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epoxy composite plates subjected to transverse impact
Composites Science and Technology 61 (2001) 135±143
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An analysis of impact force in plain-weave glass/epoxy composite plates subjected to transverse impact Jung-Kyu Kim a,*, Ki-Weon Kang b a School of Mechanical Engineering, Hanyang University, Seoul 133-791, South Korea Department of Mechanical Design and Production Engineering, Hanyang University, Seoul 133-791, South Korea
b
Received 15 September 1999; received in revised form 29 June 2000; accepted 24 August 2000
Abstract In the present study, a new analytical method is developed for predicting the impact force from the dynamic strain of composite plates subjected to transverse impact. For this, two assumptions are introduced in this study. Firstly, the impact force and dynamic strain can be separated into frequency and amplitude. Secondly, the amplitude of the impact force corresponds to that of the dynamic strain at any frequency. By applying the Rayleigh±Ritz energy method and Lagrange's principle to a rectangular plate, the governing equation is derived, and the equation is solved by an optimal design technique. To verify this analytical method, impact tests are performed on plain-weave glass/epoxy composite plates having various thicknesses. The impact forces obtained by this analytical method agree well with the experimental results for all plate thicknesses studied. Additionally, the dynamic response, such as the impact force, is greatly in¯uenced by the thickness. This behavior is attributed to the dierent energy absorbing mechanisms: the absorbed energy is governed by the plate de¯ection in a thin plate, whereas the energy is governed by the local indention in a thick plate. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Plain-weave glass/epoxy composite plate; Impact force; Dynamic strain; Green's function; Thickness eect
1. Introduction Since composite materials are susceptible to damage resulting from the impact of foreign objects, an analysis of the dynamic response of composite materials subjected to impact is necessary for the assessment of damage resistance and the design of structures [1]. To understand the dynamic response of composite materials, the accurate evaluation of impact loading history applying to composites is important, and the thickness eect is well worth investigating because the impact force and the impact resistance of composite materials are strongly in¯uenced by thickness. To analyze the dynamic response of composite materials, several researchers have proposed techniques that can determine the impact force by taking the responses of an impacted structure itself. One is to solve the problem in a frequency domain by using the fast-Fourier transform method. Doyle [2±4] evaluated the impact forces on isotropic and orthotropic plates with this * Corresponding author. Fax: +82-2-22291-6707. E-mail address: [email protected] (J.-K. Kim).
technique. The measured dynamic strain, however, needed to be properly windowed and subsequently ®ltered to eliminate the boundary eect. The other method relies on the inverse method in a time domain. In this method, when some of the structural responses are known, the impact force can be determined. This method has been successfully applied to an aluminum alloy by Yen et al. [5]. However, it is not easy to identify the adequate time interval for discretizing the convolution integral, and consequently this causes the accuracy and the eciency of a solution to depend on the used time interval signi®cantly. Moreover, it is not clear whether this method is applicable to composite materials and to thick plates for which the dynamic response is governed by the local contact deformation. The thickness of composite materials is a dominant parameter that governs the dynamic response and damage mode of impacted ones. Thus, many pieces of research have been carried out on the dynamic response and damage behavior with the thickness. For example, Kim et al. [6] investigated the eect of thickness on the impact energy by ®nite-element analysis, and Cantwell et al. [7] experimentally demonstrated that the impact
0266-3538/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00203-7
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J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
resistance is the highest at the speci®c thickness for the laminated composite. However, since the dynamic response of composite materials depends not only on the thickness but also on the type of constituent material, these studies are not likely to completely explain the eect of parameters on it. Therefore, it can be said that for various composite materials a reliable mechanism for describing the dynamic response with the thickness is not yet available and more extensive research is necessary to characterize the impact behavior of composite materials. The purpose of this study is to develop a new analytical method to predict the impact force history of plainweave glass/epoxy composite plates and to evaluate the eect of thickness on the dynamic response. For these purposes, two assumptions are introduced. Firstly, the impact force and dynamic strain can be separated into frequency and amplitude. Secondly, the two responses correspond to each other regardless of frequency. Under these assumptions, applying the Rayleigh±Ritz energy method and Lagrange's principle to the rectangular plate with thickness B, the governing equation is derived, and the equation is solved by an optimal design technique. The proposed analytical method is veri®ed by impact hammer tests and the eect of thickness on the dynamic response is examined.
