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Dynamic models for low-velocity impact damage of composite sandwich panels – Part B: Damage initiation
Composite Structures 52 (2001) 353±364
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Dynamic models for low-velocity impact damage of composite sandwich panels ± Part B: Damage initiation Michelle S. Hoo Fatt *, Kyong S. Park Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA
Abstract Equivalent single and multi degree-of-freedom systems are used to predict low-velocity impact damage of composite sandwich panels by rigid projectiles. The composite sandwich panels are symmetric and consist of orthotropic laminate facesheets and a core with constant crushing resistance. The transient deformation response of the sandwich panels subjected to impact were predicted in a previous paper, and analytical solutions for the impact force and velocity at damage initiation in sandwich panels are presented in this second paper. Several damage initiation modes are considered, including tensile and shear fracture of the top facesheet, core shear failure, and tensile failure of back facesheet. The impact failure modes are similar to static indentation failure modes, but inertial resistance and high strain rate material properties of the facesheets and core in¯uence impact damage loads. Predicted damage initiation loads and impact velocities compare well with experimental results. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Equivalent dynamic systems; Damage initiation; Shear and tensile fracture; Core shear failure
1. Introduction Equivalent single and multi degree-of-freedom systems were used to determine the transient deformation of composite sandwich panels in a companion paper [1]. Damage initiation for the same sandwich panels are proposed in this paper. Several experimental studies to characterize impact damage initiation in sandwich panels have been conducted in recent years [2±7]. C-scans, X-rays, thermography, and micrography reveal damages that include matrix cracking, ®ber failure and delaminations in facesheet, core crushing and shear failure, and debonding and facesheets and core. These recent studies suggest that the mode of damage initiation depends on support conditions, projectile noseshape as well as the geometric and material properties of the facesheet and core. Rigidly supported panels undergo only top facesheet fracture under the projectile and core shear failure, while simply supported and clamped panels may undergo tensile failure of the bottom or back facesheet in addition to top facesheet fracture and core shear failure. Impact damage of the top facesheet is similar to that
*
Corresponding author. Tel.: +1-330-972-6308; fax: +1-330-9726027. E-mail address: [email protected] (M.S. Hoo Fatt).
observed in single skin laminated composites, but the core in the sandwich panel causes facesheet damage to become more localized under the projectile. In a recent study, Roach et al. [5] found that single skin laminates absorbed more energy and were more likely to prevent projectile penetration than foam-backed laminates resting on a solid foundation. Experiments also show that blunt projectiles induce shear failure around the circumference of a plug in thick laminate facesheets, while hemispherical and cylindro-conical projectiles induce radial cracks that petal as the projectile penetrates into thin laminate facesheets [3]. Foam or honeycomb cores crush and eventually shear to form a conical plug under the projectile. Analytical solutions for the static indentation load and impact velocity to initiate damage will be presented for composite sandwich panels that are (a) rigidly supported (bottom facesheet ®xed), (b) two-sided clamped (clamped wide beam), (c) simply supported, (c) foursided clamped. The composite sandwich panels are symmetric and consist of orthotropic laminate facesheets and a core with constant crushing resistance. Except in the case of the rigidly supported sandwich panel, both local and global deformations occur. Local deformation consists of top facesheet indentation and core crushing. Global deformation consists of bending and shear of the entire panel. Rigidly supported sandwich panels have no global deformation but only local indentation.
0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 0 4 5 - 9
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Notation a Aij Aijd b C1 C1d d D1 D1d Dij Dijd Dsij Dsijd Eij Fmax Ff Gij h hk H Kc Kg Kgd Kld M0 Mij Mmax Ncr Nij P Pf Pl
length of panel laminate extensional stiness matrix dynamic laminate extensional stiness matrix width of panel static membrane stiness of laminate dynamic membrane stiness of laminate length of damage static bending stiness of laminate dynamic bending stiness of facesheet laminate bending stiness matrix dynamic laminate bending stiness matrix sandwich bending stiness matrix dynamic sandwich bending stiness matrix laminate or core stiness maximum impact force impact force at damage initiation laminate or core shear modulus facesheet thickness ply thickness core thickness constraint factor for core crushing global stiness of clamped panel dynamic global stiness of clamped panel dynamic local stiness of top facesheet (linearized) projectile mass laminate bending moments bending moments at center of sandwich panel membrane force at tensile failure laminate in-plane forces indentation force indentation force at damage initiation equivalent nonlinear spring response for top facesheet deformation
In a previous paper [1], a solution for the local indentation of the top facesheet was found by considering a laminate resting on a rigid-plastic foundation, while a solution for the global deformation was found by neglecting core crushing in the composite sandwich. The principle of minimum potential energy was used to approximate the load-indentation response. The sandwich panel was then modeled as a discrete system of lumped masses, springs and dashpots in order to derive closedform solution for the impact response of the composite sandwich panels. Lumped masses were used to represent
q qd Qd Qij R Re
static crushing strength dynamic crushing strength total dynamic core crushing strength laminate stiness matrix blunt projectile radius eective radius of hemispherical-nose projectile V0 projectile velocity w0 slope of local indentation under projectile w0cr slope of local indentation at tensile failure w0 f slope of local indentation at core shear failure x; y in-plane coordinates of sandwich panel d top facesheet deformation df facesheet indentation at damage initiation d_ initial velocity of top facesheet e strain e_ strain rate static tensile strain for facesheet failure ecr ecrd dynamic tensile fracture strain c transverse shear strain in core cf transverse shear fracture strain in core jc curvature at center of sandwich panel jij curvature of sandwich panel s13 static transverse shear failure stress of facesheet s13d dynamic transverse shear failure stress of facesheet sf static transverse shear failure stress of core sfd dynamic transverse shear failure stress of core p x Kl Kg = Kl Kg M0 frequency of vibration due to impact n extent of local indentation n:f extent of local indentation at damage time derivative d dt
the projectile mass and an eective mass of the deformed composite sandwich. Results from a static indentation analysis were used to ®nd the eective spring force and dashpot resistance. The spring and dashpot forces were adjusted with high strain rate dependent material properties of the facesheet and core. Analytical solutions were given for the transient indentation of the top facesheet and the maximum contact force between the projectile and the top facesheet. In this paper, damage initiation criteria of the same panels will be proposed. Analytical solutions to predict impact damage initiation
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will be derived based on the above discrete dynamic model. Analytical predictions of both quasi-static and impact damage will be compared to test results [3,6,8]. 2. Modes damage initiation Consider a composite sandwich panel of dimensions a b, with laminated facesheets of thickness h and core thickness H. The composite sandwich can be (a) rigidly supported, (b) two-sided clamped, (c) simply supported or (d) fully clamped around all four sides. The panel is subjected to either static indentation or low-velocity impact at the origin by a rigid, blunt cylinder of radius R and length L. Damage initiation in the composite sandwich depends primarily on the panel support conditions, projectile nose-shape and facesheet thickness. The eects of each of these are ®rst explained below. Then damage initiation criteria are proposed. 2.1. Panel support conditions Damage is initiated only in the top facesheet of rigidly supported panels. However, damage can be initiated in either the top or bottom facesheet of clamped or simply supported panels. Global panel deformation can lead to higher tensile forces in the bottom facesheet than in the top facesheet, which is also undergoing indentation. Wen et al. [3] showed experimentally that tensile fracture of the bottom facesheet of clamped composite sandwich panels consisting of GPR facesheets and very dense foam cores, occurred before shear fracture of top facesheet when the panel was subjected static indentation by a blunt cylinder. The dense core had a very high crushing resistance, thereby preventing local indentation. A large global deformation then resulted in high tensile strains in the bottom facesheet. 2.2. Projectile nose-shape Hemispherical-nose shape and cylindro-conical projectiles usually produce radial cracks emanating from the point of contact between the projectile and the facesheet. Williamson and Lagace [8] described crosshair fractures in the [0/90] graphite epoxy facesheets of composites sandwich panels undergoing impact damage by hemispherical-nose projectile. These cracks result in petal formation on the backside of the facesheet as the projectile penetrates and perforates the panel. The mode of fracture is tensile (Mode I) in the radial crack. On the other hand, blunt projectiles produce circumferential cracks and eventually, plug formation under the projectile nose. Material right under the projectile nose becomes compressed and a fracture zone develops around the periphery of the projectile. Wen et al. [3] described disc shaped cracks in the woven glass/polyes-
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ter facesheets of composite sandwich panels undergoing impact by blunt cylinders. Hemispherical nose projectiles with large diameters can also induce circumferential cracks and plugs. The large diameter hemispherical nose projectile acts like a blunt projectile with an eective radius. 2.3. Facesheet thickness When facesheet are thin compared to the lateral extent of the panel, facesheet de¯ections tend to be large (many times the facesheet thickness) and the high inplane tensile forces cause tensile cracking (Mode I). A blunt or large diameter hemispherical-nose projectile can also cause plugging in thin facesheets if the radial tensile strain reaches the tensile rupture strain of the facesheet. When the facesheet is thick, facesheet de¯ections are small (less than the facesheet thickness) and transverse shear forces in regions that surround the projectile can be very high and cause transverse shear cracking (Mode II). Following the above discussion, one can characterize damage initiation by three modes: top facesheet failure, core shear failure, bottom facesheet failure. These modes are explained below. 2.3.1. Top facesheet failure Cracking and perforation of cross-ply laminates by rigid projectiles resembles that in metal plates. For instance, plugging (circumferential cracking) in woven glass/polyester facesheets has been observed when it undergoes impact by blunt projectiles [3], while crosshair fractures (radial cracking or petalling) have been observed in carbon ®ber facesheets subjected to impact by hemispherical-nose projectiles [8]. The similarity in fracture modes suggests that failure criteria that were used in metals could be applied to the laminated top facesheet. Two failure criteria are proposed for top facesheet damage initiation in the following sections. 2.3.1.1. Shear failure. Shear failure of top facesheet occurs when the shear stress in the facesheet is equal to its shear fracture stress s13 . It is likely to occur when the facesheets are relatively thick and do not undergo very large indentation so that membrane forces are not well developed. A free-body diagram at the point of shear failure is shown in Fig. 1. The indentation load at shear failure Pf is thus given by Pf 2pRhs13 Kc pR2 q;
1
where Kc is a constraint factor, resulting from the fact that the core under the indenter is constrained by the surrounding material. Eq. (1) was ®rst proposed by Wen et al. [3]. The constraining eect gives an average indentation stress that is greater than the uniaxial compressive strength of the core. Theoretically, the
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Fig. 1. Balance of forces on blunt cylinder during shear failure.
