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Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures
Accepted Manuscript Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures P. Nagasankar, S. Balasivanandha Prabu, R. Velmurugan PII: DOI: Reference:
S0167-6636(15)00170-2 http://dx.doi.org/10.1016/j.mechmat.2015.08.002 MECMAT 2461
To appear in:
Mechanics of Materials
Received Date: Revised Date:
5 October 2014 23 June 2015
Please cite this article as: Nagasankar, P., Balasivanandha Prabu, S., Velmurugan, R., Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures, Mechanics of Materials (2015), doi: http://dx.doi.org/10.1016/j.mechmat.2015.08.002
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Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures Nagasankar P1,2 ∗, Balasivanandha Prabu S1*, Velmurugan R3 1
Department of Mechanical Engineering, College of Engineering Guindy, Anna University, Chennai - 600025, India 2
Department of Mechanical Engineering, S.A. Engineering College, Chennai - 600077, India
3
Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Abstract This paper investigates the effect of different orientations of fiber in the skins and different thicknesses of the skins and polypropylene honeycomb core (PPHC) on the transverse shear damping of the sandwich using experimental and theoretical studies. Three different sandwiches, SW1 (thickness of top and bottom FRP skin: 3.15mm each with 10mm thick PPHC), SW2 (thickness of top and bottom FRP skin: 1.575mm each with 5mm thick PPHC), SW3 (thickness of top and bottom FRP skin: 1.575mm each with 10mm thick PPHC) were fabricated for conducting experimental work. In order to study the effect of fiber orientation of the skin on the natural frequency and loss factors, five different orientations (all 0°, ±30°, ±45°, ±60° and all 90°) were considered. An impulse technique was used to calculate the natural frequency and loss factor of the composites. The natural frequency and loss factor were also computed theoretically and compared. As the in-plane load of the strong FRP skins imposes on the soft honeycomb core under dynamic condition, it causes a large transverse shear deformation on the core. This shear deformation leads to a high energy dissipation / loss factor of the sandwich. At the 0° fiber oriented sandwich, the loss factor value (η = 0.0234) becomes significantly higher without losing its natural frequency (stiffness) value (fn = 142.12 Hz), which is not so in the case of FRP composites
∗
Corresponding Author. Email addresses: [email protected] & [email protected] Tel. No.: 91-44-22357747 & Fax: 91-44-22357744. 1
having the loss factor (η) value of 0.00218 and the natural frequency value (fn) of 114.23 Hz. It is also found that the transverse shear effect and damping loss factor increases with the increase in the thicknesses of the skins and core of the sandwich.
Keywords Damping; Honeycomb structures; Laminate; Polymers; Sandwich materials; Vibration
1. Introduction Fiber reinforced polymer (FRP) composites are commonly used in automotive, aerospace, marine, and civil structures applications. Damping is an essential property of the structures when vibration control is critical to extend the service life of the structures. There are many types of damping or energy dissipation methods like active and passive (material or system) damping (Trindade 2002) and aerodynamic damping, viscous damping (Zhang et al. 2011) etc. This paper is confined to the material damping of the polymeric composite materials which can also be termed as passive damping. The damping loss factor of the polymer matrix composite (PMC) materials is about 10 to 100 times greater than that of the structural metallic materials (Yang et al. 2013). It is due to the dissipation of energy by the visco-elastic behaviour of the matrix, the fiber-matrix interface, and damage. It is also influenced by the visco-elastic layer properties (Yim et al. 2003, Meaud et al. 2013) as well as the layer orientations (Adams and Maheri 1994), interface (Chandra 2003), temperatures (Youssef and Berthelot 2006, Jen and Lin 2013), etc. The influence of all these parameters on the damping loss factor of PMC was studied by Ni and Adams (1984), Adams and Maheri (1994), Yim (1999), Yim and Gillespie (2000) and Remillat (2007). However, the studies to demonstrate the above mentioned factors on the damping loss factor of the honeycomb sandwich FRP composites have been limited. The use of a honeycomb material in a composite laminate is witnessed in aerospace structures, where weight reduction and high flexural rigidity constitute the primary concern in designing structures (Bassani et al. 2013, Wang and Yang 2000). The FRP composite has a high in-plane stiffness-to-weight ratio, whereas the honeycomb sandwich possesses a high bending stiffness-to-weight ratio, in addition to the high in-plane stiffness to weight ratio (Maheri et al. 2008).
2
As the present study focuses on the modal characteristics of polypropylene (PP) honeycomb sandwich panel, many papers related to this domain have been reviewed. Adams and Maheri (1993) have studied the variation in the specific damping capacity (SDC) with respect to shear stress amplitudes for aluminum and Nomex honeycomb sandwiches and reported that a sound honeycomb sample would experience linear damping at low stress amplitudes. The contribution of the different constituent parts in the aluminum honeycomb sandwich construction on the damping property has also been investigated by Maheri and Adams (1994) under flexural loading conditions. The findings have demonstrated a high degree of interdependence between the skins and the core, contributing to the overall damping. More recently, they (Maheri et al. 2008) have used the basic laminate theory, together with the Rayleigh–Ritz method, the finite element analysis and the experimental method, to predict damping and frequency of the honeycomb sandwich for space applications and considered the inherent damping of aluminum and carbon fibrereinforced plastic (CFRP) skins in sandwich panels. The damping effect of sandwich beams was also experimentally investigated with fine solder balls inserted in the honeycomb cells, losing the advantage of its light-weight (Wang and Yang (2000)). It was found that the damping was quite effective on measuring the reduction in amplitude from the first two modes of vibration. Yim et al. (2003) have studied the in-plane and transverse damping effect of 0° laminated composite sandwich cantilever beams with interleaving of a solid visco-elastic layer. They have also predicted the variation in the loss factor with the aspect ratio (length/thickness) of composites. Jung and Aref (2003) have combined the solid viscoelastic material with a polymer honeycomb material and studied the damping behavior under different strain and frequencies inputs. The interleaving of both the materials was expected to improve the stiffness and the energy dissipation capability when subjected to in-plane shear loading. The dynamic properties of honeycomb sandwich can also be influenced by the cell size of a honeycomb core. For an instance, Havaldar et al. (2012) have studied the effect of different cell sizes of PPHC core on the fundamental natural frequency of FRP honeycomb sandwich panels under two different boundary conditions and found that the natural 3
frequencies decreased with increase in cell sizes in both the boundary conditions. The debonding length of the sandwich composites also play vital role on their dynamic properties. For example, Idriss et al. (2013) have stated that the damping effect and stiffness of the sandwich composites were sensitive to the debonding length under static and dynamic loadings. Most importantly, there are hardly limited research reports available for the dynamic analysis (with both the frequency and damping values) of the PPHC sandwich, despite the fact that it possesses remarkable mechanical properties. Hence, this study focuses on the prediction of the dynamic characteristics of PPHC sandwich composites, theoretically and experimentally and the results have been compared. The paper has exceptionally studied the effect of the different orientations of FRP skins and the different thicknesses of the skin and core on the transverse shear damping of the sandwich with the help of appropriate equations 2. Constituent materials and their properties The matrix material consists of the low temperature curing epoxy resin with a specific gravity of 1.14 at 25°C, the solvent based high temperature curing hardener and the accelerator. The unidirectional glass fiber having a density of 2.50g/cm3 was used as the reinforcement. The polypropylene honeycomb core, possessing high strength-to-weight ratio, energy and sound absorption, and corrosion resistance, was supplied by M/s Good fellow Cambridge Limited of London. It is covered with polyester scrim on both sides, which facilitates a better bonding of the honeycomb with the surface panels. It also acts as a barrier to prevent the resin from leaking into the core cell. The transverse shear moduli and compression modulus of the core in the out of plane direction have been computed by the following Eqs. (1-3), developed by Meraghni et al. (1999). Gxzhc =
twG(1 + 2 cos θ) , 2k sin θ(cos θ + 1)
(1)
G yzhc =
twG(tw + 2k sin θ) , 2k (1 + cos θ)(tw + k sin θ)
(2)
4
Ehc =
tw E (1 + 2 cos θ) , 2k sin θ(cos θ + 1)
(3)
where k=(d/2)/sinθ, θ=2π/6=60°, k is the hexagonal cell parameter, θ is the cell angle, tw is the wall thickness with the size of 0.25 mm, d is the diameter of the cell with the size of 8 mm. G and E are the shear and Young’s modulus of PP solid materials having the values of 320MPa and 900 MPa respectively. All these parameters are clearly shown in Fig.1. 3. Fabrication of the PP honeycomb sandwich structures In order to prevent the leaking of epoxy resin into the honeycomb core, to maintain the uniform thicknesses of top and bottom skins and to obtain effective bonding between the skins and core, the lay-up has been done in two stages. It has been briefly explained in Fig. 2. Since it is carried out in two different stages and periods, the resin flows evenly and wets the skins and core uniformly. FRP of 6.3mm thickness and the three different sandwich specimens of SW1 (thickness of top and bottom FRP skins - 3.15 mm each and a thickness of core - 10 mm), SW2 (thickness of top and bottom FRP skins - 1.575 mm each and a thickness of core - 5 mm) and SW3 (thickness of top and bottom FRP skins - 1.575 mm each and a thickness of core - 10 mm) were fabricated for this study. 4. Experimental setup The Impulse technique used elsewhere (Alam et al. 2007, Havaldar et al. 2012, Senthil Kumar et al. 2014, Nagasankar et al. 2014), shown in Fig. 3, has been used to study the vibration behaviour of the specimen. One end of the laminated specimens is rigidly clamped to a support by screws, and the other end, which carries a properly positioned accelerometer, is free to vibrate like a cantilever beam. The input load is applied on the specimen, by hitting it with the instrumented impact hammer, and the output (response) is captured by the accelerometer, and read by the data acquisition card. With the help of the FFT analysis, these data are then transferred to a computer, where the natural frequencies are evaluated. Then, the loss factor can be obtained by the half power bandwidth method as given below.
