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A modiﬁed energy-balance model to predict low-velocity impact response for sandwich composites C.C. Foo a,⇑, L.K. Seah b, G.B. Chai b a b

Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore Nanyang Technological University, School of Mechanical and Aerospace Engineering, 50, Nanyang Avenue, Singapore 639798, Singapore

a r t i c l e

i n f o

Article history: Available online 24 November 2010 Keywords: Modelling Sandwich structures Energy-balance model

a b s t r a c t An energy-balance model is often used to analyze impact dynamics for composite structures. However this model tends to overestimate the peak impact load after the onset of damage since it does not account for damage initiation and propagation. In this paper, the energy-balance model is coupled with the law of conservation of momentum to extend its validity beyond the elastic response regime for a composite sandwich structure subjected to low-velocity impact. Closed-form solutions were derived for the plate’s elastic structural stiffness and the critical load at the onset of damage. The critical load was theoretically predictable by accounting for the elastic energies absorbed by the plate up to core failure. Impact test results also showed that the relative loss of the plate’s transverse stiffness after damage was directly related to the energy absorbed by the plate, which could be calculated given the damage initiation energy. The stiffnesses and the critical load were then used in the modiﬁed energy-balance model to predict transient load and deﬂection histories. Predicted results were comparable with test data, in terms of critical and peak loads, as well as the overall behaviour. This impact model is an efﬁcient design tool which can complement detailed FE simulations. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Sandwich structures with laminated facings are widely used in many applications because of their high speciﬁc bending stiffness. Despite this, the main weakness of such structures has always been the poor rigidity in the transverse direction. In particular, these structures are susceptible to low-velocity impact damage, which reduces the structural stiffness and strength [1]. One way to understand the effect of impacts is to develop a mathematical model to predict the structure’s impact response. The spring-mass models and energy-balance models are two such models that have been developed to predict the impact force history and the overall response of the structure [1–3]. In the springmass model, springs are used to represent the effective structural stiffness of the impactor-plate system. Consequently, the elastic response of the plate may then be solved from the dynamics equations of the model [1]. On the other hand, the energy-balance model considers the conservation of total energy in the system to solve for the maximum impact load [1]: the kinetic energy of the impactor is equated to the sum of the energies due to deformations. The energy-balance model assumes that the impactor becomes stationary when the structure reaches its maximum deﬂection, and ⇑ Corresponding author. E-mail address: [email protected] (C.C. Foo). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.11.008

the initial kinetic energy is used to deform the structure. In this model, one beneﬁt is that deformation energies can be quantiﬁed and identiﬁed separately [2,4]. However, unlike the spring-mass model, it only yields the maximum impact force but not the load– time history [1–3]. The simple and efﬁcient spring-mass and energy-balance models have been widely used to predict the impact force for composite laminates [5,6,8], aluminum sandwich plates [2], as well as composite sandwich structures [9,10]. But these elastic models cease to be valid after damage initiation and are unable to model damage propagation. Feraboli [11] published load-energy plots for carbon/epoxy laminates subjected to low-velocity impacts, and showed that the spring-mass model overestimates the peak impact load for the plates after the onset of damage. Similarly, the energy-balance model is inaccurate when damage initiates at higher impact energies [2,8]. Thus modiﬁed spring-mass models have been proposed to account for damage [9,11,12], but they depend strongly on unknowns that have to be determined experimentally. Here, the energy-balance model is coupled with the law of conservation of impulse-momentum to extend its validity beyond the elastic regime. Closed-form solutions were ﬁrst derived for two parameters that described the plate’s structural behaviour, namely, the plate’s elastic structural stiffness and the critical load at the onset of damage. The subsequent degradation in the plate’s stiffness

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after damage was derived using results from experimental tests. Subsequently the critical load and stiffnesses were included in the modiﬁed model to derive load and deﬂection histories for the sandwich plate. This paper is an extension to our previous work [13] which derived the critical load and the reduced stiffness from numerical ﬁnite element simulations. 2. Experimental investigation 2.1. Experimental tests We ﬁrst describe the experimental tests carried out in this study. The primary objectives of the experimental work were to guide the development of the model, and subsequently, to validate the model with test data. Both quasi-static indentation and lowvelocity impact tests were conducted on square composite sandwich plates, which were subjected to a transverse load at their centre by a hemispherical indentor/impactor [13,14], as illustrated in Fig. 1. In all tests, the specimen was clamped between two steel plates, each having a circular aperture of diameter 76.4 mm, with the midpoint of the specimen directly located underneath the indentor/ impactor. Both indentor and impactor had the same diameter of