constraint factor should increase with decreasing indenter or projectile radius and approach unity when the indenter or projectile radius is in®nite. Reddy et al. [9] carried out simple compression and indenter penetration tests on foams and found that 1:7 < Kc < 2:5. Here we take an average value of Kc 2: The same shear failure criterion can be used for impact damage by replacing the failure load with the maximum impact force at damage initiation and using dynamic material properties, i.e., Ff 2pRs13d h Kc pR2 qd ;
2
where s13d is the dynamic transverse shear strength of the facesheet and qd is the dynamic crushing strength. Nemes et al. [10] performed high strain rate experiments using a punch shear version of the split Hopkinson Bar and showed that the transverse shear strength increases with strain rate in graphite/epoxy laminate. 2.3.1.2. Tensile failure. Tensile failure occurs when the strains are equal to the tensile fracture strain ecr or when the corresponding membrane forces reach the membrane fracture force Ncr . Tensile failure is likely to occur when the facesheets are thin and de¯ections are large so that high tensile forces are developed in the facesheet. A free-body diagram of the forces on the hemisphericalnose projectile at the point of tensile failure is shown for circumferential and radial cracks in Fig. 2(a) and (b), respectively. Circumferential cracks develop under hemispherical-nose projectiles that have a large radius of curvature. A plug is expelled as in the case of a blunt projectile. A nose with a smaller radius of curvature is more likely to cause radial cracks, as in the case of cylindro-conical projectiles. For instance, cross-hair fracture patterns in orthotropic laminates, i.e., those consisting of [0/90] plies, are often observed under hemispherical-nose indenters. Such radial cracks would petal as the projectile penetrates into the facesheet.
Fig. 2. Balance of forces during tensile failure: (a) radial cracks and (b) circumferential crack.
The free-body diagram allows one to estimate the damage initiation load. The vertical components of the membrane tensile forces per unit crack length are Ncr w0cr . The area of the crushed core under the hemisphericalnose cylinder is pR2e , where Re is the eective projectile radius. The damage initiation load for the circumferential crack is given by Pf 2pRe Ncr w0cr Kc pqR2e :
3
If the radial crack length is small, one can assume that Ncr w0cr is roughly uniform. For a total damage length d, the total failure load is given by Pf dNcr w0cr Kc pqR2e :
4
If the top facesheet is modeled as a membrane and inplane deformations are neglected, the facesheet strains are given by 2 1 5 e w0 : 2 Setting this equal to the tensile fracture strain gives an expression for the slope at tensile failure p w0cr 2ecr : 6 Furthermore, the tensile membrane forces at failure are approximately Ncr A11 ecr :
7
Substituting Eqs. (6) and (7) into Eq. (3) gives p Pf 2pRe A11 ecr 2ecr Kc pqR2e 8 as the damage initiation load due to circumferential cracking. Likewise, substituting Eqs. (6) and (7) into Eq. (4) gives
M.S. Hoo Fatt, K.S. Park / Composite Structures 52 (2001) 353±364
Pf dA11 ecr
p 2ecr Kc pqR2e
9
as the damage initiation load due to radial cracking. A dynamic tensile failure criterion for impact damage is expressed in terms of the maximum impact force at damage initiation, a dynamic extensional stiness A11d , and a dynamic tensile fracture strain ecrd . The last two quantities are dynamic materials properties that can be obtained from experiments. The critical impact force at tensile failure is p Ff 2pRe A11d ecrd 2ecrd Kc pqd R2e 10 for circumferential cracks and p Ff dA11d ecrd 2ecrd Kc pqd R2e
11
for radial cracks. Eqs. (8) and (10) can also be applied to blunt projectiles. 2.3.2. Core shear failure Damage initiation of the sandwich panel can also result from sudden core shear failure while the facesheets are still intact. Core shear failure for a rigidly supported panel is shown in Fig. 3. As the honeycomb crushes under the top facesheet, the transverse shearing strain in the honeycomb surrounding the indenter approach a critical shear fracture for the honeycomb cf . This transverse shear fracture strain can be calculated from the shear strength as follows sf cf ; 12 G13 where G13 is the transverse shear modulus of the honeycomb. When the actual transverse shearing strain in the honeycomb is equal to cf , core shear failure occurs. Experiments [3] show that the core shears into a truncated cone as shown in Fig. 3. The actual shearing strain c in the honeycomb can be approximated by the shear angle. Therefore,
cf
df nf
Re
;
13
where df and nf are the de¯ection and deformation extent at core shear failure. Equilibrium relationships between d and n as well as P and d were derived for a local indentation of the facesheet into a core with constant crushing resistance (rigid-plastic) in Ref. [1]. The core should be exhibiting constant core crushing before shear failure. These equilibrium relationships can be used to predict df and nf and the load at damage initiation Pf . At core shear failure 8 128q < 255D R nf 4 ; df < h; 1 14 df h q n R4 i1=3 f : ; df > h; 9C1 ( 32 p 2D1 qdf pqR2 ; df < h; 15 Pf 8p C1 q 32 df pqR2 ; df > h; 3
15
where 16 384 7D11 7D22 4D12 8D66 ; 11 025 A11 A22 2A12 4A66 C1 8 : 45 49
D1
Dij is the laminate bending stiness and Cij is the laminate membrane stiness. The ®rst expressions in Eqs. (14) and (15) are for a plate on a rigid-plastic foundation, while the second expressions are for a membrane on a rigid-plastic foundation. If local indentations are described by a plate on a rigid-plastic foundation, the damage load is " 1=3 #1=2 32 p 255D1 4 2D1 q c Pf pqR2 : 16 15 128q f If local indentations are found by using a membrane on a rigid-plastic foundation, then the damage load is given by p 3=2 8 C1 q 9C1 c4f pqR2 : 17 Pf 3 q The corresponding impact failure loads are " 1=3 #1=2 32 p 255D1d 4 2D1d qd c pqd R2 Ff 15 128qd fd and Ff
Fig. 3. Core shear failure.
357
p 3=2 8 C1d qd 9C1d c4fd pqd R2 ; 3 qd
18
19
where D1d is the dynamic bending stiness, C1d is the dynamic membrane stiness, and cfd is the dynamic shear fracture strain. The dynamic bending and membrane stiness are given by the same expressions for the
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static bending and membrane stiness, but they are calculated from laminate bending and membrane stiness matrices that have been adjusted for high strain rate. Both qd and cfd must be obtained from experiments. 2.3.3. Bottom facesheet failure Bottom facesheet failure can occur in clamped panels if the core crushing resistance is high enough to resist local deformation or top facesheet indentation. The entire panel deforms in a shear-bending mode and failure would take place when the maximum tensile strains in the bottom facesheet are at the critical fracture strain. Maximum tensile strain occurs right under the projectile or middle of the plate, where the bending moment is a maximum. The bending moments in a rectangular sandwich plate Mxx and Myy are given by Mxx Ds11 jxx Ds12 jyy
20
and Myy Ds12 jxx Ds22 jyy ;
21
where jxx and jyy are the curvatures in the plate with respect to the x and y directions and Dsij is the bending stiness of the sandwich. At the plate center, the bending moment is a maximum. Therefore, the curvature at the plate center jc is given by jc
Mmax ; Ds12
Ds22
22
where Mmax is the maximum bending moment. Timoshenko and Woinowski-Krieger [11] give Mmax in terms of the load P as P 2a 2 ln Mmax 0:669 : 23 4p pR Therefore, the curvature at the plate center can be expressed in terms of the load, bending stiness and plate geometry. By neglecting in-plane deformation, one can directly relate the curvatures to the maximum bending strain as follows emax
H jc : 2
24
If the bottom facesheet fails when emax ecr , then a failure criterion in terms of the load at which failure would occur, can be given by substituting expression in Eqs. (22) and (23) into Eq. (24) 4pecr Ds11 Ds12 : Pf H 25 2a 0:669 h 2 ln pR 2 Bottom facesheet damage under impact load occur when the maximum impact force is given by
4pecrd Ds11d Ds12d ; Ff H 2a 0:669 h 2 ln pR 2
26
where Ds11 and Ds12 are the dynamic bending stiness of the sandwich.
