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International Journal of Impact Engineering 34 (2007) 522–543

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Response of composite sandwich panels with transverselyflexible core to low-velocity transverse impact:

A new dynamic model

K. Malekzadeha, M.R. Khalilib, R.K. Mittalc,�

aDepartment of Mechanical Engineering, Malek Ashtar University of Technology, 4th Kilameter, Makhsous RD, 1387763681 Tehran, IranbDepartment of Mechanical Engineering, K.N. Toosi University of Technology, Vafadar East Ave, 4th Tehran pars SQ., Tehran, Iran

cDepartment of Applied Mechanics, I.I.T. New Delhi 110016, India

Received 20 January 2005; accepted 16 October 2005

Available online 20 December 2005

Abstract

A new computational procedure based on improved higher order sandwich plate theory (IHSAPT) and two models

representing contact behavior between the impactor and the panel are adopted to study the low velocity impact

phenomenon of sandwich panels comprising of a transversely flexible core and laminated composite face-sheets. The

interaction between the impactor and the panel is modeled with the help of a new system having three-degrees-of-freedom,

consisting of spring–mass–damper–dashpot (SMDD) or spring–mass–damper (SMD). The effects of transverse flexibility

of the core, and structural damping are considered. The present analysis yields analytic functions describing the history of

contact force as well as the deflections of the impactor and the panel in the transverse direction. In order to determine all

components of the displacements, stresses and strains in the face-sheets and the core, a numerical procedure based on

improved higher order sandwich plate theory (IHSAPT) and Galerkin’s method is employed for modeling the layered

sandwich panel (without the impactor), while the analytic force function developed on the basis of SMDD or SMD model,

can be used for the contact force between the impactor and the panel. The contact force is considered to be distributed

uniformly over a contact patch whose size depends on the magnitude of the impact load as well as the elastic properties and

geometry of the impactor. Various boundary conditions for the sandwich panel have also been considered. Finally, the

numerical results of the analysis have been compared either with the available experimental results or with some theoretical

results.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Sandwich panel; Core flexibility; Low-velocity transverse impact; Higher-order theory; Structural damping

1. Introduction

Sandwich structures comprising of fiber-reinforced composite face sheets bonded to a lightweight corematerial are finding increasing use in a wide range of load-bearing applications. During service, these panels

ee front matter r 2005 Elsevier Ltd. All rights reserved.

mpeng.2005.10.002

ing author. Tel.: +91 11 2659 1219; fax: +91 11 2658 1119.

ess: rkmittal44@rediffmail.com (R.K. Mittal).

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Nomenclature

a; b length and width, respectively, of the sandwich panelCef effective viscous damping coefficient of the panel½Ce� matrix of viscous dampingEP effective elastic modulus of the sandwich panelEI effective elastic modulus of the impactorF static indentation forceF cr critical crushing load of the coreFdash dashpot force representing the dynamic crushing resistance of the coreF cðtÞ;F

�c ðtÞ

contact forces based on the Hertzian and linearized Hertzian contact laws, respectivelyht; hb; hc; h

thicknesses of the top and the bottom face-sheets, the core and the panel, respectivelykF stiffness of the core modeled as elastic foundationKc;K

�c coefficients of the Hertzian and linearized Hertzian contact laws, respectively

K face;Kcore;K sand effective stiffnesses of the impacted face-sheet, the core and the panel at the impactlocation

[K] stiffness matrix of sandwich panelMI mass of impactorMxx;Myy;Mxy shear and bending moments per unit length of the edgeMsand effective mass of the sandwich panelMface effective mass of the impacted face-sheet of the panel[M] mass matrix of the sandwich panelNxx;Nyy;Nxy in-plane forces per unit length of the edgeP exponent in the Hertzian contact lawq; qd static and dynamic crushing strengths of the coreqiðx; y; tÞ

contact loads over the (top or bottom) impacted face-sheet (i ¼ t;b)qmnðtÞ Fourier coefficients of the impact force distributionQx;Qy shear forces in face-sheets, per unit edge length[Q] vector of the impact forcesR contact radiusRI impactor radiust2 � t1 time interval of analysisTe kinetic energyu; v length and width of the assumed rectangular contact areauc; vc;wc displacement components of the coreu0i; v0i;w0i

displacement components of the face-sheets, (i ¼ t;b)€uc; €vc; €wc acceleration components of the core€u0i; €v0i; €w0i

acceleration components of the face-sheets, (i ¼ t;b)U e internal potential energyV e external energyV0 transverse velocity of the impactor at contactW ðx; yÞ mode shape corresponding to the fundamental natural frequency of the impacted face-sheetW nc structural damped energyðx0; y0Þ point of impact on the face-sheetzt; zb; zc normal coordinates in the mid-plane of the top and the bottom face-sheets and the core

K. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 523

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Greek letters

aðtÞ contact indentationamax maximum contact indentationbi terms representing parts of the viscous damping effectsd variational operatordT deflection under the impactor in static load-indentation testD0ðtÞ;D1ðtÞ;D2ðtÞ transverse displacement functions in discrete dynamic systems (SMD and SMDD

models)Zst effective structural loss factor of the panelnP effective Poisson’s ratio of the sandwich panelnI effective Poisson’s ratio of the impactorrt;rb;rcmaterial densities of the face-sheets and the coresczz normal stress in the coretcxz; t

cyz shear stresses in the core

o11 fundamental natural frequency (rad/s) of the panelcx;cy rotation of the normal section of the face-sheet along y and x axes

K. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543524

may encounter low-velocity impacts caused by runway stones, hails, tool drop, tire blowout debris, etc.Although extensive research has been devoted to the impact behavior of composite laminates in general [1–9],the work on sandwich structures is somewhat limited. In this context, the work of Ambur and Cruz [10] maybe mentioned in which a local–global analysis was carried out to determine the contact force and displacementof the panel. In deriving closed-form solution for the impact response of the composite sandwich panels, thecomposite structures have sometimes been modeled as a discrete dynamic system with equivalent masses,springs and damper. For instance, Shivakumar et al. [11] presented a two-degrees-of-freedom model thatconsisted of four springs for bending, shear, membrane and contact rigidities for predicting the impactresponse of a circular plate. They calculated on the basis of this model, the contact force and contact durationfor low-velocity impact on circular laminates. Other two-degrees-of-freedom models for impact response ofcomposite plates include those by Sjoblom et al. [12] and Lal [13]. Caprino et al. [14] used a single degree-of-freedom system to analyze drop weight impact tests on glass/polyester sandwich panels. Anderson [15]described an investigation using single degree-of-freedom model for large mass impact on composite sandwichlaminates. The stiffness parameters of the model were derived from the results of a three-dimensional quasi-static contact analysis of a rigid sphere indenting a multi-layered sandwich laminate. Gong and Lam [16] useda spring–mass model having two degrees-of-freedom in order to determine the history of contact forceproduced during impact. They included structural damping also in their model.

As noted above, much effort has been made to analyze composite plates subjected to low-velocity impactusing a discrete spring–mass dynamic system. However, none of those studies considered the effects oftransverse flexibility and structural damping of the core of sandwich panels on their transient response. Thus,an analytical procedure that includes the transverse flexibility and structural damping of the core material andgives accurate results for impact behavior of sandwich panels has not yet been dealt with.

Complete impact analysis of sandwich panels having a core of elastic foam has been reported by Nemes andSimmonds [17] and Lee et al. [18]. Davies [19] carried out experimental study of impact on different sandwichpanels with laminated face-sheets subjected to low velocity impact. Recently, a complete review of the subjectof impact on sandwich structures was carried out by Abrate [20]. Olsson and McManus [21] set up ananalytical model for the indentation of sandwich panels, while Hoo Fatt and Park [22] obtained analyticsolutions for the transient deformation response of sandwich panels. Olsson [23] also gave an engineeringmethod for predicting the impact response and damage in sandwich panels. Various approaches have beenproposed for the analysis of impact response of sandwich panels. The classical method decouples the local andglobal responses and ignores any interaction between the two. The first-order shear deformation theory(FSDT) and higher order shear deformation theory (HSDT) do not consider the flexibility of the core intransverse direction, and the interaction between face-sheets and the soft flexible core is neglected [24,25].

