epoxy foam sandwich panels under quasi-static indentation

Engineering Fracture Mechanics 67 (2000) 329±344

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Graphite/epoxy foam sandwich panels under quasi-static indentation T. Anderson, E. Madenci * Department of Aerospace and Mechanical Engineering, University of Arizona, Aero Building 119, Tucson, AZ 85721, USA Received 26 August 1999; received in revised form 22 May 2000; accepted 24 June 2000

Abstract This study investigates the force±indentation response of sandwich panels subjected to a rigid spherical indentor. The sandwich panels are made of graphite/epoxy face sheets with a polymethacrylimide foam core. A three-dimensional analytical solution method is developed to determine the complete stress and displacement ®elds in a sandwich panel, as well as the contact pressure arising from static indentation by a rigid sphere. Unlike the usual assumption of a Hertzian-type contact pressure distribution, the sphereÕs unknown contact area and pressure distribution due to indentation are obtained as part of the solution by utilizing an iterative solution method leading to the contact force±indentation relation. Analytical predictions are validated by performing quasi-static indentation experiments. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Sandwich; Construction; Contact force; Indentation

1. Introduction Composite sandwich panels are increasingly being utilized as primary load-carrying components in aircraft and aerospace structures. These sandwich panels may encounter low-velocity impacts, such as tooldrop, runway stones, and tire blowout debris. Even though a visual examination of the impacted surface may reveal very little damage, signi®cant damage might exist between the face sheet and the core [1]. This type of damage leads to substantial reduction of the compressive and bending strengths of the sandwich construction [2]. Along with experimental investigations, analytical and computational models have been developed to gain a better understanding of the e€ects of low-velocity impact on composite sandwich constructions. A comprehensive summary of previous analyses can be found in a review article by Abrate [3]. The presence of transverse deformation and general material orthotropy, coupled with the transient surface contact loading, renders the analysis rather complex. A recent experimental investigation by Ferri and Sankar [4] has revealed that the contact force±indentation relations for a quasi-static test and a low-velocity impact *

Corresponding author. Fax: +1-520-621-8191. E-mail address: [email protected] (E. Madenci).

0013-7944/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 6 6 - 7

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event are virtually equivalent. Therefore, an alternative to modeling the transient impact phenomenon is to model it as quasi-static indentation by a rigid sphere. In this regard, many previous analyses utilized the Hertzian contact law to establish the relationship between the depth of indentation and the contact area for a speci®ed contact force. However, this contact law is inappropriate for establishing the contact force± indentation relationship for sandwich panels because it cannot account for anisotropy of the face sheets, their relative thickness with respect to that of the core, and the di€erence in moduli between the face sheets and the core. Although the contact force±indentation relationship can be established experimentally, it requires a new indentation test for each di€erent combination of material properties, face sheets and core, and their layups. Measuring the contact region and pressure distribution experimentally is a dicult if not impossible task as any type of measuring device that is introduced to measure these quantities will alter their distributions. Therefore, this study presents an analytical model providing the three-dimensional stress and displacement ®elds, as well as the contact pressure and its region, for a sandwich panel indented by a rigid sphere. As required for veri®cation, quasi-static indentation tests were performed on graphite/epoxy foam sandwich panels. The ®delity of this model is established by comparing the predicted and measured force± indentation relationships. Subsequent sections describe the analysis method, experimental investigation, and the comparison of predictions and measurements. 2. Analytical modeling The analytical model concerns the determination of the complete stress and displacement ®elds in a ®nite-geometry sandwich construction subjected to indentation by a rigid sphere. The extent of the contact region and the contact pressure arising from the indentation are also determined as part of the solution. The geometry, loading, and reference frame of the sandwich panel are illustrated in Fig. 1. The length and width of the rectangular panel are denoted by a and b, and its thickness by h. The location of contact between the sphere and the panel is at …x0 ; y0 ; 0†. The position of the interfaces in reference to the upper surface of the panel is speci®ed by zk , as shown in Fig. 2. The thickness of the kth layer is given by tk ˆ zk ÿ zkÿ1 . The face sheets and core are comprised of homogeneous, elastic, and specially orthotropic materials. When the material and reference coordinate systems coincide, the constitutive relationship for the kth layer is represented by 9k 8 9k 2 3k 8 exx > rxx > S11 S12 S13 0 0 0 > > > > > > > >e > >r > > > > 6 S12 S22 S23 0 0 0 7 > yy > yy > > > > 7> 6 = = < 7 6 0 0 7 rzz S13 S23 S33 0 zz ˆ6 ; …1† 6 0 0 7 0 0 S44 0 > > > > cyz > > ryz > 7> 6 > > > > > > > 4 0 0 0 0 S55 0 5 > > > > > > cxz > > rxz > ; ; : : cxy rxy 0 0 0 0 0 S66 where rij and eij are the components of the stress and strain tensors, respectively, and Sij represents the compliance matrix with nine independent material constants. The boundary conditions along the edges of the panel are representative of roller supports, and the layers are treated as perfectly bonded with continuous traction and displacements. The edge boundary conditions can be expressed as vk ˆ wk ˆ 0; k