3. Impact damage initiation criteria When the maximum impact force is equal to the above critical impact forces, damage initiates. Expressions for the maximum impact force were given in the previous paper [1]. They are repeated here convenience. (i) Rigidly supported panels. The following nonlinear dierential is solved numerically M0 d Pl d Qd 0;
27
where M0 is the projectile mass, Pl d is the equivalent spring force and Qd pR2 qd is the dynamic core crushing resistance. The ordinary dierential equation is _ solved with initial conditions d 0 V0 and d 0 0, where V0 is the initial velocity of the projectile.The maximum impact force occurs at tmax when d_ 0, and is given by Fmax
max : M0 d t
28
(ii) Simply supported and clamped panels. The equivalent nonlinear spring force is ®rst linearized Pl d Kld d;
29
where Kld is the linearized dynamic local spring constant. Then, a closed-form expression for the maximum impact force can be derived as M0 Kgd Kld x Q2 x2 q d_20 Kld d Fmax ; 30 Kgd Kld 2 2 Qd x d_0 Kld where s Kld Kgd and x Kld Kgd M0
d_0
V0 Kgd Kgd Kld
and Kgd is the dynamic global spring constant. Setting equal Ff to Fmax in the above equations allows one to calculate a critical impact velocity for impact damage.
4. Comparison with experimental results The analytical solutions are applied to test data from three independent studies [3,6,8]. Composite sandwich panels with AS4/3501-6 Carbon/Epoxy facesheets and Nomex honeycomb core and hemispherical nose-shape indenters/projectiles were used in the ®rst two experimental studies, while E-glass WR cloth reinforced
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laminate (GRP) facesheets and Divinycell H130 foam core was used in the last experimental study. 4.1. Rigidly supported and two-sided clamped composite sandwich panels with AS4/3501-6 carbon/epoxy facesheets and Nomex honeycomb core Williamson and Lagace [8] performed a series of static indentation and low-velocity impact tests on rigidly supported and two-sided clamped composite sandwich panels made with AS4/3501-6 carbon/epoxy facesheet with Nomex honeycomb cores. The indenter and projectile were hemispherical-nose tups or cylinders made from case-hardened steel (rigid compared to the sandwich panel). Material and geometric properties of the facesheet, core and projectile are given in Table 1. Material properties for the woven [0/90] plies could not be found and the laminate facesheet was assumed to consist of separate 0° and 90° plies. The actual stiness and strength properties of the laminate facesheets would be slightly dierent than those shown in Table 1. Williamson and Lagace found that damage was initiated by top facesheet failure in all the tests. Core shear failure would occur only after the indenter/projectile penetrated through the top facesheet. Furthermore, the tensile strain in the bottom facesheet of the two-sided clamped panels was not high enough to cause bottom Table 1 Material properties of the rigidly supported and two-sided clamped C/E composite sandwich panels and indenter/projectile Facesheet
Hercules AW193-PW prepreg consisting of AS4 ®bers in a 3501-6 matrix a a 102 102 mm2 : square panel dimensions a b 203 89 mm2 : wide beam dimensions hk 0:175 mm: ply thickness qf 1617:3 kg/m3 : mass density Ply Stiness E11 142 GPa: longitudinal stiness E22 9:8 GPa: transverse stiness G12 7:1 GPa: in-plane shear modulus m12 0:3: Poisson's ratio Ply Strength cr 0:0112: static tensile failure strain s13 97 MPa: out-of-plane shear strength
Core
HRH 10 1/8±3.0 Nomex honeycomb (Ciba-Geigy) qc 48 kg/m3 : density d 3:2 mm: cell diameter H 25:4 mm: core thickness q 1:389 MPa: crushing resistance
Indenter/ projectile
Hemispherical-nose cylinder made of case-hardened steel D 12:7, 25.4 and 38.1 mm: diameter L 660 mm: length M0 1:53, 1.61 and 1.69 kg: masses
359
facesheet damage before top facesheet damage or core shear failure. Cross-hair fractures were observed in the top facesheets. The cross-hair patterns are typical of an orthotropic laminate that has failed due to high in-plane tensile strains. Radial cracks run parallel to the ®ber direction, [0/90] in these tests. Tensile failure was the likely mode of damage because of the hemispherical-nose shape of the tup and the large facesheet de¯ections or membrane forces that are induced in the relatively thin top facesheet. Cylindro-conical tups will also produce the same cross-hair fractures, which will eventually lead to petaling when the projectile perforates the facesheet. 4.1.1. Static damage initiation Damage is a local phenomenon that depends only on the local indentation of the top facesheet. Therefore, rigidly supported and two-sided clamped sandwich panels made of the same materials should fail at the same load. In predicting either static and impact damage, an eective radius of 0.4R for the hemisphericalnose cylinder was used. The indentation force at damage initiation based on the tensile failure was predicted for sandwiches with three dierent facesheets thickness using Eq. (9). The analytical predictions are compared to experimental results in Fig. 4. The failure load depends on the size of the damage length, which is also shown in the diagram. Except for the sandwich with [0/90] facesheets, the analytical predictions for the damage initiation load using Eq. (9) are within 10% of tests results. Notice that the damage load for sandwiches with approximately the same damage length are the same for the rigidly supported and twosided clamped cases. Therefore, top facesheet damage initiation is indeed a local phenomenon. 4.1.2. Impact damage initiation In order to ®nd the dynamic strength of the facesheet and core, strain rates in the facesheet and core had to be found. The maximum value of the strain rate in the facesheet during impact damage initiation was approximated by dividing the tensile failure strain by the time it takes to reach maximum impact force, tmax . The strain rates for the low velocity impact tests were computed using static stiness and tensile failure strain and are found to be no greater than e_ 4 s 1 . It was dicult to ®nd tensile test data of carbon/epoxy laminates at this particular strain rate. However, Harding et al. [12] found that the in-plane stiness of woven carbon/epoxy laminates increased only by 17%, while the failure strain is about the same at rates of about 290 per s. Therefore, it would appear that at these low strain rates, the inplane stiness of the woven carbon/epoxy laminate facesheets were about the same as the static values. It was assumed that dynamic material properties for AS4/ 3501-6 carbon/epoxy laminate are equivalent to those for static indentation as follows:
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Fig. 4. Failure load vs. facesheet thickness for rigidly supported and two-sided clamped C/E composite sandwich indented by 25.4-mm diameter hemispherical-nose cylinder.
C1d C1 , Aijd Aij , ecrd ecr ,
dynamic membrane stiness of carbon/ epoxy, dynamic extensional stiness of carbon/ epoxy, dynamic tensile fracture strain of carbon/ epoxy.
Goldsmith et al. [13] found that the dynamic crushing strength of Nomex honeycomb was always 10% greater than the static crushing strength in low velocity impacts and independent impact velocity. Therefore, qd 1:1q,
dynamic crushing strength of Nomex honeycomb.
Critical velocities at damage initiation for sandwiches with the three dierent facesheet thickness were calculated from Eqs. (11) and (30) for the two-sided clamped panel. Unlike static damage initiation, the maximum impact force and critical impact velocity at damage depend on the boundary conditions. The analytical solutions are shown together with experimental results in Fig. 5. Except for the [0/90] facesheets, the predicted failure loads were about 20% lower than test results. 4.2. Simply-supported composite sandwich panels with AS4/3501-6 carbon/epoxy facesheets and Nomex honeycomb core Herup and Palzotto [2] conducted a series of static indentation and low-velocity impact tests on simply
supported composite sandwich panels with AS4/3501-6 carbon/epoxy facesheet with Nomex honeycomb cores. The indenter and projectile were hemispherical-nose tups or cylinders made from case-hardened steel. The panels were simply supported. The material and geometric properties of the facesheet, core and projectile are given in Table 2. Herup and Palazotto de®ned impact damage initiation as the ®rst drop in the quasi-static load or the dynamic contact load between the projectile and the top facesheet. C-scans revealed that the sandwich underwent multiple delaminations and matrix cracking in the top facesheet as well as core crushing and shear fracture. None of the panels exhibited bottom facesheet failure. The ®rst load drop corresponded to core shear failure. This is in contrast to the experiments by Williamson and Lagace [8], where damage initiation was due to top facesheet fracture. 4.2.1. Static damage initiation The static load at the instant of core shear failure was calculated from Eq. (16) for several composite sandwich panels with dierent facesheet thickness. The transverse shear fracture strain for HRH-10-1/8-4.0 Nomex honeycomb cf was calculated from Ref. [14] as 2.9%. The analytical predictions of the failure load are compared to the experimental results in Fig. 6. The analytical predictions are within 25% of the tests results. Eq. (16) seems to be most accurate for the thicker facesheets. This is because the shear fracture zone becomes more
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361
Fig. 5. Failure velocity vs. facesheet thickness for two-sided clamped C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
localized in sandwiches with thicker facesheets and the approximation of the shear angle in Eq. (13) is better for more localized deformations.
Table 2 Material properties of simply supported C/E composite sandwich panels and indenter/projectile Facesheet
AS4/3501-6 graphite±epoxy a a 178 178 mm2 : square panel dimensions hk 0:0635 mm: ply thickness qf 1632 kg/m3 : mass density
4.2.2. Impact damage initiation The strain rates that were calculated for the AS4/ 3501-6 facesheets and Nomex honeycomb were similar to those in the experiments by Williamson and Lagace [8]. Therefore, one can also assume the following dynamic material properties: D1d D1 , Dijd Dij ,
dynamic bending stiness of carbon/ epoxy, dynamic ¯exural stiness of carbon/ epoxy,
Ply Stiness E11 144:8 GPa: longitudinal stiness E22 9:7 GPa: transverse stiness G12 7:1 GPa: in-plane shear modulus m12 0:3: Poisson's ratio Ply Strength cr 0:0145: static tensile failure strain s13 120:7 MPa: out-of-plane shear strength Core
HRH 10 1/8±4.0 Nomex honeycomb (Ciba-Geigy) qc 64 kg/m3 : density d 3:2 mm: cell diameter H 12:7 mm: core thickness q 3:83 MPa: crushing resistance
Indenter/projectile
Hemispherical-nose cylinder (Case-hardened steel) D 25:4 mm: diameter M0 3:48 kg: mass
Fig. 6. Static failure load vs. facesheet thickness for simply supported C/E composite sandwich indented by 25.4-mm diameter hemisphericalnose cylinder.