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 525

Most recently, the higher-order sandwich plate theory (HSAPT) has been used [26–29]. For a soft core, thevertical flexibility of the core must be taken into account since this flexibility of the core influences the stressand displacement fields in the face sheets. The equations describing the static behavior of sandwich panels witha soft core have been obtained in Ref. [28] by applying the principle of virtual work. In the presentspring–mass–damper model, equivalent stiffnesses of the impacted top face-sheet and the sandwich panel havebeen obtained from the static analysis of sandwich panel based on an improved higher-order sandwich platetheory (IHSAPT) as used in [28].

Recently, Malekzadeh et al. [30,31] introduced FSDT for face-sheets and a non-linear variation of verticalacceleration in the core, instead of linear acceleration variation as is commonly assumed. With thismodification, solutions were obtained for damped and undamped free vibrations of simply supportedsandwich panels based on improved higher-order sandwich plate theory (IHSAPT). Another important step inthe solution of the impact problem is the contact law which gives the relationship between the impact forceand the indentation of the target surface. For isotropic homogeneous linear elastic bodies, the use of Hertz’scontact law is well established [1–6] when the indentation is much smaller than the plate thickness. But forsandwich panels, the face sheets are stiff and often anisotropic while the core is flexible as compared to the facesheets. Therefore, the Hertz’s indentation law is not applicable in the present case because of thisinhomogeneity.

Spring–mass models are used extensively to analyze the dynamics of impact. An analytical procedure thatincludes the transverse flexibility and structural damping of the core of sandwich panels has not yet been dealtwith. In this paper, a new three-degrees-of-freedom (TDOF) springs–masses–damper (SMD) model isproposed to predict the contact force history for composite sandwich panels with transversely flexible core. Inorder to determine all components of the displacements, stresses and strains in the face-sheets and the core, acommercial FEM software or a specially developed numerical program like ours which is based on Galerkin’smethod, can be employed only for modeling the layered sandwich panel (without the impactor), while theforce function presented in this paper can be used to determine the contact force between the impactor and thepanel. In this paper, the full dynamic effect which includes the horizontal vibrations as well as the rotaryinertia, in addition to the normal mass inertia of the core and face-sheets is considered.

2. Formulation of the problem

The rectangular sandwich flat panel studied in this paper is composed of two FRP composite laminated andsymmetric face sheets and a core, as shown in Fig. 1 where coordinates and sign conventions are also shown.The assumptions used in the present analysis are those encountered in linear elastic small deformationtheories. The face sheets are considered as ordinary thin or thick plates. The stacking sequences of laminate intop and bottom face sheets are assumed to be symmetric. The core behavior follows the assumption which has

Fig. 1. Sandwich composite panel with laminated face sheets. Panel coordinates and panel dimensions are also shown. t, b, and c mean the

top and the bottom face-sheets and the core, respectively.

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been adopted by many researchers for the honeycomb type of core [28]. The core has shear resistance but isfree of in-plane normal and shear stresses. This assumption is practically valid for a foam core, since elasticmodulus and flexural rigidity are, respectively, about three and two orders smaller than those of the facesheets. The core is assumed to behave in a linear elastic manner with small deformations, although its heightmay change and its transverse plane takes a nonlinear shape after deformation. The top and bottom facesheets and the core are assumed to be perfectly bonded, i.e. there is no relative displacement between the coreand the adjacent face sheets at the interface. The impactor is assumed to have a spherical/hemispherical shapeand is made of an elastic material with high stiffness in comparison with the transverse stiffness of panel. Thefriction between panel and impactor is assumed to be negligible. In this work, the composite sandwich panelcan be simply supported (SSSS) or fully clamped (CCCC) around all four edges of top and bottom face sheets.Alternatively, the boundary conditions of the top and bottom face-sheets may be prescribed separately. Forexample, the four edges of the bottom sheet may be simply supported while the four edges of the top sheet areclamped (CTBS). The effects of secondary contact loadings are assumed to be negligible. Therefore, thecontact force acts only over the impacted surface of the panel during the first contact. Various stages of theproblem formulation are discussed below.

2.1. The contact forces

The contact loads qi ði ¼ t;bÞ are assumed to be represented by a double Fourier series expansion and areseparable into functions of time and position as follows:

qiðx; y; tÞ ¼X1m¼1

X1n¼1

qmnðtÞ sinðamxÞ sinðbnyÞ. (1)

The Fourier coefficients qmn(t), depend on the nature of the load distribution, for example, if the load isassumed to be concentrated at the impact point (x0, y0), then

qmnðtÞ ¼4F cðtÞ

absinðamx0Þ sinðbny0Þ, (2)

where am ¼ mp=a and bn ¼ np=b.A concentrated load generally results in a singularity in stress and bending moment distributions at the

point of the load application. Instead, the actual load distribution obtained in contact problems may be used.However, in order to simulate actual contact conditions, one may note the following.

If the indentation due to impact load is much smaller than the plate thickness, Hertz’s contact theory [32]may be used for isotropic homogeneous media. For an anisotropic medium Sveklo’s contact theory [33] maybe used. Since this theory has many similarities with the Hertz’s theory, it will be referred to as Hertz–Sveklotheory in the following.

According to Hertz’s theory, the contact patch between the contacting sphere and the target medium iscircular and the load distribution is hemispherical. The radius of the circular patch, R(t), depends on thecontact load in the following manner:

RðtÞ ¼ 1:109:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF cðtÞ:RI

EI

3

s. (3)

On the other hand, for an orthotropic target medium, such as an aligned fiber-reinforced panel, Hertz–Sveklotheory predicts an elliptical contact patch, whose semi-major and semi-minor axes are also influencedby the magnitude of the contact load. These axes are proportional to ½F cðtÞ�

1=3, as in Eq. (3). A detailedanalysis of carbon fiber-reinforced epoxy plates subjected to impact by steel spheres was carriedout by Frischbier [34] using Hertz–Sveklo theory. It was shown that for these plates, the contact patchwas elliptical of low eccentricity. The ratio of semi-minor to semi-major axes was nearly 0.88, unless thefiber volume fraction is very high (470%). In view of this observation, the contact patch may beapproximated as circular.

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Since Hertz–Sveklo theory also predicts a parabolic load distribution, the following contact load distributionmay be used in the present case.

qiðx; y; tÞ ¼3F cðtÞ

2pR21�ðx� x0Þ

2þ ðy� y0Þ

2

R2

� �1=2. (4)

Unfortunately, this impact load distribution [Eq. (4)] does not result in closed form expressions for the Fouriercoefficients. Instead, numerical integration has to be resorted to due to the non-linearity of the integral. For agiven F cðtÞ the integral can be solved numerically with the help of commercial software and the Fouriercoefficients qmnðtÞ obtained for various values of m and n. Since the impact force itself is generally unknown forthe impact problem and many iterations are needed, as will be shown later, for its determination, it is desirableto have an explicit expression for the Fourier coefficients. Therefore, a further simplification of the contactload distribution was sought.

One possibility, used initially by Dobyns [35] and later used often, is to assume a small rectangular patchwith a uniform contact load distribution over it. This assumption simplifies the analysis very much and theFourier coefficients are obtained as given below

qmnðtÞ ¼16F cðtÞ

p2mnuvsinðamx0Þ sinðbny0Þ sinðamu=2Þ sinðbnv=2Þ, (5)

where u and v are the dimensions of the rectangular patch. Since this patch is an approximation of the circularpatch, as discussed above, it has been assumed that u ¼ v ¼

ffiffiffipp

R. This ensures that the rectangular patch hasthe same area as the circular patch. Therefore, in place of Eq. (5), the following equation was used for Fouriercoefficients:

qmnðtÞ ¼16F cðtÞ

p3mnR2sinðamx0Þ sinðbny0Þ sin am

ffiffiffipp

R

2

� �sin bn

ffiffiffipp

R

2

� �, (6)

where R and F cðtÞ are related through Eq. (3).A comparison of qmnðtÞ based on Eq. (6) with those obtained for a circular patch using numerical integration

showed a close agreement between the two. This can be explained in view of the fact that the size of the contactpatch is much smaller than the size of the plate. Therefore, in further analysis, Eq. (6) was adopted.