k

u ˆ w ˆ 0;

rkxx ˆ 0 for x ˆ 0; x ˆ a;

0 6 y 6 b;

rkyy

0 6 x 6 a;

ˆ0

for y ˆ 0; y ˆ b;

…2†

where u, v, and w represent the displacement components in the x, y, and z directions, respectively, and k denotes the layer of the sandwich construction. These edge boundary conditions can be classi®ed as simply

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331

Fig. 1. Sandwich plate subjected to a rigid sphere.

Fig. 2. Identi®cation of the layers and their position in relation to the reference frame.

supported (type S2) as suggested by Jones [5]. The z ˆ 0 surface of the panel is subjected only to loading by the rigid sphere, with the other surface being traction free. The unknown loading arising from the indentation is represented by p…x; y†, and the continuity of traction and displacement components across the layers is enforced explicitly. By applying the variational principle to ReissnerÕs [6] functional as suggested by Noor and Burton [7], the governing Euler±Lagrange equations for each layer are derived as k k k k k k rxx ‡ S12 ryy ‡ S13 rzz ÿ uk;x ˆ 0; S11 k k k k k k rxx ‡ S22 ryy ‡ S23 rzz ÿ vk;y ˆ 0; S12 k k k k k k rxx ‡ S23 ryy ‡ S33 rzz ÿ wk;z ˆ 0; S13 k k ryz ÿ vk;z ÿ wk;y ˆ 0; S44 k k rzx S55 k k S66 rxy

rkab;b

ÿ uk;z ÿ uk;y

ˆ 0;

ÿ wk;x ÿ vk;x

ˆ 0; ˆ 0;

a; b ˆ x; y; z:

…3†

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These governing equations are reduced to a system of ordinary di€erential equations by representing the stress and displacement components for the kth layer in terms of a Fourier series as 8 k 9 8 9 < uk …x; y; z† = X 1 X 1 < umn …z† cos am x sin bn y = vk …z† sin am x cos bn y ; vk …x; y; z† ˆ ; : k ; mˆ1 nˆ1 : mn wkmn …z† sin am x sin bn y w …x; y; z†

…4a†

8 k 9 8 k 9 1 X 1 < rxxmn …z† = < rxx …x; y; z† = X rk …x; y; z† ˆ rkyymn …z† sin am x sin bn y; : ; : yy ; mˆ1 nˆ1 rkzzmn …z† rkzz …x; y; z†

…4b†

8 k 9 8 k 9 1 X 1 < ryzmn …z† sin am x cos bn y = < ryz …x; y; z† = X rk …x; y; z† ˆ rk …z† cosam x sin bn y : kxz ; mˆ1 nˆ1 : xzmn ; rxy …x; y; z† rkxymn …z† sin am x sin bn y

…4c†

in which rkxxmn ; rkyymn ; . . . ; wkmn are unknown auxiliary functions for each m and n, am ˆ mp=a and bn ˆ np=b. The Fourier series representation of the unknown loading function, arising from indentation, is of the form: p…x; y† ˆ