362
qd 1:1q, cfd 1:3cf ,
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dynamic crushing strength of Nomex honeycomb, dynamic shear fracture strain of Nomex honeycomb.
The 30% increase in the dynamic shear fracture strain of Nomex honeycomb was assumed to be the same as that in aluminum honeycomb [15]. The maximum impact load at the instant of core shear failure was calculated from Eq. (18) for several composite sandwich panels with dierent facesheet thickness. The analytical predictions of the failure load are compared to the experimental results in Fig. 7. The analytical predictions are again within 25% of the tests results and Eq. (18) appear to be most accurate for the thicker facesheets. The variation of both static and impact failure loads with facesheet thickness are plotted in Fig. 8. In all cases the impact damage load is higher than the static failure load. This is primarily due to the fact that the crushing and shear strength of the core increase with strain rates. Material properties of the C/E facesheet are almost unchanged. Herup and Palazotto [2] noted that the dierences between the static and impact failure loads were slightly larger for sandwiches with the thicker facesheets. They attributed this tendency to the increase of inertia associated with a thicker facesheet. The failure load predicted by Eq. (18) does not include the inertial resistance of the facesheet because it was assumed to be negligible compared to the mass of the projectile. 4.3. Composite sandwich panels with E-glass WR cloth reinforced facesheets and divinycell H130 foam core
Fig. 8. Comparison of static and impact failure load for simply supported C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
clamped composite sandwich panels made of E-glass WR cloth reinforced laminate (GRP) facesheets and Divinycell H130 foam cores. The GRP composite sandwich panels were clamped along four edges, but inplane were not constrained. The material properties for the facesheet and core are given in Table 3. As in the Table 3 Material properties of the clamped GRP sandwich panel and indenter/ projectile Facesheet laminate
Wen et al. [3] conducted a series of static indentation, impact and perforation tests with blunt cylinders and
E-glass/polyester laminate a a 200 200 mm2 850 850 mm2 : square panel dimensions hk 0:29 mm: ply thickness qf 1650 kg/m3 : mass density Ply Stiness E11 24:4 GPa: longitudinal stiness E22 6:87 GPa: transverse stiness G12 2:89 GPa: in-plane shear modulus m12 0:32: Poisson's ratio Ply Strength cr 0:021: static tensile failure strain s13 45 MPa: out-of-plane shear strength
Core
Divinycell H130 foam core (Barracuda) qc 130 kg/m3 : mass density H 25:4 mm: core thickness Stiness Ec 175 MPa: Young's modulus Gc 50 MPa: shear modulus mc 0:3: Poisson's ratio Strength q 2:5 MPa: crushing resistance sc 2 MPa: out-of-plane shear strength
Indenter/projectile Fig. 7. Impact failure load vs. facesheet thickness for simply supported C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
Blunt cylinder made of case-hardened steel D 10:5±50 mm: diameter M0 17:9 g±42.5 kg: mass
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case of the C/E laminate, all the material properties for the woven [0/90] plies could not be found and the laminate facesheet was assumed to consist of separate 0° and 90° plies. Test results for damage initiation due to low velocity impact were not available, but damage initiation loads for the panels undergoing static indentation was plentiful. Damage initiation was attributed to shear failure in the top facesheet by the blunt cylinder/projectile. Shear failure was the likely mode of damage initiation because the facesheets were relatively thick. When the facesheet is thick, local indentations are small and no appreciable membrane force can develop in the facesheet. Furthermore, core material right under a blunt cylinder is compressed and a localized shear zone develops around the compressed region. Core material being compressed under the blunt cylinder is also constrained by the adjacent honeycomb cells. The interaction of the blunt projectile and facesheet thus makes shear failure a likely mode of damage. Shear failure is a local phenomenon and should be independent of panel size and boundary conditions. As described in Eq. (1), the shear failure load should increase parabolically with indenter radius and increase linearly with facesheet thickness. Failure loads based on shear fracture of the top facesheet were calculated from Eq. (1). The analytical results are compared to test results for varying projectile radius and facesheet thickness in Figs. 9 and 10, respectively. The test results shown in Fig. 9 are for three dierent panel sizes, but despite this, the parabolic function that describes how the failure load varies with indenter radius ®ts the experimental data rather well. The analytical predictions are within 10% of experimental results. Therefore, the shear failure load was independent of panel size. Furthermore, the linear function describing how the failure load varies with facesheet thickness also ®ts the experi-
363
Fig. 10. Failure load vs. facesheet thickness for rigidly supported and four-sided clamped GRP composite sandwich indented by 50-mm diameter blunt cylinder.
mental data well in Fig. 10. The analytical predictions are within 20% of experimental results. Test results for rigidly supported panels were compared to the clamped panel results in Fig. 10. The shear failure loads for both support conditions are approximately the same. Therefore, the shear failure load was independent of boundary conditions. In an eort to induce tensile failure of bottom facesheet due to global bending, Wen et al. [3] used denser PVC foam core, HCP100 with a nominal density of 400 kg/m3 , in the composite sandwich with [0/90]6 or 3.25 mm thick E-glass/polyester facesheets. The panel was indented by the 50-mm diameter blunt cylinder and failed by tensile failure of bottom facesheet at a load of 57 kN. The bottom facesheet tensile failure load calculated from Eq. (25) was equal to 69.5 kN, which about 22% higher than the experimental results. 5. Conclusion
Fig. 9. Failure load vs. indenter radius for four-sided clamped GRP composite sandwich with [0/90]6 laminates indented by blunt cylinder.
Equivalent single and multi degree-of-freedom systems were used to predict the transient deformations of composite sandwich panels under impact loading in a previous paper, and analytical solutions for low-velocity impact damage of the same composite sandwich panels were given in this paper. The composite sandwich panels were symmetric and consisted of orthotropic laminate facesheets and a core with constant crushing resistance. The initial mode of impact damage depended on the panel support conditions, projectile nose-shape and geometric and material properties of facesheet and core. Particular damage initiation modes that were discussed in this paper include fracture of the top facesheet, core shear failure, as well as tensile failure of back facesheet. Fracture patterns in the top facesheet resembled some of those in metals, such as shear plugging by blunt
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projectiles and petalling by hemispherical-nose projectiles. The impact failure modes were similar to static indentation failure modes, but material properties of the facesheets and core had to be adjusted for the in¯uence of high strain rate. Predicted damage initiation loads and impact velocities compared well to experimental results. Acknowledgements This work was supported by the Ohio Aerospace Institute under the 1999 OAI Core Collaborative Research program. The research was made possible through collaboration with GE Aircraft Engines, RollsRoyce Allison, BFGoodrich Aerospace and the NASA Glenn Research Center. The authors wish to thank Dr. P.A. Lagace, Dr. A.N. Palazotto, and Dr. H.M. Wen for the use of their experimental results. References [1] Hoo Fatt MS, Park KS. Dynamic models for low-velocity impact damage of composite sandwich panels ± Part A, Deformation. Compos. Struct. 2001;52(3±4):335±51. [2] Palazotto AN, Gummadi LNB, Vaidya UK, Herup EJ. Lowvelocity impact damage characteristics of Z-®ber sandwich panels ± an experimental study. Compos Struct 1999;43:275±88. [3] Wen HW, Reddy TY, Reid SR, Soden PD. Indentation, penetration and perforation of composite laminates and sandwich panels under quasi-static and projectile loading. Key Eng Mater 1998;141±143:501±52. [4] Roach AM, Jones N, Evans KE. The penetration energy of sandwich panel elements under static and dynamic loading. Part I. Compos Struct 1998;42:119±34.
[5] Roach AM, Jones N, Evans KE. The penetration energy of sandwich panel elements under static and dynamic loading. Part II. Compos Struct 1998;42:135±52. [6] Herup EJ, Palazotto AN. low-velocity impact damage initiation in graphite/epoxy/Nomex honeycomb-sandwich plates. Compos Sci Technol 1997;57:1581±98. [7] Ferri R, Sankar BV. Static indentation and low-velocity impact tests on sandwich plates. In: Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition, vol. 55, 1997; Dallas. p. 485±90. [8] Williamson JE, Lagace PA. Response mechanism in the impact of graphite/epoxy honeycomb sandwich panels. In: Proceedings of the Eighth ASC Technical Conference, Cleveland, OH; 1993. p. 287±97. [9] Reddy TY, Soden PD, Reid SR, Sadighi M. Impact response of thick composite laminates and sandwich panels. Collaborative Research Program on the Cost-Eective Use of Fibre-Reinforced Composite Oshore, Marintech Research, Phase 1, Report No. CP04, 1991. [10] Nemes JA, Eskandari H, Rakitch L. Eect of laminate parameters on penetration of graphite/epoxy composites. Int J Impact Eng 1998;21(1/2):97±112. [11] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. New York: McGraw-Hill; 1961. [12] Harding J, Li YL, Saka K, Taylor MEC. Characterization of the impact strength of woven carbon ®bre/epoxy laminates. In: Proceedings of the International Conference on Mechanical Properties of Materials at High Rates of Strain, Oxford; 1989. p. 403±10. [13] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. Int J Impact Eng 1991;12(2):241±62. [14] Hexcel Cooperation. Mechanical Properties of Hexcel Honeycomb Materials. TBS 120, Dublin, CA; 1988. [15] Goldsmith W, Louie DL. Axial perforation of aluminum honeycombs by projectiles. Int J Solids Struct 1995;32(8/9): 1017±46.