η=
f1 − f 2 , 2 fn 5
(4)
where, f1 and f2= Band width at the half-power points of resonant peak for nth mode fn= Natural frequency 5. Theoretical prediction of the loss factors and natural frequencies
The loss factor of the sandwich, ηsw or the specific damping capacity, Ψsw is usually defined by the ratio of the dissipated energy to the maximum stored energy. In the sandwich panel, the strain energy generated and dissipated due to the in-plane deformations is attributable solely to the FRP skins, and the same due to the out-of plane deformations to the PPHC core. Yim et al.' s (2003) theoretical approach which was used for the prediction of the loss factor of FRP sandwich (interleaved with solid visco-elastic layer) has been applied here to find out the loss factor of the PPHC sandwiches, under different orientations of the fiber from Eqs. (5) - (10). ηsw =
∆W x + ∆W y + ∆W xy + ∆Wxz + ∆W yz Ψ sw ∆W , = = 2π 2 πW Wb + W xz + W yz
(5)
The dissipated energy in X-direction is calculated as follows: l
∆Wx = ∫ 2 0
h2
∫
l h2
πηL σ x ε x dzdx = 2 πηL ∫
hc /2
∫
(6)
σ x ε x dzdx,
0 hc /2
l h2 2π 2 * * * * * C11 Q11 + C12 Q12 + C16 Q16 z 2 dz , = *2 ∫ M 1 dx ηL ∫ m 2 m 2 C11 + mnC16 I 0 hc 2
(
)(
)
(7)
where l is the length of the beam, I* the normalizing factor which equals hf 3/12, hf -the thickness of the FRP skins, hc -the thickness of the core, h -the thickness of the sandwich, ηL the longitudinal loss factor, [Cij*] the normalized flexural compliance, [Qij] the stiffness
of the lamina, M1=P x, the bending moment under point loading or M1=1/2 wx2, the bending moment under distributed loading, and m=cos θ; n=sin θ. Similarly, the dissipated energies in Y and XY- directions are calculated by Eqs. (89).
6
l h2 2π 2 * * * * * C11 Q11 + C12 Q12 + C16 Q16 z 2 dz , ∆Wy = *2 ∫ M1 dx ηT ∫ n 2 n 2C11 − mnC16 I 0 hc 2
(
∆Wxy =
)(
)
h 2 l 2π 2 * 2 2 * M dx C16 η 1 LT ∫ mn 2 mnC11 − m − n *2 ∫ I 0 hc 2
(
) ) (C
(
* 11Q11
(8)
* * Q12 + C16 Q16 z 2 dz , (9) + C12
)
where ηT is the transverse loss factor, and ηLT is the longitudinal or the in-plane shear loss factor. The bending strain energy of the FRP beam is calculated from Eq. 10. l
Wb =
∫ M 1k1dx = 0
* C11 I*
l
∫ M 1 dx 2
.
(10)
0
The loss factors (ηL, ηT, ηLT) and their respective moduli (EL, ET, GLT) of the FRP have been estimated from Eqs. (11 -16) used by Saravanos and Chamis (1989). E L = V f E L( f ) + Vm E L( m ) , ηL =
(11)
η(Lf ) E L( f )V f + η(Lm ) E L( m )Vm EL( f )V f + E L( m )Vm
(
)
( m) T
ET = 1 − V f E
ηT = η(T f ) V f
(
(12)
V f ET( m )
(
Vf
) EE
T (m) T
G LT + η (LTm ) 1 − V f (f) G LT
(
)
,
(13)
,
(14)
E (m ) 1 − V f 1 − T( f ) ET
ET + ηT( m ) 1 − V f (f) ET
η LT = η(LTf ) V f
G LT = 1 −
+
,
) GG
LT (m) LT
,
V f G L( mT )
G L( mT ) + 1−
(15) ,
(16)
G (m) V f 1 − L( Tf ) G LT
where the superscripts f and m represent the fiber and matrix respectively, the subscripts L, T and LT the properties of the moduli and loss factors in the longitudinal, transverse, longitudinal-transverse directions, and Vf and Vm the volume fractions of the fiber and matrix. 7
Taking into account the values of loss factors and poisson's ratio of the fiber taken from the reference (Saravanos and Chamis 1989), the other properties of the matrix and FRP specimen with respect to the direction of the fiber have been measured with the help of mechanical and dynamic tests. The results are tabulated in Table 1. It is learnt from the present theoretical and experimental studies, that the out-of plane deformations on the PPHC core can be easily generated in the XZ and YZ directions, by the in-plane stresses of the FRP skins under impact loading i.e., the in-plane load of the FRP skin is very much greater than that of the polypropylene honeycomb core. Yim et al. (2003), in their study, have considered both the in-plane and transverse shear stresses and their related energy terms only in the XZ direction for the 0°oriented FRP specimen. But in the present work, as the thickness of specimens is very large and the PPHC core is lighter (a mere density value of 80 kg/m3) and less stiff, the same have been considered in both the directions with respect to fiber orientations and thicknesses of skins and core. Since, the in-plane load of FRP skin at the 90° fiber orientation is negligible in the XZ direction, the transverse shear/energy terms have become zero at this orientation and direction. Similarly, the same at the 0° fiber orientation is negligible in the YZ direction; the same terms become zero at this orientation and direction. According to Yim et al. (2003), the in-plane stresses of the FRP can be evaluated as follows; σ(xt ) =
zM 1 2 zM m ( C11* Q11( t ) + C12* Q12( t ) + C16* Q16( t ) ) = * 1 F1 , * I I
(17)
σ (yt ) =
zM 1 2 * ( t ) zM n ( C11Q11 + C12* Q12( t ) + C16* Q16( t ) ) = * 1 F2 , * I I
(18)
σ (xyt ) =
zM 1 zM * ( − mn ) ( C11* Q11( t ) + C12* Q12( t ) + C16 Q16( t ) ) = * 1 F3 , * I I
(19)
where F1, F2 and F3 are constants stated in the square bracket of the above equations. In order to calculate the transverse shear stresses (σxz(t) and σyz(t)), the in-plane stresses stated above can be substituted in the following equilibrium equations. (t ) ∂σ(xt ) ∂σ xy ∂σ(xzt ) + + = 0, ∂x ∂y ∂z
∂σ(xyt ) ∂x
+
∂σ(yt ) ∂y
+
∂σ(yzt ) ∂z
= 0,
(t ) ∂σ(xzt ) ∂σ yz ∂σ(zt ) + + = 0. ∂x ∂y ∂z
The transverse shear stresses, as per Yim et al. (2003), are calculated as follows; 8
(20)
σ(xzt )
1 =− * I
zt
∫
F1
−h 2
dM 1 zdz. dx
(21)
According to the beam theory, the transverse shear terms are obtained from the following equations.
σ(xzt )
σ(yzt )
Q ( b) =− * I Q ( b) =− * I
zt
∫
F1 zdz
(22)
F2 zdz.
(23)
−h 2 zt
∫ −h 2
where, Q(b) = P = bending force in the point loading or Q(b)=wx in the distributed loading. Since the transverse shear terms of PP honeycomb core are influenced by the effect of the in-plane load of all lamina of the top and bottom FRP skins, the above equations have been suitably modified for the polypropylene honeycomb sandwich as given below. z
σ xz
Q ( b ) sk = − * 2 ∫ F1 zdz , I −h 2
σ yz
Q ( b ) sk = − * 2 ∫ F2 zdz. I −h 2
(24)
z
(25)
Then, their shear strain energies have been calculated from Eqs. (26) & (27):
σ 2xz 1 Wxz = ∫ dv = 2Gxzhc Gxzhc vol 1 Wyz = G yzhc
l
l
hc /2
∫0 ∫
σ2xz bdzdx ,
(26)
0
hc /2
∫0 ∫
σ2yz bdzdx.
(27)
0
Similarly, the dissipated energies due to the transverse shear stress can be obtained as: ηxz ∆Wxz = 2π hc G xz hc
∆W yz
η yz = 2 π hc G yz hc
l
hc /2
∫0 ∫ 0
l
hc /2
∫0 ∫ 0
σ2xz bdzdx ,
(28)
σ2yz bdzdx ,
(29)
9
where ηxzhc , η yzhc - the transverse shear loss factors of PPHC core in XZ and YZ directions, Gxzhc , G yzhc - the transverse shear modulus of PPHC core in XZ and YZ directions, b , l -
breadth , length of the skins and core (b=25mm and l=200mm). Considering the five energy terms (Wx, Wy, Wxy, Wxz, Wyz) related to both the inplane stresses of the skins and the transverse shear stresses of the core, the loss factors of the sandwich have been calculated from Eq. (5). But in the case of FRP, it has been calculated from the first three energy terms of their in-plane stresses as per Eq. (30).
η frp =
∆ Wx + ∆ W y + ∆W xy Wb
.
(30)
Normally, the natural frequency with the transverse shear deformations is neglected for the modal analysis of the slender laminated plates/beams, whereas in the case of the frequency of the thick sandwich beams, the natural frequency with both the flexural and transverse shear deformations are incorporated. Using the Dunkerlay method cited by Blevins (1979), the natural frequency of the same has been predicted comprehensively, considering the effect of both the skins and the core. The equation is given as follows: 1 f s2w
=
1 1 , + 2 fF f S2
(31)
where fF , fS - the frequencies of the flexural and shear deformations respectively. It is necessary to predict them separately for the FRP skins and the PPHC core. The frequency of flexural deformations can be evaluated from the following equation. f F = f F( sk ) + f F( hc ) ,
(32)
where f F( sk ) , f F( hc ) - the flexural frequencies of the skins and core respectively. The frequency of the first term in Eq. (32) can be calculated by Eq. (37), using the input values of the moduli (Ex, Ey, Gxy), Poisson's ratio (νxy), and mass per unit area of the plate (γ). As it was initially developed for computing only the frequency of the laminates with unidirectional fibers, the method of computation has been slightly modified for the 10
laminates with differently oriented fibers, i.e., the transformation relations (given in Eqs. (33) - (36)) have been used to convert their input values from the fiber coordinate system to the laminate coordinate system. n2 2 m2 n 2 , 1 m2 2 2 2 = m − n ν + n − m ν + ( ( LT ) TL ) E x EL ET GLT
(33)
m2 m2n2 , 1 n2 2 = ( n − m2νLT ) + ( m2 − n2νTL ) + Ey EL ET GLT
(34)
2
( m2 − n2 ) , 1 4m2n2 4m2n2 = (1 +νLT ) + (1 + νTL ) + Gxy EL ET GLT
(35)
νxy
(36)
Ex
=
ν yx Ey
=
m2 2 n2 m2n2 , m νLT − n2 ) + ( n2νTL − m2 ) + ( EL ET GLT
where, Ex, Ey, Gxy, νxy, νyx and EL, ET, GLT, νLT, νTL are the longitudinal modulus, transverse modulus, shear modulus and Poisson’s ratio of the laminate co-ordinate system and the fiber co-ordinate system respectively. As the orthotropic axes align with the plate axes of a uniform rectangular plate, which spans x = 0 to l and y = 0 to b, the flexural frequency of the skins as per Blevins (1979) is stated as follows: 4 π G14 D x G2 D y 2 H 1 H 2 D xy 4 Dk ( J 1 J 2 − H 1 H 2 ) = + + + 1 4 b4 l 2b2 l 2 b2 2γ 2 l
f F( sk )
1
2
; i = 1, 2,.. j = 1, 2, .. , (37)
where Dx, Dy, Dk and Dxy are orthotropic constants and defined by the plate analysis as: E x h 3f
Dx =
Dy =
Dk =
,
12 (1 − ν xy ν yx ) E y h 3f
,
12 (1 − ν xy ν yx )
Gxy h 3f 12
(38)
,
Dxy = Dx ν yx + 2Dk
,
11
G, H, J = the dimensionless parameters, directly provided by Blevins (1979), i.e., G1=0.597,
H1=-0.0870 and J1=0.47, h f = thickness of the composite plate, and γ =
mass/unit area of the plate. Since the cantilever beam has one end fixed and other ends free, the dimensionless parameters, G2= H2 = J2 = 0. The flexural frequency of the PPHC core is stated as; 1/2
f
( hc ) F
λ 2 Ehc I hc = ; i = 1, 2..., 2πl 2 m
(39)
where λ=1.8751 is the Blevins's parameter depending on the boundary conditions, Ehc, Ihc, l, b and hc are the elastic modulus, moment of inertia, length, width and thickness of the core respectively, Ihc=bhc3/12 and m=bµhc is the mass of the core/unit length and µ is its density, having the value of 0.08g.cm-3.