2.2. The improved high order sandwich plate theory (IHSAPT)

These equations were derived by Frostig [28], using the principle of virtual work and have been used bypresent authors with some modifications for the dynamic analysis of sandwich plates [30]. They have beenfurther modified by incorporating FSDT for face sheets and assuming a nonlinear variation of verticalacceleration of the core instead of the usual assumption of linear variations. The full dynamic effects (i.e.besides the mass inertia of the core, both the horizontal vibrations and rotary inertia of the core and face-sheets are also included) are considered in the analysis. The governing equations and the boundary conditionsare derived using Hamilton’s principle which requires that

dZ t2

t1

½Ue þ V e þW nc � Te�dt ¼ 0. (7)

By carrying out the variational procedure, the equations of motion are obtained and are given below(see Ref. [30])

N txx;x þN t

xy;y þ tcxzðzc ¼ 0Þ ¼ I0t €u0t þ b1,

Nbxx;x þNb

xy;y � tcxzðzc ¼ cÞ ¼ I0b €u0b þ b2,

Qtx;x þQt

y;y þ qt þ sczzðzc ¼ 0Þ ¼ I0t €w0t þ b3,

Qbx;x þQb

y;y þ qb � sczzðzc ¼ cÞ ¼ I0b €w0b þ b4,

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543528

N tyy;y þN t

xy;x þ tcyzðzc ¼ 0Þ ¼ I0t €v0t þ b5,

Nbyy;y þNb

xy;x � tcyzðzc ¼ cÞ ¼ I0b €v0b þ b6,

Mtxx;x þMt

xy;y �Qtx þ tcxzðzc ¼ 0Þ:ðht=2Þ ¼ I2t €cxt þ b7,

Mbxx;x þMb

xy;y �Qbx þ tcxzðzc ¼ cÞ:ðhb=2Þ ¼ I2b €cxb þ b8,

Mtxy;x þM t

yy;y �Qty þ tcyzðzc ¼ 0Þ:ðht=2Þ ¼ I2t €cyt þ b9,

Mbxy;x þMb

yy;y �Qby þ tcyzðzc ¼ cÞ:ðhb=2Þ ¼ I2b €cyb þ b10,

tcxz;z ¼ rc €uc; tcyz;z ¼ rc €vc; tcxz;x þ tcyz;y þ sczz;z ¼ rc €wc. ð8Þ

The boundary conditions are obtained in Appendix A. Further, I jt and I jb are the moments of inertia for theupper and lower face sheets. tcxz, t

cyz (zc ¼ 0, c) and sczzðzc ¼ 0; cÞ are the shear and the vertical normal stresses

at the upper and the lower interfaces between the core and the face sheets, respectively, Qjx and Qj

y aredistributed shear forces per unit length of the edge (j ¼ t, b) in the x- and y-directions, respectively. Using thelast thee equations of the set of Eqs. (8), constitutive law of the core material and the continuity conditions forthe top and the bottom face-sheets, the analytical relations for the normal stress as well as the horizontal andvertical displacements in the core were derived [30].

These relations are given in Appendix B. bi (i ¼ 1–10) are the terms representing parts of equivalent viscousdamping effects and are given in Appendix C.

The impact solution for a rectangular panel with simply supported boundary conditions at the top andbottom face sheets is assumed to be in the following form:

u0jðx; y; tÞ

v0jðx; y; tÞ

w0jðx; y; tÞ

cxjðx; y; tÞ

cyjðx; y; tÞ

tcxzðx; y; tÞ

tcyzðx; y; tÞ

26666666666664

37777777777775¼X1m¼1

X1n¼1

u0jmnðtÞ: cosðamxÞ: sinðbnyÞ

v0jmnðtÞ: sinðamxÞ: cosðbnyÞ

w0jmnðtÞ: sinðamxÞ: sinðbnyÞ

A0jmnðtÞ: cosðamxÞ: sinðbnyÞ

B0jmnðtÞ: sinðamxÞ: cosðbnyÞ

TcxmnðtÞ: cosðamxÞ: sinðbnyÞ

TcymnðtÞ: sinðamxÞ: cosðbnyÞ

2666666666664

3777777777775. (9)

The above double Fourier series functions satisfy the boundary conditions of panel, i.e. simply supported onall edges. However, when all edges are clamped, the functions cosðamxÞ and cosðbnyÞ in series expansions of cxj

and cyj , respectively, are replaced by sinðamxÞ and sinðbnyÞ.In Eq. (9) u0jmn; v0jmn;w0jmn;A0jmn;B0jmn;Tcxmn and T cymn are time dependent unknown Fourier coefficients,

m and n are, respectively, the half wave numbers in x and y directions and j ¼ t, b, where t and b mean the topand the bottom face sheets. The dynamic equations of motion in terms of deformation and rotationcomponents and shear stresses in the core are derived by using the field equations alongwith the constitutiverelations and the governing equations (8). Then by applying the Galerkin method, the governing equations arereduced to the following system of ordinary differential equations:

½M�f€wg þ ½Ce�f_wg þ ½K �fwg ¼ fQg. (10)

Therefore, the problem of impact on a sandwich panel reduces to the standard structural response equation. InEq. (10), [M] is a (10mn)� (10mn) square symmetric mass matrix, [K] is a (10mn)� (10mn) square symmetricstiffness matrix, [Ce] is a (10mn)� (10mn) square damping matrix and {Q} is a (10mn)� 1 vector of impactforces. The system damping is simulated by proportional viscous damping terms. Eq. (10) can be readilysolved with a suitable numerical integration procedure [30].

The symmetric mass and stiffness coefficients for simply-supported rectangular sandwich panels are listed inAppendix D. For the case of general dynamic analysis, the vector fwðtÞg½ð10�m�nÞ;1� contains ten sets of timedependent unknowns: u0t, v0t, u0b, v0b, w0t, w0b, cx, cy, t

cxz and tcyz.

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2.3. Effective contact stiffness

The contact force between impactor and the target has often been assumed to be known. However, inpractice, the contact force is the result of contact deformation between the impactor and the target and shouldbe evaluated. The effective stiffness Kc can be obtained by the experimental static load-indentation test [22], orit can also be estimated by the following modified analytical methods.

2.3.1. The modified Hertzian contact stiffness of impacted sandwich panels with elastic flexible core

Note that the solution suggested below is only applicable prior to fiber fracture, core crushing(plastic deformation) or the penetration of the panel. Let D1ðtÞ;D2ðtÞ and D0ðtÞ represent the transversedisplacements at the load point of the sandwich flat panel in the impacted top face sheet, bottom facesheet (see Fig. 2) and that of the impactor, respectively, at any time t during impact. The contactdeformation is

aðtÞ ¼ D0ðtÞ � D1ðtÞ. (11)

The contact force between the impactor and the sandwich panel during the impact is assumed to be governedby the nonlinear Hertzian contact law of the form

F cðtÞ ¼ KcaP, (12)

where Kc and P can be obtained by static indentation tests or they can also be estimated by Hertzian contacttheory [9]. However, for a complete characterization of the contact behavior including the unloading phase,the contact law must be determined experimentally. As Eq. (12) can be highly nonlinear, seeking an analyticsolution for the contact force might pose a formidable task. The present approach, as in Ref. [16], employs aneffective contact stiffness K�c and the assumption of linear relationship between equivalent contact force andcontact deformation. This relationship is given below

F�c ðtÞ ¼ K�ca ¼ K�c ½D0ðtÞ � D1ðtÞ�. (13)

Fig. 2. Equivalent three-degrees-of-freedom models of the panel and impactor system: (a) Model I: spring–mass–damper; (b) Model II:

spring–mass–damper–dashpot.