1 X 1 X mˆ1 nˆ1

pmn sin am x sin bn y:

…5†

Substituting for the stress and displacement components in terms of their Fourier series representation permits the governing equations corresponding to a speci®c m and n for each layer to be recast in matrix form as 

M11 M21

M12 M22

k 

F T

k



0 ‡ 0

0 N22

k 

T

oF=oz oT=oz

k

  0 ˆ ; 0

…6†

T

where Fk ˆ f rxxmn ryymn rxymn g and Tk ˆ f ryzmn rzxmn rzzmn umn vmn wmn g. The explicit de®nitions of Mk11 , Mk12 , Mk22 , and Nk22 are given in Appendix A. The matrix representation of the governing di€erential equations permits the expression of the vector Fk (containing the in-plane stress coecients) in terms of the vector Tk (containing the displacement and out-of-plane stress coecients). Substituting for Fk in this equation results in a coupled system of ®rst-order ordinary di€erential equations, oTk ‡ Kk Tk ˆ 0 oz

ÿ1

with Kk ˆ Nk22

h

i T ÿ1 ÿ Mk12 Mk11 Mk12 ‡ Mk22 :

…7†

By de®ning Tk ˆ Qk Rk with Qk being the transformation matrix of eigenvectors, the system of equations is uncoupled as oRk ‡ Kk Rk ˆ 0 oz

…8†

in which Kk is a diagonal matrix composed of the eigenvalues. Using the procedure developed by Mal [8], the solution to the uncoupled system of equations is written as

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2

eÿk1 z 6 0 6 6 0 k k k R …z† ˆ E …z†C ˆ 6 6 0 6 4 0 0

0 eÿk2 z 0 0 0 0

0 0

eÿk3 z 0 0 0

0 0 0 ek1 z 0 0

0 0 0 0 ek 2 z 0

3 k 8 9k C1 > 0 > > > > > > C2 > 0 7 > > 7> = < > C 0 7 3 7 : C4 > 0 7 > > 7> > > C > > > 0 5> > ; : 5> C6 ek3 z

333

…9†

Using the transformation matrix, Qk , the solution for the out-of-plane stress and displacement coecients becomes Tk …z† ˆ Qk Ek …z†Ck :

…10† k

Decomposing the vector T in the form:  k V…z† k T …z† ˆ U…z† T

…11†

T

with Vk ˆ ‰ ryzmn rzxmn rzzmn Š and Uk ˆ ‰ umn vmn wmn Š permits the equations for the stress and displacement coecients at the k ÿ 1 and k interfaces of layer k (Fig. 2) to be rewritten as k   k  k  k Et 0 C‡ Q11 Q12 V…zkÿ1 † ˆ ; …12a† 0 I Q21 Q22 Cÿ U…zkÿ1 † 

V…zk † U…zk †

k



Q11 ˆ Q21

Q12 Q22

k 

I 0

0 Et

k 

C‡ Cÿ

k ;

…12b†

where Qij are the sub-matrices of the transformation matrix Q. The vectors Ck‡ and Ckÿ contain the unknown coecients consistent with the partitioning of the matrix Qk for each layer. The matrix Et is de®ned as 2 kt 3k e1 0 0 …13† Ekt ˆ 4 0 ek2 t 0 5 0 0 ek3 t with tk equal to the thickness of the kth layer. The boundary conditions at the z ˆ z0 and z ˆ zN surfaces can be expressed as 8 9 < 0 = 0 ˆ Q111 E1t C1‡ ‡ Q112 C1ÿ ; V1 …z0 † ˆ : ; pmn 8 9 <0= VN …zN † ˆ 0 ˆ QN11 CN‡ ‡ QN12 ENt CNÿ : : ; 0

…14a†

…14b†

A recursive relationship is then established to enforce the continuity of out-of-plane stresses and displacements between the k and k ‡ 1 interfaces as k‡1 k‡1 k‡1 C‡ ÿ Qk‡1 Qk11 Ck‡ ‡ Qk12 Ekt Ckÿ ÿ Qk‡1 11 Et 12 Cÿ ˆ 0; k‡1 k‡1 k‡1 C‡ ÿ Qk‡1 Qk21 Ck‡ ‡ Qk22 Ekt Ckÿ ÿ Qk‡1 21 Et 22 Cÿ ˆ 0:

…15†

With this relationship, the boundary and continuity conditions are rewritten, forming the algebraic equations, to determine the unknown layer coecients Ck‡ and Ckÿ . The solution to this system of algebraic

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equations leads to the out-of-plane stress and displacement coecients through the substitution of Ck into Eq. (10). The in-plane stress coecients are then obtained from their relation to Tk . The process of determining the stress and displacement coecients is repeated for each value of m and n in the Fourier series representation. In order to solve for the stress and displacement ®elds in a sandwich construction subjected to a spherical indentor, the solution method involves simultaneous solution of both the unknown contact area, X, and contact pressure distribution, p…x; y†. This is accomplished by adopting the method and notation suggested by Wu and Yen [9] in conjunction with the solution method outlined above. The deformation on the upper surface of the panel is of the form: w…x; y; 0† ˆ wmn sin am x sin bn y

…16†

due to a distributed pressure, with unit amplitude represented as p…x; y† ˆ sin am x sin bn y;

…17†

where wmn are the Fourier coecients. Based on the principle of superposition, the deformation due to an arbitrary external loading, p…x; y† ˆ

1 X 1 X mˆ1 nˆ1

pmn sin am x sin bn y

…18†

leads to w…x; y; 0† ˆ

1 X 1 X pmn wmn sin am x sin bn y:

…19†

mˆ1 nˆ1

Substituting for pmn , the transverse displacement on the upper surface becomes   Z 1 X 1 X 4 wmn p…n; g† sin am n sin bn g dn dg sin am x sin bn y; w…x; y; 0† ˆ ab X mˆ1 nˆ1 or

…20†

Z w…x; y; 0† ˆ

X

G…x; y; n; g†p…n; g† dn dg

…21†

in which G…x; y; n; g† is GreenÕs function representing the displacement at the …x; y† coordinate location due to a unit load applied at …n; g†. Describing the pro®le of the rigid spherical indentor with radius R on the surface as q …22† f …x; y† ˆ w0 ÿ R ‡ R2 ÿ …x ÿ x0 †2 ‡ …y ÿ y0 †2 ; where w0 is the out-of-plane displacement at the initial contact point …x0 ; y0 †, and requiring that the indentor conforms to the surface of the panel within the contact region lead to Z G…x; y; n; g†p…n; g† dn dg: …23† f …x; y† ˆ X

A numerical procedure similar to the one used by Wu and Yen [9] is employed for the solution of this equation. As shown in Fig. 3, an overly large initial estimate of the contact region is discretized into N rectangular patches with dimensions s  t. If the center of the patch is at …xj ; yj † and the pressure over the patch is assumed to be constant, Eq. (23) becomes

T. Anderson, E. Madenci / Engineering Fracture Mechanics 67 (2000) 329±344

335

Fig. 3. Assumed contact region and its discretization.

f …x; y† ˆ

N X

Z pj

jˆ1

yj ‡…t=2† yj ÿ…t=2†

Z

xj ‡…s=2†

xj ÿ…s=2†

G…x; y; n; g†dndg;

…24†

which, after integration, is expressed as f …x; y† ˆ

N 1 X 1 a s b t X 16 X wmn m n sin sin sin am x sin bn y sin am xj sin bn yj : p j p2 jˆ1 mˆ1 nˆ1 mn 2 2

…25†

When the indentor pro®le f …x; y† is evaluated at each patch, it results in an N  N system of linear equations of the form: f …xi ; yi † ˆ

N X

Kij pj ;

…26†

jˆ1

where 1 X 1 a s 16 X wmn m sin sin Kij ˆ 2 p mˆ1 nˆ1 mn 2



bn t 2

 sin am xi sin bn yi sin am xj sin bn yj :