∅13.1 mm

Steel indentor/impactor

13.1 mm, and the impactor mass was 2.65 kg. A spherical indentor was chosen in this work because this shape is the most widely-discussed in the literature. The absence of sharp edges in the indentor also reduces the chances of discontinuities in the deformation proﬁle on the indented skin, which may further complicate any analysis. The static indentation tests were conducted using the Instron 5500R test system operating under displacement control at a constant cross-head speed of 0.5 mm/min. For the drop-weight impact tests, the Instron Dynatup 8250 impact testing machine was used. The plates were made of FibreduxÒ 913C-HTA carbon/epoxy skins bonded to HexWebÒ A1 Nomex honeycombs. The test specimens, which measured 100 mm by 100 mm each, were hand-laid and cured in-house. The material properties of the laminates and honeycombs are listed in Tables 2 and 3. In total there were 20 conﬁgurations of sandwich plates loaded under quasi-static indentation, and 11 conﬁgurations impacted at various energies (Table 1). 2.2. Low-velocity impact response A typical load–time history and load–displacement plot for the composite sandwich subjected to low-velocity impact is shown in Fig. 2. Initially, the impact force increases in a sinusoidal-like manner with time and linearly with the displacement of the impactor (with slope K0), as also observed in Refs. [6,8,11]. As the force increases up to a critical value (P1), there is a sudden load drop indicating the onset of damage. This is followed by a subsequent increase to the maximum load at a lower stiffness, represented

Honeycomb core Laminated facesheets

Table 2 Material properties for FibreduxÒ 913C-HTA carbon– epoxy laminates.

Thickness of facesheet, hf Core height, h c ∅76.4 mm

hf

Clamp plate 100 mm Fig. 1. Schematic setup of the clamped sandwich plate in the static and impact tests.

Property

Value

Longitudinal modulus, E11 (GPa) Transverse modulus, E22 (GPa) Out-of-plane modulus, E33 (GPa) Poisson’s ratios, m12 and m13 Poisson’s ratio, m23 Shear moduli, G12 and G13 (GPa) Shear modulus, G23 (GPa) Density, q (kg/m3)

150 9.5 9.5 0.263 0.458 5.43 3.26 1100

Table 1 List of composite sandwich specimens used in the static indentation and low-velocity impact tests.

a b

Specimen identity

Skin stacking sequencea

Core cell size (mm)

Core height (mm)

Loading rate (m/s)

C1/3/15 C1/6/15 C1/13/15

[0/90/0/90/0]s [0/90/0/90/0]s [0/90/0/90/0]s

3 6 13

15 15 15

C1/3/20 C1/6/20 C1/13/20 C1/3/25 C1/6/25 C1/13/25 C2/3/15 C2/13/15 C3/3/15 C3/6/15 C3/13/15 C3/3/20 C3/6/20 C3/13/20 C3/3/25 C3/6/25 C3/13/25

[0/90/0/90/0]s [0/90/0/90/0]s [0/90/0/90/0]s [0/90/0/90/0]s [0/90/0/90/0]s [0/90/0/90/0]s [+45/45/0/90/0]s [+45/45/0/90/0]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s [+45/02/90/02/45]s

3 6 13 3 6 13 3 13 3 6 13 3 6 13 3 6 13

20 20 20 25 25 25 15 15 15 15 15 20 20 20 25 25 25

QSb; 1.18, 1.65 QS; 1.65 QS; 1.18; 1.65; 1.98 QS; 1.65 QS; 1.65 QS; 1.65 QS; 1.65 QS; 1.65 QS; 1.65 QS; 1.18 QS; 1.18, 1.65 QS QS QS QS QS QS QS QS QS

Each lamina ply has a nominal thickness of 0.125 mm. ‘QS’ refers to quasi-static indentation test.

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C.C. Foo et al. / Composite Structures 93 (2011) 1385–1393 Table 3 Material properties of HexWebÒ A1 Nomex honeycomb cores with nominal core density of 64 kg/m3.

a

Honeycomb conﬁguration

hc (mm)

E33c (MPa)

G23c (MPa)

G13c (MPa)

Elastic strain energy per unit area,a U (kJ/m2)

3–15 3–20 3–25 6–15 6–20 6–25 13–15 13–20 13–25

15 20 25 15 20 25 15 20 25

201 181 181 150 169 173 139 209 200

63 63 63 55 55 55 55 55 55

35 35 35 33 33 33 32 32 32

0.992 ± 0.142 1.095 ± 0.163 0.968 ± 0.095 1.045 ± 0.079 0.992 ± 0.099 1.306 ± 0.113 0.976 ± 0.071 0.948 ± 0.162 1.115 ± 0.151

Measured values given in mean ± standard deviation, with nine samples for each core.