The transverse shear frequencies of the core are evaluated in the XZ and YZ directions by the following equations.
f
( hc ) S XZi
1/2 λˆ i KGxzhc λˆ i KG yzhc ( hc ) = , f = SYZi 2 πl µ 2 πl µ
1/2
,
(2i − 1)π 10(1 + ν) λˆ i = , K= , 2 12 + 11ν
(40)
(41)
where λˆ i is the Blevins's dimensionless parameter [defined depending on the boundary conditions], i is the mode number, K is the shear co-efficient, ν is the poisson's ratio, l is the length of the beam. The resultant of the shear frequencies given in the above Eq. (40) is calculated as follows: f s = f S(i hc ) =
2
( f ) +( f ) ( hc ) S XZi
( hc ) SYZi
2
.
(42)
The value of the above equation has been substituted along with the result of Eq. (32) in Eq. (31) to calculate the frequency of the sandwiches SW1, SW2 and SW3. As this 12
study has also compared the same with the frequency of FRP, Eq. (37) alone has been used for the corresponding calculation. 6. Results and discussions
Despite the fact that there are various damping mechanisms, like the influence of temperatures, fiber orientations, interfaces, etc., the large damping loss factor of the FRP can be achieved only at the expense of the stiffness/the natural frequency. But, in this study, it is noted that the values of loss factor increase with the increase in the frequency, after interleaving the honeycomb core into the FRP skins. The above said values have been evaluated under different orientations of fibers and thicknesses of the FRP skins and PP honeycomb core of sandwich specimens. In Figs. 4 (a - c) and 6 (a - c), the variation in the experimental and theoretical results of the FRP and honeycomb sandwiched FRP panels respectively, with the different orientations and thicknesses, are shown with the error bars. The deviations between the experimental and theoretical results may be due to various factors, such as small hairline cracks, tiny blow holes, fine impurities, improper bonding at the interface between the fibers and the matrix, or at the interface between the FRP skins and the PPHC core. Hence, the interface plays very important role in quantifying these error. (The extensive study of the interface damping effect / three phase damping (fiber + matrix + interface) model has been elaborately carried out by Nagasankar et al. (2014)). Since the two phase damping (fiber + matrix) model is considered in this study, the error may be due to the damping of the interfaces. These variations are observed to be higher at 0° fiber orientation for both the FRP and sandwich specimens (Fig. 4(a-c)). This higher value of interface damping is caused mainly by the larger value of Young’s modulus (Ex) obtained at 0° fiber orientation. (Ref. Eq. 19 of the reference (Nagasankar et al. 2014)). On examining the deviations between the experimental and theoretical results of FRP and sandwich specimens (Fig. 4(a-c)), the sandwich specimens show the large variations. The reason behind this is due to the additional interface effects or the relative movement between the layers of two different materials of FRP skins and PPHC core under dynamic condition apart from the interface effects between the laminae and the interface effects between the fibers and the matrix of FRP skins. Though the FRP skin materials are 13
very much compatible and bonded with PPHC core and its polyester scrim materials, there may be some discontinuities/tiny holes/hairline cracks existing between the skins and core or between the fiber and matrix. When the specimen is subjected to the in-plane load, the energy dissipation occurs due to the displacement of the irregularities or discontinuities at the interface between the skins and core. This kind of energy dissipation at the interfaces of FRP composites can be referred from the article (Chandra, 2003). Since the sandwich (SW1) has more number of layers in their skins than those of the sandwiches (SW2 and SW3), the energy dissipation due to these discontinuities at the interfaces would be more. Hence, the interface damping/error of SW1 is comparatively higher. When comparing to the loss factors, the errors of natural frequencies of the FRP and the sandwiches SW1, SW2 and SW3 are lower and almost consistent under all the orientations of fiber (Fig. 6(a-c)). The errors observed in the values of loss factors and natural frequencies which do not affect the dynamic behaviour of the sandwiches significantly may be permitted. Comparing those results of the two categories of specimens (FRP and SW1) after hitting them by hammer, the loss factor and natural frequency of the sandwich (SW1) are observed to be higher than that of the FRP at all the layups shown in the Figs. 4(a) and 6(a) respectively. The reason behind the higher loss factor of the sandwich is quite interesting, and is only due to the significant amount of the transverse shear deformation/stress of the soft PPHC core, generated by the larger in-plane load imposed by the stronger FRP skins on the core. It is due to the fact that the soft PPHC core material having a lower density value of 0.08 g.cm-3 and the lower shear modulus values of GXZhc, GYZhc (obtained from Eqs. 1 and 2) is easily deformed in the transverse direction by the stronger FRP skins having the larger density values of 1.855 g.cm-3 and the larger modulus values of Ex = 38.21 GPa, Ey = 7.34 GPa and Gxy = 2.814 GPa at 0° fiber orientation. This large shear deformation of the soft PPHC core leads to produce the higher transverse shear energy dissipations (∆Wxz, ∆Wyz given in Eq. 5) of the sandwich, which in turn, increases the total loss factor (ηsw) of
the sandwich given in Eq. 5. It is also noted from Fig. 4 (a), that the loss factors of the FRP curve has an upward slope i.e., it increases with the increase in the fiber orientations, whereas in the case of 14
honeycomb interleaved FRP structure, the loss factors of the sandwich curve (SW1) has a downward slope i.e., it decreases with the increase in the orientations of fibers. As the loss factor of the sandwich is comparatively high at the 0° oriented fibers, the difference between the loss factor of both FRP and sandwich curves appears to be very large at this orientation, and gradually lowers at the other higher orientations. It is due to the highest inplane load produced by the 0° oriented FRP skins when compared to the in-plane loads of other oriented FRP skins (modulus values of the other oriented skins gradually decrease with increase in their fiber orientations as per the transformation Eqs. 33-36) on the honeycomb core, which then easily get deformed in the transverse direction and generate comparatively large damping loss factor. Hence, from the above discussion, it is understood that the large deviation of the loss factors at 0° fiber orientation between SW1 and FRP specimens is mainly due to the large transverse shear deformation of SW 1 (the large transverse shear deformation is due to the higher in-plane load of fiber reinforced skins) and lower bending deformation of FRP specimens (it is due to its higher bending stiffness). But, this result is reversed at 90° fiber orientation, i.e., the slight deviation/convergence of the loss factors between SW1 and FRP specimens at this orientation is due to the lower transverse shear deformation of SW 1, dissipating less energy/loss factor (it is due to the negligible in-plane load of fiber reinforced skins, besides the addition of the core does not produce additional damping.) and the large bending deformation of FRP dissipating more energy/loss factor (it is due to its lower bending stiffness) occur at 90° fiber orientation., their loss factor values have been slightly deviated/converged. Apart from the large loss factor values, we can also get maximum natural frequency/stiffness at 0° fiber orientation. For an instance, the natural frequency at the 0° fiber orientation of the sandwich SW1 is about 140 Hz which is comparatively higher than that (85 Hz) of 90° fiber orientation as shown in Fig. 6(a). The upward or downward slope of the sandwich curves, intended mainly for the loss factors, depends on the thicknesses of the negligible weighed honeycomb core and the strong FRP skins. In order to prove this, the analysis has also been extended to the other two different thicknesses of sandwiches, SW2 and SW3 (their sandwich curves are shown in Fig. 4(b)). As the thicknesses of both the skins and core are smaller in the sandwich, 15
SW2, the transverse shear effect of core (such as the shear modulus values of GXZhc, GYZhc obtained from Eqs. 1 and 2) gets reduced resulting in the lower loss factor values at the 0° and 30° orientations and the larger values at the other orientations of fibers i.e., as the inplane effect of the structure dominates their transverse shear properties when those thicknesses are reduced, the lower loss factor of the sandwich is caused at the lower orientations of the fibers as like a normal FRP structure producing the same effect at the lower orientations of the fibers. When the thickness of core alone is increased into the sandwich, SW3, the transverse shear effect becomes more at the 0° oriented fibers. The variation of loss factors with respect to the different thicknesses of skin and core is also presented for the same orientation of fibers in Fig. 5. It is understood that the larger the thickness of the soft honeycomb core the greater is the transverse shear deformation. Due to which, larger energy is dissipated at the maximum thickness of the core. The type of materials like Nomex paper, Polypropylene and aluminum having different densities (say 0.031 g.cm-3, 0.08 g.cm-3 and 0.192 g.cm-3 respectively) and different strength, stiffness and moduli of GXZhc, GYZhc obtained (Eqs. 1 and 2) used for making the honeycomb core and FRP skins also influence the transverse shear loss factor, besides the different orientations and thicknesses of the sandwiches. The natural frequencies of sandwich, SW1 are also higher than those of the FRP and other sandwiches SW2 and SW3 at all the fiber orientations as shown in Figs. 6 (a & b). It is due to the large thickness of the sandwich, SW1 causing more bending stiffness/natural frequency than that of the FRP and other sandwiches, SW2 and SW3. Hence, the natural frequency of the sandwiches also increases with the thickness of the specimens. The variation of the natural frequencies with the different thicknesses of the skin and core is also reported for the 0° oriented fibers in Fig.7. The deviations between the theoretical and experimental results may be caused by various factors, like improper bonding, hairline cracks, tiny blow holes, fine impurities, etc., at the interface between the fiber and the matrix, or the interface between the FRP skins and the PPHC core. 7. Conclusions
This research has focused mainly on the study of the dynamic behaviour of the
16
PPHC sandwich under different fiber orientations, and different thicknesses of core and skins by the theoretical and experimental methods and their results have been compared. 1. As the transverse shear attributed by the PPHC core (10mm thickness) is maximum due to the high in-plane load of the 0° fiber oriented FRP skins (3.15 mm thicknesses of each top and bottom FRP skins), the large damping loss factor is obtained at this orientation and thickness. Moreover, the sandwich with the same fiber orientation also gives the maximum natural frequency/stiffness. Hence, 0° fiber oriented sandwich with the large thickness is considered as the best choice among the different orientations of the sandwich having varying stiffness levels. 2. It is found that the damping loss factor of the sandwich with the '3.15mm skin and 10mm core thicknesses' is larger than that of pure FRP under the different fiber orientations. The damping loss factor of the pure FRP normally increases only at the expense of natural frequency or stiffness, whereas in the case of the above said sandwich the damping loss factor increases with 20 % higher natural frequency than the pure FRP. 3. When the loss factor and the natural frequencies are compared among the different thicknesses of the skins and core (i.e., different sandwiches with '3.15mm and 10mm', '1.575mm and 5mm' and '1.575 mm and 10mm' thicknesses of both skins and core respectively), the parameters of the sandwich with 3.15mm skin and 10mm core thicknesses are observed to be high due to the maximum thicknesses of both the skins and core materials of the above three categories of sandwiches. As the maximum thickness of the PPHC core undergo greater transverse shear deformation, the above said sandwich produces higher damping loss factors. Acknowledgements
The authors wish to acknowledge the support of M/s ROTO Polymers Pvt. Ltd of Chennai, Atalon (DEWETRON-India) of Sriperumbudur, CIPET of Chennai and IIT Madras of Chennai. References
Adams, R.D., Maheri, M.R., 1993. The dynamic shear properties of structural honeycomb materials. Compos. Sci. Technol. 47, 15-23. 17
Adams, R.D., Maheri, M.R., 1994. Dynamic flexural properties of anisotropic fibrous composite beams. Compos. Sci. Technol. 50, 497-514. Alam, M.S., Wahab, M.A., Jenkins, C.H., 2007. Mechanics in naturally compliant structures. Mech. Mater. 39 (2), 145–160. Bassani, P., Biffi, C.A., Carnevale, M., Lecis, N., Previtali, B., Conte, A.L., 2013. Passive damping of slender and light structures. Mater. Design 45, 88–95. Blevins, R.D., 1979. Formulas for Natural Frequency and Mode Shape. Krieger Publishing Company, Malabar, 1-492.