In these, p(x, y) is the lateral loading on the plate, w(x, y) the lateral displacement, and N0x, N0y, and N0xy the in-plane loadings. The kinetic energy of a plate is expressed as the surface integrals of the related mass and velocities ph T 2
a b 0 0
: wdydx
5
: where {} denotes the derivative with respect to time and is the density of the plate. In the Rayleigh±Ritz energy method, the assumed displacement mode shapes are used to transform the spatially continuous system to the discrete system of modal amplitudes [10]. Therefore, the governing equation of the plate, which is expressed in terms of the transformed system of modal amplitudes, can be derived by the minimization of the plate potential and kinetic energy. If the displacements are assumed to be separable into x, y and z, the planar and lateral displacements can be expressed as u x; y; z; t z x x; y; t z
N1 X N2 X Aij tXi x; x Yj y
v x; y; z; t z y x; y; t z
i
2. Dynamic analysis model
w x; y; t
The governing equation of equilibrium is deduced by the minimization of potential and kinetic energy. For the rectangular plate with the thickness B and in-plane size ab, the bending and shearing strain energies are given as the surface integral forms Ub
1 2
a b
ks Us 2
0 0
fgT Dfgdydx
a b 0 0
1
f gT Af gdydx
2
where {} and { } denote the plate curvature and shear strain in the Reissner±Mindlin plate theory [8], respectively and s is the shear correction factor [9]. Moreover, the external work on the plate is the sum of the lateral loading work W1 on the plate and the inplane loading work W2 at the mid-plane as follows: a b p x; y; tw x; y; tdydx W1 ÿ
3
0 0
1 W2 2
a b 0 0
w; w;
x y
"
N0x N0xy
N0xy N0y
#
w; w;
x y
j
i
N1 X N2 X j
Bij tXi xYj y;
N1 X N2 X Cij tXi xYj y
y
6
j
i
where Xi(x) and Yj(y) are the beam functions determined from the appropriate boundary conditions, and Aij(t), Bij(t) and Cij(t) are the time-varying modal amplitudes which must be determined. N1 and N2 are the number of beam functions in the x and y directions, respectively. The governing equation expressed in terms of modal amplitudes can be obtained by applying the following Lagrange's equation. d @T @T @V 0 : ÿ dt @rj @rj @rj
7
where V denotes the potential energy, which is de®ned as Ub+Usÿ(W1+W2) and rj is the modal amplitude. Therefore, the following equation can be developed by substituting Eqs. (1)±(6) into Eq. (7)
0 0 0 M22
r1 r2
K11 K21
K12 K22
r1 r2
0 P
8
dydx:
4
where {r1} denotes the column vector containing Aij(t) and Bij(t), and {r2} is the column vector composed of
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Cij(t). The mass matrix [M22] is a diagonal matrix due to the orthogonality characteristics of the beam functions. In addition, the components of the stiness matrix, [Kij], are obtained by integrating the assumed beam functions and their derivatives over the plate surface. The ®nal governing equation of the composite plate can be obtained by applying the static condensation scheme to Eq. (8) as follows: M22 fr2 g bK22 cfr2 g fPg
9
fr1 g ÿK11 ÿ1 K12 fr2 g
10
where K22 K22 ÿ K12 T K11 ÿ1 K12 :
3. Numerical analysis The eigenmode expansion method is introduced to solve Eq. (9). Let Eq. (9) be of Lank K, and !1, !2,. . .,!k and {e1}, {e2},. . .,{ek} be the eigenvalues and eigenvectors of the homogeneous equation, respectively. The unknown vector {r2}, therefore, can be expressed by linear superposition of the eigernvectors as follows: fr2 g
K X aj fej g
11
j1
where each aj is the function of time and can be obtained by solving each of the uncoupled equations as follows: mj a j kj aj p t fej gT fHg ;
j 1; 2; . . . ; K
12