…27†

The linear system of equations is solved for the unknown pressures, pj . Because the initial contact region is assumed to be larger than the true contact region and the panel is required to conform to the surface of the indentor within the contact region, the patches on the periphery are in tension. All tensile patches are then removed from the contact region, and the equation is solved again for the new contact region. This procedure is repeated until only compressive pressure patches remain and the approximate contact region is determined. The validity of this procedure is established by considering the experimental and analytical studies provided by Tan and Sun [10] and Wu and Yen [9], respectively, for a monolithic composite laminate with a lay-up of [0/45/0/ÿ45/0]2S . The 50:8  50:8  2:7 mm3 laminate is loaded with a rigid steel sphere 19.1 mm in diameter. The comparison of the results is presented in Fig. 4. The analytical predictions are based on the material properties for the orthotropic face sheet material speci®ed by Exx ˆ 80:2 GPa, Eyy ˆ 18:5 GPa, Ezz ˆ 8:36 GPa, Gxy ˆ 15:7 GPa, Gyz ˆ 3:52 GPa,

336

T. Anderson, E. Madenci / Engineering Fracture Mechanics 67 (2000) 329±344

Fig. 4. Contact force±indentation relation for a composite laminate.

Gxz ˆ 4:99 GPa, mxy ˆ 0:62, myz ˆ 0:27, and mxz ˆ 0:018. Also, in Tan and SunÕs [10] experiment, the specimens were supported with clamped boundary conditions whereas the formulations of Wu and Yen [9] and the present analysis permit only simply supported edges. However, the di€erences in the localized contact force±indentation relationship created by the di€erent boundary conditions appear to be minimal, as apparent from the results of Wu and Yen [9] and Tan and Sun [10]. Along with the assessment of the loading function, p…x; y†, arising from the indentation and the contact area, X, this solution method also provides the complete stress and displacement ®elds for failure prediction essential for damage-tolerant design.

3. Experimental investigation To examine the contact force±indentation behavior, two di€erent sandwich panels were fabricated with graphite/epoxy (LTM45EL/CF0111) face sheets (Advanced Composites Group, Inc.) and Rohacellä 110WF foam (R ohm, GmBH). Rohacell foam is a closed-cell polymethacrylimide foam. The panels were fabricated to the manufacturer's speci®cations without an additional adhesive layer. The specimens were then sectioned into 76:2  76:2 mm2 samples using a diamond-coated abrasive cutting wheel. The ®rst panel con®guration consists of 0.0158 mm face sheets with the lay-up [02 /902 /02 ], and the second panel con®guration has face sheets of half the thickness and a lay-up of [0/90/0]. The 12.7 mm-thick Rohacell 110WF foam core is assumed to be isotropic and a thin resin-rich layer whose behavior is dominated by the properties of the neat resin is assumed to exist between the face sheets and the core. This resin-rich

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337

Fig. 5. Schematic of the testing ®xture for static indentation.

layer is 0.01 mm thick and has a YoungÕs modulus of E ˆ 2:7 MPa and PoissonÕs ratio of m ˆ 0:42. The overall sandwich specimen dimensions are 76:2  76:2 mm2 . The ®xture utilized for the static indentation tests is similar to that used by Tan and Sun [10], except for the simply supported end conditions. The sandwich specimens are supported in a simply supported manner for better comparison with the analysis. With this testing ®xture, shown in Fig. 5, only the relative motion between the top and bottom surfaces is measured, thereby eliminating the e€ects of machine and ®xture compliance. The diameters of the spherical indentor and the simple support rollers are 25.4 and 3.175 mm, respectively. The sandwich panel is centered over the support rollers, which are located 69.85 mm apart. Indentation tests were performed on an electromechanical testing frame. The load and indentation measurements were recorded throughout the loading increments up to core crushing. The digital indicating device used to measure the depth of indentation has a resolution of 0.001 mm and an accuracy of 0.001 mm. Six di€erent sandwich specimens of each face-sheet con®guration were tested in the static indentation ®xture. These results are presented in Figs. 6 and 7. As apparent in these ®gures, the contact-force relation has a bi-linear behavior. The initial linear response ceases at load levels of approximately 300 and 750 N for the thin and thick face sheets, respectively. The divergence from the initial linear response at these load levels occurs possibly because of the onset of material damage in the form of core crushing.