(a)

(b)

Fig. 3. Normalised energy plot for 73 composite sandwich specimens subjected to low-velocity impact. Fig. 2. Schematic of a typical (a) load–displacement plot and (b) load–time plot for a composite structure under impact.

by the slope of the load–displacement curve (Kdam), before rebounding occurs. Similar ﬁndings have also been reported in previous studies [6,7]. In the literature, it is well-accepted that most of the energy absorbed by the plate during impact is dissipated in the form of damage modes, and the extent of damage is reﬂected in the reduction of the plate’s stiffness. Here we present an approach to deduce the reduced stiffness Kdam of a plate at a particular impact energy Eimp based on the absorbed energy Eabs and the damage initiation energy U1. This energy U1 is deﬁned as the area of the load–displacement curve up to the critical load P1, while the absorbed energy Eabs is the area enclosed between the loading and unloading portions of the load–displacement curve. The impact energy refers to the incident kinetic energy of the impactor. We ﬁrst plot the normalized absorbed energy ratio Eabs/U1 against the inverse of normalized impact energy ratio U1/Eimp in Fig. 3 for 73 sandwich composites of various core thicknesses and cell sizes with two different laminate orientations, impacted at various velocities (Table 1). This group of specimens suffered barely visible impact damage (BVID) in the form of permanent indentation on the impacted skin without skin fracture. The power regression curve in Fig. 3 is the best-ﬁtted curve based on the leastsquares method, and the equation of the curve can be rewritten to give:

0:2042 Eabs U1 ¼ 0:3494 Eimp Eimp

ð1Þ

Eq. (1) relates the absorbed energy to the critical energy for damage initiation and the impact energy for these specimens. Accordingly, a plate with a lower U1 absorbs more energy at a given impact energy level. Eq. (1) is only valid for 0.048 < U1/Eimp < 0.244 based on available data; these limits could be extended to a wider range under more testing. This empirical energy equation, which is different from others proposed in the literature (see Refs. [6,15–17]), includes U1 which deﬁnes the onset of damage and characterizes the impact damage resistance of the sandwich plate. With the empirical relationship, the energy absorbed by the structure at a given impact energy can be predicted simply by determining U1, which is rate-independent, as also observed in Refs. [11,18]. Next we plot the ratio of the plate’s stiffness after and before impact (Kdam/K0) as a function of the absorbed energy ratio (Eabs/ Eimp) in Fig. 4 for 11 sandwich conﬁgurations of various cores and skins impacted at various Eimp (Table 1). The reduced stiffness Kdam was determined as the ﬁnal slope of the load–displacement curve just prior to unloading. The experimental points in Fig. 4 are approximated by a straight line,

K dam Eabs ¼1 K0 Eimp

ð2Þ

This equation implies that the plate’s reduced stiffness is a function of the recoverable energy at the end of the impact event, and is identical to the one identiﬁed by Lifshitz et al. [7] for carbon ﬁbre reinforced plastic beams. Note also that the ratio Kdam/K0 for these plates lies mainly in the range of 0.4–0.6. Consequently, Eqs. (1) and (2) allow us to infer the reduced stiffness of the impacted plate by simply knowing the value of the critical energy for damage initiation U1.

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where a0 and a are the transverse deﬂection and the radius of region of local indentation on the top skin, respectively (Fig. 5). This indentation proﬁle approximates the lowest mode of vibration for a clamped circular plate [27], where the shape of the indentation proﬁle is symmetrical about its centre. According to Abrate [1], laminated facesheets with more than 6 plies and a symmetric lay-up can be modelled as orthotropic plates. Thus, under axisymmetrical bending, the strain energy of the elastic circular clamped facesheet can be expressed as [28,29]

U1 ¼

32pDf a20 3a2

ð4Þ

where an equivalent bending stiffness of the orthotropic facesheet, Df, is given by

Df ¼ Fig. 4. Relative reduction in the stiffness of the plate as a function of relative loss in impact energy.

