequencies for the instability regions The equation of motion of a facing subject to the stress resultant Nx ¼ N0 þ N cos vt coincides with the Mathieu equation for an infinite beam supported by an elastic foundation [11]
rf W;tt þ 21W;t " # 2 p 4 p þ Df þk 2 ðN0 þ N cos vtÞ W ¼ 0 l l
ð1Þ
where the wrinkling motion is represented by wf ¼ W sinðpx=lÞ; l is an unknown length of the wrinkle, 1 is a damping coefficient, and W ¼ WðtÞ: The stiffness of the facing relative to its own middle axis is denoted by Df ; the mass of the unit surface area of the facing is rf : In Eq. (1), the compressive stress resultant is positive. Eq. (1) can be represented in the form
N0 þ N cos vt 0 2 W;tt þ 21 W;t þ V 1 2 W ¼0 Ncr
ð2Þ
where 10 ¼ 1=rf ; and the squared wrinkling frequency and ‘critical’ wrinkling load are given by
V2 ¼
p l
4
Df þ k
rf
Ncr ¼
p l
2
Df þ k
l p
2
ð3Þ
If the effect of damping is neglected, the lowest driving frequency corresponding to the boundary of the principal instability region is obtained in the form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u " 2 uk ðN þ 0:5NÞ 0 ð4Þ v ¼ 2t 12 rf 4Df k Note that the length of the wrinkle is rffiffiffiffiffi Df l ¼ 2p N
ð5Þ
The principal instability region is usually most dangerous in practical applications since it corresponds to the broadest range of driving frequencies and it is less affected by damping. At the frequencies that are higher than the value given by Eq. (4) the facing develops wrinkling vibrations, while at lower frequencies it remains stable. The regions of dynamic instability corresponding to various lengths l occur at all frequencies exceeding the boundary defined by Eq. (4). The presence of damping requires a much higher applied load necessary to realize higher and combination regions of dynamic instability, as compared to the principal region. Therefore, the analysis will be concerned with the principal instability region. 2.1.2. Effect of damping on the principal instability region The critical damping for the principal instability region of the Mathieu equation is obtained from the solution of Bolotin [11]. Transformations yield vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 2 N0 ffi 101 ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ u ! u t p 2 l 2 4 u kl Df þ k 4trf Df 1 þ 4 l p p Df The principal region of dynamic instability can be realized if damping is smaller than the value corresponding to Eq. (6), while in the case of a higher damping the motion remains stable. It is immediately evident that the critical damping approaches zero, if the wrinkle wave is long ðl ! 1Þ: However, if this length is very short, ðl ! 0Þ; the minimum damping that eliminates the possibility of instability is available as N 10min ¼ pffiffiffiffiffiffi 4 rf D f
ð7Þ
Note that the minimum value of damping obtained according to Eq. (7) is independent of the static stress resultant. This is anticipated in a geometrically linear formulation.
V. Birman / Composites: Part B 35 (2004) 665–672
If damping is not sufficient to neutralize the principal instability region, the boundaries of this region can approximately be determined from
2
1 þ N 2 v
2Ncr
4V 2
10 v
sffiffiffiffiffiffiffiffiffiffiffi
N
2
V 12 0
Ncr
¼0 2 v
2 4V2
10 v 2 sffiffiffiffiffiffiffiffiffiffiffi N V2 1 2 0 Ncr 12
N 2Ncr
ð8Þ
2.2. Analysis of facing dynamic stability based on the Plantema model of the core This model yielded more conservative results than either elastic foundation or Hoff’s models of the core in case of denser core [10]. According to the approach of Plantema [3] it is assumed that the core deflections under the wrinkle decay exponentially with the distance from the interface between the facing and core wc ¼ W e2rz sin
px l
ð9Þ
where r is an unknown parameter and z is counted from the facing –core interface. The solution is obtained by the Rayleigh method. The strain energy components coincide with those for the static case. In particular, the strain energy of the wrinkling facing is Uf ¼
Df ðl p4 ðwf;xx Þ2 dx ¼ 3 Df W 2 2 0 4l
ð10Þ
The energy of the core is obtained as 1 ðl ðh 2 1 ðl ðh 2 scz dz dx þ t dz dx 2Ec 0 0 2Gc 0 0 cz ! Gc p2 W 2 ¼ Ec rl þ rl 8
Uc ¼
ð11Þ
where h is a depth of the section of the core that is affected by wrinkling. The energy of the dynamic stress resultant applied to the facing under consideration is 1 ðl UN ¼ 2 ðN þ N cos vtÞðwf;x Þ2 dx 2 0 0 ¼2
p2 ðN þ N cos vtÞW 2 4l 0
ð12Þ
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The kinetic energy of the affected facing and the core is 1 ðl 0 1 ðl ðh rf ðwf;t Þ2 dx þ g ðw Þ2 dz dx T¼ 2 0 2 0 0 c c;t g W;t2 l ð13Þ ¼ r0f þ c 4 2r where r0f is the mass of the facing per unit length of the front of the wrinkle, and gc is the mass density (mass per unit volume) of the core. The application of the Rayleigh method to the total energy P ¼ Uf þ Uc þ UN 2 T yields the Mathieu equation 4 g p Er G p 2 rf þ c W;tt þ Df þ c þ c l 2 2r 2r l 2 p 2 ðN0 þ N cos vtÞ W ¼ 0 ð14Þ l where both r and l are unknown. The boundaries of the principal dynamic instability region corresponding to the motion described by Eq. (14) are defined by
v2 ¼4
Df
4 2 p Er G p 2 p 1 p 2 þ c þ c 2N0 ^ N l 2 2r l l 2 l gc rf þ 2r
ð15Þ
The envelope of the principal instability region similar to that given by Eq. (4) can be obtained by minimizing the smaller squared frequency in Eq. (15) with respect to r and l: However, this minimization results in prohibitively complicated equations that deny exact solution. A convenient estimate of a relative contribution of inertias of the facings and core can be obtained by adopting the value of r from the solution of the static wrinkling problem, i.e. sffiffiffiffiffiffiffiffi 2 2 3 p Gc r¼ ð16Þ 8Df Ec For example, if we assume that Ec <2:6Gc and Df ¼ Ef tf3 =12 (in the case of a panel wrinkling forming a cylindrical surface, the average facing modulus Ef should be replaced with the corresponding reduced stiffness Q11 ), pffiffiffiffiffiffiffi r¼ð1:3=tf Þ 3 Ec =Ef : Accordingly, the denominator in the right-hand side of Eq. (15) becomes sffiffiffiffi! gc 3 Ef ð17Þ r¼ rf 1þ 2:6gf Ec where gf is the mass density of the material of the facing. The ratio r=rf is close to unity in numerous applications, indicating that the contribution of the inertia of the core is relatively small and it can be disregarded. In this case, the minimization of the squared frequency given by Eq. (15) with respect to r yields the result available from the static
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pffiffiffiffiffiffiffiffi wrinkling analysis, i.e. r¼ðp=lÞ Gc =Ec : Subsequently, using this expression in Eq. (15) and minimizing the resulting equation with respect to the length of the wrinkle results in the cubic equation for this length that is affected by the amplitude of the stress resultant N: h i pffiffiffiffiffiffiffi ðp Ec Gc Þl3 2 2p2 ðN0 ^0:5NÞ l2 þ16p4 Df ¼0 ð18Þ The envelope of dynamic stability can be obtained by solving Eq. (18) for prescribed values of N: Subsequently, the corresponding frequency is available from Eq. (15).
the in-plane equilibrium equation (equilibrium of stresses in the x-direction). However, as shown below, this equation can be satisfied at each coordinate z ¼ constant in the integral sense. The facings are assumed to be in the state of plane stress, while the core is two-dimensional, i.e. the stresses in both x and z directions are accounted for. Contrary to the models of the core based on the first-order shear-deformation theory, normal stresses in the thickness direction are included in the analysis. Following the discussion of Eq. (17), the inertia of the core is neglected.