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The effective stiffness K�c can be estimated with the help of the following relationship [16]:

K�c ¼ffiffiffipp

GPþ 1

2

� � 2GP

2þ 1

� �þ

ffiffiffipp

GPþ 1

2

� �

4G2 P

2þ 1

� �þ pG2 Pþ 1

2

� � aP�1maxKc, (14)

where GðxÞ is the gamma function. For an impactor of small mass ðMsand=MI410Þ, amax was estimated byGong and Lam [16], as given below

amax ¼Msand:MI

Msand þMI

� �1=ðPþ1ÞPþ 1

2

� �1=ðPþ1Þ

ðV 20=KcÞ

1=ðPþ1Þ. (15)

Many experimental studies have shown that P is equal to 1.5 for laminated composite targets. Also, Hertzianindentation ðP ¼ 1:5Þ dominates at small loads and for thick face-sheets and stiff cores. For most cases ofinterest in sandwich structures, the Hertzian indentation is negligible and the combined action of softening dueto core crushing and stiffening due to membrane/bending effects in face-sheets results in a nearly linear load-indentation relationship, implying that P � 1. In Hertzian indentation [1], the contact stiffness Kc can beestimated using

Kc ¼ ð43ÞER

1=2I ;

1

E¼

1� n2IEIþ

1� n2PEp

. (16)

Because the elastic moduli in z direction of all laminas are identical, with some simplifications, i.e. quasi-isotropic assumption and by neglecting the effect of in-plane moduli, EP and nP can be calculated. An initialestimate can be obtained using the rule-of-mixtures in z direction of the panel [31].

The modified contact law with a proper reduction in the elastic constant of the core provides reasonableprediction for low-speed impact behavior of composite sandwich structures [31]. It has been found that theproposed contact law can predict the measured contact forces and the contact duration for most cases,especially for thick or stiff flat sandwich panels.

2.3.2. The contact stiffness of impacted sandwich panels assuming rigid plastic flexible core in contact zone

Many analytical methods for determining the local deformation assume a Hertzian contact [17–20].However, the local deformation considered here causes transverse deflections of the entire top face sheet andcore crushing, so that the Hertzian contact laws are inappropriate for finding the local indentation response.Hoo Fatt and Park [22] divided the whole impact indentation process into three stages and applied threedifferent mathematical models to the corresponding stages. These stages are: (I) plate on an elastic foundation;(II) plate on a rigid-plastic foundation; (III) Membrane on a rigid-plastic foundation. The load-indentationresponse has been obtained for the stage (II) by using the principle of minimum potential energy and is givenas follows [22]:

F ¼32ffiffiffiffiffiffiffiffi255p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2D1dT q

pþ pqR2

I , (17)

where

D1 ¼16 384

11 025ð7D11 þ 7D22 þ 4D12 þ 8D66Þ. (18)

Dij (i, j ¼ 1; 2; 6) are coefficients of bending stiffness matrix of the panel. D1 is the bending stiffness of theorthotropic face sheet alone. The first nonlinear term of Eq. (17) is the resistance due to the face sheet bendingand crushing of honeycomb/foam outside the contact area of indentor, while the second term is dueto crushing of core under the indentor (see Ref. [22]). The effective stiffness K�c can be obtained by linearizingEq. (17).

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 531

2.4. The new proposed procedure for low-velocity impact analysis

In this study the equivalent three-degrees-of-freedom (TDOF) spring–mass–damper–dashpot (SMDD) andspring–mass–damper (SMD) models as well as a new procedure are proposed and used to predict the low-velocity impact response of a composite sandwich panel having a core that can be either transversely stiff orflexible (see Figs. 2a and b). In Figs. 2a and b, the core is assumed to be elastic or rigid plastic (in contactzone), respectively. When the indentation is very small and core crushing is elastic (i.e. transverse strain is lessthan the elastic limit eec), the impact response is found by considering the model I (Fig. 2a). As the face sheetindentation becomes larger and core crushing is rigid plastic (in the contact zone), impact response is foundusing the model II (Fig. 2b).

In Figs. 2a and b the boundary conditions of the two face-sheets are assumed independently. Thus, theequivalent stiffness of the impacted face-sheet K face is connected to the fixed reference frame. After obtainingthe contact force history analytically, using SMD or SMDD models, a commercial FEM software or otherspecially developed numerical program like ours (based on Fourier series and Galerkin’s methods) can beemployed only for modeling the layered sandwich panel (without the impactor) in order to determine allcomponents of displacement, strain and stress in face sheets as well as the core. The analytic force functionpresented in this paper can be implemented to handle the contact force between the impactor and the panel.Therefore, the problem of impact on the sandwich structures can be simplified to solving a standard structuralresponse equation for a known impact loading.

2.4.1. Low-velocity impact response of sandwich panel with elastic flexible core (SMD model)

The equations of motion of the three-degree-of-freedom system (model I—Fig. 2a) are as follows:

MI€D0 þ K�c ðD0 � D1Þ ¼ 0,

Mface€D1 þ K faceD1 þ KcoreðD1 � D2Þ þ K�c ðD1 � D0Þ ¼ 0,

Msand€D2 þ K sandD2 þ KcoreðD2 � D1Þ þ Cef

_D2 ¼ 0. ð19Þ

The effective structural loss factor for a two-layer composite beam has been proposed by Gong and Lam [16]and their procedure has been modified for the calculation of the effective structural loss factor Zst for athree-layer sandwich composite sheet. Then the following effective viscous damping coefficient Cef can beobtained [16]:

Cef ¼ ZstK sand

o11. (20)

Local deflection prior to core crushing at the critical crush load F cr is modeled using small deflection theoryfor a plate on an elastic foundation, which is given in Ref. [23]. The effective compressive stiffness of elasticflexible core was given as follows [23]:

Kcore ¼ 8ffiffiffiffiffiffiffiffiffiffiffikFD�f

p(21)

where D�f is the effective stiffness of the impacted face sheet given as follows:

D�f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD11D22ðgþ 1Þ=2

p(22)

and

g ¼ ðD12 þ 2D66Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD11D22

p. (23)

The elastic region of the core is modeled as a Winkler foundation, i.e. without the out-of-plane shear stiffness.Such a core is in a uniaxial stress state with stresses and displacements related through the elastic foundationstiffness [23].

For a core thickness less than

hcmax � hf32

27

� �4Q�f3Ec

� �1=3

where Q�f ¼12D�f

h3f

(24)

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the deviation from a uniaxial stress state is small [23]. In Eq. (24) hf is the thickness of the impacted face sheet.The foundation stiffness in thick cores is not uniquely defined, since displacements and stresses cannot besimultaneously matched to the proper three-dimensional solution. The available solution for isotropic foamcores may also be applied for honeycombs, since they have a similar relation between the average out-of planeshear modulus and Young’s modulus [23]. For the case of thick cores, Eq. (25) below gives the foundationstiffness kF which results in a good agreement for deflections while for thin cores, Eq. (26) gives goodagreement for stresses and deflections

kF ¼Ec

hc

where hc ¼ hc=1:38 for hcphcmax and hc ¼ 2hcmax for hc4hcmax, (25)

hc ¼27

64

� �2

2hcmax for hc4hcmax. (26)

The above classification of hc is considered in the calculation of kF and kcore, respectively.The equivalent stiffness of impacted face sheets for impact at center or off-center can be calculated from

static analysis of laminated composite face sheets with arbitrary boundary conditions by using the analyticalmethod [25] based on the first shear deformation theory (FSDT). In the special case of central impact, theequivalent stiffness of face sheets can be obtained as follows:

K face ¼ o211Mface, (27)

where o11 is the fundamental natural frequency parameter of face sheets that can be calculated from freevibration analysis of laminated composite plates [25]. The equivalent stiffness of the impacted sandwich panelwith soft flexible core with an arbitrary location of impact point can be calculated from static/free vibrationanalysis of laminated composite sandwich panels with simply supported or fully clamped boundary conditionsbased on improved higher-order sandwich plate theory (IHSAPT) [30]. In special case of impact at center, theequivalent stiffness of sandwich panel can be obtained as follows:

K sand ¼ o211Msand, (28)

where o11 is the fundamental natural frequency parameter of sandwich panel that can be calculated from thefree vibration analysis of sandwich composite panel [30]. The effective mass of impacted face sheet can bedetermined from Ref. [16]

Mface ¼ rtht

ZAP

ZW 2ðx; yÞdAP (29)

in which dAP is the differential surface area of the target face sheet. The system of ordinary differentialequation (19) can be solved using the following initial conditions:

D0ðt ¼ 0Þ ¼ 0; D1ðt ¼ 0Þ ¼ 0; D2ðt ¼ 0Þ ¼ 0,

_D0ðt ¼ 0Þ ¼ V0; _D1ðt ¼ 0Þ ¼ 0; _D2ðt ¼ 0Þ ¼ 0. ð30Þ

By applying the equivalent damping concept due to Gong and Lam [16], the eigenvalue equation can beobtained. Therefore,

ðMIM faceM�

sandÞo6 � ½KgbcM faceMI þ KgccM

�

sandMI þ K�cM faceM�

sand�o4

þ ½MIðKgccKgbc � K2coreÞ þ K�c ðKgbcMface þ KgccM

�

sandÞ � K�c2M�

sand�o2

þ ½K�c2Kgbc � K�c ðKgccKgbc � K2

coreÞ� ¼ 0, ð31Þ

where

Kgcc ¼ K face þ Kcore þ K�c ,

Kgbc ¼ K sand þ Kcore; M�

sand ¼ Msandð1� ZstÞ. ð32Þ

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Eq. (31) can be solved analytically as follows:

½o2�1;2;3 ¼ 2ffiffiffiffiffiffiffiffi�Lp

cosðyþ 2npÞ �B3

3B1

� �; n ¼ 0; 1; 2,

L ¼3B1

ffiffiffiffiffiffiB5

p� B2

3

9B21

; cosð3yÞ ¼9B1B3

ffiffiffiffiffiffiB5

p� 27B2

1

ffiffiffiffiffiffiB7

3p� 2B3

3

54B31

ffiffiffiffiffiffiffiffiffiffi�L3p , ð33Þ

where

B1 ¼ ðMIM faceM�

sandÞ,

B3 ¼ ½KgbcMfaceMI þ KgccM�

sandMI þ K�cMfaceM�

sand�,

B5 ¼ ½MIðKgccKgbc � K2coreÞ þ K�c ðKgbcM face þ KgccM

�

sandÞ � K�c2M�

sand�,

B7 ¼ ½K�c2Kgbc � K�c ðKgccKgbc � K2

coreÞ�. ð34Þ

Then, the analytic functions of dynamic deflections and impact force can be obtained as follows:

D0 ¼ c1f10 sinðo1tÞ þ c2f

20 sinðo2tÞ þ c3f

30 sinðo3tÞ, (35a)

D1 ¼ c1 sinðo1tÞ þ c2 sinðo2tÞ þ c3 sinðo3tÞ, (35b)

D2 ¼ c1f12 sinðo1tÞ þ c2f

22 sinðo2tÞ þ c3f

32 sinðo3tÞ, (35c)

F�c ðtÞ ¼ K�c ½c1ðf10 � 1Þ sinðo1tÞ þ c2ðf

20 � 1Þ sinðo2tÞ þ c3ðf

30 � 1Þ sinðo3tÞ�, (35d)

where

fi0 ¼

K�cK�c �MIo2

i

; fi2 ¼

Kcore

Kgbc � M�

sando2i

; i ¼ 1; 2; 3, (36)

c1 ¼�V0ðf

22 � f3

2Þ

o1½ðf20 � f1

0Þðf12 � f3

2Þ � ðf22 � f1

2Þðf10 � f3

0Þ�,

c2 ¼V 0ðf

12 � f3

2Þ

o2½ðf20 � f1

0Þðf12 � f3

2Þ � ðf22 � f1

2Þðf10 � f3

0Þ�,

c3 ¼�V0ðf

12 � f2

2Þ

o3½ðf20 � f1

0Þðf12 � f3

2Þ � ðf22 � f1

2Þðf10 � f3

0Þ�. ð37Þ

The general solution procedure proposed above can be used for three classes of impact problems based onOlsson’s classification [23]. (1) Large mass impacts ðMI=M face42Þ, (2) small mass impacts ðMI=M faceo0:2Þ,and (3) medium mass impacts ð0:2oMI=Mfaceo2Þ.

2.4.2. Low-velocity impact of sandwich panel with rigid-plastic flexible core in contact zone (SMDD-model)

The quasi-static force-indentation response of a rigidly supported panel can be described in terms of anonlinear spring force and a constant force dashpot. The response of this model is given by Eq. (17). Theeffective stiffness K�c can be obtained by linearizing Eq. (17). For impact loading, additional resistance will beproduced by the inertia of the impactor and deforming face sheets and the core. The inertia of the deformedcore is negligible when compared to that of the face sheets and the impactor [22]. For dynamic analysis, D1

and q in Eq. (17) are replaced by D1d and qd, respectively, where these quantities are the dynamic counterpartsof D1 and q. The dynamic bending stiffness is given by the same expression as for the static bending stiffness,but it is calculated from a laminate bending stiffness matrix that has been adjusted for high strain rate tests[22]. The dynamic crushing resistance qd is also measured from tests on core materials. The constant forcedashpot represents the dynamic crushing resistance of the core and is given by

Fdash ¼ pR2qd. (38)

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The equations of motion of the SMDD system (model II) are as follows:

MI€D0 þ K�c ðD0 � D1Þ ¼ 0,

Mface€D1 þ ðK face þ K�c ÞD1 � K�cD0 þ Fdash ¼ 0,

Msand€D2 þ K sandD2 � Fdash þ Cef

_D2 ¼ 0. ð39Þ

The dynamic response of the impacted panel is described by the system of equations (39). The third equation isdecoupled and independent, but the first two equations are coupled. The first two equations of the set ofEqs. (39) can be rewritten as follows:

D1 ¼MI

K�c

� �€D0 þ D0, (40)

ðM faceMIÞ

_ _ _

D0 þ ½ðK face þ K�c ÞMI þMface:K�c �€D0 þ ðK faceK

�c ÞD0 þ K�c Fdash ¼ 0. (41)

Using Eqs. (30) and (40), the above stated fourth-order ordinary differential equation can be solvedanalytically with the following initial conditions:

D0ð0Þ ¼ 0; _Dð0Þ ¼ V 0; €D0ð0Þ ¼ 0;

_ _ _

D0ð0Þ ¼ �V0

MIK�c . (42)

Therefore, the impactor deflection function is obtained analytically as follows:

D0ðtÞ ¼ c1ea1t þ c2e

a2t þ c3ea3t þ c4ea4t �

Fdash

K face, (43)

where ai ði ¼ 1; 2; 3; 4Þ are the analytical roots of following polynomial equation:

ðM faceMIÞa4 þ ½ðK face þ K�c ÞMI þ M face:K

�c �a

2 þ ðK faceK�c Þ ¼ 0 (44)

and the coefficients ci ði ¼ 1; 2; 3; 4Þ can be obtained using Eqs. (42)–(44). Then, the general contact forcefunction can be obtained as follows:

F�c ðtÞ ¼ �MI€D0ðtÞ ¼ �MIðc1a

21ea1t þ c2a

22ea2t þ c3a

23ea3t þ c4a

24e

a4tÞ. (45)

The above general procedure can be used for the solution of low-velocity impact case involving an impactor ofarbitrary mass. Also, for the special case of large mass impact, the fourth-order ordinary differential equation(41) can be reduced and simplified to a second-order ordinary differential equation yielding a solution in theform of trigonometric functions.