U2 ¼

3.1. Elastic stiffness We consider the elastic response of the plate prior to damage. By decoupling the local and global responses and ignoring any interaction between them, the local and global stiffnesses can be determined separately [20–22]. 3.1.1. Local indentation The principle of minimum total potential energy is used here to derive the elastic local stiffness, Kloc, of a clamped circular sandwich panel indented by a spherical indentor at its middle (Fig. 5). The top skin is modelled as an elastic plate subjected to a concentrated load, which rests on the core modelled as an elastic Winkler foundation. The shear resistance between the laminate and the core, as well as membrane stretching of the skin, is ignored for small indentation. This approach is similar to those adopted in previous studies [21,23–25]. The elastic modulus of the foundation kc, which has the dimensions of force per unit surface area of plate per unit deﬂection, is related to the modulus of the core in the out-of-plane direction E33c and the thickness of the core hc by kc = E33c/hc [1,23,26]. The localized indentation area is assumed to be clamped at its boundary [10], with the proﬁle of the local indentation represented by

2 r2 aðrÞ ¼ a0 1 2 a

ð3Þ

Fig. 5. A clamped rigidly-supported circular sandwich plate indented by a spherical indentor.

ð5Þ

in which Dij are terms from the laminate bending stiffness matrix. The strain energy due to the deformation of the elastic foundation is given as [27]

3. Analytical formulation This section presents the derivation of the elastic stiffness (K0) and the critical load at damage initiation (P1). These two parameters, along with the reduced stiffness of the plate after damage (Kdam) deduced earlier in Section 2.2 from test data, are then incorporated into the modiﬁed energy-balance model to predict the low-velocity impact response of sandwich plates. We assume the sandwich plate essentially deforms in a quasi-static manner under low-velocity impact [2,13,19].

1 ½3D11 þ 2ðD12 þ 2D66 Þ þ 3D22 8

Z 2p Z 0

0

a

1 pkc a2 a20 kc a2 rdrdh ¼ 10 2

ð6Þ

The work done by the contact load P is then

W ¼ P ðaÞr¼0 ¼ Pa0

ð7Þ

By considering the total potential energy, P = U1 + U2 W, and applying the minimising condition, oP/@a, it yields

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 320Df a¼ 3kc

ð8Þ

Again, applying oP/@ a0 and substituting Eq. (8), the localized load– indentation relationship is derived. Accordingly the elastic local stiffness of a rigidly-supported circular sandwich plate with orthotropic skins is

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K loc ¼ 12:98 Df kc

ð9Þ

The local stiffness is directly proportional to the foundation modulus of the core and the bending stiffness of the top skin. As observed by many investigators [1,19,21,30,31], the initial load varies linearly with indentation prior to core failure, which is also conﬁrmed by Eq. (9). 3.1.2. Global deﬂection When the sandwich plate is clamped around its edges, it experiences both local and global deformation. Global deformation, w, refers to the bending and shearing of the entire sandwich plate. Because the core is much thicker than the skins, the membrane stretching of the skins and the core are assumed to be negligible [10]. Thus, the load sustained by the plate is related to this global deﬂection [5] by

P ¼ K glo w0

ð10Þ

where Kglo = KbKs/(Kb + Ks) is the effective global stiffness due to bending stiffness, Kb, and shear stiffness, Ks. The derivation of the bending stiffness and shear stiffness hereafter follows closely to that adopted by Zhou and Stronge [10]. The shear stiffness of the plate, Ks, is derived by dividing the central uniformly distributed pressure load, which acts over a contact area with radius Rc, by the shear deﬂection at the centre of the plate to give [10]

Ks ¼

4pGc ðhf þ hc Þ2 hc ð1 þ 2 lnðRp =Rc ÞÞ

ð11Þ

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where Gc is the average out-of-plane shear modulus of the core; hc is the thickness of the core; Rp and Rc are the outer and contact radii of the plate, respectively. An initial value of Rc is required to obtain qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ks, where Rc ¼ 2a0 Ri a20 by geometrical relation in which Ri is the radius of the indentor [1]. However the effective shear stiffness is insensitive to this assumed contact radius [10,13]. Thus the mean value for the function, ln (Rp/Rc), for the range of a0 from 0 to Ri was used; it was calculated to be 2.07. Using the classical plate theory, the effective bending stiffness of the clamped sandwich plate, Kb, is derived as [5,27,29]

Kb ¼

16pD0sw

ð12Þ

R2p

where D0sw is the equivalent bending stiffness of the clamped sandwich plate [28,29]. Using Eq. (5), D0sw can be determined by replacing the equivalent Dij terms from the [D] stiffness matrix of the sandwich panel. The bending stiffnesses of sandwich panels with thin stiff facesheets on a soft core are given as [21,23]