2.3. Analysis of the facing wrinkling instability based on the simplified elasticity model
2.3.1. Analysis of deformations and stresses in the core As indicated above, the analysis is conducted using static equilibrium equations for the core (standard notation)
The present approach is based on the assumption that wrinkling occurs simultaneously in both facings. This should be expected in the case of a thin or relatively stiff core (the loss of strength may also be the actual form of failure in such situations). However, even if the core is thick and compliant, the facing opposite to the wrinkling facing should be affected by wrinkling. The case where the amplitudes of the forces applied to each facing are different can also be analyzed using the theory introduced in this section. The mode shape of wrinkling deformations in a sandwich beam subjected to dynamic loads applied to both facings is depicted in Fig. 1. It is assumed that although the amplitudes of wrinkling deformations in the facings may be different, the size of the wrinkles, i.e. their length l; is the same. The result that immediately follows from this assumption is related to neglecting in-plane deformations within the core. Note that the assumption that in-plane deformations are negligible has been adopted in a number of solutions, such as those based on the elastic foundation model of the core, Hoff’s and Plantema’s theories and the recent theory of Vonach and Rammerstorfer. Neglecting in-plane deformations in the facings, one cannot expect to satisfy
sx;x þ txz;z ¼ 0
ð19Þ
txz;x þ sz;z ¼ 0 where the stresses are the following functions of the strains
sx ¼ Q11 1x þ Q13 1z
ð20Þ
sz ¼ Q31 1x þ Q33 1z txz ¼ Ggxz In Eq. (20), Qij denote reduced stiffnesses of the core ðQ31 ¼ Q13 Þ and G ¼ Gc is the core shear modulus. The stiffnesses in the x and z directions may differ, enabling us to incorporate the case of orthotropic cores in the analysis. The strains in the core are the following functions of deformations 1x ¼ uc;x
ð21Þ
1z ¼ wc;z
gxz ¼ uc;z þ wc;x where uc is the displacement in the x-direction. According to the assumption introduced above, in-plane displacements of the core are neglected in Eq. (21). Then the substitution of simplified Eq. (21) into Eq. (20) and subsequently, into Eq. (19) yields the following equations of elastic equilibrium: wc;xz ¼ 0
ð22Þ
Q33 wc;zz þ Gwc;xx ¼ 0 The solution of the second Eq. (22) can be obtained in the form of a product of two functions: w ¼ XðxÞZðzÞ
Fig. 1. The model of wrinkling used in the simplified elasticity approach. (a) Beam or panel subject to dynamic compression applied to both facings. (b) The mode shape of wrinkling. The orientation of the wrinkles can be symmetric or antisymmetric, but their amplitudes can be different. The length of the wrinkles in the facings is identical.
ð23Þ
The substitution of Eq. (23) into the second Eq. (22) and the separation of variables yield Q33
Z;zz X ¼ 2G ;xx ¼ C Z X
where C is a constant.
ð24Þ
V. Birman / Composites: Part B 35 (2004) 665–672
The solution of Eq. (24) is
Eqs. (10) – (13). Accordingly
Z ¼ A1 sinh lz þ A2 cosh lz
ð25Þ
Uf ¼
X ¼ A3 sin f lx þ A4 cos f lx pffiffiffiffiffiffiffiffi where ffiffiffiffiffiffiffiffiAi are constants of integration, l ¼ C=Q33 ; f ¼ p Q33 =G: The constants of integration have to be specified from the continuity conditions at the core – facing interfaces. Therefore, at this point, it is necessary to address the issue of assumed wrinkling modes in both facings. The sinusoidal mode shape of wrinkles is assumed in both facings, i.e. wi ¼ Wi sin
Uc ¼
1 ðl ðhc 2 1 ðl ðhc 2 sz dz dx þ t dz dx 2E 0 0 2G 0 0 xz
UN ¼ 2
ð26Þ
where i ¼ 1; 2 and the indices 1 and 2 refer to two facings, as shown in Fig. 1. As indicated above, though both facings exhibit identical mode shape of instability, the amplitudes of the wrinkles can be different. The continuity of deflections along the core – facing interface z ¼ 0 dictates A4 ¼ 0
As follows from the last condition (27), constant C can be determined in terms of the wrinkle width l: The continuity of deflections along the second core – facing interface z ¼ hc yields W2 2 W1 cosh lhc ; W0 sinh lhc
ð28Þ
2 ðl 1X ðw ; Þ2 ðN0i þ Ni cos vtÞdx 2 i¼1 0 i x 2 p2 X W 2 ðN þ Ni cos vtÞ 4l i¼1 i 0i
0
0
wc;xz dx ¼ 0
ð33Þ
0 h
l c 1ðð gc ðwc;t Þ2 dz dx þ 2
ð34Þ
0 0
where r0fi is the mass of the corresponding facing per unit length of the front of the wrinkle and E ¼ Q33 is the stiffness of the core in the thickness direction. The evaluation of integrals Eqs. (32) and (34) yields a 2 b 2 W þ W2 þ cW1 W2 2 1 2 d _2 e _2 _ 1W _2 T¼ W þ W þ gW 2 1 2 2 Uc ¼
px l
a ¼ aðlÞ ¼ ð29Þ
ð35Þ
p2 G ½ðs1 þ s2 Þtanh22 ðlhc Þ þ ðs2 þ s1 Þ 2l 2 2s3 tanh21 ðlhc Þ
It is noted that the first equilibrium condition (22) is satisfied in the integral sense for an arbitrary 0 , z , hc ; i.e. ðl
ð32Þ
where
Accordingly, the deflection within the core is wc ¼ ðW 0 sinh lz þ W1 cosh lzÞsin
ð31Þ
l l 1ð 0 1ð 0 2 T¼ rf1 ðw1;t Þ dx þ rf2 ðw2;t Þ2 dx 2 2
ð27Þ
p fl ¼ l
A1 A3 ¼
2 2 X X Dfi ðl p4 ðwi;xx Þ2 dx ¼ D W2 3 fi i 2 4l 0 i¼1 i¼1
¼2
px l
A2 A3 ¼ W 1 ;
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b ¼ bðlÞ ¼
p2 G ½ðs1 þ s2 Þsinh22 ðlhc Þ 2l
c ¼ cðlÞ ¼
p2 G ½s sinh21 ðlhc Þ 2 ðs1 þ s2 Þtanh21 ðlhc Þ 2l 3
ð30Þ
The variables that will be specified in the subsequent analysis include the length of the wrinkle l; and the amplitudes of the wrinkle in each of the facings W1 ; W2 : In conclusion of this section, it is noted that the present approach is referred to as a simplified elasticity model because of the assumptions introduced above, i.e. neglecting in-plane displacements in the core and its inertia, and the satisfaction of the in-plane equilibrium equation in the integral, rather than exact, sense. 2.3.2. Analysis of the dynamic wrinkling instability Referring to the formulation of the problem shown in Fig. 1, the total energy of the system consists of the contributions given by modified equations similar to
ð36Þ
sinh21 ðlhc Þ d ¼ dðlÞ ¼
r0f1 l gl þ ½s2 tanh22 ðlhc Þ þ s1 2 s3 tanh21 ðlhc Þ 2 2
e ¼ eðlÞ ¼
r0f2 l gl þ s2 sinh22 ðlhc Þ 2 2
g ¼ gðlÞ ¼
gl sinh21 ðlhc Þ½s3 2 2s2 tanh21 ðlhc Þ 4