Then, the analytic time dependent response functions for deflections and impact force for large-mass impactcan be obtained as follows:

D0ðtÞ ¼V0

osinðotÞ þ

Fdash

K face½cosðotÞ � 1�, (46a)

D1ðtÞ ¼V 0

o�

MIV0oK�c

� �sinðotÞ þ

Fdash

K face1�

MIo2

K�c

� �� �cosðotÞ �

Fdash

K face, (46b)

D2ðtÞ ¼Fdash

K sand½1� cosðotÞ�, (46c)

F�c ðtÞ ¼ �MI€D0ðtÞ ¼MIo V 0 sinðotÞ þ

FdashoK face

cosðotÞ

� �, (46d)

where

o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK�cK face

MIðK face þ K�c Þ

s; o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiK sand

M�

sand

s. (47)

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 535

The proposed spring–mass–damper–dashpot (SMDD) and spring–mass–damper (SMD) models facilitate thegeneral formulation of the analytic impact force functions for various categories of impact (i.e. small-massimpact, large-mass impact and medium-mass impact conditions), when the flexible core deforms according toelastic or rigid plastic behaviors. Based on the impact state, the derived force function (see Eqs. (35d), (45) and(46d)) is incorporated in the dynamic system of equations of the panel (see Eq. (8)).

2.5. Verification of results and discussions

The characteristics of the proposed models in predicting the transient impact response of orthotropic,laminated composite sandwich panels are evaluated. In order to validate the present formulation, theresults obtained from the present method are compared with those already reported by other authorsin literature.

2.5.1. Numerical accuracy of the approach for an isotropic plate subjected to low-velocity impact

To verify the present analysis, the results obtained for the case of an isotropic plate subjected to low velocityimpact are compared with those reported by others [7,9]. Consider an isotropic rectangular steel plate, 0.2mlong, 0.2m wide and 0.008m thick, subjected to an impact at its center by a 20mm diameter steel ball strikingwith a velocity of 1m/s.

The material properties of the plate and the impactor are: E ¼ 200GPa;r ¼ 7971 kg=m3, n ¼ 0:3 and theloss factor Z ¼ 0:02. The contact force histories calculated by Wu et al. [7], Gong et al. [9], and the presentanalysis are shown in Fig. 3. It is observed that the force–time curve obtained from present three-degrees-of-freedom spring–mass–damper model (SMD-Model) and previous impact force functions are almost the same.

2.5.2. Numerical accuracy of the solution procedure for sandwich panels subjected to low-velocity impact

In this section, three examples are considered. The results are compared with other analytical andexperimental results for sandwich panels with edge support to verify the accuracy of the procedure.

2.5.2.1. Example 1: Dynamic response of sandwich panel subjected to low-velocity large-mass impact. Thepanel is simply supported and made of AS4/3501-6 carbon/epoxy face sheets and Nomex honeycomb core(HRH 10 1/8-4.0) with crushing strength q ¼ 3:83MPa. The local indentation before damage initiation wasabout 0.7mm which is small compared to the face sheet thickness which is approximately 2mm.

The value of critical strain eec at which the Nomex core begins to exhibit nonlinear elastic behavior is0.02mm/mm [22]. Therefore, the 12.7mm thick core should deform more than 0.24mm before it can bemodeled as plastic. In impact analysis, the dynamic crushing strength of Nomex honeycomb qd is equal to 1.1q

[22]. The material and geometrical properties of impactor are as follows:Geometry: Hemispherical-nosed cylinder (case-hardened steel): V 0 ¼ 1:42m=s; d ¼ 25:4mm;MI ¼ 3:48 kg.

Fig. 3. Contact force history of steel panel impacted by steel ball.

ARTICLE IN PRESS

Table 1

Material properties of sandwich panels

Examples Example 1: Ref. [22] Example 2: Ref. [23] Example 3: Ref. [10]

Properties Face Core Face Core Face Core

E11 (GPa) 144.8 0.0804 12.5 0.092 153 0.000689

E22 (GPa) 9.7 0.0804 12.5 0.092 8.96 0.000689

E33 (GPa) 9.7 1.005 8.0 0.092 8.96 0.4756

G12 (GPa) 7.1 0.0322 4.0 0.0325 5.1 0.000275

G13 (GPa) 7.1 0.1206 4.0 0.0325 3.79 0.2754

G23 (GPa) 3.76 0.0758 4.0 0.0325 3.3 0.0965

n12 0.3 0.25 0.25 0.41 0.29 0.25

n13 0.3 0.02 0.5 0.41 0.29 0.03

n33 0.3 0.02 0.5 0.41 0.29 0.03

r ðkg=m3Þ 1632 64 1630 80 1650 88.14

hc (mm) — 12.7 — 22 — 9.525

Ply thickness

(mm) 0.0635 — 1.1 — 0.3 —

a (mm) 178 178 800 800 241.3 241.3

b (mm) 178 178 800 800 114.3 114.3

Material AS4/3501-6 HRH 10 1/8-4 Glass-Polyester H80/PVC IM7/5260 Glass-Phenolic

Fig. 4. Contact force histories of the impacted composite sandwich panel and the effects of various boundary conditions on contact force

history.

K. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543536

The material properties of the core and the face sheets of example 1 are shown in Table 1. The stackingsequences of 32 plies in the top and the bottom face sheets are [0/90]8s. The contact force history is comparedwith the analytical and experimental results in Fig. 4. Also, the effects of boundary conditions on contact forcehistory are shown in Fig. 4. There is a negligible difference in the phase and magnitude of contact force resultsfor a fully clamped and simply supported boundary conditions. Further, the contact force result for asandwich panel with top and bottom face sheets clamped on all edges (CCCC) is the same as the contact forceresult for a sandwich panel with fully clamped boundary in top face sheet and simply supported in bottom facesheet (CTSB). Therefore, the effects of the boundary conditions of bottom face sheet on contact force historyare negligible. These observations indicate that the impact response of a sandwich panel is a very localizedphenomenon as compared to that of a monolithic plate. This may be explained by the fact that most of theimpact energy is taken up by the flexible core during its deformation in the region close to the impact point.For the clamped case, contact force magnitude is a little larger, and duration is smaller than for the simplysupported case. Contact force history for simply supported panel obtained from the present method is in goodagreement with that obtained from the experiments [22]. The theoretical predictions based on SMDD model,shown in Fig. 4, indicate that the largest error in the value of maximum contact force obtained from thepresent analysis vis-a-vis the measured experimental values [22] is about 4.76% which corresponds to theimpactor velocity, V 0 ¼ 1:42m=s. A possible reason for this small discrepancy may be the non-inclusion of theeffects of loading rate in case of both the honeycomb constitutive model as well as the linear elastic model usedfor the face sheets and the bond layer.

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 537

2.5.2.2. Example 2: Dynamic response of simply supported composite sandwich panel subjected to low-velocity

small-mass impact. In this example, the numerical results are compared with the results of impactexperiments on sandwich panels with edge supports [19]. This example has been selected as explicitmeasurement of deflections on the back face and is available for comparison with the results of present work.The details of the sandwich construction are presented in the column labeled as example 2 of Table 1. Thesandwich panel was hit at 7.67m/s by a 2.5 kg impactor with 12.5mm tup radius. The impactor/panel massratio of 0.44 puts the problem in the category of small mass impact response. The ply thickness is 1.1mm andthickness of each face-sheet is 2.2mm (hb ¼ ht ¼ 2:2mm).

For the Galerkin’s method based on Fourier series, the optimum number of terms for summation isdetermined by the required accuracy and computational efficiency. Accuracy obviously increases with moreterms, though not very significantly beyond some finite number. Practical considerations, like the computationtime, limit the accuracy further. Fig. 5 shows the convergence of the solution with increasing number of termsof the Fourier series. It was observed that the difference in deflection response for m, n415 (225 terms) is verysmall. The size of time step ðDtÞ is generally chosen to be a fraction of the period corresponding to the highestfrequency of the vibrating structure to ensure stability of the numerical solution process. Dt is commonly takento be equal to 0:1Tk, where Tk ¼ 2p=ok is the time period corresponding to the circular frequency ok, and k isthe mode number of the highest frequency. A comparison of the contact force and deflection results obtainedby the present method with analytical and experimental results obtained from [19,23] are, respectively, shown

Fig. 5. Effect of varying the number of terms of the Fourier series included in the solution for transverse deflection of the panel.