www.elsevier.com/locate/compstruct
Dynamic models for low-velocity impact damage of composite sandwich panels ± Part B: Damage initiation Michelle S. Hoo Fatt *, Kyong S. Park Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA
Abstract Equivalent single and multi degree-of-freedom systems are used to predict low-velocity impact damage of composite sandwich panels by rigid projectiles. The composite sandwich panels are symmetric and consist of orthotropic laminate facesheets and a core with constant crushing resistance. The transient deformation response of the sandwich panels subjected to impact were predicted in a previous paper, and analytical solutions for the impact force and velocity at damage initiation in sandwich panels are presented in this second paper. Several damage initiation modes are considered, including tensile and shear fracture of the top facesheet, core shear failure, and tensile failure of back facesheet. The impact failure modes are similar to static indentation failure modes, but inertial resistance and high strain rate material properties of the facesheets and core in¯uence impact damage loads. Predicted damage initiation loads and impact velocities compare well with experimental results. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Equivalent dynamic systems; Damage initiation; Shear and tensile fracture; Core shear failure
1. Introduction Equivalent single and multi degree-of-freedom systems were used to determine the transient deformation of composite sandwich panels in a companion paper [1]. Damage initiation for the same sandwich panels are proposed in this paper. Several experimental studies to characterize impact damage initiation in sandwich panels have been conducted in recent years [2±7]. C-scans, X-rays, thermography, and micrography reveal damages that include matrix cracking, ®ber failure and delaminations in facesheet, core crushing and shear failure, and debonding and facesheets and core. These recent studies suggest that the mode of damage initiation depends on support conditions, projectile noseshape as well as the geometric and material properties of the facesheet and core. Rigidly supported panels undergo only top facesheet fracture under the projectile and core shear failure, while simply supported and clamped panels may undergo tensile failure of the bottom or back facesheet in addition to top facesheet fracture and core shear failure. Impact damage of the top facesheet is similar to that
*
Corresponding author. Tel.: +1-330-972-6308; fax: +1-330-9726027. E-mail address: [email protected] (M.S. Hoo Fatt).
observed in single skin laminated composites, but the core in the sandwich panel causes facesheet damage to become more localized under the projectile. In a recent study, Roach et al. [5] found that single skin laminates absorbed more energy and were more likely to prevent projectile penetration than foam-backed laminates resting on a solid foundation. Experiments also show that blunt projectiles induce shear failure around the circumference of a plug in thick laminate facesheets, while hemispherical and cylindro-conical projectiles induce radial cracks that petal as the projectile penetrates into thin laminate facesheets [3]. Foam or honeycomb cores crush and eventually shear to form a conical plug under the projectile. Analytical solutions for the static indentation load and impact velocity to initiate damage will be presented for composite sandwich panels that are (a) rigidly supported (bottom facesheet ®xed), (b) two-sided clamped (clamped wide beam), (c) simply supported, (c) foursided clamped. The composite sandwich panels are symmetric and consist of orthotropic laminate facesheets and a core with constant crushing resistance. Except in the case of the rigidly supported sandwich panel, both local and global deformations occur. Local deformation consists of top facesheet indentation and core crushing. Global deformation consists of bending and shear of the entire panel. Rigidly supported sandwich panels have no global deformation but only local indentation.
0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 0 4 5 - 9
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Notation a Aij Aijd b C1 C1d d D1 D1d Dij Dijd Dsij Dsijd Eij Fmax Ff Gij h hk H Kc Kg Kgd Kld M0 Mij Mmax Ncr Nij P Pf Pl
length of panel laminate extensional stiness matrix dynamic laminate extensional stiness matrix width of panel static membrane stiness of laminate dynamic membrane stiness of laminate length of damage static bending stiness of laminate dynamic bending stiness of facesheet laminate bending stiness matrix dynamic laminate bending stiness matrix sandwich bending stiness matrix dynamic sandwich bending stiness matrix laminate or core stiness maximum impact force impact force at damage initiation laminate or core shear modulus facesheet thickness ply thickness core thickness constraint factor for core crushing global stiness of clamped panel dynamic global stiness of clamped panel dynamic local stiness of top facesheet (linearized) projectile mass laminate bending moments bending moments at center of sandwich panel membrane force at tensile failure laminate in-plane forces indentation force indentation force at damage initiation equivalent nonlinear spring response for top facesheet deformation
In a previous paper [1], a solution for the local indentation of the top facesheet was found by considering a laminate resting on a rigid-plastic foundation, while a solution for the global deformation was found by neglecting core crushing in the composite sandwich. The principle of minimum potential energy was used to approximate the load-indentation response. The sandwich panel was then modeled as a discrete system of lumped masses, springs and dashpots in order to derive closedform solution for the impact response of the composite sandwich panels. Lumped masses were used to represent
q qd Qd Qij R Re
static crushing strength dynamic crushing strength total dynamic core crushing strength laminate stiness matrix blunt projectile radius eective radius of hemispherical-nose projectile V0 projectile velocity w0 slope of local indentation under projectile w0cr slope of local indentation at tensile failure w0 f slope of local indentation at core shear failure x; y in-plane coordinates of sandwich panel d top facesheet deformation df facesheet indentation at damage initiation d_ initial velocity of top facesheet e strain e_ strain rate static tensile strain for facesheet failure ecr ecrd dynamic tensile fracture strain c transverse shear strain in core cf transverse shear fracture strain in core jc curvature at center of sandwich panel jij curvature of sandwich panel s13 static transverse shear failure stress of facesheet s13d dynamic transverse shear failure stress of facesheet sf static transverse shear failure stress of core sfd dynamic transverse shear failure stress of core p x Kl Kg = Kl Kg M0 frequency of vibration due to impact n extent of local indentation n:f extent of local indentation at damage time derivative d dt
the projectile mass and an eective mass of the deformed composite sandwich. Results from a static indentation analysis were used to ®nd the eective spring force and dashpot resistance. The spring and dashpot forces were adjusted with high strain rate dependent material properties of the facesheet and core. Analytical solutions were given for the transient indentation of the top facesheet and the maximum contact force between the projectile and the top facesheet. In this paper, damage initiation criteria of the same panels will be proposed. Analytical solutions to predict impact damage initiation
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will be derived based on the above discrete dynamic model. Analytical predictions of both quasi-static and impact damage will be compared to test results [3,6,8]. 2. Modes damage initiation Consider a composite sandwich panel of dimensions a b, with laminated facesheets of thickness h and core thickness H. The composite sandwich can be (a) rigidly supported, (b) two-sided clamped, (c) simply supported or (d) fully clamped around all four sides. The panel is subjected to either static indentation or low-velocity impact at the origin by a rigid, blunt cylinder of radius R and length L. Damage initiation in the composite sandwich depends primarily on the panel support conditions, projectile nose-shape and facesheet thickness. The eects of each of these are ®rst explained below. Then damage initiation criteria are proposed. 2.1. Panel support conditions Damage is initiated only in the top facesheet of rigidly supported panels. However, damage can be initiated in either the top or bottom facesheet of clamped or simply supported panels. Global panel deformation can lead to higher tensile forces in the bottom facesheet than in the top facesheet, which is also undergoing indentation. Wen et al. [3] showed experimentally that tensile fracture of the bottom facesheet of clamped composite sandwich panels consisting of GPR facesheets and very dense foam cores, occurred before shear fracture of top facesheet when the panel was subjected static indentation by a blunt cylinder. The dense core had a very high crushing resistance, thereby preventing local indentation. A large global deformation then resulted in high tensile strains in the bottom facesheet. 2.2. Projectile nose-shape Hemispherical-nose shape and cylindro-conical projectiles usually produce radial cracks emanating from the point of contact between the projectile and the facesheet. Williamson and Lagace [8] described crosshair fractures in the [0/90] graphite epoxy facesheets of composites sandwich panels undergoing impact damage by hemispherical-nose projectile. These cracks result in petal formation on the backside of the facesheet as the projectile penetrates and perforates the panel. The mode of fracture is tensile (Mode I) in the radial crack. On the other hand, blunt projectiles produce circumferential cracks and eventually, plug formation under the projectile nose. Material right under the projectile nose becomes compressed and a fracture zone develops around the periphery of the projectile. Wen et al. [3] described disc shaped cracks in the woven glass/polyes-
355
ter facesheets of composite sandwich panels undergoing impact by blunt cylinders. Hemispherical nose projectiles with large diameters can also induce circumferential cracks and plugs. The large diameter hemispherical nose projectile acts like a blunt projectile with an eective radius. 2.3. Facesheet thickness When facesheet are thin compared to the lateral extent of the panel, facesheet de¯ections tend to be large (many times the facesheet thickness) and the high inplane tensile forces cause tensile cracking (Mode I). A blunt or large diameter hemispherical-nose projectile can also cause plugging in thin facesheets if the radial tensile strain reaches the tensile rupture strain of the facesheet. When the facesheet is thick, facesheet de¯ections are small (less than the facesheet thickness) and transverse shear forces in regions that surround the projectile can be very high and cause transverse shear cracking (Mode II). Following the above discussion, one can characterize damage initiation by three modes: top facesheet failure, core shear failure, bottom facesheet failure. These modes are explained below. 2.3.1. Top facesheet failure Cracking and perforation of cross-ply laminates by rigid projectiles resembles that in metal plates. For instance, plugging (circumferential cracking) in woven glass/polyester facesheets has been observed when it undergoes impact by blunt projectiles [3], while crosshair fractures (radial cracking or petalling) have been observed in carbon ®ber facesheets subjected to impact by hemispherical-nose projectiles [8]. The similarity in fracture modes suggests that failure criteria that were used in metals could be applied to the laminated top facesheet. Two failure criteria are proposed for top facesheet damage initiation in the following sections. 2.3.1.1. Shear failure. Shear failure of top facesheet occurs when the shear stress in the facesheet is equal to its shear fracture stress s13 . It is likely to occur when the facesheets are relatively thick and do not undergo very large indentation so that membrane forces are not well developed. A free-body diagram at the point of shear failure is shown in Fig. 1. The indentation load at shear failure Pf is thus given by Pf 2pRhs13 Kc pR2 q;
1
where Kc is a constraint factor, resulting from the fact that the core under the indenter is constrained by the surrounding material. Eq. (1) was ®rst proposed by Wen et al. [3]. The constraining eect gives an average indentation stress that is greater than the uniaxial compressive strength of the core. Theoretically, the
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Fig. 1. Balance of forces on blunt cylinder during shear failure.