4. Sandwich laminate analysis The complete analysis of a sandwich panel subjected to quasi-static transverse loading by a rigid sphere is performed. A carbon/epoxy face sheet with lay-up [02 /902 /02 ] is bonded to a foam core. The sandwich

338

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Fig. 6. Measured contact force±indentation relation for the panel con®guration [0/90/0/110WF/0/90/0].

panel is 76:2  76:2 mm2 and is supported by the roller-type boundary conditions. The material properties and thickness of the face sheet and the core are presented in Table 1. Also included in this analysis is a thin, 0.01 mm adhesive layer between the face sheet and the core. This layer is not intended to actually model a layer of pure epoxy resin but rather a thin layer whose properties are dominated by the epoxy resin. The material properties of the layer given in Table 1 are assigned the values of the neat epoxy resin of the face sheet. As with any Fourier series representation, the solution is obtained by truncating the series. The maximum number of terms or values of m and n for this analysis, or mmax and nmax , are both equal to 290. This number is not only chosen to allow convergence in the Fourier series solution for the stress and displacement components, but also to ensure a well-conditioned problem. The system of linear equations found in Eq. (26) is only well conditioned and invertable if the smallest wavelength in the Fourier series is smaller than the smallest patch size. The out-of-plane displacement of the rigid sphere prescribed in the analysis is 0.004 mm. The corresponding level of load created by this displacement is 734 N, a level at which damage was seen to occur during the quasi-static indentation experiments. The initial oversized contact region, located at the center of the panel, was estimated to be 6-mm square and was discretized into a 19  19 grid of equally sized square patches. Six iterations were required at this level of load to determine the approximate contact area. With the stress and strain ®elds de®ned for the sandwich panel, subsequent failure analyses may be performed. Failure within composite sandwich panels subjected to low-velocity impact is an extremely complex phenomenon with many failure modes occurring simultaneously. Although many di€erent failure criteria exist, no single criterion will ever accurately describe all modes of failure. As core/face sheet

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339

Fig. 7. Measured contact force±indentation relation for the panel con®guration [02 /902 /02 /110WF/02 /902 /02 ].

Table 1 Material properties for the Type III sandwich laminate Material property

Thick face sheet

High density foam core

Adhesive layer

E1 E2 E3 G12 G23 G13 m12 m23 m13 Thickness

54 GPa 54 GPa 4.84 GPa 3.16 GPa 1.78 GPa 1.78 GPa 0.06 0.313 0.313 1.584 mm

180 MPa 180 MPa 180 MPa 70 MPa 70 MPa 70 MPa 0.286 0.286 0.286 12.7 mm

2.7 GPa 2.7 GPa 2.7 GPa 0.951 GPa 0.951 GPa 0.951 GPa 0.42 0.42 0.42 0.01 mm

delamination signi®cantly a€ects the residual strength of a laminate, a simple criterion that establishes delamination in the present analysis is desired. The speci®c energy criterion proposed by Gillemot [11] is selected to determine failure at the interface between the core and the face sheet. This interactive criterion will determine failure by comparing the strain energy at a point to that of the critical value of the strain energy of the matrix material. If the threshold is surpassed, failure has occurred at the core/face sheet interface. As strain energy density will be utilized as the failure criterion, it is also determined throughout the sandwich laminate.

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Fig. 8. Delamination area (in grey) at the core/face sheet interface and at the center of the sandwich laminate.

Using the data provided by the material manufacturer, the critical strain energy density of the neat resin, de®ned by   dW 1 ˆ rf ef …28† dV crit 2 was determined to be 176.7 KPa. The ultimate tensile stress (rf ) is 31 MPa and ef , the ultimate tensile strain, is equal to 0.0114. If the failure mechanism is assumed to be delamination at the core/face sheet interface, and the failure criterion is based on this critical level of strain energy density, the resulting delamination area for the given load is predicted to be approximately 2 mm, as shown in Fig. 8. Because the thickness of the resin layer is so small, the delamination area remains the same through its thickness. As with all analytical predictions, this damage region needs subsequent experimental veri®cation.