Chandra, R., 2003. A study of damping in fiber-reinforced composites, J. Vib. Control 262, 475-496. Havaldar, S.S., Sharma, R.S., Antony, A.P.M.D., Bangaru, M., 2012. Effect of Cell Size on the Fundamental Natural Frequency of FRP Honeycomb Sandwich Panels. J. Miner. Mater. Charact. Eng. (11), 653-660. Idriss, M., Mahi, A.E.I., Assarar, M., Guerjouma, R.E.I., 2013. Damping analysis in cyclic fatigue loading of sandwich beams with debonding. Compos. Part B-Eng. 44, 597-603. Jen, Y.M., Lin, H.B., 2013. Temperature-dependent monotonic and fatigue bending strengths of adhesively bonded aluminum honeycomb sandwich beams. Mater. Design 45, 393–406. Jiang, C.H., Chang, Y.H., Kam, T.Y., 2014. Optimal design of rectangular composite flat-panel sound radiators considering excitation location. Compos. Struct. 108, 65-76. Jung, W.Y., Aref, A.J. 2003. A combined honeycomb and solid viscoelastic material for structural damping applications. Mech. Mater. 35 (8), 831–844. Maheri, M,R., Adams, R.D., Hugon, J., 2008. Vibration damping in sandwich panels. J. Mater. Sci. 43, 6604-6618. Maheri, M.R., Adams, R.D., 1994. Steady state flexural vibration damping honeycomb sandwich beams. Compos. Sci. Technol. 52, 333-347. Meaud, J., Sain, T., Hulbert, G.M., Waas, A.M., 2013. Analysis and optimal design of layered composites with high stiffness and high damping. Int. J. Solids Struct. 50 18
(9), 1342–1353 Meraghni, F., Desrumaux, F., Benzeggagh, M.L., 1999. Mechanical behaviour of cellular core for structural sandwich panels. Compos. Part A-Appl. S. 30, 767-779. Nagasankar, P., Balasivanandha Prabu, S., Velmurugan, R., 2014. The effect of the strand diameter on the damping characteristics of fiber reinforced polymer matrix composites: Theoretical and Experimental study’, Int. J. Mech. Sci. (39): 279-288. Ni, R.G., Adams, R.D., 1984. A rational method for obtaining the dynamic mechanical properties of laminate for predicting the stiffness and damping of laminated plates and beams. Composites 15 (3), 193-199. Remillat. C., 2007. Damping mechanism of polymers filled with elastic particles, Mech. Mater. 39, 525–537. Saravanos, D.A., Chamis, C.C., 1989. Unified Micromechanics of Damping for Unidirectional Fiber Reinforced Composites. NASA TM-102107, National Aeronautics and Space Administration, Cleveland, 1-25. Senthil Kumar, K., Siva, I., Jeyaraj, P., Winowlin, Jappes J.T., Amico, S.C., Rajini, N., 2014. Synergy of fiber length and content on free vibration and damping behavior of natural fiber reinforced polyester composite beams. Mater. Design 56, 379-386. Trindade, M.A., Benjeddon, A., 2002. Hybrid Active-Passive Damping Treatments using Visco-elastic and Piezoelectric materials: Review and Assessment. J. Vib. Control 8(6), 699-745. Wang, B., Yang, M., 2000. Damping of honeycomb sandwich beams. J. Mater. Process Tech. 105, 67-72. Yang, J., Xiong, J., Ma, L., Wang, B., Zhang, G., Wu, L., 2013. Vibration and damping characteristics of hybrid carbon fiber composite pyramidal truss sandwich panels with viscoelastic layers. Compos. Struct. 106, 570-580. Yim, J.H., 1999. A damping analysis of composite laminates using the closed form expression for the basic damping of Poisson's ratio. Compos. Struct. 46(4), 405-411. Yim, J.H., Cho, S.Y., Seo, Y.J., Jang, B.Z., 2003. A study on material damping of 0° laminated composite sandwich cantilever beams with a viscoelastic layer. Compos. Struct. 60(4), 367-374. 19
Yim, J.H., Gillespie Jr, J.W., 2000. Damping characteristics of 0° & 90° AS4/3501-6 unidirectional laminates including the transverse shear effect. Compos. Struct. 50(3), 217-225. Youssef, S., Berthelot, J.M., 2006. Temperature effect on the damping properties of unidirectional glass fibre composites. Compos. Part B-Eng. 37, 346-355. Zhang, H., Xie, X., Zhao, J.L., 2011. Parametric Vibration of Carbon fiber reinforced plastic cables with damping effects in long-span cable-stayed bridges. J. Vib. Control 17 (14), 2117- 2130.
List of Tables and Figures
Fig. 1. Honeycomb cell structure and its parameters. Fig. 2. Two Stage Fabricating Method. Fig. 3. Impulse testing setup with the half power bandwidth method. Fig. 4. Variations of the theoretical and experimental loss factors for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3). (a) FRP vs. SW1; (b) FRP vs.SW2; (c) FRP vs. SW3 Fig. 5. Variations of the theoretical and experimental loss factors with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels. Fig. 6. Variations of the theoretical and experimental frequencies for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3). (a) FRP vs. SW1; (b) FRP vs.SW2; (c) FRP vs. SW3 Fig. 7. Variations of the theoretical and experimental frequencies with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
Table 1. Mechanical properties of the FRP, PPHC, fiber and matrix materials 20
Figure(s)
tw k
Cell diameter, d
c
hc Fig. 1. Honeycomb cell structure and its parameters.
Additional Weight
Semi-finished Bottom skin Sandwich
Top Mould PP Honeycomb
Top mould with additional weight
skinn
Bottom skin
Bottom skin
skinSkin
Bottom Mould (a) First stage curing done on the four staked up layers along with the honeycomb under the mould pressure for 8 hours
(b) Semi-finished sandwich taken from the first stage and inverted upside down.
Fig. 2. Two Stage Fabricating Method.
21
(c) Second stage curing done on the four staked up layers along with the inverted semi-finished sandwich under the mould pressure for another 8 hours.
Figure(s)
Fixed end
Impact Hammer
PP honeycomb Specimen
Accelerometer
Amplitude maximum, Amax Amplitude factor = Amax/√2
PC Frequency f1
Data Acquisition Card
Fig. 3. Impulse testing setup with the half power bandwidth method.
(a) FRP vs. SW1
22
fn
f2
(b) FRP vs. SW2
(c) FRP vs. SW3 Fig. 4. Variations of the theoretical and experimental loss factors for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3).
23
Figure(s)
Fig. 5. Variations of the theoretical and experimental loss factors with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
(a) FRP vs. SW1
24
(b) FRP vs. SW2
(c) FRP vs. SW3 Fig. 6. Variations of the theoretical and experimental frequencies for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1,SW2 & SW3). 25
Figure(s)
Fig. 7. Variations of the theoretical and experimental frequencies with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
26
Table 1. Mechanical properties of the FRP, PPHC, fiber and matrix materials
Materials
Longi -tudinal modulus (GPa)
Transverse modulus (GPa)
Shear modulus (GPa)
Longi tudina l loss factor %
Trans- Longi verse tudina loss l shear factor loss factor % %
FRP
38.21
7.34
2.814
-
-
-
PPHC
-
-
-
-
-
-
Fiber Matrix
72.5 2.737
72.5 2.737
30.20 1.013
0.175* 0.175* 0.095* 1.398 1.398 1.569
Transverse shear loss factor %
Poisson 's ratio
Density g.cm-3
-
0.304
1.855
-
0.08
0.2* 0.351
-
0.0488 ηxz = ηyz 0.095* 1.569
* Values were taken from the reference (Saravanos and Chamis 1989 )
21
Highlights • The analytical derivation and expressions have been used to study the damping characteristics of the Polypropylene honeycomb structure with different fiber orientations (0°, 30°, 45°, 60°, 90°) and thicknesses. • The damping loss factor of the sandwich structure is increased by the large in-plane load of the FRP skins at 0° fiber orientation on the transverse shear of the soft honeycomb core. • The damping loss factor also increases with the increase in thicknesses of the skins and core in the sandwich structure.