constraint factor should increase with decreasing indenter or projectile radius and approach unity when the indenter or projectile radius is in®nite. Reddy et al. [9] carried out simple compression and indenter penetration tests on foams and found that 1:7 < Kc < 2:5. Here we take an average value of Kc 2: The same shear failure criterion can be used for impact damage by replacing the failure load with the maximum impact force at damage initiation and using dynamic material properties, i.e., Ff 2pRs13d h Kc pR2 qd ;
2
where s13d is the dynamic transverse shear strength of the facesheet and qd is the dynamic crushing strength. Nemes et al. [10] performed high strain rate experiments using a punch shear version of the split Hopkinson Bar and showed that the transverse shear strength increases with strain rate in graphite/epoxy laminate. 2.3.1.2. Tensile failure. Tensile failure occurs when the strains are equal to the tensile fracture strain ecr or when the corresponding membrane forces reach the membrane fracture force Ncr . Tensile failure is likely to occur when the facesheets are thin and de¯ections are large so that high tensile forces are developed in the facesheet. A free-body diagram of the forces on the hemisphericalnose projectile at the point of tensile failure is shown for circumferential and radial cracks in Fig. 2(a) and (b), respectively. Circumferential cracks develop under hemispherical-nose projectiles that have a large radius of curvature. A plug is expelled as in the case of a blunt projectile. A nose with a smaller radius of curvature is more likely to cause radial cracks, as in the case of cylindro-conical projectiles. For instance, cross-hair fracture patterns in orthotropic laminates, i.e., those consisting of [0/90] plies, are often observed under hemispherical-nose indenters. Such radial cracks would petal as the projectile penetrates into the facesheet.
Fig. 2. Balance of forces during tensile failure: (a) radial cracks and (b) circumferential crack.
The free-body diagram allows one to estimate the damage initiation load. The vertical components of the membrane tensile forces per unit crack length are Ncr w0cr . The area of the crushed core under the hemisphericalnose cylinder is pR2e , where Re is the eective projectile radius. The damage initiation load for the circumferential crack is given by Pf 2pRe Ncr w0cr Kc pqR2e :
3
If the radial crack length is small, one can assume that Ncr w0cr is roughly uniform. For a total damage length d, the total failure load is given by Pf dNcr w0cr Kc pqR2e :
4
If the top facesheet is modeled as a membrane and inplane deformations are neglected, the facesheet strains are given by 2 1 5 e w0 : 2 Setting this equal to the tensile fracture strain gives an expression for the slope at tensile failure p w0cr 2ecr : 6 Furthermore, the tensile membrane forces at failure are approximately Ncr A11 ecr :
7
Substituting Eqs. (6) and (7) into Eq. (3) gives p Pf 2pRe A11 ecr 2ecr Kc pqR2e 8 as the damage initiation load due to circumferential cracking. Likewise, substituting Eqs. (6) and (7) into Eq. (4) gives
M.S. Hoo Fatt, K.S. Park / Composite Structures 52 (2001) 353±364
Pf dA11 ecr
p 2ecr Kc pqR2e
9
as the damage initiation load due to radial cracking. A dynamic tensile failure criterion for impact damage is expressed in terms of the maximum impact force at damage initiation, a dynamic extensional stiness A11d , and a dynamic tensile fracture strain ecrd . The last two quantities are dynamic materials properties that can be obtained from experiments. The critical impact force at tensile failure is p Ff 2pRe A11d ecrd 2ecrd Kc pqd R2e 10 for circumferential cracks and p Ff dA11d ecrd 2ecrd Kc pqd R2e
11
for radial cracks. Eqs. (8) and (10) can also be applied to blunt projectiles. 2.3.2. Core shear failure Damage initiation of the sandwich panel can also result from sudden core shear failure while the facesheets are still intact. Core shear failure for a rigidly supported panel is shown in Fig. 3. As the honeycomb crushes under the top facesheet, the transverse shearing strain in the honeycomb surrounding the indenter approach a critical shear fracture for the honeycomb cf . This transverse shear fracture strain can be calculated from the shear strength as follows sf cf ; 12 G13 where G13 is the transverse shear modulus of the honeycomb. When the actual transverse shearing strain in the honeycomb is equal to cf , core shear failure occurs. Experiments [3] show that the core shears into a truncated cone as shown in Fig. 3. The actual shearing strain c in the honeycomb can be approximated by the shear angle. Therefore,
cf
df nf
Re
;
13
where df and nf are the de¯ection and deformation extent at core shear failure. Equilibrium relationships between d and n as well as P and d were derived for a local indentation of the facesheet into a core with constant crushing resistance (rigid-plastic) in Ref. [1]. The core should be exhibiting constant core crushing before shear failure. These equilibrium relationships can be used to predict df and nf and the load at damage initiation Pf . At core shear failure 8 128q < 255D R nf 4 ; df < h; 1 14 df h q n R4 i1=3 f : ; df > h; 9C1 ( 32 p 2D1 qdf pqR2 ; df < h; 15 Pf 8p C1 q 32 df pqR2 ; df > h; 3
15
where 16 384 7D11 7D22 4D12 8D66 ; 11 025 A11 A22 2A12 4A66 C1 8 : 45 49
D1
Dij is the laminate bending stiness and Cij is the laminate membrane stiness. The ®rst expressions in Eqs. (14) and (15) are for a plate on a rigid-plastic foundation, while the second expressions are for a membrane on a rigid-plastic foundation. If local indentations are described by a plate on a rigid-plastic foundation, the damage load is " 1=3 #1=2 32 p 255D1 4 2D1 q c Pf pqR2 : 16 15 128q f If local indentations are found by using a membrane on a rigid-plastic foundation, then the damage load is given by p 3=2 8 C1 q 9C1 c4f pqR2 : 17 Pf 3 q The corresponding impact failure loads are " 1=3 #1=2 32 p 255D1d 4 2D1d qd c pqd R2 Ff 15 128qd fd and Ff
Fig. 3. Core shear failure.
357
p 3=2 8 C1d qd 9C1d c4fd pqd R2 ; 3 qd
18
19
where D1d is the dynamic bending stiness, C1d is the dynamic membrane stiness, and cfd is the dynamic shear fracture strain. The dynamic bending and membrane stiness are given by the same expressions for the
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static bending and membrane stiness, but they are calculated from laminate bending and membrane stiness matrices that have been adjusted for high strain rate. Both qd and cfd must be obtained from experiments. 2.3.3. Bottom facesheet failure Bottom facesheet failure can occur in clamped panels if the core crushing resistance is high enough to resist local deformation or top facesheet indentation. The entire panel deforms in a shear-bending mode and failure would take place when the maximum tensile strains in the bottom facesheet are at the critical fracture strain. Maximum tensile strain occurs right under the projectile or middle of the plate, where the bending moment is a maximum. The bending moments in a rectangular sandwich plate Mxx and Myy are given by Mxx Ds11 jxx Ds12 jyy
20
and Myy Ds12 jxx Ds22 jyy ;
21
where jxx and jyy are the curvatures in the plate with respect to the x and y directions and Dsij is the bending stiness of the sandwich. At the plate center, the bending moment is a maximum. Therefore, the curvature at the plate center jc is given by jc
Mmax ; Ds12
Ds22
22
where Mmax is the maximum bending moment. Timoshenko and Woinowski-Krieger [11] give Mmax in terms of the load P as P 2a 2 ln Mmax 0:669 : 23 4p pR Therefore, the curvature at the plate center can be expressed in terms of the load, bending stiness and plate geometry. By neglecting in-plane deformation, one can directly relate the curvatures to the maximum bending strain as follows emax
H jc : 2
24
If the bottom facesheet fails when emax ecr , then a failure criterion in terms of the load at which failure would occur, can be given by substituting expression in Eqs. (22) and (23) into Eq. (24) 4pecr Ds11 Ds12 : Pf H 25 2a 0:669 h 2 ln pR 2 Bottom facesheet damage under impact load occur when the maximum impact force is given by
4pecrd Ds11d Ds12d ; Ff H 2a 0:669 h 2 ln pR 2
26
where Ds11 and Ds12 are the dynamic bending stiness of the sandwich.