Fig. 6. Comparsion of the results of the present analysis with the analytical and experimental results: (a) contact force; (b) transverse

deflection of the bottom face sheet.

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in Figs. 6a and b. The contact force and deflection histories obtained from the present method are in fairlygood agreement with those obtained in Ref. [23]. Any small disagreement between the two sets of results maybe attributed to some uncertainty in the properties of the core material as used in the present analysis. Also,large deformation of the top face-sheet due to core crushing has not been considered in our analysis.

Further, the difference between our results and theoretical results [23] may be due to the different sets ofassumption, e.g. an infinite plate was considered and the effects of rotary inertia and transverse shear wereneglected in Ref. [23] while we considered a finite plate with rotary inertia and shear effects, etc. Also, it wasnoted that the variation of contact area with respect to time consumes a lot of computer time during dynamicanalysis of structures and its effect on the impact response of sandwich panels was so small that it can beneglected.

Also, in Fig. 6, the effects of structural damping on contact force and central deflection of panel are shown.The loss factor of core was assumed to be 0.1 and loss factors of face sheets were neglected in computation.Maximum induced vertical shear stresses in the core tcxz; t

cyz and maximum in-plane shear stresses txy in the top

and the bottom face sheets on the section at y ¼ b=2 and parallel to the x-axis are shown in Fig. 7a. This figureshows that the shear stresses in the top face sheet txy are larger than the shear stresses in the bottom face sheetand in the core. Also, Fig. 7b shows the maximum normal stress in the mid-plane of the core on the section aty ¼ b=2 parallel to the x-axis.

2.5.2.3. Example 3: Dynamic response of simply supported composite sandwich panel subjected to low-velocity

medium-mass impact. The panel considered is simply supported and is fabricated with graphite-bismaleimide(IM7/5260) face sheets and glass fiber-reinforced phenolic honeycomb core. The stacking sequence of plies inthe top face sheet and the bottom face sheet is ½ð45=0=� 45=90=0Þ2=90�s. The details of this sandwichconstruction are presented in the column ‘example 3’ of Table 1. The panel is 114.3mm wide and 241.3mmlong. Its exposed area was impacted by means of a dropped weight with 12.7mm diameter hemispherical tupðMI=Msand ¼ 1:602Þ at an energy level of 2.7138 J, which is below the damage threshold level to generatecontact force and surface-strain results [10] that are used for validating the present approach. Bottom surfacestrain for this sandwich panel are presented in Fig. 8. The strain profile so obtained is compared with theexperimental and analytical results of Ref. [10] in which the face sheet is treated as a plate on an elasticfoundation. The strain results, based on present SMD model (model I) are in good agreement with theexperimental results. The results obtained from SMD model are closer to experimental results than thoseobtained from the plate on elastic foundation model. This close agreement of back-surface strain with

Fig. 7. Maximum induced normal and shear stresses in the core and the face sheets along the section y ¼ b=2 and parallel to x-axis: (a)

maximum shear stresses in the mid-plane of the core and the face sheets; (b) maximum normal stress in the core.

ARTICLE IN PRESS

Fig. 9. Central deflections of the top and the bottom face sheets and indentation of impacted panel.

Fig. 8. Variation of strain on the lower surface of the impacted panel at a point opposite to the impact point.

Fig. 10. Normal stress (sczz)-profile in the mid-plane of the core with increasing contact time during impact: (a) along the section at

x ¼ a=2 and parallel to y-axis; (b) along the section at y ¼ b=2 and parallel to x-axis.

K. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 539

experimental results reflects that the through-thickness deformation and the transverse flexibility of the core ofpanel which influence the back-surface strain have been appropriately accounted for in the present analysisbased on the improved high order sandwich plate theory (IHSAPT). The central deflection histories of top andbottom face sheets and the total indentation of the panel are shown in Fig. 9. The figure illustrates that the

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543540

compressibility and transverse flexibility of the core cause core indentation and therefore, the Hertzianindentation of top face sheet is relatively small as compared to the core indentation. The indentation of core isalmost equal to the difference of the top and the bottom transverse deflections that are shown in Fig. 9. Fig. 10shows the profiles of normal stress sczz in the mid-plane of the core with increasing contact time. Fig. 10a showsthe normal stress on a section at x ¼ a/2 and parallel to y-axis while Fig. 10b shows the normal stress on asection at y ¼ b/2 and parallel to x-axis. Maximum normal stress is obtained at the impact point, as expected.

3. Conclusions

A new computational method based on the improved higher order sandwich plate theory (IHSAPT) for facesheets has been introduced to analyze transverse low velocity impact on sandwich panels caused by a sphericalimpactor. In this method the transverse flexibility of the core has been taken into account, as is necessary forfoam cores or non-metallic honeycomb cores. For the solution procedure the response functions of the panelare represented by double Fourier series and then the set of the governing partial differential equations reducesto a set of ordinary differential equations. The interaction between the panel and the impactor is representedby one of the two models (SMD or SMDD) of three degrees of freedom, depending on the behavior of the coreunder transverse load. The solution procedure has been validated by comparing its results with those obtainedeither experimentally or analytically by other researchers. Three examples were considered. In one case, thepredictions of the present theory for the history of the contact force were found to be in close agreement withexperimental measurements of contact force, in spite of some uncertainty about the physical properties of thecore material as used in the analysis.

For the other two examples, the comparison of the results of the present work with those available in theliterature was made and found to be quite good, notwithstanding the difference in assumptions. The examplesconsidered also showed that the effects of structural damping (of face sheets as well as the core) are very smallin the low-velocity impact analysis, although the present method is capable of accounting for these effects.Similarly, in the present method the variability of the contact patch size with the change in impact load (aspredicted by Hertz–Sveklo contact theory) was not neglected. However, for the low velocity impact case, thisvariability has only a small effect on the response of the sandwich panel as long as the contact patch does notreduce to a point.

It was also observed that when the core is soft and flexible, the solution of the low-velocity impact problemis quite insensitive to the type of boundary conditions, which may be prescribed independently on the top andbottom face sheets. Therefore, in such cases, the impact phenomenon is very much localized. This may beattributed to the large proportion of the impact energy being absorbed by the core in deformation or crushing.The present analysis is based on linear elastic or viscoelastic behavior of the face sheets and the core. Hence,the deformations are assumed to be small. The in-plane stresses in the core are also neglected. However, thereis no restriction on the ratio of the impactor mass and the panel mass, i.e. the analysis can be applied to allthree categories of mass ratio as defined by Olsson.

Computationally, the present method is very economical as compared to other numerical procedures, suchas F.E.M. In other methods, the contact force history has to be obtained as a part of the solution and thisinvolves a lot of computation effort. However, the F.E.M. gives a detailed information about the distributionsof stresses and strains in the face sheets and the core, separately. This is not so with the present procedure.Instead, the contact force obtained by the present procedure is used as an input for further analysis of thepanel, like the deflection under the point of impactor and the failure prediction of the core, etc.