5. Comparison of predictions and measurements Analyses were performed on the sandwich panel con®gurations considered for testing. The contact force±indentation results of each analysis are displayed in Fig. 9. As expected, the sandwich panel with the thick face sheet has a higher sti€ness than the panel with the thin face sheet. A power-law contact force± indentation relationship of the form: F ˆ C1 aC2

…29†

proves to ®t the analytical predictions rather well. In this relationship, F is the contact force, a is the depth of indentation, and C1 and C2 are coecients speci®c to each panel con®guration. Based on a curve-®tting technique, the values of these coecients are determined to be C1 ˆ 2107 and C2 ˆ 1:186 for the panel con®guration [0/90/0/110WF/0/90/0], and C1 ˆ 4160 and C2 ˆ 1:209 for the panel con®guration [02 /902 /02 /

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341

Fig. 9. Predicted contact force±indentation relations for the panel con®gurations [0/90/0/110WF/0/90/0] and [02 /902 /02 /110WF/02 /902 / 02 ].

110WF/02 /902 /02 ]. The curve ®ts have correlation coecients of 0.99998 and 0.99994 for the thick face sheet and thin face sheet con®gurations, respectively. Since the analysis contains no means for incorporating material damage, the analytical predictions overpredict the sti€ness subsequent to damage initiation observed in the experiments. In other words, the measured bi-linear behavior is not captured in the analytical predictions. Comparison of the predictions with the measurements taken during the initial loading before signi®cant material damage occurs is shown in Figs. 10 and 11. It is evident that the predictions are in agreement with the experimental results within the linear material response or up to the onset of material damage.

6. Conclusions This study presents an analytical method based on the three-dimensional elasticity theory to establish the contact force±indentation relations for composite sandwich constructions subjected to rigid spherical indentors. The contact area and the contact pressure distribution between the rigid sphere and the sandwich panel are determined as part of the solution. The accuracy of the predictions is established through comparisons with experimental measurements of graphite/epoxy foam sandwich panels. As demonstrated by the results, the predictions from the present analysis are in remarkable agreement with the experimental results until damage initiation in the face sheets and core.

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Fig. 10. Predicted and measured contact force±indentation relation for the [0/90/0/110WF/0/90/0] specimens before the onset of damage.

With the complete stress and strain ®elds, the strain energy density criterion can be utilized to predict the onset of damage. In this investigation, a delamination area of 2 mm is predicted. Validation of failure predictions requires the application of non-destructive inspection techniques, which is beyond the scope of the present study. The results from this study will serve as a basis for developing other numerical solution methods with a wide range of applicability. The analytical solutions will also serve as a benchmark solution for the validation of more-simpli®ed numerical and computational studies. In particular, the accuracy of new and existing ®nite elements used to model composite plates and shells could be determined for the cases of contact and low-velocity impact. Appendix A. De®nitions of Mk11 , Mk12 , Mk22 , and Nk22 2

S11 Mk11 ˆ 4 S21 0

S12 S22 0 2

kT

Mk12 ˆ M21

0 ˆ 40 0

3k 0 0 5; S66 0 S13 0 S23 0 0

am 0 ÿbn

0 bn ÿam

3k 0 05 ; 0

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343

Fig. 11. Predicted and measured contact force±indentation relation for the [02 /902 /02 /110WF/02 /902 /02 ] specimens before the onset of damage.

2

S44 6 0 6 6 0 Mk22 ˆ 6 6 0 6 4 0 ÿbn 2

0 60 6 60 Nk22 ˆ 6 60 6 41 0

0 0 0 1 0 0

0 S55 0 0 0 ÿam 0 0 0 0 0 1

0 0 S33 0 0 0

0 ÿ1 0 0 0 0

0 0 0 0 0 0

ÿ1 0 0 0 0 0

0 0 0 0 0 0

3k ÿbn ÿam 7 7 0 7 7; 0 7 7 0 5 0

3 0 0 7 7 ÿ1 7 7: 0 7 7 0 5 0

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