2 E1f hf ðhc þ hf Þ2 6 Dij ¼ 4 v 12f E2f 2ð1 v 12f v 21f Þ 0

v 12f E2f

0

E2f

0

0

G12f ð1 v 12f v 21f Þ

3 7 5 ð13Þ

where the subscripts f and c refer to the properties of facesheet and core, respectively. The elastic constants of the laminates were calculated using the composite laminate theory. Consequently, the elastic stiffness of the sandwich panel, K0, is then,

1 1 1 ¼ þ K 0 K loc K glo

ð14Þ

3.2. Onset of damage At the onset of damage (P1 in Fig. 2), there is a substantial drop in the structural stiffness, and core failure has been previously identiﬁed to occur at that point [19,32–34]. Core materials are expected to substantially affect the damage initiation characteristics of sandwich panels because they generally have lower mechanical properties than skins due to their lower density [31,35]. Several studies have attempted to predict the critical load at this state. The theoretical load at the onset of core yielding in foam-cored sandwich structures has been investigated in Refs. [26,36], where the core is modelled as a homogenous material. In honeycomb sandwich structures, Zheng et al. [37] proposed a failure criterion based on the core compressive yield strength to model damage. Similarly, Castanie and colleagues [38] modelled the honeycomb core as a grid of nonlinear springs and used a crushing law which was empirically derived from ﬂatwise compression tests on bare honeycombs to model damage. Olsson [21,39] presented an explicit expression for the critical load for core crushing by using small deﬂection theory for a plate resting on an elastic foundation. Here, the load at failure initiation (P1) is predicted by considering the elastic energy absorbed by the plate up to the onset of core damage. Under indentation loading, the core in the sandwich plate is subjected to both shear and out-of-plane compressive stresses. However Castanie et al. [38] examined the behaviour of Nomex honeycomb core under compression, and found that the compression load is supported mainly by these vertical edges in the hexagonal cell. Neglecting the inﬂuence of shear loading, they modelled the honeycomb core as a grid of nonlinear springs located exactly at the vertical edges, and accurately predicted the indentation of these sandwich structures. Moreover they found that the inﬂuence of the shear force is negligible for spherical indentors [38]. These results suggest that, for the current aim of predicting the load at

initial failure, we can neglect the inﬂuence of shear loading and consider the compression behaviour of the core under indentation. First, the strain energy absorbed by the core at initial failure under local indentation is determined. Flatwise compression tests were carried out on bare Nomex cores using the Instron 5500R machine [40]. Three different core sizes (9, 33, and 60 cells) were considered for each core conﬁguration listed in Table 1, with three test samples for each core size. Fig. 6 shows a typical load–displacement curve obtained from the ﬂatwise compression tests. Initially, the compressive load increased linearly due to the elastic bending of the thin cell walls until a critical load, PPk, was reached, and core failure is assumed to initiate at this point. The area under the load– displacement curve up to the peak load is the energy absorbed by the core at initial failure, UPk. The elastic strain energy absorbed by the core per unit area at initial failure is deﬁned as U = UPk/Acore where Acore is the effective crushing area of the core (Table 3). Consequently, under local indentation, the energy absorbed by the core up to initial damage, U th core , is estimated by multiplying the planar core damage area by U. The core damage region which is assumed to be circular is characterized by an average diameter 2Rcr, where Rcr is a function of the indentor’s radius (Rcr = bRi). The constant, b, depends on the materials of the top skin and core under concern. Accordingly, the energy absorbed by the core up to initial damage under local indentation U th core is determined as 2 2 2 U th core ¼ pRcr U ¼ pb Rind U

ð15Þ

Here, we ignore the energy to deform the surrounding cell walls that have not yet failed, which is a reasonable assumption since the initial damage is expected to be highly localized in the vicinity of the indentor. Next, the strain energy of the top skin, which is assumed to be elastic at the onset of damage, U th tfs , is calculated. The interaction between the core and the top skin, as well as the deformation of the bottom facesheet, is ignored. This strain energy comprises the bending and membrane stretching energies (Ub and Um). For an undamaged facesheet, its bending energy Ub is expressed in Eq. (4). Similarly the membrane stretching energy Um for a clamped circular plate with vf = 0.3 can be simpliﬁed as [10,13]

U m ¼ 2:59pDf

!

a40 2

a2 hf

ð16Þ

Accordingly, the local indentation ath 0 and the radius of the local indentation area ath are required to determine the elastic energies th U th b and U m at the onset of damage. Note that the radius of the local

Fig. 6. A load–displacement curve for compressive test on 3–15 bare Nomex honeycombs with 60 cells.