Appendix A

The boundary conditions at the edges of upper and lower face sheet are:

Nixxðx ¼ 0 or x ¼ aÞ ¼ aN

i

xx or u0iðx ¼ 0 or x ¼ aÞ ¼ 0

Nixyðx ¼ 0 or x ¼ aÞ ¼ aN

i

xy or v0iðx ¼ 0 or x ¼ aÞ ¼ 0

Niyyðy ¼ 0 or y ¼ bÞ ¼ aN

i

yy or v0iðy ¼ 0 or y ¼ bÞ ¼ 0

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543 541

Nixyðy ¼ 0 or y ¼ bÞ ¼ aN

i

xy or u0iðy ¼ 0 or y ¼ bÞ ¼ 0

Mixxðx ¼ 0 or x ¼ aÞ ¼ 0 or cxiðx ¼ 0 or x ¼ aÞ ¼ 0

Miyyðy ¼ 0 or y ¼ bÞ ¼ 0 or cyiðy ¼ 0 or y ¼ bÞ ¼ 0

Mixyðy ¼ 0 or y ¼ bÞ ¼ 0 or cxiðy ¼ 0 or y ¼ bÞ ¼ 0

Mixyðx ¼ 0 or x ¼ aÞ ¼ 0 or cyiðx ¼ 0 or x ¼ aÞ ¼ 0

Mixy atððx ¼ 0 or x ¼ aÞ and ðy ¼ 0 or y ¼ bÞÞ ¼ 0

w0iððx ¼ 0 or x ¼ aÞ and ðy ¼ 0 or y ¼ bÞÞ ¼ 0

Qiyðy ¼ 0 or y ¼ bÞ ¼ 0 or w0iðy ¼ 0 or y ¼ bÞ ¼ 0

Qixðx ¼ 0 or x ¼ aÞ ¼ 0 or w0iðx ¼ 0 or x ¼ aÞ ¼ 0

i ¼ t;b

Subscripts and superscripts t and b mean the top and bottom face sheets. The boundary conditions at theedges of the core at zc ¼ z, read: tcxz(x ¼ 0 or x ¼ a) ¼ 0 or wc(( x ¼ 0 or x ¼ a), z) ¼ 0; tcyz(y ¼ 0 ory ¼ b) ¼ 0 or wc(( y ¼ 0 or y ¼ b), z) ¼ 0.

Appendix B

Using the results obtained in [31] analytically, the vertical normal stress and the vertical deformationsthrough the depth of the core are as follows:

sczzðx; y; zcÞ ¼ �ðtcxz;x þ tcyz;yÞ

2ð2zc � cÞ þ

ðw0b � w0tÞEc

c

þ rc €w0bz2c2c�

c

6

� �� €w0t

z2c2c� zc þ

c

3

� �� �,

wcðx; y; zcÞ ¼ �ðtcxz;x þ tcyz;yÞ

2Ecðz2c � c:zcÞ þ ðw0b � w0tÞ:

zc

cþ w0t

þrc6Ec

€w0bz3cc� czc

� �� €w0t

z3cc� 3z2c þ 2czc

� �� �.

Also, the in-plane displacements of the core in x- and y-directions through the depth of the core are as follows:

ucðx; y; zcÞ ¼tcxz:zc

Gc�ðtcxz;xx þ tcyz;yxÞð3cz2c � 2z3cÞ

12Ec�ðw0b;x � w0t;xÞ:z2c

2c� w0t;x:zc

þrc

24Ec

€w0b;x 2cz2c �z4cc

� �þ €w0t;x 4cz2c þ

z4cc� 4z3c

� �� �þ ðht=2Þ:cxt þ u0t,

vcðx; y; zcÞ ¼tcyz:zc

Gc�ðtcxz;xy þ tcyz;yyÞð3cz2c � 2z3cÞ

12Ec�ðw0b;y � w0t;yÞ:z2c

2c� w0t;y:zc

þrc

24Ec

€w0b;y 2cz2c �z4cc

� �þ €w0t;y 4cz2c þ

z4cc� 4z3c

� �� �þ ðht=2Þ:cyt þ v0t.

Appendix C

The terms representing parts of equivalent viscous damping effects are as follows:

bc ¼Ce

ða� bÞðhc=hÞ

b1 ¼ �bcð _u0t=3þ _u0b=6� hb: _cxb=12þ ht: _cxt=6Þ,

b2 ¼ �bcð _u0b=3þ _u0t=6� hb: _cxb=6þ ht: _cxt=12Þ,

ARTICLE IN PRESSK. Malekzadeh et al. / International Journal of Impact Engineering 34 (2007) 522–543542

b3 ¼ �bcð _w0t=3þ _w0b=6Þ,

b4 ¼ �bcð _w0b=3þ _w0t=6Þ,

b5 ¼ �bcð_v0t=3þ _v0b=6� hb: _cyb=12þ ht: _cyt=6Þ,

b6 ¼ �bcð_v0b=3þ _v0t=6� hb: _cyb=6þ ht: _cyt=12Þ,

b7 ¼ �bcð _u0t:ht=6þ _u0b:ht=12� hb:hb: _cxb=24þ h2t :_cxt=12Þ,

b8 ¼ �bcð� _u0t:hb=12� _u0b:hb=6þ h2b:_cxb=12� ht:hb: _cxt=24Þ,

b9 ¼ �bcð_v0t:ht=6þ _v0b:ht=12� hb:hb: _cyb=24þ h2t :_cyt=12Þ,

b10 ¼ �bcð� _v0t:hb=12� _v0b:hb=6þ h2b:_cyb=12� ht:hb: _cyt=24Þ.

Appendix D

Symmetric stiffness matrix-coefficients in the free vibration eigenvalue equation:

K11mn ¼ At66b

2n þ At

11a2m; K12mn ¼ ðA

t12 þ At

66Þambn; K1;11mn ¼ �1,

K22mn ¼ At22b

2n þ At

66a2m; K2;12mn ¼ �1; K33mn ¼ Ab

66b2n þ Ab

11a2m;

K34mn ¼ ðAb12 þ Ab

66Þambn; K3;11mn ¼ 1; K44mn ¼ Ab22b

2n þ Ab

66a2m;

K4;12mn ¼ 1; K55mn ¼ kAt55a

2m þ kAt

44b2n þ R11; K56mn ¼ �R11;

K57mn ¼ kAt55am; K59mn ¼ kAt

44bn; K5;11mn ¼ amhc=2;

K5;12mn ¼ bnhc=2; K66mn ¼ kAb55a

2m þ kAb

44b2n þ R11; K68mn ¼ kAb

55am;

K6;10mn ¼ kAb44bn; K6;11mn ¼ amhc=2; K6;12mn ¼ bnhc=2;

K77mn ¼ Dt66b

2n þDt

11a2m þ kAt

55; K79mn ¼ ðDt12 þDt

66Þambn; K7;11mn ¼ �ht=2;

K88mn ¼ Db66b

2n þDb

11a2m þ kAb

55; K8;10mn ¼ ðDb12 þDb

66Þambn; K8;11mn ¼ �hb=2;

K99mn ¼ Dt66a

2m þDt

22b2n þ kAt

44; K9;12mn ¼ �ht=2; K10;11mn ¼ �hb=2;

K10;10mn ¼ Db66a

2m þDb

22b2n þ kAb

44; K11;11mn ¼ �ðR12 þ R13a2mÞ; K11;12mn ¼ �ðR13ambnÞ;

K12;12mn ¼ �ðR12 þ R13b2nÞ:

Except for the above elements and their symmetric counterparts, the other elements of the complex stiffnessmatrix are zero.

Symmetric mass matrix-coefficients:

M11 ¼ R5, M13 ¼ R2, M17 ¼ 2R3, M18 ¼ �R1, M22 ¼ R5, M24 ¼ R2, M29 ¼ 2R3,M2;10 ¼ �R1, M33 ¼ R4, M37 ¼ R3, M38 ¼ �2R1, M44 ¼ R4, M49 ¼ R3, M4;10 ¼ �2R1,

M55 ¼ R5, M56 ¼ R2, M66 ¼ R4, M77 ¼ R9 þ I2t, M78 ¼ �R7, M88 ¼ R10 þ I2b, M99 ¼ R9 þ I2t,M9;10 ¼ �R7, M10;10 ¼ R10 þ I2b,

where

R4 ¼ I0b þmc=3;R9 ¼ mch2t =12;R12 ¼ hc=Gc; R13 ¼ h3

c=ð12EcÞ;R14 ¼ mch2c=ð24EcÞ;

R5 ¼ I0t þmc=3;R11 ¼ Ec=hc, R7 ¼ mchbht=24;R9 ¼ mch2t =12;R10 ¼ mch

2b=12;

R1 ¼ mchb=12;R2 ¼ mc=6;R3 ¼ mcht=12:

Except for the above elements and their symmetric counterparts, the other elements of the mass matrix arezero.

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