3. Impact damage initiation criteria When the maximum impact force is equal to the above critical impact forces, damage initiates. Expressions for the maximum impact force were given in the previous paper [1]. They are repeated here convenience. (i) Rigidly supported panels. The following nonlinear dierential is solved numerically M0 d Pl d Qd 0;
27
where M0 is the projectile mass, Pl d is the equivalent spring force and Qd pR2 qd is the dynamic core crushing resistance. The ordinary dierential equation is _ solved with initial conditions d 0 V0 and d 0 0, where V0 is the initial velocity of the projectile.The maximum impact force occurs at tmax when d_ 0, and is given by Fmax
max : M0 d t
28
(ii) Simply supported and clamped panels. The equivalent nonlinear spring force is ®rst linearized Pl d Kld d;
29
where Kld is the linearized dynamic local spring constant. Then, a closed-form expression for the maximum impact force can be derived as M0 Kgd Kld x Q2 x2 q d_20 Kld d Fmax ; 30 Kgd Kld 2 2 Qd x d_0 Kld where s Kld Kgd and x Kld Kgd M0
d_0
V0 Kgd Kgd Kld
and Kgd is the dynamic global spring constant. Setting equal Ff to Fmax in the above equations allows one to calculate a critical impact velocity for impact damage.
4. Comparison with experimental results The analytical solutions are applied to test data from three independent studies [3,6,8]. Composite sandwich panels with AS4/3501-6 Carbon/Epoxy facesheets and Nomex honeycomb core and hemispherical nose-shape indenters/projectiles were used in the ®rst two experimental studies, while E-glass WR cloth reinforced
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laminate (GRP) facesheets and Divinycell H130 foam core was used in the last experimental study. 4.1. Rigidly supported and two-sided clamped composite sandwich panels with AS4/3501-6 carbon/epoxy facesheets and Nomex honeycomb core Williamson and Lagace [8] performed a series of static indentation and low-velocity impact tests on rigidly supported and two-sided clamped composite sandwich panels made with AS4/3501-6 carbon/epoxy facesheet with Nomex honeycomb cores. The indenter and projectile were hemispherical-nose tups or cylinders made from case-hardened steel (rigid compared to the sandwich panel). Material and geometric properties of the facesheet, core and projectile are given in Table 1. Material properties for the woven [0/90] plies could not be found and the laminate facesheet was assumed to consist of separate 0° and 90° plies. The actual stiness and strength properties of the laminate facesheets would be slightly dierent than those shown in Table 1. Williamson and Lagace found that damage was initiated by top facesheet failure in all the tests. Core shear failure would occur only after the indenter/projectile penetrated through the top facesheet. Furthermore, the tensile strain in the bottom facesheet of the two-sided clamped panels was not high enough to cause bottom Table 1 Material properties of the rigidly supported and two-sided clamped C/E composite sandwich panels and indenter/projectile Facesheet
Hercules AW193-PW prepreg consisting of AS4 ®bers in a 3501-6 matrix a a 102 102 mm2 : square panel dimensions a b 203 89 mm2 : wide beam dimensions hk 0:175 mm: ply thickness qf 1617:3 kg/m3 : mass density Ply Stiness E11 142 GPa: longitudinal stiness E22 9:8 GPa: transverse stiness G12 7:1 GPa: in-plane shear modulus m12 0:3: Poisson's ratio Ply Strength cr 0:0112: static tensile failure strain s13 97 MPa: out-of-plane shear strength
Core
HRH 10 1/8±3.0 Nomex honeycomb (Ciba-Geigy) qc 48 kg/m3 : density d 3:2 mm: cell diameter H 25:4 mm: core thickness q 1:389 MPa: crushing resistance
Indenter/ projectile
Hemispherical-nose cylinder made of case-hardened steel D 12:7, 25.4 and 38.1 mm: diameter L 660 mm: length M0 1:53, 1.61 and 1.69 kg: masses
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facesheet damage before top facesheet damage or core shear failure. Cross-hair fractures were observed in the top facesheets. The cross-hair patterns are typical of an orthotropic laminate that has failed due to high in-plane tensile strains. Radial cracks run parallel to the ®ber direction, [0/90] in these tests. Tensile failure was the likely mode of damage because of the hemispherical-nose shape of the tup and the large facesheet de¯ections or membrane forces that are induced in the relatively thin top facesheet. Cylindro-conical tups will also produce the same cross-hair fractures, which will eventually lead to petaling when the projectile perforates the facesheet. 4.1.1. Static damage initiation Damage is a local phenomenon that depends only on the local indentation of the top facesheet. Therefore, rigidly supported and two-sided clamped sandwich panels made of the same materials should fail at the same load. In predicting either static and impact damage, an eective radius of 0.4R for the hemisphericalnose cylinder was used. The indentation force at damage initiation based on the tensile failure was predicted for sandwiches with three dierent facesheets thickness using Eq. (9). The analytical predictions are compared to experimental results in Fig. 4. The failure load depends on the size of the damage length, which is also shown in the diagram. Except for the sandwich with [0/90] facesheets, the analytical predictions for the damage initiation load using Eq. (9) are within 10% of tests results. Notice that the damage load for sandwiches with approximately the same damage length are the same for the rigidly supported and twosided clamped cases. Therefore, top facesheet damage initiation is indeed a local phenomenon. 4.1.2. Impact damage initiation In order to ®nd the dynamic strength of the facesheet and core, strain rates in the facesheet and core had to be found. The maximum value of the strain rate in the facesheet during impact damage initiation was approximated by dividing the tensile failure strain by the time it takes to reach maximum impact force, tmax . The strain rates for the low velocity impact tests were computed using static stiness and tensile failure strain and are found to be no greater than e_ 4 s 1 . It was dicult to ®nd tensile test data of carbon/epoxy laminates at this particular strain rate. However, Harding et al. [12] found that the in-plane stiness of woven carbon/epoxy laminates increased only by 17%, while the failure strain is about the same at rates of about 290 per s. Therefore, it would appear that at these low strain rates, the inplane stiness of the woven carbon/epoxy laminate facesheets were about the same as the static values. It was assumed that dynamic material properties for AS4/ 3501-6 carbon/epoxy laminate are equivalent to those for static indentation as follows:
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Fig. 4. Failure load vs. facesheet thickness for rigidly supported and two-sided clamped C/E composite sandwich indented by 25.4-mm diameter hemispherical-nose cylinder.
C1d C1 , Aijd Aij , ecrd ecr ,
dynamic membrane stiness of carbon/ epoxy, dynamic extensional stiness of carbon/ epoxy, dynamic tensile fracture strain of carbon/ epoxy.
Goldsmith et al. [13] found that the dynamic crushing strength of Nomex honeycomb was always 10% greater than the static crushing strength in low velocity impacts and independent impact velocity. Therefore, qd 1:1q,
dynamic crushing strength of Nomex honeycomb.
Critical velocities at damage initiation for sandwiches with the three dierent facesheet thickness were calculated from Eqs. (11) and (30) for the two-sided clamped panel. Unlike static damage initiation, the maximum impact force and critical impact velocity at damage depend on the boundary conditions. The analytical solutions are shown together with experimental results in Fig. 5. Except for the [0/90] facesheets, the predicted failure loads were about 20% lower than test results. 4.2. Simply-supported composite sandwich panels with AS4/3501-6 carbon/epoxy facesheets and Nomex honeycomb core Herup and Palzotto [2] conducted a series of static indentation and low-velocity impact tests on simply
supported composite sandwich panels with AS4/3501-6 carbon/epoxy facesheet with Nomex honeycomb cores. The indenter and projectile were hemispherical-nose tups or cylinders made from case-hardened steel. The panels were simply supported. The material and geometric properties of the facesheet, core and projectile are given in Table 2. Herup and Palazotto de®ned impact damage initiation as the ®rst drop in the quasi-static load or the dynamic contact load between the projectile and the top facesheet. C-scans revealed that the sandwich underwent multiple delaminations and matrix cracking in the top facesheet as well as core crushing and shear fracture. None of the panels exhibited bottom facesheet failure. The ®rst load drop corresponded to core shear failure. This is in contrast to the experiments by Williamson and Lagace [8], where damage initiation was due to top facesheet fracture. 4.2.1. Static damage initiation The static load at the instant of core shear failure was calculated from Eq. (16) for several composite sandwich panels with dierent facesheet thickness. The transverse shear fracture strain for HRH-10-1/8-4.0 Nomex honeycomb cf was calculated from Ref. [14] as 2.9%. The analytical predictions of the failure load are compared to the experimental results in Fig. 6. The analytical predictions are within 25% of the tests results. Eq. (16) seems to be most accurate for the thicker facesheets. This is because the shear fracture zone becomes more
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Fig. 5. Failure velocity vs. facesheet thickness for two-sided clamped C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
localized in sandwiches with thicker facesheets and the approximation of the shear angle in Eq. (13) is better for more localized deformations.
Table 2 Material properties of simply supported C/E composite sandwich panels and indenter/projectile Facesheet
AS4/3501-6 graphite±epoxy a a 178 178 mm2 : square panel dimensions hk 0:0635 mm: ply thickness qf 1632 kg/m3 : mass density
4.2.2. Impact damage initiation The strain rates that were calculated for the AS4/ 3501-6 facesheets and Nomex honeycomb were similar to those in the experiments by Williamson and Lagace [8]. Therefore, one can also assume the following dynamic material properties: D1d D1 , Dijd Dij ,
dynamic bending stiness of carbon/ epoxy, dynamic ¯exural stiness of carbon/ epoxy,
Ply Stiness E11 144:8 GPa: longitudinal stiness E22 9:7 GPa: transverse stiness G12 7:1 GPa: in-plane shear modulus m12 0:3: Poisson's ratio Ply Strength cr 0:0145: static tensile failure strain s13 120:7 MPa: out-of-plane shear strength Core
HRH 10 1/8±4.0 Nomex honeycomb (Ciba-Geigy) qc 64 kg/m3 : density d 3:2 mm: cell diameter H 12:7 mm: core thickness q 3:83 MPa: crushing resistance
Indenter/projectile
Hemispherical-nose cylinder (Case-hardened steel) D 25:4 mm: diameter M0 3:48 kg: mass
Fig. 6. Static failure load vs. facesheet thickness for simply supported C/E composite sandwich indented by 25.4-mm diameter hemisphericalnose cylinder.