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deformation zone a, as presented in Eq. (8), is valid only for the elastic indentation before core damage. Therefore it is reasonable to estimate the radius of the local indentation zone on the facesheet at the onset of damage, ath, using Eq. (8). Although the indentation is assumed to be circular here, in reality, the indentation is elliptical in an orthotropic plate because the elastic constants of the plate are direction-dependent. However, according to Olsson [41], this effect is small: he cited a previous numerical solution which showed that the axis ratio of the ellipse was only 1.07 for E1f/E2f = 14.3. Next, to calculate the local indentation ath 0 , assume that the load varies linearly with the local indentation prior to initial damage [10,31], such that P = Kloca0. As such, the energy due to local indentation in the contact region is

Uc ¼

Z

a0

0

1 P2 Pda0 ¼ K loc a20 ¼ 2 2K loc

ð17Þ

This energy is then equated to the energies sustained by the core and the top facesheet under local indentation,

U c ¼ U core þ ðU b þ U m Þtfs

ð18Þ

Substituting Eqs. (4), (15), (16) and (17) into Eq. (18) yields

32pDf a20 1 a2 K loc a20 ¼ pb2 R2ind U þ 1 þ 0:244 20 2 2 3a hf

losses, the total work done by the impact load on the plate at any instant t is equal to the change in kinetic energy of the impactor at that instant,

U c þ U bs ¼

P2 1 ¼ Mimp V 2imp VðtÞ2 2:K 0 2

ð23Þ

where M imp denotes the mass of the impactor, V imp is the impact velocity and V(t) refers to the velocity of the impactor at time t. The impact load is also a function of time, i.e., P = P(t). By the law of conservation of impulse-momentum,

Mimp ðV imp VðtÞÞ ¼

Z

t

Pdt

ð24Þ

0

The load and velocity histories are then solved using Eqs. (23) and (24). Subsequently, the deﬂection of the impactor is obtained by integrating the velocity history. Once the load reached the critical load P1, the elastic stiffness K0 is degraded to the reduced stiffness Kdam to account for damage. Fig. 7 illustrates the calculation

(a)

! ð19Þ

By solving Eq. (19), four roots would be obtained, of which the smallest positive root is the indentation at the onset of failure ath 0 . Note that, if ath 0 < hf , the strain energy for the top skin is mainly due to bending [27]; by ignoring the membrane stretching energy, Eq. (19) reduces to a simple quadratic equation. Similarly, again assuming that the load varies linearly with the global deformation (w0) prior to initial damage [10,31], i.e., P = Kglow0, the energy due to global deformation in the form of bending and shear deformations is

U bs ¼

Z

w0

Pdw0 ¼ 0

1 P2 K glo w20 ¼ 2 2K glo

ð20Þ

Previous studies have shown that there is a critical impact force, P1 at the onset of damage, that does not depend on the impact energy [4,8,11,42]. Given that the damage initiation energy, Uth, is the sum of Uc and Ubs up to that instant (Eqs. (17) and (20)), the critical load can be expressed as

P1 ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2K 0 U th

ð21Þ

This result is analogous to the peak impact load Ppeak derived in References [1,11], which use an elastic spring-mass model to show that,

Ppeak ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2K 0 Eimp

ð22Þ

(b)

where Eimp is the impact energy and K0 is the structural stiffness prior to damage. However Eq. (22) ceases to work beyond the elastic regime. If the impact energy is lower than the threshold energy, no damage will occur and the peak load can be predicted by Eq. (22). Conversely, if the impact energy exceeds this threshold value, damage will initiate. 3.3. Impact model In this analysis, the energy-balance model is modiﬁed to derive the load and deﬂection histories for the impacted sandwich plate. The maximum impact load and maximum plate deﬂection are assumed to occur when the velocity of the impactor becomes zero. By the law of conservation of energy, assuming no other energy

Fig. 7. Calculation procedure for impact model: (a) ﬂowchart; and (b) approximation of integral in Eq. (24) using trapezoidal rule in representative load–time plot.