22
S0167-6636(15)00170-2 http://dx.doi.org/10.1016/j.mechmat.2015.08.002 MECMAT 2461
To appear in:
Mechanics of Materials
Received Date: Revised Date:
5 October 2014 23 June 2015
Please cite this article as: Nagasankar, P., Balasivanandha Prabu, S., Velmurugan, R., Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures, Mechanics of Materials (2015), doi: http://dx.doi.org/10.1016/j.mechmat.2015.08.002
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Role of different fiber orientations and thicknesses of the skins and the core on the transverse shear damping of polypropylene honeycomb sandwich structures Nagasankar P1,2 ∗, Balasivanandha Prabu S1*, Velmurugan R3 1
Department of Mechanical Engineering, College of Engineering Guindy, Anna University, Chennai - 600025, India 2
Department of Mechanical Engineering, S.A. Engineering College, Chennai - 600077, India
3
Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Abstract This paper investigates the effect of different orientations of fiber in the skins and different thicknesses of the skins and polypropylene honeycomb core (PPHC) on the transverse shear damping of the sandwich using experimental and theoretical studies. Three different sandwiches, SW1 (thickness of top and bottom FRP skin: 3.15mm each with 10mm thick PPHC), SW2 (thickness of top and bottom FRP skin: 1.575mm each with 5mm thick PPHC), SW3 (thickness of top and bottom FRP skin: 1.575mm each with 10mm thick PPHC) were fabricated for conducting experimental work. In order to study the effect of fiber orientation of the skin on the natural frequency and loss factors, five different orientations (all 0°, ±30°, ±45°, ±60° and all 90°) were considered. An impulse technique was used to calculate the natural frequency and loss factor of the composites. The natural frequency and loss factor were also computed theoretically and compared. As the in-plane load of the strong FRP skins imposes on the soft honeycomb core under dynamic condition, it causes a large transverse shear deformation on the core. This shear deformation leads to a high energy dissipation / loss factor of the sandwich. At the 0° fiber oriented sandwich, the loss factor value (η = 0.0234) becomes significantly higher without losing its natural frequency (stiffness) value (fn = 142.12 Hz), which is not so in the case of FRP composites
∗
Corresponding Author. Email addresses: [email protected] & [email protected] Tel. No.: 91-44-22357747 & Fax: 91-44-22357744. 1
having the loss factor (η) value of 0.00218 and the natural frequency value (fn) of 114.23 Hz. It is also found that the transverse shear effect and damping loss factor increases with the increase in the thicknesses of the skins and core of the sandwich.
Keywords Damping; Honeycomb structures; Laminate; Polymers; Sandwich materials; Vibration
1. Introduction Fiber reinforced polymer (FRP) composites are commonly used in automotive, aerospace, marine, and civil structures applications. Damping is an essential property of the structures when vibration control is critical to extend the service life of the structures. There are many types of damping or energy dissipation methods like active and passive (material or system) damping (Trindade 2002) and aerodynamic damping, viscous damping (Zhang et al. 2011) etc. This paper is confined to the material damping of the polymeric composite materials which can also be termed as passive damping. The damping loss factor of the polymer matrix composite (PMC) materials is about 10 to 100 times greater than that of the structural metallic materials (Yang et al. 2013). It is due to the dissipation of energy by the visco-elastic behaviour of the matrix, the fiber-matrix interface, and damage. It is also influenced by the visco-elastic layer properties (Yim et al. 2003, Meaud et al. 2013) as well as the layer orientations (Adams and Maheri 1994), interface (Chandra 2003), temperatures (Youssef and Berthelot 2006, Jen and Lin 2013), etc. The influence of all these parameters on the damping loss factor of PMC was studied by Ni and Adams (1984), Adams and Maheri (1994), Yim (1999), Yim and Gillespie (2000) and Remillat (2007). However, the studies to demonstrate the above mentioned factors on the damping loss factor of the honeycomb sandwich FRP composites have been limited. The use of a honeycomb material in a composite laminate is witnessed in aerospace structures, where weight reduction and high flexural rigidity constitute the primary concern in designing structures (Bassani et al. 2013, Wang and Yang 2000). The FRP composite has a high in-plane stiffness-to-weight ratio, whereas the honeycomb sandwich possesses a high bending stiffness-to-weight ratio, in addition to the high in-plane stiffness to weight ratio (Maheri et al. 2008).
2
As the present study focuses on the modal characteristics of polypropylene (PP) honeycomb sandwich panel, many papers related to this domain have been reviewed. Adams and Maheri (1993) have studied the variation in the specific damping capacity (SDC) with respect to shear stress amplitudes for aluminum and Nomex honeycomb sandwiches and reported that a sound honeycomb sample would experience linear damping at low stress amplitudes. The contribution of the different constituent parts in the aluminum honeycomb sandwich construction on the damping property has also been investigated by Maheri and Adams (1994) under flexural loading conditions. The findings have demonstrated a high degree of interdependence between the skins and the core, contributing to the overall damping. More recently, they (Maheri et al. 2008) have used the basic laminate theory, together with the Rayleigh–Ritz method, the finite element analysis and the experimental method, to predict damping and frequency of the honeycomb sandwich for space applications and considered the inherent damping of aluminum and carbon fibrereinforced plastic (CFRP) skins in sandwich panels. The damping effect of sandwich beams was also experimentally investigated with fine solder balls inserted in the honeycomb cells, losing the advantage of its light-weight (Wang and Yang (2000)). It was found that the damping was quite effective on measuring the reduction in amplitude from the first two modes of vibration. Yim et al. (2003) have studied the in-plane and transverse damping effect of 0° laminated composite sandwich cantilever beams with interleaving of a solid visco-elastic layer. They have also predicted the variation in the loss factor with the aspect ratio (length/thickness) of composites. Jung and Aref (2003) have combined the solid viscoelastic material with a polymer honeycomb material and studied the damping behavior under different strain and frequencies inputs. The interleaving of both the materials was expected to improve the stiffness and the energy dissipation capability when subjected to in-plane shear loading. The dynamic properties of honeycomb sandwich can also be influenced by the cell size of a honeycomb core. For an instance, Havaldar et al. (2012) have studied the effect of different cell sizes of PPHC core on the fundamental natural frequency of FRP honeycomb sandwich panels under two different boundary conditions and found that the natural 3
frequencies decreased with increase in cell sizes in both the boundary conditions. The debonding length of the sandwich composites also play vital role on their dynamic properties. For example, Idriss et al. (2013) have stated that the damping effect and stiffness of the sandwich composites were sensitive to the debonding length under static and dynamic loadings. Most importantly, there are hardly limited research reports available for the dynamic analysis (with both the frequency and damping values) of the PPHC sandwich, despite the fact that it possesses remarkable mechanical properties. Hence, this study focuses on the prediction of the dynamic characteristics of PPHC sandwich composites, theoretically and experimentally and the results have been compared. The paper has exceptionally studied the effect of the different orientations of FRP skins and the different thicknesses of the skin and core on the transverse shear damping of the sandwich with the help of appropriate equations 2. Constituent materials and their properties The matrix material consists of the low temperature curing epoxy resin with a specific gravity of 1.14 at 25°C, the solvent based high temperature curing hardener and the accelerator. The unidirectional glass fiber having a density of 2.50g/cm3 was used as the reinforcement. The polypropylene honeycomb core, possessing high strength-to-weight ratio, energy and sound absorption, and corrosion resistance, was supplied by M/s Good fellow Cambridge Limited of London. It is covered with polyester scrim on both sides, which facilitates a better bonding of the honeycomb with the surface panels. It also acts as a barrier to prevent the resin from leaking into the core cell. The transverse shear moduli and compression modulus of the core in the out of plane direction have been computed by the following Eqs. (1-3), developed by Meraghni et al. (1999). Gxzhc =
twG(1 + 2 cos θ) , 2k sin θ(cos θ + 1)
(1)
G yzhc =
twG(tw + 2k sin θ) , 2k (1 + cos θ)(tw + k sin θ)
(2)
4
Ehc =
tw E (1 + 2 cos θ) , 2k sin θ(cos θ + 1)
(3)
where k=(d/2)/sinθ, θ=2π/6=60°, k is the hexagonal cell parameter, θ is the cell angle, tw is the wall thickness with the size of 0.25 mm, d is the diameter of the cell with the size of 8 mm. G and E are the shear and Young’s modulus of PP solid materials having the values of 320MPa and 900 MPa respectively. All these parameters are clearly shown in Fig.1. 3. Fabrication of the PP honeycomb sandwich structures In order to prevent the leaking of epoxy resin into the honeycomb core, to maintain the uniform thicknesses of top and bottom skins and to obtain effective bonding between the skins and core, the lay-up has been done in two stages. It has been briefly explained in Fig. 2. Since it is carried out in two different stages and periods, the resin flows evenly and wets the skins and core uniformly. FRP of 6.3mm thickness and the three different sandwich specimens of SW1 (thickness of top and bottom FRP skins - 3.15 mm each and a thickness of core - 10 mm), SW2 (thickness of top and bottom FRP skins - 1.575 mm each and a thickness of core - 5 mm) and SW3 (thickness of top and bottom FRP skins - 1.575 mm each and a thickness of core - 10 mm) were fabricated for this study. 4. Experimental setup The Impulse technique used elsewhere (Alam et al. 2007, Havaldar et al. 2012, Senthil Kumar et al. 2014, Nagasankar et al. 2014), shown in Fig. 3, has been used to study the vibration behaviour of the specimen. One end of the laminated specimens is rigidly clamped to a support by screws, and the other end, which carries a properly positioned accelerometer, is free to vibrate like a cantilever beam. The input load is applied on the specimen, by hitting it with the instrumented impact hammer, and the output (response) is captured by the accelerometer, and read by the data acquisition card. With the help of the FFT analysis, these data are then transferred to a computer, where the natural frequencies are evaluated. Then, the loss factor can be obtained by the half power bandwidth method as given below.