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V. Birman / Composites: Part B 35 (2004) 665–672
The following notation was introduced in these equations: 1 sinhðlhc Þcoshðlhc Þ s1 ¼ þ hc ð37Þ 2 l 1 sinhðlhc Þcoshðlhc Þ s2 ¼ 2 hc 2 l s3 ¼
coshðlhc Þ 2 1 l
The Lagrange equation, i.e. d ›T ›T ›ðUf þ Uc þ UN Þ þ ¼0 2 _ dt ›Wi ›W i ›W i
ð38Þ
yields the following set of coupled Mathieu-type equations d2 f C 2 þ ðE 2 aA 2 bB cos vtÞf ¼ 0 dt
ð39Þ
where the notation is chosen to comply with that in the monograph of Bolotin [11]. The vectors and matrices introduced in Eq. (39) are: " # ( ) d g W1 f ¼ C¼ ð40Þ g e W2 ½E 2 aA 3 ! 2 p4 p2 N c D þa2 7 6 7 6 2l3 f1 2l 01 7 6 7 ¼6 !7 6 4 2 7 6 p p 5 4 c N02 D þb2 3 f1 2l 2l "
bB ¼
N1 0 0
N2
#
p2 2l
According to Bolotin [11], the boundaries of the principal region of instability of a system described by Eq. (39) can be approximately determined from the determinant # !
"
p4 p2 p2 d 2
N ^ N 2 v Df1 þ a 2
4 2l 01 4l 1 2l3
g
c 2 v2
4
3. Numerical examples Eq. (4) can be represented in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v s0 þ 0:5s 2 ¼ 12 ð42Þ scr 2Vmin pffiffiffiffiffi ¼ N=hf (hf is the where Vmin ¼ k=rf ; s0 ¼ N0 =hf ; psffiffiffiffiffi thickness of the facing), and scr ¼ 2 Df k=hf : This modified equation enables us to show a ‘master curve’ for arbitrary facing and core materials and thicknesses. This curve shown in Fig. 2 results in an important conclusion about the probability of dynamic wrinkling in engineering sandwich structures. To comprehend this conclusion, it is necessary to analyze the values of 2Vmin found for typical balsa and Divinycell cores listed in Table 1. Obviously, high driving frequencies that would be close to the values 2Vmin in Table 1 are unlikely in most applications. The driving frequency corresponding to the principal instability region declines and approaches the practical range when the following function of the static and dynamic applied stresses approaches the value corresponding to the static wrinkling stress, according to the Winkler model of the core, i.e. pffiffiffiffiffi 2 Df k s0 þ 0:5s ! ð43Þ hf pffiffiffiffiffi An estimate of the magnitude of 2 Df k=hf is presented in Table 2 for sandwich beams with the facings manufactured from a quasi-isotropic chopped glass fiber vinyl-ester (QC-8700) with the fiber volume fraction equal to 46%. This material, recommended for its excellent fatigue properties, has Ef ¼ 26:9 GPa and gf ¼ 2000 kg=m3 (the subscript ‘f’ refers to the facing, rather than the fiber). Note that the results reflected in the table should be compared to the compressive strength of the facings that is close to 260 MPa. Accordingly, the results are shown in the table only for the case where the stress does not exceed the strength of the facing. The second type of facings that was also considered is manufactured from a balanced cross-ply carbon/epoxy AS4/3501-6 with Ef ¼ 76:2 GPa and gf ¼ 1580 kg=m3 :
g
c 2 v2
4
" #
¼ 0 ! 4 2 2 p p p e
N02 ^ N2 2 v2
Df2 þ b 2 3 4 2l 4l 2l
Note that Eq. (41) contains the constant applied stress resultants N01 ; N02 ; the amplitudes of applied stress resultants N1 ; N2 ; the driving frequency v and the length of the wrinkling wave l: Therefore, it is possible to determine the lower boundary of the principal dynamic instability region (the ‘smallest’ combination of N0i ; Ni corresponding to a prescribed driving frequency) by varying l:
ð41Þ
Obviously, these facings require an even higher applied stress to achieve wrinkling within a realistic range of the driving frequencies. It can be concluded that dynamic wrinkling can only become dangerous in configurations where the thickness of the facing represents a small fraction of that of the core and the stiffness of the core is small. Otherwise, the loss of strength is the dominant mode of
V. Birman / Composites: Part B 35 (2004) 665–672
671
Table 2 pffiffiffiffiffi The magnitude of s0 þ 0:5s ¼ 2 Df k=hf (MPa) for sandwich beams with the facings manufactured from a quasi-isotropic chopped glass fiber vinylester (QC-8700) Ratio hf =hc
Fig. 2. Master curve of the boundary of the principal instability region according to the Winkler core model. Nondimensional stress is ðs0 þ pffiffiffiffiffi 0:5sÞ=scr where scr ¼ 2 Df k=hf ; nondimensional frequency is pffiffiffiffiffi v=ð2 k=rf Þ ¼ v=2Vmin :
failure. In general, dynamic stability of the facing should be checked if the sum of the static and one-half amplitude of the dynamic stress approaches the static wrinkling value. A different model than the Winkler model of the core used in these examples, including those presented in the paper, may affect numerical results, but it is unlikely to alter the conclusions. It is interesting to identify the magnitude of damping that is sufficient to prevent dynamic wrinkling. It can easily be shown using Eq. (7) that the minimum damping coefficient necessary to achieve this goal is given by rffiffiffiffiffi g 1cr ¼ 0:866s f ð44Þ Ef For example, for glass/vinyl-ester facing, this yields 1cr ¼ 0:236 £ 1023 s; while for a carbon/epoxy facing, 1cr ¼ 0:125 £ 1023 s In these equations, the stress is measured in MPa, while the damping coefficients are in Ns/m3. The value of the critical damping coefficient can be related to the loss factor of the composite facing that can be determined following the methodology of Birman and Byrd [12]. It is important to justify the approach adopted in the simplified elasticity model that was based on neglecting the effect of the inertia of the core. Therefore, the ratio r=rf given by Eq. (17) is shown in Table 3. The actual effect of the inertia of the core is even smaller than the ratios shown in Table 3 since the frequencies calculated by Eq. (15) are inversely proportional to a square root of the corresponding Table 1 Properties of core materials considered in examples and 2Vmin for a 20-mm core Material
Ec (MPa) gc (kg/m3) 2Vmin (1/s)
Balsa
280.8 96 1.5 £ 105
Divinylcell H-30
H-45
H-100
H-160
20.0 36 4.0 £ 104
40.0 48 5.7 £ 104
105.0 100 9.2 £ 104
170.0 160 11.7 £ 105
The facings are manufactured from a quasi-isotropic chopped glass fiber vinyl-ester (QC-8700). The thickness of the facings is 2.5 mm.