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procedure for the impact model. The integral on the right-hand side of Eq. (24) is approximated by the area under the load–time curve using the trapezoidal rule. No unloading is considered. 4. Results and discussion This section presents the results of the model which is used to predict the impact force history of composite sandwich structures subjected to low-velocity impact. Table 4 shows the predicted elastic stiffness (K0) compared against test results for 20 conﬁgurations of sandwich plates loaded under quasi-static indentation (Table 1). Despite the simple deﬁnition of the foundation modulus used in our approach here, the predicted values of K0 are reasonable, within 19% of the test data. For the same core conﬁguration, the plates with thicker skins (C3 types) exhibited greater elastic stiffnesses due to the increase in ﬂexural rigidity, transverse shear resistance, and local contact resistance. Similarly, the plates with thicker cores were stiffer. The damage initiation threshold load P1 of the composite sandwich plate (Table 4) is predicted based on the assumption that core damage occurs at the onset of damage for the plate. This is also in accordance with previous ﬁnite element results [13] which indicated that core damage does occur either at or very near to P1. The predicted loads are within 22% of the test data. The critical loads increased signiﬁcantly when the skin thickness was increased. This load is an important parameter because it measures the plates ability to resist damage. Since sandwich plates with thicker skins have higher damage initiation loads, we would expect lower absorbed energies for these plates at a given impact energy (Eq. (1)). The distinct load drop observed in the load–time response at this critical point appears to be speciﬁc for the composite plates, with similar observations reported elsewhere for composite laminates [8,11,42] and sandwich structures [9,19,32]. However no load drop was observed for aluminum sandwich plates in our previous results [43], which could be due to the different plasticity of the material. Moreover contrary to the linear contact law observed for these composite sandwich plates, the contact load P for the aluminum sandwich plate is related to the local indentation a0 by the power law, P ¼ K loc an0 , where n is some constant. As shown in [43], the local and global stiffnesses could be determined by explicit FE quasi-static analyses. Table 4 Comparison of predicted and experimental values (mean ± standard deviation) for elastic stiffnesses and damage initiation threshold loads for composite sandwich plates loaded by indentation. Sandwich conﬁguration C1/3/15 C1/6/15 C1/13/15 C1/3/20 C1/6/20 C1/13/20 C1/3/25 C1/6/25 C1/13/25 C2/3/15 C2/13/15 C3/3/15 C3/6/15 C3/13/15 C3/3/20 C3/6/20 C3/13/20 C3/3/25 C3/6/25 C3/13/25

Elastic stiffness K0 (kN/mm)

Damage initiation load P1 (kN)

Predicted

Experiment

Predicted

Experiment

1.46 1.30 1.27 1.60 1.48 1.53 1.71 1.60 1.64 1.46 1.27 1.74 1.55 1.53 1.98 1.81 1.85 2.16 2.00 2.04

1.28 ± 0.02 1.40 ± 0.03 1.37 ± 0.08 1.39 ± 0.09 1.45 ± 0.08 1.44 ± 0.04 1.49 ± 0.03 1.53 ± 0.04 1.48 ± 0.07 1.40 ± 0.04 1.30 ± 0.01 1.71 ± 0.18 1.76 ± 0.16 1.91 ± 0.04 1.80 ± 0.07 1.90 ± 0.10 1.94 ± 0.08 1.91 ± 0.14 2.07 ± 0.04 1.81 ± 0.25

1.17 1.12 1.06 1.12 1.04 1.08 0.99 1.14 1.09 1.17 1.06 1.66 1.58 1.42 1.58 1.48 1.52 1.41 1.61 1.55

1.06 ± 0.08 0.98 ± 0.06 1.04 ± 0.06 1.01 ± 0.01 1.00 ± 0.07 1.00 ± 0.04 1.03 ± 0.01 1.00 ± 0.07 1.08 ± 0.06 0.96 ± 0.01 1.09 ± 0.07 1.51 ± 0.01 1.53 ± 0.07 1.46 ± 0.09 1.45 ± 0.19 1.41 ± 0.13 1.39 ± 0.06 1.44 ± 0.03 1.55 ± 0.15 1.52 ± 0.10

Fig. 8. Illustration of interaction between hemispherical indentor and the sandwich plate assuming direct shear through [31].