η=
f1 − f 2 , 2 fn 5
(4)
where, f1 and f2= Band width at the half-power points of resonant peak for nth mode fn= Natural frequency 5. Theoretical prediction of the loss factors and natural frequencies
The loss factor of the sandwich, ηsw or the specific damping capacity, Ψsw is usually defined by the ratio of the dissipated energy to the maximum stored energy. In the sandwich panel, the strain energy generated and dissipated due to the in-plane deformations is attributable solely to the FRP skins, and the same due to the out-of plane deformations to the PPHC core. Yim et al.' s (2003) theoretical approach which was used for the prediction of the loss factor of FRP sandwich (interleaved with solid visco-elastic layer) has been applied here to find out the loss factor of the PPHC sandwiches, under different orientations of the fiber from Eqs. (5) - (10). ηsw =
∆W x + ∆W y + ∆W xy + ∆Wxz + ∆W yz Ψ sw ∆W , = = 2π 2 πW Wb + W xz + W yz
(5)
The dissipated energy in X-direction is calculated as follows: l
∆Wx = ∫ 2 0
h2
∫
l h2
πηL σ x ε x dzdx = 2 πηL ∫
hc /2
∫
(6)
σ x ε x dzdx,
0 hc /2
l h2 2π 2 * * * * * C11 Q11 + C12 Q12 + C16 Q16 z 2 dz , = *2 ∫ M 1 dx ηL ∫ m 2 m 2 C11 + mnC16 I 0 hc 2
(
)(
)
(7)
where l is the length of the beam, I* the normalizing factor which equals hf 3/12, hf -the thickness of the FRP skins, hc -the thickness of the core, h -the thickness of the sandwich, ηL the longitudinal loss factor, [Cij*] the normalized flexural compliance, [Qij] the stiffness
of the lamina, M1=P x, the bending moment under point loading or M1=1/2 wx2, the bending moment under distributed loading, and m=cos θ; n=sin θ. Similarly, the dissipated energies in Y and XY- directions are calculated by Eqs. (89).
6
l h2 2π 2 * * * * * C11 Q11 + C12 Q12 + C16 Q16 z 2 dz , ∆Wy = *2 ∫ M1 dx ηT ∫ n 2 n 2C11 − mnC16 I 0 hc 2
(
∆Wxy =
)(
)
h 2 l 2π 2 * 2 2 * M dx C16 η 1 LT ∫ mn 2 mnC11 − m − n *2 ∫ I 0 hc 2
(
) ) (C
(
* 11Q11
(8)
* * Q12 + C16 Q16 z 2 dz , (9) + C12
)
where ηT is the transverse loss factor, and ηLT is the longitudinal or the in-plane shear loss factor. The bending strain energy of the FRP beam is calculated from Eq. 10. l
Wb =
∫ M 1k1dx = 0
* C11 I*
l
∫ M 1 dx 2
.
(10)
0
The loss factors (ηL, ηT, ηLT) and their respective moduli (EL, ET, GLT) of the FRP have been estimated from Eqs. (11 -16) used by Saravanos and Chamis (1989). E L = V f E L( f ) + Vm E L( m ) , ηL =
(11)
η(Lf ) E L( f )V f + η(Lm ) E L( m )Vm EL( f )V f + E L( m )Vm
(
)
( m) T
ET = 1 − V f E
ηT = η(T f ) V f
(
(12)
V f ET( m )
(
Vf
) EE
T (m) T
G LT + η (LTm ) 1 − V f (f) G LT
(
)
,
(13)
,
(14)
E (m ) 1 − V f 1 − T( f ) ET
ET + ηT( m ) 1 − V f (f) ET
η LT = η(LTf ) V f
G LT = 1 −
+
,
) GG
LT (m) LT
,
V f G L( mT )
G L( mT ) + 1−
(15) ,
(16)
G (m) V f 1 − L( Tf ) G LT
where the superscripts f and m represent the fiber and matrix respectively, the subscripts L, T and LT the properties of the moduli and loss factors in the longitudinal, transverse, longitudinal-transverse directions, and Vf and Vm the volume fractions of the fiber and matrix. 7
Taking into account the values of loss factors and poisson's ratio of the fiber taken from the reference (Saravanos and Chamis 1989), the other properties of the matrix and FRP specimen with respect to the direction of the fiber have been measured with the help of mechanical and dynamic tests. The results are tabulated in Table 1. It is learnt from the present theoretical and experimental studies, that the out-of plane deformations on the PPHC core can be easily generated in the XZ and YZ directions, by the in-plane stresses of the FRP skins under impact loading i.e., the in-plane load of the FRP skin is very much greater than that of the polypropylene honeycomb core. Yim et al. (2003), in their study, have considered both the in-plane and transverse shear stresses and their related energy terms only in the XZ direction for the 0°oriented FRP specimen. But in the present work, as the thickness of specimens is very large and the PPHC core is lighter (a mere density value of 80 kg/m3) and less stiff, the same have been considered in both the directions with respect to fiber orientations and thicknesses of skins and core. Since, the in-plane load of FRP skin at the 90° fiber orientation is negligible in the XZ direction, the transverse shear/energy terms have become zero at this orientation and direction. Similarly, the same at the 0° fiber orientation is negligible in the YZ direction; the same terms become zero at this orientation and direction. According to Yim et al. (2003), the in-plane stresses of the FRP can be evaluated as follows; σ(xt ) =
zM 1 2 zM m ( C11* Q11( t ) + C12* Q12( t ) + C16* Q16( t ) ) = * 1 F1 , * I I
(17)
σ (yt ) =
zM 1 2 * ( t ) zM n ( C11Q11 + C12* Q12( t ) + C16* Q16( t ) ) = * 1 F2 , * I I
(18)
σ (xyt ) =
zM 1 zM * ( − mn ) ( C11* Q11( t ) + C12* Q12( t ) + C16 Q16( t ) ) = * 1 F3 , * I I
(19)
where F1, F2 and F3 are constants stated in the square bracket of the above equations. In order to calculate the transverse shear stresses (σxz(t) and σyz(t)), the in-plane stresses stated above can be substituted in the following equilibrium equations. (t ) ∂σ(xt ) ∂σ xy ∂σ(xzt ) + + = 0, ∂x ∂y ∂z
∂σ(xyt ) ∂x
+
∂σ(yt ) ∂y
+
∂σ(yzt ) ∂z
= 0,
(t ) ∂σ(xzt ) ∂σ yz ∂σ(zt ) + + = 0. ∂x ∂y ∂z
The transverse shear stresses, as per Yim et al. (2003), are calculated as follows; 8
(20)
σ(xzt )
1 =− * I
zt
∫
F1
−h 2
dM 1 zdz. dx
(21)
According to the beam theory, the transverse shear terms are obtained from the following equations.
σ(xzt )
σ(yzt )
Q ( b) =− * I Q ( b) =− * I
zt
∫
F1 zdz
(22)
F2 zdz.
(23)
−h 2 zt
∫ −h 2
where, Q(b) = P = bending force in the point loading or Q(b)=wx in the distributed loading. Since the transverse shear terms of PP honeycomb core are influenced by the effect of the in-plane load of all lamina of the top and bottom FRP skins, the above equations have been suitably modified for the polypropylene honeycomb sandwich as given below. z
σ xz
Q ( b ) sk = − * 2 ∫ F1 zdz , I −h 2
σ yz
Q ( b ) sk = − * 2 ∫ F2 zdz. I −h 2
(24)
z
(25)
Then, their shear strain energies have been calculated from Eqs. (26) & (27):
σ 2xz 1 Wxz = ∫ dv = 2Gxzhc Gxzhc vol 1 Wyz = G yzhc
l
l
hc /2
∫0 ∫
σ2xz bdzdx ,
(26)
0
hc /2
∫0 ∫
σ2yz bdzdx.
(27)
0
Similarly, the dissipated energies due to the transverse shear stress can be obtained as: ηxz ∆Wxz = 2π hc G xz hc
∆W yz
η yz = 2 π hc G yz hc
l
hc /2
∫0 ∫ 0
l
hc /2
∫0 ∫ 0
σ2xz bdzdx ,
(28)
σ2yz bdzdx ,
(29)
9
where ηxzhc , η yzhc - the transverse shear loss factors of PPHC core in XZ and YZ directions, Gxzhc , G yzhc - the transverse shear modulus of PPHC core in XZ and YZ directions, b , l -
breadth , length of the skins and core (b=25mm and l=200mm). Considering the five energy terms (Wx, Wy, Wxy, Wxz, Wyz) related to both the inplane stresses of the skins and the transverse shear stresses of the core, the loss factors of the sandwich have been calculated from Eq. (5). But in the case of FRP, it has been calculated from the first three energy terms of their in-plane stresses as per Eq. (30).
η frp =
∆ Wx + ∆ W y + ∆W xy Wb
.
(30)
Normally, the natural frequency with the transverse shear deformations is neglected for the modal analysis of the slender laminated plates/beams, whereas in the case of the frequency of the thick sandwich beams, the natural frequency with both the flexural and transverse shear deformations are incorporated. Using the Dunkerlay method cited by Blevins (1979), the natural frequency of the same has been predicted comprehensively, considering the effect of both the skins and the core. The equation is given as follows: 1 f s2w
=
1 1 , + 2 fF f S2
(31)
where fF , fS - the frequencies of the flexural and shear deformations respectively. It is necessary to predict them separately for the FRP skins and the PPHC core. The frequency of flexural deformations can be evaluated from the following equation. f F = f F( sk ) + f F( hc ) ,
(32)
where f F( sk ) , f F( hc ) - the flexural frequencies of the skins and core respectively. The frequency of the first term in Eq. (32) can be calculated by Eq. (37), using the input values of the moduli (Ex, Ey, Gxy), Poisson's ratio (νxy), and mass per unit area of the plate (γ). As it was initially developed for computing only the frequency of the laminates with unidirectional fibers, the method of computation has been slightly modified for the 10
laminates with differently oriented fibers, i.e., the transformation relations (given in Eqs. (33) - (36)) have been used to convert their input values from the fiber coordinate system to the laminate coordinate system. n2 2 m2 n 2 , 1 m2 2 2 2 = m − n ν + n − m ν + ( ( LT ) TL ) E x EL ET GLT
(33)
m2 m2n2 , 1 n2 2 = ( n − m2νLT ) + ( m2 − n2νTL ) + Ey EL ET GLT
(34)
2
( m2 − n2 ) , 1 4m2n2 4m2n2 = (1 +νLT ) + (1 + νTL ) + Gxy EL ET GLT
(35)
νxy
(36)
Ex
=
ν yx Ey
=
m2 2 n2 m2n2 , m νLT − n2 ) + ( n2νTL − m2 ) + ( EL ET GLT
where, Ex, Ey, Gxy, νxy, νyx and EL, ET, GLT, νLT, νTL are the longitudinal modulus, transverse modulus, shear modulus and Poisson’s ratio of the laminate co-ordinate system and the fiber co-ordinate system respectively. As the orthotropic axes align with the plate axes of a uniform rectangular plate, which spans x = 0 to l and y = 0 to b, the flexural frequency of the skins as per Blevins (1979) is stated as follows: 4 π G14 D x G2 D y 2 H 1 H 2 D xy 4 Dk ( J 1 J 2 − H 1 H 2 ) = + + + 1 4 b4 l 2b2 l 2 b2 2γ 2 l
f F( sk )
1
2
; i = 1, 2,.. j = 1, 2, .. , (37)
where Dx, Dy, Dk and Dxy are orthotropic constants and defined by the plate analysis as: E x h 3f
Dx =
Dy =
Dk =
,
12 (1 − ν xy ν yx ) E y h 3f
,
12 (1 − ν xy ν yx )
Gxy h 3f 12
(38)
,
Dxy = Dx ν yx + 2Dk
,
11
G, H, J = the dimensionless parameters, directly provided by Blevins (1979), i.e., G1=0.597,
H1=-0.0870 and J1=0.47, h f = thickness of the composite plate, and γ =
mass/unit area of the plate. Since the cantilever beam has one end fixed and other ends free, the dimensionless parameters, G2= H2 = J2 = 0. The flexural frequency of the PPHC core is stated as; 1/2
f
( hc ) F
λ 2 Ehc I hc = ; i = 1, 2..., 2πl 2 m
(39)
where λ=1.8751 is the Blevins's parameter depending on the boundary conditions, Ehc, Ihc, l, b and hc are the elastic modulus, moment of inertia, length, width and thickness of the core respectively, Ihc=bhc3/12 and m=bµhc is the mass of the core/unit length and µ is its density, having the value of 0.08g.cm-3.