0.02 0.05 0.10 0.15 0.20
Core Balsa
H-30
H-40
H-100
H-160
F F F F F
84.7 133.9 189.4 231.9 267.8
119.8 189.4 267.8 F F
194.1 F F F F
246.9 F F F F
Stress levels exceeding 260 MPa result in the loss of strength (‘F’ in the table).
ratios. Therefore, the assumption neglecting the inertia of the core adopted in the simplified elasticity model is justified, at least for the material systems considered in this paper.
4. Conclusions The paper illustrates the approach to the analysis of wrinkling dynamic instability of facings of composite sandwich beams and large aspect ratio wide panels vibrating forming a cylindrical surface. The solution concentrates on the principal dynamic instability region since higher regions are less likely to be realized due to the effect of damping. The solution of the dynamic wrinkling problem is shown using three different models. Each model is modified to incorporate dynamic effects. One of these models, i.e. the simplified elasticity model, has not been published prior to this paper. This model enables us to account for deformations of both facings, without an assumption that they are symmetric or antisymmetric. Numerical results were generated using the solution based on the Winkler elastic foundation model of the core. As follows from these results, dynamic wrinkling can occur only if the frequency of the driving force is very large. Dynamic instability can occur in a more practical range of driving frequencies if a sum of the static component and one-half amplitude of the dynamic component of the applied stress approaches the static wrinkling value. However, in many situations, such large stress amplitudes can result in the loss of strength of the facing. Based on the analysis, it is anticipated that dynamic wrinkling may be Table 3 The ratio r=rf for sandwich configurations considered in examples (Eq. 17) Core
Balsa
H-30
H-45
H-100
H-160
Case 1 Case 2
1.084 1.151
1.076 1.137
1.081 1.145
1.122 1.219
1.166 1.298
Case 1, quasi-isotropic chopped fiber glass/vinyl-ester facings; Case 2, balanced cross-ply carbon/epoxy facings (AS4/3501-6).
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V. Birman / Composites: Part B 35 (2004) 665–672
dangerous only in the case of a very thin facing supported by a light core. It is possible that the Plantema model and the simplified elasticity model outlined in the paper may affect the numerical results, as compared to those generated by the Winkler model, but the qualitative conclusions are unlikely to be altered.
Acknowledgements This work was supported by the Office of Naval Research Program on Composites for Marine Structures (Contract ONR N00014-00-1-0323). The Program Manager is Dr Yapa D.S. Rajapakse.
References [1] Gough GS, Elam CF, deBruyne NA. The stabilization of a thin sheet by a continuous supporting medium. J R Aeronaut Soc 1940;44: 12–43.
[2] Hoff NJ, Mautner SE. The buckling of sandwich-type panels. J Aeronaut Sci 1945;12:285–97. [3] Plantema FJ. Sandwich construction. New York: Wiley; 1966. [4] Goodier JN, Neou IM. The evaluation of theoretical compression in sandwich plates. J Aeronaut Sci 1951;18:649–57. [5] Allen HG. Analysis and design of structural sandwich panels. Oxford: Pergamon Press; 1969. [6] Fagerberg L. Wrinkling in sandwich panels for marine applications. Lucentiate Thesis. Report 2001-17. Royal Institute of Technology, Stockholm, Sweden; 2001. [7] Vonach WK, Rammerstorfer FG. The effect of in-plane core stiffness on the wrinkling behavior of thick sandwiches. Acta Mech 2000;141: 1–10. [8] Vonach WK, Rammerstorfer FG. A general approach to the wrinkling instability of sandwich plates. Struct Engng Mech 2001;12:363– 76. [9] Kardomateas GA. Wrinkling of wide sandwich panels/beams by an elasticity approach. In: Vinson JR, Rajapakse YDS, Carlsson LA, editors. Proceedings of the Sixth International Conference on Sandwich Structures. Boca Raton: CRC Press; 2003. p. 142 –53. [10] Birman V, Bert CW. Wrinkling of composite-facing sandwich panels under biaxial loading. J Sandwich Struct Mater 2004; in press. [11] Bolotin VV. The dynamic stability of elastic systems. San Francisco: Holden-Day; 1964. [12] Birman V, Byrd LW. Analytical evaluation of damping in composite and sandwich structures. AIAA J 2002;40:1638–43.