In order to determine the energy absorbed by the core under local indentation up to initial damage, the size of the planar damage area in the core, as measured by the radius Rcr, has to be known. Rcr was earlier assumed to be a function of the indentor’s radius Ri. One simple method to obtain Rcr is to estimate the contact area at the skin–core interface by examining the interaction between the indentor and the honeycomb cells underneath. Following Zhou et al. [31], the contact area at the skin–core interface could be estimated by assuming that the direct action of the indentor sheared through the top skins at an angle of 45° (Fig. 8). Accordingly, given that the total displacements at initial damage were approximately 1.0 mm in the experiments, Rcr for the plates with skin thicknesses of 1.25 mm and 1.75 mm were calculated to be about 4.75 mm and 5.25 mm (b ’ 0.73 and 0.80), respectively. Previously, Turk and Hoo Fatt [25] assumed an effective radius of 0.4Ri in a similar attempt to calculate the energy dissipated in crushing the honeycomb under a hemispherical indentor for a composite sandwich plate consisting of graphite/epoxy laminates and Nomex honeycomb. The constant b appears to be unique for various types of sandwich plates which comprise different cores and laminates. Finally, the impact model was used to derive the response of the composite sandwich plates subjected to low-velocity impact. Fig. 9 show the experimental and predicted load time histories and load– deﬂection histories for four composite sandwich plates subjected to low-velocity impacts. In the load–time curves, the load initially increased linearly up to the critical load P1 and then dropped suddenly. Subsequently the load increased to a maximum at a reduced stiffness. The predicted results are comparable with test data, in terms of the critical and peak loads, as well as the overall behaviour. This approach demonstrates the capability of the modiﬁed energy-balance model to reproduce the low-velocity impact response of a sandwich plate by using just three parameters (K0, P1, and Kdam). Unloading was not considered for the impacts in Fig. 9 because the contact law for the unloading phase must be determined empirically for these cases. Beyond the onset of damage, the loading and unloading phases are signiﬁcantly different due to the substantial amount of energy dissipated in damage [1]. Nevertheless the modiﬁed energy-balance model is also capable of predicting the entire response for purely elastic impacts. Fig. 10 shows the predicted response of Plate C2/13/15 impacted at 0.334 J, before the damage threshold is reached. The analytical result is compared with results from a ﬁnite element simulation because it was not possible to obtain a purely elastic response even at the lowest drop height in the current experimental setup. Details of the FE model can be found in our previous publication [13]. Both results are comparable, with the analytical response stiffer by the numerical response by 5%. However the most notable difference lies in the solution runtime: the FE analysis took a few hours on a home PC with a PentiumÒ 4 Processor while the analytical calculation re-

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Fig. 9. Load–time and load–deﬂection histories for four composite sandwich plates under low-velocity impact. Bold lines refer to numerical predictions whereas thinner lines are representative experimental curves.

quired a few minutes. Due to the elastic impact, loading and unloading followed the same paths with no hysteresis loop (Fig 10); no energy was dissipated due to damage. The load–time history also showed a half-sine wave which is representative of elastic impacts [11]. The good correlation between predicted results and test data also implies that the quasi-static assumption adopted for the energy-balance model is valid here. In the literature, some researchers have suggested to use impact velocity or impactor mass as the criterion to determine whether an impact event can be considered quasi-static [1,44,45]. The suggested upper limit for impact velocity ranges from 10 to 20 m/s for typical composite materials [44]. Likewise, an impactor that has a mass at least 10 times greater than its target is recommended to ensure quasi-static impact response [45].

cores. Further studies could examine this dependence in greater detail. In addition, impact test results showed that the relative loss of the plates transverse stiffness after damage Kdam/K0 was related to the relative loss in impact energy. The energy absorbed during impact could be estimated using a single empirical equation, which also highlighted the inverse relationship between the absorbed energy and the damage initiation energy. Consequently the residual stiffness of the damaged structure could be evaluated quickly.

(a)

5. Conclusions A model is developed to characterize the elastic response and impact damage initiation of idealized composite sandwich panels under quasi-static indentation, in order to predict the low-velocity impact response of such structures. Closed-form solutions are provided to theoretically predict the elastic stiffness K0 and the load at the onset of damage P1. The critical load was theoretically predicted by accounting for the elastic energy absorbed by the plate up to the point of core failure. At that load, the corresponding damage initiation energy U1 consists of the energy due to bending and shear deformations Ubs and the energy due to localized indentation Uc. Here the energy absorbed by the core under local indentation up to initial damage was quantiﬁed based on empirical observations. It is expected that this energy would depend strongly on the diameter of the indentor, as well as the material properties of the constituent skins and

(b)

Fig. 10. Simulated linear elastic impact response Plate C2/13/15 prior to damage.

C.C. Foo et al. / Composite Structures 93 (2011) 1385–1393

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