The transverse shear frequencies of the core are evaluated in the XZ and YZ directions by the following equations.
f
( hc ) S XZi
1/2 λˆ i KGxzhc λˆ i KG yzhc ( hc ) = , f = SYZi 2 πl µ 2 πl µ
1/2
,
(2i − 1)π 10(1 + ν) λˆ i = , K= , 2 12 + 11ν
(40)
(41)
where λˆ i is the Blevins's dimensionless parameter [defined depending on the boundary conditions], i is the mode number, K is the shear co-efficient, ν is the poisson's ratio, l is the length of the beam. The resultant of the shear frequencies given in the above Eq. (40) is calculated as follows: f s = f S(i hc ) =
2
( f ) +( f ) ( hc ) S XZi
( hc ) SYZi
2
.
(42)
The value of the above equation has been substituted along with the result of Eq. (32) in Eq. (31) to calculate the frequency of the sandwiches SW1, SW2 and SW3. As this 12
study has also compared the same with the frequency of FRP, Eq. (37) alone has been used for the corresponding calculation. 6. Results and discussions
Despite the fact that there are various damping mechanisms, like the influence of temperatures, fiber orientations, interfaces, etc., the large damping loss factor of the FRP can be achieved only at the expense of the stiffness/the natural frequency. But, in this study, it is noted that the values of loss factor increase with the increase in the frequency, after interleaving the honeycomb core into the FRP skins. The above said values have been evaluated under different orientations of fibers and thicknesses of the FRP skins and PP honeycomb core of sandwich specimens. In Figs. 4 (a - c) and 6 (a - c), the variation in the experimental and theoretical results of the FRP and honeycomb sandwiched FRP panels respectively, with the different orientations and thicknesses, are shown with the error bars. The deviations between the experimental and theoretical results may be due to various factors, such as small hairline cracks, tiny blow holes, fine impurities, improper bonding at the interface between the fibers and the matrix, or at the interface between the FRP skins and the PPHC core. Hence, the interface plays very important role in quantifying these error. (The extensive study of the interface damping effect / three phase damping (fiber + matrix + interface) model has been elaborately carried out by Nagasankar et al. (2014)). Since the two phase damping (fiber + matrix) model is considered in this study, the error may be due to the damping of the interfaces. These variations are observed to be higher at 0° fiber orientation for both the FRP and sandwich specimens (Fig. 4(a-c)). This higher value of interface damping is caused mainly by the larger value of Young’s modulus (Ex) obtained at 0° fiber orientation. (Ref. Eq. 19 of the reference (Nagasankar et al. 2014)). On examining the deviations between the experimental and theoretical results of FRP and sandwich specimens (Fig. 4(a-c)), the sandwich specimens show the large variations. The reason behind this is due to the additional interface effects or the relative movement between the layers of two different materials of FRP skins and PPHC core under dynamic condition apart from the interface effects between the laminae and the interface effects between the fibers and the matrix of FRP skins. Though the FRP skin materials are 13
very much compatible and bonded with PPHC core and its polyester scrim materials, there may be some discontinuities/tiny holes/hairline cracks existing between the skins and core or between the fiber and matrix. When the specimen is subjected to the in-plane load, the energy dissipation occurs due to the displacement of the irregularities or discontinuities at the interface between the skins and core. This kind of energy dissipation at the interfaces of FRP composites can be referred from the article (Chandra, 2003). Since the sandwich (SW1) has more number of layers in their skins than those of the sandwiches (SW2 and SW3), the energy dissipation due to these discontinuities at the interfaces would be more. Hence, the interface damping/error of SW1 is comparatively higher. When comparing to the loss factors, the errors of natural frequencies of the FRP and the sandwiches SW1, SW2 and SW3 are lower and almost consistent under all the orientations of fiber (Fig. 6(a-c)). The errors observed in the values of loss factors and natural frequencies which do not affect the dynamic behaviour of the sandwiches significantly may be permitted. Comparing those results of the two categories of specimens (FRP and SW1) after hitting them by hammer, the loss factor and natural frequency of the sandwich (SW1) are observed to be higher than that of the FRP at all the layups shown in the Figs. 4(a) and 6(a) respectively. The reason behind the higher loss factor of the sandwich is quite interesting, and is only due to the significant amount of the transverse shear deformation/stress of the soft PPHC core, generated by the larger in-plane load imposed by the stronger FRP skins on the core. It is due to the fact that the soft PPHC core material having a lower density value of 0.08 g.cm-3 and the lower shear modulus values of GXZhc, GYZhc (obtained from Eqs. 1 and 2) is easily deformed in the transverse direction by the stronger FRP skins having the larger density values of 1.855 g.cm-3 and the larger modulus values of Ex = 38.21 GPa, Ey = 7.34 GPa and Gxy = 2.814 GPa at 0° fiber orientation. This large shear deformation of the soft PPHC core leads to produce the higher transverse shear energy dissipations (∆Wxz, ∆Wyz given in Eq. 5) of the sandwich, which in turn, increases the total loss factor (ηsw) of
the sandwich given in Eq. 5. It is also noted from Fig. 4 (a), that the loss factors of the FRP curve has an upward slope i.e., it increases with the increase in the fiber orientations, whereas in the case of 14
honeycomb interleaved FRP structure, the loss factors of the sandwich curve (SW1) has a downward slope i.e., it decreases with the increase in the orientations of fibers. As the loss factor of the sandwich is comparatively high at the 0° oriented fibers, the difference between the loss factor of both FRP and sandwich curves appears to be very large at this orientation, and gradually lowers at the other higher orientations. It is due to the highest inplane load produced by the 0° oriented FRP skins when compared to the in-plane loads of other oriented FRP skins (modulus values of the other oriented skins gradually decrease with increase in their fiber orientations as per the transformation Eqs. 33-36) on the honeycomb core, which then easily get deformed in the transverse direction and generate comparatively large damping loss factor. Hence, from the above discussion, it is understood that the large deviation of the loss factors at 0° fiber orientation between SW1 and FRP specimens is mainly due to the large transverse shear deformation of SW 1 (the large transverse shear deformation is due to the higher in-plane load of fiber reinforced skins) and lower bending deformation of FRP specimens (it is due to its higher bending stiffness). But, this result is reversed at 90° fiber orientation, i.e., the slight deviation/convergence of the loss factors between SW1 and FRP specimens at this orientation is due to the lower transverse shear deformation of SW 1, dissipating less energy/loss factor (it is due to the negligible in-plane load of fiber reinforced skins, besides the addition of the core does not produce additional damping.) and the large bending deformation of FRP dissipating more energy/loss factor (it is due to its lower bending stiffness) occur at 90° fiber orientation., their loss factor values have been slightly deviated/converged. Apart from the large loss factor values, we can also get maximum natural frequency/stiffness at 0° fiber orientation. For an instance, the natural frequency at the 0° fiber orientation of the sandwich SW1 is about 140 Hz which is comparatively higher than that (85 Hz) of 90° fiber orientation as shown in Fig. 6(a). The upward or downward slope of the sandwich curves, intended mainly for the loss factors, depends on the thicknesses of the negligible weighed honeycomb core and the strong FRP skins. In order to prove this, the analysis has also been extended to the other two different thicknesses of sandwiches, SW2 and SW3 (their sandwich curves are shown in Fig. 4(b)). As the thicknesses of both the skins and core are smaller in the sandwich, 15
SW2, the transverse shear effect of core (such as the shear modulus values of GXZhc, GYZhc obtained from Eqs. 1 and 2) gets reduced resulting in the lower loss factor values at the 0° and 30° orientations and the larger values at the other orientations of fibers i.e., as the inplane effect of the structure dominates their transverse shear properties when those thicknesses are reduced, the lower loss factor of the sandwich is caused at the lower orientations of the fibers as like a normal FRP structure producing the same effect at the lower orientations of the fibers. When the thickness of core alone is increased into the sandwich, SW3, the transverse shear effect becomes more at the 0° oriented fibers. The variation of loss factors with respect to the different thicknesses of skin and core is also presented for the same orientation of fibers in Fig. 5. It is understood that the larger the thickness of the soft honeycomb core the greater is the transverse shear deformation. Due to which, larger energy is dissipated at the maximum thickness of the core. The type of materials like Nomex paper, Polypropylene and aluminum having different densities (say 0.031 g.cm-3, 0.08 g.cm-3 and 0.192 g.cm-3 respectively) and different strength, stiffness and moduli of GXZhc, GYZhc obtained (Eqs. 1 and 2) used for making the honeycomb core and FRP skins also influence the transverse shear loss factor, besides the different orientations and thicknesses of the sandwiches. The natural frequencies of sandwich, SW1 are also higher than those of the FRP and other sandwiches SW2 and SW3 at all the fiber orientations as shown in Figs. 6 (a & b). It is due to the large thickness of the sandwich, SW1 causing more bending stiffness/natural frequency than that of the FRP and other sandwiches, SW2 and SW3. Hence, the natural frequency of the sandwiches also increases with the thickness of the specimens. The variation of the natural frequencies with the different thicknesses of the skin and core is also reported for the 0° oriented fibers in Fig.7. The deviations between the theoretical and experimental results may be caused by various factors, like improper bonding, hairline cracks, tiny blow holes, fine impurities, etc., at the interface between the fiber and the matrix, or the interface between the FRP skins and the PPHC core. 7. Conclusions
This research has focused mainly on the study of the dynamic behaviour of the
16
PPHC sandwich under different fiber orientations, and different thicknesses of core and skins by the theoretical and experimental methods and their results have been compared. 1. As the transverse shear attributed by the PPHC core (10mm thickness) is maximum due to the high in-plane load of the 0° fiber oriented FRP skins (3.15 mm thicknesses of each top and bottom FRP skins), the large damping loss factor is obtained at this orientation and thickness. Moreover, the sandwich with the same fiber orientation also gives the maximum natural frequency/stiffness. Hence, 0° fiber oriented sandwich with the large thickness is considered as the best choice among the different orientations of the sandwich having varying stiffness levels. 2. It is found that the damping loss factor of the sandwich with the '3.15mm skin and 10mm core thicknesses' is larger than that of pure FRP under the different fiber orientations. The damping loss factor of the pure FRP normally increases only at the expense of natural frequency or stiffness, whereas in the case of the above said sandwich the damping loss factor increases with 20 % higher natural frequency than the pure FRP. 3. When the loss factor and the natural frequencies are compared among the different thicknesses of the skins and core (i.e., different sandwiches with '3.15mm and 10mm', '1.575mm and 5mm' and '1.575 mm and 10mm' thicknesses of both skins and core respectively), the parameters of the sandwich with 3.15mm skin and 10mm core thicknesses are observed to be high due to the maximum thicknesses of both the skins and core materials of the above three categories of sandwiches. As the maximum thickness of the PPHC core undergo greater transverse shear deformation, the above said sandwich produces higher damping loss factors. Acknowledgements
The authors wish to acknowledge the support of M/s ROTO Polymers Pvt. Ltd of Chennai, Atalon (DEWETRON-India) of Sriperumbudur, CIPET of Chennai and IIT Madras of Chennai. References
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Adams, R.D., Maheri, M.R., 1994. Dynamic flexural properties of anisotropic fibrous composite beams. Compos. Sci. Technol. 50, 497-514. Alam, M.S., Wahab, M.A., Jenkins, C.H., 2007. Mechanics in naturally compliant structures. Mech. Mater. 39 (2), 145–160. Bassani, P., Biffi, C.A., Carnevale, M., Lecis, N., Previtali, B., Conte, A.L., 2013. Passive damping of slender and light structures. Mater. Design 45, 88–95. Blevins, R.D., 1979. Formulas for Natural Frequency and Mode Shape. Krieger Publishing Company, Malabar, 1-492.