Dynamic wrinkling in sandwich beams Victor Birman* Engineering Education Center, University of Missouri-Rolla, 8001 Natural Bridge Road, St Louis, MO 63121, USA Received 2 June 2003; accepted 5 August 2003 Available online 9 April 2004
Abstract Facings of sandwich structures employed in typical applications are often subject to parametric periodic loading. Such loading can cause local dynamic instability of the facings, i.e. large-amplitude small wavelength lateral vibrations. This phenomenon, called in the paper dynamic wrinkling, may result in fatigue damage or immediate failure. The problem of dynamic wrinkling of the facings is analyzed in the present paper for sandwich beams and for large aspect ratio wide panels that vibrate forming a cylindrical surface. The solution is obtained for the case of a relatively thick or compliant core where the Winkler elastic foundation model of the core is applicable. In addition, the problem is formulated as an extension of the Plantema core model that may be preferable for thinner and stiffer cores. In addition, a new simplified elasticity model is introduced in the paper that is based on the assumption that both facings experience simultaneous and interactive dynamic wrinkling instability. Numerical results shown for the elastic foundation model include the criterion for the onset of dynamic wrinkling and the critical value of the damping coefficient of the facing that is sufficient to prevent such wrinkling. As follows from these results, dynamic wrinkling is unlikely in most engineering applications, except for the case in which the maximum stresses in the facing approach the static wrinkling value. q 2004 Elsevier Ltd. All rights reserved. Keywords: Sandwich beams and panels
1. Introduction Although the problem of wrinkling of the facings of sandwich structures was first addressed by Gough et al. [1], it is still far from being resolved. Numerous theories of wrinkling under static loads have been published, including the Winkler-type elastic foundation model [1], the models of Hoff and Mautner [2], Plantema [3], Goodier and Neou [4], and Allen [5] and others. Recent studies of various aspects of static wrinkling include the work of Fagerberg [6], Vonach and Rammerstorfer [7,8], Kardomateas [9], Birman and Bert [10]. A comprehensive review of extensive literature on static wrinkling is outside the scope of the present paper. In addition to wrinkling under static loads, the facings of sandwich structures can experience wrinkling under dynamic in-plane (parametric) periodic-in-time stresses. In marine sandwich structures such parametric dynamic stresses can be caused by vibration of the supporting stiffeners (kinematic excitation) generated by vibrating * Tel.: þ1-3145-165436; fax: þ1-3145-165434. E-mail address: [email protected] (V. Birman). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.08.012
machinery supported by these stiffeners. Propeller-induced vibrations in the stern section of a vessel represent another potential source of dynamic parametric loads. In the case where wrinkling is caused by parametric dynamic loads, it is characterized by large-amplitude local vibrations of the facing. Such vibrations may cause immediate failure or fatigue damage of the facing or the facing – core interface. The present paper illustrates the solution of the problem of dynamic wrinkling of sandwich beams or wide panels that vibrate forming a cylindrical surface using three different theories. The first theory is a dynamic version of the approach suggested by Gough et al. [1], who represented the effect of the core by the Winkler elastic foundation. The second method is a modification of the Plantema approach [3], accounting for dynamic effects. Finally, ‘the simplified theory of elasticity model’ enables us to fully account for the interaction of both facings that experience a simultaneous and interrelated wrinkling. Numerical examples presented in the paper are based on the Winkler model of the core that was shown conservative and suitable for the analysis of sandwich structures with a relatively thick and compliant core [10].
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2. Analysis The study of dynamic wrinkling is based on the theory of dynamic stability discussed in numerous references; in particular, the formulae for the regions of dynamic instability and for the critical damping coefficient employed below are obtained by modifying the corresponding equations in the monograph of Bolotin [11]. Note that these are approximate equations and the simplifications utilized to derive them are not discussed here for brevity (the reader is referred to the above-mentioned book of Bolotin for details). The accuracy of approximations used in this paper is justified in applications where the amplitude of the dynamic in-plane load represents a fraction of the static wrinkling load. 2.1. Analysis of the facing wrinkling instability based on the Winkler elastic foundation model of the core This model is acceptable if the core is relatively light. For example, in the case of finite aspect ratio sandwich panels, the model provided conservative estimates of the wrinkling stress for balsa and Divinylcell cores of grades H-30, H-45, H-100 and H-160 in numerous sandwich configurations [10]. If the facings wrinkle simultaneously as a result of an applied in-plane dynamic load forming a symmetric pattern relative to the middle plane, the modulus of the foundation provided by the core is given by k ¼ 2Ec =hc where Ec and hc are the modulus of the core in the z-direction and its thickness, respectively. 2.1.1. Boundary frequencies for the instability regions The equation of motion of a facing subject to the stress resultant Nx ¼ N0 þ N cos vt coincides with the Mathieu equation for an infinite beam supported by an elastic foundation [11]
rf W;tt þ 21W;t " # 2 p 4 p þ Df þk 2 ðN0 þ N cos vtÞ W ¼ 0 l l
ð1Þ
where the wrinkling motion is represented by wf ¼ W sinðpx=lÞ; l is an unknown length of the wrinkle, 1 is a damping coefficient, and W ¼ WðtÞ: The stiffness of the facing relative to its own middle axis is denoted by Df ; the mass of the unit surface area of the facing is rf : In Eq. (1), the compressive stress resultant is positive. Eq. (1) can be represented in the form
N0 þ N cos vt 0 2 W;tt þ 21 W;t þ V 1 2 W ¼0 Ncr
ð2Þ
where 10 ¼ 1=rf ; and the squared wrinkling frequency and ‘critical’ wrinkling load are given by
V2 ¼
p l
4
Df þ k
rf
Ncr ¼
p l
2
Df þ k
l p
2
ð3Þ
If the effect of damping is neglected, the lowest driving frequency corresponding to the boundary of the principal instability region is obtained in the form vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u " 2 uk ðN þ 0:5NÞ 0 ð4Þ v ¼ 2t 12 rf 4Df k Note that the length of the wrinkle is rffiffiffiffiffi Df l ¼ 2p N
ð5Þ
The principal instability region is usually most dangerous in practical applications since it corresponds to the broadest range of driving frequencies and it is less affected by damping. At the frequencies that are higher than the value given by Eq. (4) the facing develops wrinkling vibrations, while at lower frequencies it remains stable. The regions of dynamic instability corresponding to various lengths l occur at all frequencies exceeding the boundary defined by Eq. (4). The presence of damping requires a much higher applied load necessary to realize higher and combination regions of dynamic instability, as compared to the principal region. Therefore, the analysis will be concerned with the principal instability region. 2.1.2. Effect of damping on the principal instability region The critical damping for the principal instability region of the Mathieu equation is obtained from the solution of Bolotin [11]. Transformations yield vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 2 N0 ffi 101 ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ u ! u t p 2 l 2 4 u kl Df þ k 4trf Df 1 þ 4 l p p Df The principal region of dynamic instability can be realized if damping is smaller than the value corresponding to Eq. (6), while in the case of a higher damping the motion remains stable. It is immediately evident that the critical damping approaches zero, if the wrinkle wave is long ðl ! 1Þ: However, if this length is very short, ðl ! 0Þ; the minimum damping that eliminates the possibility of instability is available as N 10min ¼ pffiffiffiffiffiffi 4 rf D f
ð7Þ
Note that the minimum value of damping obtained according to Eq. (7) is independent of the static stress resultant. This is anticipated in a geometrically linear formulation.