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qd 1:1q, cfd 1:3cf ,
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dynamic crushing strength of Nomex honeycomb, dynamic shear fracture strain of Nomex honeycomb.
The 30% increase in the dynamic shear fracture strain of Nomex honeycomb was assumed to be the same as that in aluminum honeycomb [15]. The maximum impact load at the instant of core shear failure was calculated from Eq. (18) for several composite sandwich panels with dierent facesheet thickness. The analytical predictions of the failure load are compared to the experimental results in Fig. 7. The analytical predictions are again within 25% of the tests results and Eq. (18) appear to be most accurate for the thicker facesheets. The variation of both static and impact failure loads with facesheet thickness are plotted in Fig. 8. In all cases the impact damage load is higher than the static failure load. This is primarily due to the fact that the crushing and shear strength of the core increase with strain rates. Material properties of the C/E facesheet are almost unchanged. Herup and Palazotto [2] noted that the dierences between the static and impact failure loads were slightly larger for sandwiches with the thicker facesheets. They attributed this tendency to the increase of inertia associated with a thicker facesheet. The failure load predicted by Eq. (18) does not include the inertial resistance of the facesheet because it was assumed to be negligible compared to the mass of the projectile. 4.3. Composite sandwich panels with E-glass WR cloth reinforced facesheets and divinycell H130 foam core
Fig. 8. Comparison of static and impact failure load for simply supported C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
clamped composite sandwich panels made of E-glass WR cloth reinforced laminate (GRP) facesheets and Divinycell H130 foam cores. The GRP composite sandwich panels were clamped along four edges, but inplane were not constrained. The material properties for the facesheet and core are given in Table 3. As in the Table 3 Material properties of the clamped GRP sandwich panel and indenter/ projectile Facesheet laminate
Wen et al. [3] conducted a series of static indentation, impact and perforation tests with blunt cylinders and
E-glass/polyester laminate a a 200 200 mm2 850 850 mm2 : square panel dimensions hk 0:29 mm: ply thickness qf 1650 kg/m3 : mass density Ply Stiness E11 24:4 GPa: longitudinal stiness E22 6:87 GPa: transverse stiness G12 2:89 GPa: in-plane shear modulus m12 0:32: Poisson's ratio Ply Strength cr 0:021: static tensile failure strain s13 45 MPa: out-of-plane shear strength
Core
Divinycell H130 foam core (Barracuda) qc 130 kg/m3 : mass density H 25:4 mm: core thickness Stiness Ec 175 MPa: Young's modulus Gc 50 MPa: shear modulus mc 0:3: Poisson's ratio Strength q 2:5 MPa: crushing resistance sc 2 MPa: out-of-plane shear strength
Indenter/projectile Fig. 7. Impact failure load vs. facesheet thickness for simply supported C/E composite sandwich impacted by 25.4-mm diameter hemispherical-nose cylinder.
Blunt cylinder made of case-hardened steel D 10:5±50 mm: diameter M0 17:9 g±42.5 kg: mass
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case of the C/E laminate, all the material properties for the woven [0/90] plies could not be found and the laminate facesheet was assumed to consist of separate 0° and 90° plies. Test results for damage initiation due to low velocity impact were not available, but damage initiation loads for the panels undergoing static indentation was plentiful. Damage initiation was attributed to shear failure in the top facesheet by the blunt cylinder/projectile. Shear failure was the likely mode of damage initiation because the facesheets were relatively thick. When the facesheet is thick, local indentations are small and no appreciable membrane force can develop in the facesheet. Furthermore, core material right under a blunt cylinder is compressed and a localized shear zone develops around the compressed region. Core material being compressed under the blunt cylinder is also constrained by the adjacent honeycomb cells. The interaction of the blunt projectile and facesheet thus makes shear failure a likely mode of damage. Shear failure is a local phenomenon and should be independent of panel size and boundary conditions. As described in Eq. (1), the shear failure load should increase parabolically with indenter radius and increase linearly with facesheet thickness. Failure loads based on shear fracture of the top facesheet were calculated from Eq. (1). The analytical results are compared to test results for varying projectile radius and facesheet thickness in Figs. 9 and 10, respectively. The test results shown in Fig. 9 are for three dierent panel sizes, but despite this, the parabolic function that describes how the failure load varies with indenter radius ®ts the experimental data rather well. The analytical predictions are within 10% of experimental results. Therefore, the shear failure load was independent of panel size. Furthermore, the linear function describing how the failure load varies with facesheet thickness also ®ts the experi-
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Fig. 10. Failure load vs. facesheet thickness for rigidly supported and four-sided clamped GRP composite sandwich indented by 50-mm diameter blunt cylinder.
mental data well in Fig. 10. The analytical predictions are within 20% of experimental results. Test results for rigidly supported panels were compared to the clamped panel results in Fig. 10. The shear failure loads for both support conditions are approximately the same. Therefore, the shear failure load was independent of boundary conditions. In an eort to induce tensile failure of bottom facesheet due to global bending, Wen et al. [3] used denser PVC foam core, HCP100 with a nominal density of 400 kg/m3 , in the composite sandwich with [0/90]6 or 3.25 mm thick E-glass/polyester facesheets. The panel was indented by the 50-mm diameter blunt cylinder and failed by tensile failure of bottom facesheet at a load of 57 kN. The bottom facesheet tensile failure load calculated from Eq. (25) was equal to 69.5 kN, which about 22% higher than the experimental results. 5. Conclusion
Fig. 9. Failure load vs. indenter radius for four-sided clamped GRP composite sandwich with [0/90]6 laminates indented by blunt cylinder.
Equivalent single and multi degree-of-freedom systems were used to predict the transient deformations of composite sandwich panels under impact loading in a previous paper, and analytical solutions for low-velocity impact damage of the same composite sandwich panels were given in this paper. The composite sandwich panels were symmetric and consisted of orthotropic laminate facesheets and a core with constant crushing resistance. The initial mode of impact damage depended on the panel support conditions, projectile nose-shape and geometric and material properties of facesheet and core. Particular damage initiation modes that were discussed in this paper include fracture of the top facesheet, core shear failure, as well as tensile failure of back facesheet. Fracture patterns in the top facesheet resembled some of those in metals, such as shear plugging by blunt
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projectiles and petalling by hemispherical-nose projectiles. The impact failure modes were similar to static indentation failure modes, but material properties of the facesheets and core had to be adjusted for the in¯uence of high strain rate. Predicted damage initiation loads and impact velocities compared well to experimental results. Acknowledgements This work was supported by the Ohio Aerospace Institute under the 1999 OAI Core Collaborative Research program. The research was made possible through collaboration with GE Aircraft Engines, RollsRoyce Allison, BFGoodrich Aerospace and the NASA Glenn Research Center. The authors wish to thank Dr. P.A. Lagace, Dr. A.N. Palazotto, and Dr. H.M. Wen for the use of their experimental results. References [1] Hoo Fatt MS, Park KS. Dynamic models for low-velocity impact damage of composite sandwich panels ± Part A, Deformation. Compos. Struct. 2001;52(3±4):335±51. [2] Palazotto AN, Gummadi LNB, Vaidya UK, Herup EJ. Lowvelocity impact damage characteristics of Z-®ber sandwich panels ± an experimental study. Compos Struct 1999;43:275±88. [3] Wen HW, Reddy TY, Reid SR, Soden PD. Indentation, penetration and perforation of composite laminates and sandwich panels under quasi-static and projectile loading. Key Eng Mater 1998;141±143:501±52. [4] Roach AM, Jones N, Evans KE. The penetration energy of sandwich panel elements under static and dynamic loading. Part I. Compos Struct 1998;42:119±34.
[5] Roach AM, Jones N, Evans KE. The penetration energy of sandwich panel elements under static and dynamic loading. Part II. Compos Struct 1998;42:135±52. [6] Herup EJ, Palazotto AN. low-velocity impact damage initiation in graphite/epoxy/Nomex honeycomb-sandwich plates. Compos Sci Technol 1997;57:1581±98. [7] Ferri R, Sankar BV. Static indentation and low-velocity impact tests on sandwich plates. In: Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition, vol. 55, 1997; Dallas. p. 485±90. [8] Williamson JE, Lagace PA. Response mechanism in the impact of graphite/epoxy honeycomb sandwich panels. In: Proceedings of the Eighth ASC Technical Conference, Cleveland, OH; 1993. p. 287±97. [9] Reddy TY, Soden PD, Reid SR, Sadighi M. Impact response of thick composite laminates and sandwich panels. Collaborative Research Program on the Cost-Eective Use of Fibre-Reinforced Composite Oshore, Marintech Research, Phase 1, Report No. CP04, 1991. [10] Nemes JA, Eskandari H, Rakitch L. Eect of laminate parameters on penetration of graphite/epoxy composites. Int J Impact Eng 1998;21(1/2):97±112. [11] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. New York: McGraw-Hill; 1961. [12] Harding J, Li YL, Saka K, Taylor MEC. Characterization of the impact strength of woven carbon ®bre/epoxy laminates. In: Proceedings of the International Conference on Mechanical Properties of Materials at High Rates of Strain, Oxford; 1989. p. 403±10. [13] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. Int J Impact Eng 1991;12(2):241±62. [14] Hexcel Cooperation. Mechanical Properties of Hexcel Honeycomb Materials. TBS 120, Dublin, CA; 1988. [15] Goldsmith W, Louie DL. Axial perforation of aluminum honeycombs by projectiles. Int J Solids Struct 1995;32(8/9): 1017±46.