Chandra, R., 2003. A study of damping in fiber-reinforced composites, J. Vib. Control 262, 475-496. Havaldar, S.S., Sharma, R.S., Antony, A.P.M.D., Bangaru, M., 2012. Effect of Cell Size on the Fundamental Natural Frequency of FRP Honeycomb Sandwich Panels. J. Miner. Mater. Charact. Eng. (11), 653-660. Idriss, M., Mahi, A.E.I., Assarar, M., Guerjouma, R.E.I., 2013. Damping analysis in cyclic fatigue loading of sandwich beams with debonding. Compos. Part B-Eng. 44, 597-603. Jen, Y.M., Lin, H.B., 2013. Temperature-dependent monotonic and fatigue bending strengths of adhesively bonded aluminum honeycomb sandwich beams. Mater. Design 45, 393–406. Jiang, C.H., Chang, Y.H., Kam, T.Y., 2014. Optimal design of rectangular composite flat-panel sound radiators considering excitation location. Compos. Struct. 108, 65-76. Jung, W.Y., Aref, A.J. 2003. A combined honeycomb and solid viscoelastic material for structural damping applications. Mech. Mater. 35 (8), 831–844. Maheri, M,R., Adams, R.D., Hugon, J., 2008. Vibration damping in sandwich panels. J. Mater. Sci. 43, 6604-6618. Maheri, M.R., Adams, R.D., 1994. Steady state flexural vibration damping honeycomb sandwich beams. Compos. Sci. Technol. 52, 333-347. Meaud, J., Sain, T., Hulbert, G.M., Waas, A.M., 2013. Analysis and optimal design of layered composites with high stiffness and high damping. Int. J. Solids Struct. 50 18
(9), 1342–1353 Meraghni, F., Desrumaux, F., Benzeggagh, M.L., 1999. Mechanical behaviour of cellular core for structural sandwich panels. Compos. Part A-Appl. S. 30, 767-779. Nagasankar, P., Balasivanandha Prabu, S., Velmurugan, R., 2014. The effect of the strand diameter on the damping characteristics of fiber reinforced polymer matrix composites: Theoretical and Experimental study’, Int. J. Mech. Sci. (39): 279-288. Ni, R.G., Adams, R.D., 1984. A rational method for obtaining the dynamic mechanical properties of laminate for predicting the stiffness and damping of laminated plates and beams. Composites 15 (3), 193-199. Remillat. C., 2007. Damping mechanism of polymers filled with elastic particles, Mech. Mater. 39, 525–537. Saravanos, D.A., Chamis, C.C., 1989. Unified Micromechanics of Damping for Unidirectional Fiber Reinforced Composites. NASA TM-102107, National Aeronautics and Space Administration, Cleveland, 1-25. Senthil Kumar, K., Siva, I., Jeyaraj, P., Winowlin, Jappes J.T., Amico, S.C., Rajini, N., 2014. Synergy of fiber length and content on free vibration and damping behavior of natural fiber reinforced polyester composite beams. Mater. Design 56, 379-386. Trindade, M.A., Benjeddon, A., 2002. Hybrid Active-Passive Damping Treatments using Visco-elastic and Piezoelectric materials: Review and Assessment. J. Vib. Control 8(6), 699-745. Wang, B., Yang, M., 2000. Damping of honeycomb sandwich beams. J. Mater. Process Tech. 105, 67-72. Yang, J., Xiong, J., Ma, L., Wang, B., Zhang, G., Wu, L., 2013. Vibration and damping characteristics of hybrid carbon fiber composite pyramidal truss sandwich panels with viscoelastic layers. Compos. Struct. 106, 570-580. Yim, J.H., 1999. A damping analysis of composite laminates using the closed form expression for the basic damping of Poisson's ratio. Compos. Struct. 46(4), 405-411. Yim, J.H., Cho, S.Y., Seo, Y.J., Jang, B.Z., 2003. A study on material damping of 0° laminated composite sandwich cantilever beams with a viscoelastic layer. Compos. Struct. 60(4), 367-374. 19
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List of Tables and Figures
Fig. 1. Honeycomb cell structure and its parameters. Fig. 2. Two Stage Fabricating Method. Fig. 3. Impulse testing setup with the half power bandwidth method. Fig. 4. Variations of the theoretical and experimental loss factors for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3). (a) FRP vs. SW1; (b) FRP vs.SW2; (c) FRP vs. SW3 Fig. 5. Variations of the theoretical and experimental loss factors with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels. Fig. 6. Variations of the theoretical and experimental frequencies for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3). (a) FRP vs. SW1; (b) FRP vs.SW2; (c) FRP vs. SW3 Fig. 7. Variations of the theoretical and experimental frequencies with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
Table 1. Mechanical properties of the FRP, PPHC, fiber and matrix materials 20
Figure(s)
tw k
Cell diameter, d
c
hc Fig. 1. Honeycomb cell structure and its parameters.
Additional Weight
Semi-finished Bottom skin Sandwich
Top Mould PP Honeycomb
Top mould with additional weight
skinn
Bottom skin
Bottom skin
skinSkin
Bottom Mould (a) First stage curing done on the four staked up layers along with the honeycomb under the mould pressure for 8 hours
(b) Semi-finished sandwich taken from the first stage and inverted upside down.
Fig. 2. Two Stage Fabricating Method.
21
(c) Second stage curing done on the four staked up layers along with the inverted semi-finished sandwich under the mould pressure for another 8 hours.
Figure(s)
Fixed end
Impact Hammer
PP honeycomb Specimen
Accelerometer
Amplitude maximum, Amax Amplitude factor = Amax/√2
PC Frequency f1
Data Acquisition Card
Fig. 3. Impulse testing setup with the half power bandwidth method.
(a) FRP vs. SW1
22
fn
f2
(b) FRP vs. SW2
(c) FRP vs. SW3 Fig. 4. Variations of the theoretical and experimental loss factors for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1, SW2 & SW3).
23
Figure(s)
Fig. 5. Variations of the theoretical and experimental loss factors with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
(a) FRP vs. SW1
24
(b) FRP vs. SW2
(c) FRP vs. SW3 Fig. 6. Variations of the theoretical and experimental frequencies for the different fiber orientations and different thicknesses of the FRP and sandwiches (SW1,SW2 & SW3). 25
Figure(s)
Fig. 7. Variations of the theoretical and experimental frequencies with different thicknesses of the skins and cores for the 0° oriented fibers of FRP and sandwich panels.
26
Table 1. Mechanical properties of the FRP, PPHC, fiber and matrix materials
Materials
Longi -tudinal modulus (GPa)
Transverse modulus (GPa)
Shear modulus (GPa)
Longi tudina l loss factor %
Trans- Longi verse tudina loss l shear factor loss factor % %
FRP
38.21
7.34
2.814
-
-
-
PPHC
-
-
-
-
-
-
Fiber Matrix
72.5 2.737
72.5 2.737
30.20 1.013
0.175* 0.175* 0.095* 1.398 1.398 1.569
Transverse shear loss factor %
Poisson 's ratio
Density g.cm-3
-
0.304
1.855
-
0.08
0.2* 0.351
-
0.0488 ηxz = ηyz 0.095* 1.569
* Values were taken from the reference (Saravanos and Chamis 1989 )
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Highlights • The analytical derivation and expressions have been used to study the damping characteristics of the Polypropylene honeycomb structure with different fiber orientations (0°, 30°, 45°, 60°, 90°) and thicknesses. • The damping loss factor of the sandwich structure is increased by the large in-plane load of the FRP skins at 0° fiber orientation on the transverse shear of the soft honeycomb core. • The damping loss factor also increases with the increase in thicknesses of the skins and core in the sandwich structure.
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