V. Birman / Composites: Part B 35 (2004) 665–672
If damping is not sufficient to neutralize the principal instability region, the boundaries of this region can approximately be determined from
2
1 þ N 2 v
2Ncr
4V 2
10 v
sffiffiffiffiffiffiffiffiffiffiffi
N
2
V 12 0
Ncr
¼0 2 v
2 4V2
10 v 2 sffiffiffiffiffiffiffiffiffiffiffi N V2 1 2 0 Ncr 12
N 2Ncr
ð8Þ
2.2. Analysis of facing dynamic stability based on the Plantema model of the core This model yielded more conservative results than either elastic foundation or Hoff’s models of the core in case of denser core [10]. According to the approach of Plantema [3] it is assumed that the core deflections under the wrinkle decay exponentially with the distance from the interface between the facing and core wc ¼ W e2rz sin
px l
ð9Þ
where r is an unknown parameter and z is counted from the facing –core interface. The solution is obtained by the Rayleigh method. The strain energy components coincide with those for the static case. In particular, the strain energy of the wrinkling facing is Uf ¼
Df ðl p4 ðwf;xx Þ2 dx ¼ 3 Df W 2 2 0 4l
ð10Þ
The energy of the core is obtained as 1 ðl ðh 2 1 ðl ðh 2 scz dz dx þ t dz dx 2Ec 0 0 2Gc 0 0 cz ! Gc p2 W 2 ¼ Ec rl þ rl 8
Uc ¼
ð11Þ
where h is a depth of the section of the core that is affected by wrinkling. The energy of the dynamic stress resultant applied to the facing under consideration is 1 ðl UN ¼ 2 ðN þ N cos vtÞðwf;x Þ2 dx 2 0 0 ¼2
p2 ðN þ N cos vtÞW 2 4l 0
ð12Þ
667
The kinetic energy of the affected facing and the core is 1 ðl 0 1 ðl ðh rf ðwf;t Þ2 dx þ g ðw Þ2 dz dx T¼ 2 0 2 0 0 c c;t g W;t2 l ð13Þ ¼ r0f þ c 4 2r where r0f is the mass of the facing per unit length of the front of the wrinkle, and gc is the mass density (mass per unit volume) of the core. The application of the Rayleigh method to the total energy P ¼ Uf þ Uc þ UN 2 T yields the Mathieu equation 4 g p Er G p 2 rf þ c W;tt þ Df þ c þ c l 2 2r 2r l 2 p 2 ðN0 þ N cos vtÞ W ¼ 0 ð14Þ l where both r and l are unknown. The boundaries of the principal dynamic instability region corresponding to the motion described by Eq. (14) are defined by
v2 ¼4
Df
4 2 p Er G p 2 p 1 p 2 þ c þ c 2N0 ^ N l 2 2r l l 2 l gc rf þ 2r
ð15Þ
The envelope of the principal instability region similar to that given by Eq. (4) can be obtained by minimizing the smaller squared frequency in Eq. (15) with respect to r and l: However, this minimization results in prohibitively complicated equations that deny exact solution. A convenient estimate of a relative contribution of inertias of the facings and core can be obtained by adopting the value of r from the solution of the static wrinkling problem, i.e. sffiffiffiffiffiffiffiffi 2 2 3 p Gc r¼ ð16Þ 8Df Ec For example, if we assume that Ec <2:6Gc and Df ¼ Ef tf3 =12 (in the case of a panel wrinkling forming a cylindrical surface, the average facing modulus Ef should be replaced with the corresponding reduced stiffness Q11 ), pffiffiffiffiffiffiffi r¼ð1:3=tf Þ 3 Ec =Ef : Accordingly, the denominator in the right-hand side of Eq. (15) becomes sffiffiffiffi! gc 3 Ef ð17Þ r¼ rf 1þ 2:6gf Ec where gf is the mass density of the material of the facing. The ratio r=rf is close to unity in numerous applications, indicating that the contribution of the inertia of the core is relatively small and it can be disregarded. In this case, the minimization of the squared frequency given by Eq. (15) with respect to r yields the result available from the static
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pffiffiffiffiffiffiffiffi wrinkling analysis, i.e. r¼ðp=lÞ Gc =Ec : Subsequently, using this expression in Eq. (15) and minimizing the resulting equation with respect to the length of the wrinkle results in the cubic equation for this length that is affected by the amplitude of the stress resultant N: h i pffiffiffiffiffiffiffi ðp Ec Gc Þl3 2 2p2 ðN0 ^0:5NÞ l2 þ16p4 Df ¼0 ð18Þ The envelope of dynamic stability can be obtained by solving Eq. (18) for prescribed values of N: Subsequently, the corresponding frequency is available from Eq. (15).
the in-plane equilibrium equation (equilibrium of stresses in the x-direction). However, as shown below, this equation can be satisfied at each coordinate z ¼ constant in the integral sense. The facings are assumed to be in the state of plane stress, while the core is two-dimensional, i.e. the stresses in both x and z directions are accounted for. Contrary to the models of the core based on the first-order shear-deformation theory, normal stresses in the thickness direction are included in the analysis. Following the discussion of Eq. (17), the inertia of the core is neglected.
2.3. Analysis of the facing wrinkling instability based on the simplified elasticity model
2.3.1. Analysis of deformations and stresses in the core As indicated above, the analysis is conducted using static equilibrium equations for the core (standard notation)
The present approach is based on the assumption that wrinkling occurs simultaneously in both facings. This should be expected in the case of a thin or relatively stiff core (the loss of strength may also be the actual form of failure in such situations). However, even if the core is thick and compliant, the facing opposite to the wrinkling facing should be affected by wrinkling. The case where the amplitudes of the forces applied to each facing are different can also be analyzed using the theory introduced in this section. The mode shape of wrinkling deformations in a sandwich beam subjected to dynamic loads applied to both facings is depicted in Fig. 1. It is assumed that although the amplitudes of wrinkling deformations in the facings may be different, the size of the wrinkles, i.e. their length l; is the same. The result that immediately follows from this assumption is related to neglecting in-plane deformations within the core. Note that the assumption that in-plane deformations are negligible has been adopted in a number of solutions, such as those based on the elastic foundation model of the core, Hoff’s and Plantema’s theories and the recent theory of Vonach and Rammerstorfer. Neglecting in-plane deformations in the facings, one cannot expect to satisfy
sx;x þ txz;z ¼ 0
ð19Þ
txz;x þ sz;z ¼ 0 where the stresses are the following functions of the strains
sx ¼ Q11 1x þ Q13 1z
ð20Þ
sz ¼ Q31 1x þ Q33 1z txz ¼ Ggxz In Eq. (20), Qij denote reduced stiffnesses of the core ðQ31 ¼ Q13 Þ and G ¼ Gc is the core shear modulus. The stiffnesses in the x and z directions may differ, enabling us to incorporate the case of orthotropic cores in the analysis. The strains in the core are the following functions of deformations 1x ¼ uc;x
ð21Þ
1z ¼ wc;z
gxz ¼ uc;z þ wc;x where uc is the displacement in the x-direction. According to the assumption introduced above, in-plane displacements of the core are neglected in Eq. (21). Then the substitution of simplified Eq. (21) into Eq. (20) and subsequently, into Eq. (19) yields the following equations of elastic equilibrium: wc;xz ¼ 0
ð22Þ
Q33 wc;zz þ Gwc;xx ¼ 0 The solution of the second Eq. (22) can be obtained in the form of a product of two functions: w ¼ XðxÞZðzÞ
Fig. 1. The model of wrinkling used in the simplified elasticity approach. (a) Beam or panel subject to dynamic compression applied to both facings. (b) The mode shape of wrinkling. The orientation of the wrinkles can be symmetric or antisymmetric, but their amplitudes can be different. The length of the wrinkles in the facings is identical.
ð23Þ
The substitution of Eq. (23) into the second Eq. (22) and the separation of variables yield Q33
Z;zz X ¼ 2G ;xx ¼ C Z X
where C is a constant.
ð24Þ
V. Birman / Composites: Part B 35 (2004) 665–672
The solution of Eq. (24) is
Eqs. (10) – (13). Accordingly
Z ¼ A1 sinh lz þ A2 cosh lz
ð25Þ
Uf ¼
X ¼ A3 sin f lx þ A4 cos f lx pffiffiffiffiffiffiffiffi where ffiffiffiffiffiffiffiffiAi are constants of integration, l ¼ C=Q33 ; f ¼ p Q33 =G: The constants of integration have to be specified from the continuity conditions at the core – facing interfaces. Therefore, at this point, it is necessary to address the issue of assumed wrinkling modes in both facings. The sinusoidal mode shape of wrinkles is assumed in both facings, i.e. wi ¼ Wi sin
Uc ¼
1 ðl ðhc 2 1 ðl ðhc 2 sz dz dx þ t dz dx 2E 0 0 2G 0 0 xz
UN ¼ 2
ð26Þ
where i ¼ 1; 2 and the indices 1 and 2 refer to two facings, as shown in Fig. 1. As indicated above, though both facings exhibit identical mode shape of instability, the amplitudes of the wrinkles can be different. The continuity of deflections along the core – facing interface z ¼ 0 dictates A4 ¼ 0
As follows from the last condition (27), constant C can be determined in terms of the wrinkle width l: The continuity of deflections along the second core – facing interface z ¼ hc yields W2 2 W1 cosh lhc ; W0 sinh lhc
ð28Þ
2 ðl 1X ðw ; Þ2 ðN0i þ Ni cos vtÞdx 2 i¼1 0 i x 2 p2 X W 2 ðN þ Ni cos vtÞ 4l i¼1 i 0i
0
0
wc;xz dx ¼ 0
ð33Þ
0 h
l c 1ðð gc ðwc;t Þ2 dz dx þ 2
ð34Þ
0 0
where r0fi is the mass of the corresponding facing per unit length of the front of the wrinkle and E ¼ Q33 is the stiffness of the core in the thickness direction. The evaluation of integrals Eqs. (32) and (34) yields a 2 b 2 W þ W2 þ cW1 W2 2 1 2 d _2 e _2 _ 1W _2 T¼ W þ W þ gW 2 1 2 2 Uc ¼
px l
a ¼ aðlÞ ¼ ð29Þ
ð35Þ
p2 G ½ðs1 þ s2 Þtanh22 ðlhc Þ þ ðs2 þ s1 Þ 2l 2 2s3 tanh21 ðlhc Þ
It is noted that the first equilibrium condition (22) is satisfied in the integral sense for an arbitrary 0 , z , hc ; i.e. ðl
ð32Þ
where
Accordingly, the deflection within the core is wc ¼ ðW 0 sinh lz þ W1 cosh lzÞsin
ð31Þ
l l 1ð 0 1ð 0 2 T¼ rf1 ðw1;t Þ dx þ rf2 ðw2;t Þ2 dx 2 2
ð27Þ
p fl ¼ l
A1 A3 ¼
2 2 X X Dfi ðl p4 ðwi;xx Þ2 dx ¼ D W2 3 fi i 2 4l 0 i¼1 i¼1
¼2
px l
A2 A3 ¼ W 1 ;
669
b ¼ bðlÞ ¼
p2 G ½ðs1 þ s2 Þsinh22 ðlhc Þ 2l
c ¼ cðlÞ ¼
p2 G ½s sinh21 ðlhc Þ 2 ðs1 þ s2 Þtanh21 ðlhc Þ 2l 3
ð30Þ
The variables that will be specified in the subsequent analysis include the length of the wrinkle l; and the amplitudes of the wrinkle in each of the facings W1 ; W2 : In conclusion of this section, it is noted that the present approach is referred to as a simplified elasticity model because of the assumptions introduced above, i.e. neglecting in-plane displacements in the core and its inertia, and the satisfaction of the in-plane equilibrium equation in the integral, rather than exact, sense. 2.3.2. Analysis of the dynamic wrinkling instability Referring to the formulation of the problem shown in Fig. 1, the total energy of the system consists of the contributions given by modified equations similar to
ð36Þ
sinh21 ðlhc Þ d ¼ dðlÞ ¼
r0f1 l gl þ ½s2 tanh22 ðlhc Þ þ s1 2 s3 tanh21 ðlhc Þ 2 2
e ¼ eðlÞ ¼
r0f2 l gl þ s2 sinh22 ðlhc Þ 2 2
g ¼ gðlÞ ¼
gl sinh21 ðlhc Þ½s3 2 2s2 tanh21 ðlhc Þ 4
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V. Birman / Composites: Part B 35 (2004) 665–672
The following notation was introduced in these equations: 1 sinhðlhc Þcoshðlhc Þ s1 ¼ þ hc ð37Þ 2 l 1 sinhðlhc Þcoshðlhc Þ s2 ¼ 2 hc 2 l s3 ¼
coshðlhc Þ 2 1 l
The Lagrange equation, i.e. d ›T ›T ›ðUf þ Uc þ UN Þ þ ¼0 2 _ dt ›Wi ›W i ›W i
ð38Þ
yields the following set of coupled Mathieu-type equations d2 f C 2 þ ðE 2 aA 2 bB cos vtÞf ¼ 0 dt
ð39Þ
where the notation is chosen to comply with that in the monograph of Bolotin [11]. The vectors and matrices introduced in Eq. (39) are: " # ( ) d g W1 f ¼ C¼ ð40Þ g e W2 ½E 2 aA 3 ! 2 p4 p2 N c D þa2 7 6 7 6 2l3 f1 2l 01 7 6 7 ¼6 !7 6 4 2 7 6 p p 5 4 c N02 D þb2 3 f1 2l 2l "
bB ¼
N1 0 0
N2
#
p2 2l
According to Bolotin [11], the boundaries of the principal region of instability of a system described by Eq. (39) can be approximately determined from the determinant # !
"
p4 p2 p2 d 2
N ^ N 2 v Df1 þ a 2
4 2l 01 4l 1 2l3
g
c 2 v2
4
3. Numerical examples Eq. (4) can be represented in the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v s0 þ 0:5s 2 ¼ 12 ð42Þ scr 2Vmin pffiffiffiffiffi ¼ N=hf (hf is the where Vmin ¼ k=rf ; s0 ¼ N0 =hf ; psffiffiffiffiffi thickness of the facing), and scr ¼ 2 Df k=hf : This modified equation enables us to show a ‘master curve’ for arbitrary facing and core materials and thicknesses. This curve shown in Fig. 2 results in an important conclusion about the probability of dynamic wrinkling in engineering sandwich structures. To comprehend this conclusion, it is necessary to analyze the values of 2Vmin found for typical balsa and Divinycell cores listed in Table 1. Obviously, high driving frequencies that would be close to the values 2Vmin in Table 1 are unlikely in most applications. The driving frequency corresponding to the principal instability region declines and approaches the practical range when the following function of the static and dynamic applied stresses approaches the value corresponding to the static wrinkling stress, according to the Winkler model of the core, i.e. pffiffiffiffiffi 2 Df k s0 þ 0:5s ! ð43Þ hf pffiffiffiffiffi An estimate of the magnitude of 2 Df k=hf is presented in Table 2 for sandwich beams with the facings manufactured from a quasi-isotropic chopped glass fiber vinyl-ester (QC-8700) with the fiber volume fraction equal to 46%. This material, recommended for its excellent fatigue properties, has Ef ¼ 26:9 GPa and gf ¼ 2000 kg=m3 (the subscript ‘f’ refers to the facing, rather than the fiber). Note that the results reflected in the table should be compared to the compressive strength of the facings that is close to 260 MPa. Accordingly, the results are shown in the table only for the case where the stress does not exceed the strength of the facing. The second type of facings that was also considered is manufactured from a balanced cross-ply carbon/epoxy AS4/3501-6 with Ef ¼ 76:2 GPa and gf ¼ 1580 kg=m3 :
g
c 2 v2
4
" #
¼ 0 ! 4 2 2 p p p e
N02 ^ N2 2 v2
Df2 þ b 2 3 4 2l 4l 2l
Note that Eq. (41) contains the constant applied stress resultants N01 ; N02 ; the amplitudes of applied stress resultants N1 ; N2 ; the driving frequency v and the length of the wrinkling wave l: Therefore, it is possible to determine the lower boundary of the principal dynamic instability region (the ‘smallest’ combination of N0i ; Ni corresponding to a prescribed driving frequency) by varying l:
ð41Þ
Obviously, these facings require an even higher applied stress to achieve wrinkling within a realistic range of the driving frequencies. It can be concluded that dynamic wrinkling can only become dangerous in configurations where the thickness of the facing represents a small fraction of that of the core and the stiffness of the core is small. Otherwise, the loss of strength is the dominant mode of
V. Birman / Composites: Part B 35 (2004) 665–672
671
Table 2 pffiffiffiffiffi The magnitude of s0 þ 0:5s ¼ 2 Df k=hf (MPa) for sandwich beams with the facings manufactured from a quasi-isotropic chopped glass fiber vinylester (QC-8700) Ratio hf =hc
Fig. 2. Master curve of the boundary of the principal instability region according to the Winkler core model. Nondimensional stress is ðs0 þ pffiffiffiffiffi 0:5sÞ=scr where scr ¼ 2 Df k=hf ; nondimensional frequency is pffiffiffiffiffi v=ð2 k=rf Þ ¼ v=2Vmin :
failure. In general, dynamic stability of the facing should be checked if the sum of the static and one-half amplitude of the dynamic stress approaches the static wrinkling value. A different model than the Winkler model of the core used in these examples, including those presented in the paper, may affect numerical results, but it is unlikely to alter the conclusions. It is interesting to identify the magnitude of damping that is sufficient to prevent dynamic wrinkling. It can easily be shown using Eq. (7) that the minimum damping coefficient necessary to achieve this goal is given by rffiffiffiffiffi g 1cr ¼ 0:866s f ð44Þ Ef For example, for glass/vinyl-ester facing, this yields 1cr ¼ 0:236 £ 1023 s; while for a carbon/epoxy facing, 1cr ¼ 0:125 £ 1023 s In these equations, the stress is measured in MPa, while the damping coefficients are in Ns/m3. The value of the critical damping coefficient can be related to the loss factor of the composite facing that can be determined following the methodology of Birman and Byrd [12]. It is important to justify the approach adopted in the simplified elasticity model that was based on neglecting the effect of the inertia of the core. Therefore, the ratio r=rf given by Eq. (17) is shown in Table 3. The actual effect of the inertia of the core is even smaller than the ratios shown in Table 3 since the frequencies calculated by Eq. (15) are inversely proportional to a square root of the corresponding Table 1 Properties of core materials considered in examples and 2Vmin for a 20-mm core Material
Ec (MPa) gc (kg/m3) 2Vmin (1/s)
Balsa
280.8 96 1.5 £ 105
Divinylcell H-30
H-45
H-100
H-160
20.0 36 4.0 £ 104
40.0 48 5.7 £ 104
105.0 100 9.2 £ 104
170.0 160 11.7 £ 105
The facings are manufactured from a quasi-isotropic chopped glass fiber vinyl-ester (QC-8700). The thickness of the facings is 2.5 mm.
0.02 0.05 0.10 0.15 0.20
Core Balsa
H-30
H-40
H-100
H-160
F F F F F
84.7 133.9 189.4 231.9 267.8
119.8 189.4 267.8 F F
194.1 F F F F
246.9 F F F F
Stress levels exceeding 260 MPa result in the loss of strength (‘F’ in the table).
ratios. Therefore, the assumption neglecting the inertia of the core adopted in the simplified elasticity model is justified, at least for the material systems considered in this paper.
4. Conclusions The paper illustrates the approach to the analysis of wrinkling dynamic instability of facings of composite sandwich beams and large aspect ratio wide panels vibrating forming a cylindrical surface. The solution concentrates on the principal dynamic instability region since higher regions are less likely to be realized due to the effect of damping. The solution of the dynamic wrinkling problem is shown using three different models. Each model is modified to incorporate dynamic effects. One of these models, i.e. the simplified elasticity model, has not been published prior to this paper. This model enables us to account for deformations of both facings, without an assumption that they are symmetric or antisymmetric. Numerical results were generated using the solution based on the Winkler elastic foundation model of the core. As follows from these results, dynamic wrinkling can occur only if the frequency of the driving force is very large. Dynamic instability can occur in a more practical range of driving frequencies if a sum of the static component and one-half amplitude of the dynamic component of the applied stress approaches the static wrinkling value. However, in many situations, such large stress amplitudes can result in the loss of strength of the facing. Based on the analysis, it is anticipated that dynamic wrinkling may be Table 3 The ratio r=rf for sandwich configurations considered in examples (Eq. 17) Core
Balsa
H-30
H-45
H-100
H-160
Case 1 Case 2
1.084 1.151
1.076 1.137
1.081 1.145
1.122 1.219
1.166 1.298
Case 1, quasi-isotropic chopped fiber glass/vinyl-ester facings; Case 2, balanced cross-ply carbon/epoxy facings (AS4/3501-6).
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dangerous only in the case of a very thin facing supported by a light core. It is possible that the Plantema model and the simplified elasticity model outlined in the paper may affect the numerical results, as compared to those generated by the Winkler model, but the qualitative conclusions are unlikely to be altered.
Acknowledgements This work was supported by the Office of Naval Research Program on Composites for Marine Structures (Contract ONR N00014-00-1-0323). The Program Manager is Dr Yapa D.S. Rajapakse.
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