where mj={ej}T [M22] {ej}, kj=mj!j2 and K is equal to N1N2. Thus, the modal amplitude Cij(t), contained in vector {r2}, can be obtained by solving Eq. (12) and the values of Aij(t) and Bij(t) can be determined by substituting the results into Eq. (10). Consequently, the dynamic response of the plate at a certain point (x1,y1) can be expressed in terms of these modal amplitudes as follows: " x1 ; y1 ; z; t ÿ x1 ; y1 ; z; t ÿz bP T K11 ÿ1 K12 fej gcbfej gT fHgc mj
t 0
K X
gj t
sin !j t=!j ; !j 6 0 !j 0 !j ;
137
14
where represents x or y. {} is the column vector, which can be obtained from the relationship between displacements and beam functions, Eq. (6). Thus, if the dynamic strain is given in Eq. (13), one can ®nd the force function at any position of the plate. Since an analytical solution is still not available for Eq. (13), Yen et al. [5] discretized the convolution integral in the time domain to develop an approximate numerical solution. By separating the concerned time period into n of the same intervals and by using the linear interpolation technique, Eq. (13) is transformed into the following algebraic equation: 8 9 2 3 8p 9 G1 ÿ1 > > 1> > > > > > > = < ÿ2 > 6 G2 G1 7 < p2 = 6 7 ÿz4 .. 15 .. .. .. 5 > .. > > > . . . > > ; > : . > > . > ; : Gn Gnÿ1 G1 l pn ÿn where Green's function, G means the transient strain response at (x1,y1) by a unit impulse force applied at (x0,y0) and l indicates the number of used strain gage. Yen et al. [11] reported Eq. (15) is very eective for determining the impact force of an aluminum plate and can be used to predict the impact force by using an incomplete strain response, which often occurs in the service condition of a structure. However, it is not easy to identify the adequate time interval, n for discretizing the convolution integral. Consequently the accuracy and eciency of the solution depend greatly on the used time interval. To overcome this shortcoming, it is assumed that the impact force can be separated into frequency and amplitude. Thus, the impact force can be approximately expressed as the following Mth-order Fourier series. p t
M X
pi t sin fi t
16
i1
where fi and pi are the frequency and the amplitude of impact force, respectively. By substituting Eq. (16) into Eq. (13), we can obtain a new equation with regard to the impact force and strain response as follows: " x1 ; y1 ; z; t ÿz
M X K X Gj x1 ; y1 ; t; x0 ; y0 pi i1 j1
j1
17
p t ÿ gj d 13
where G is Green's function, which is expressed as follows:
138
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Gj x1 ; y1 ; t; x0 ; y0 T K11 ÿ1 K12 fej g fej gT fHg sin wj tfi ÿ sin fi twj : mj wj f2i ÿ w2j 18 Obviously, Eq. (17) can be used to determine the impact force from the dynamic strain response. However, due to the complicated form of Green's function in Eq. (18), the impact force cannot be determined by an analytical method. Therefore, the optimization technique is adopted to approximately obtain the solution in this study. To establish an object function used in an optimization technique, the following assumptions are made. Firstly, the dynamic strain can be separated into frequency and amplitude such as in Eq. (19). Secondly, the frequency characteristics of impact force and dynamic strain are identical and the amplitudes of the two responses at any frequency correspond to each other. Based on these assumptions, the object function and constraint can be de®ned as Eqs. (20) and (21) " t
M X
"i t sin fi t
19
i1
Objective
( L X M X l1 i1
subjected to p t
K X "i z Gj pi
) 20
j1
M X pi t sin fi t40
250 mm in length and 40 mm in width, as shown in Fig. 1. Various thicknesses (B=2.3, 3.0, 4.0, 5.0 and 6.6 mm) were prepared to identify the thickness eect on the dynamic response. The specimens were clamped on two opposite edges and were left free on the other two edges. 4.2. Impact hammer test An impact hammer is generally used for evaluating the dynamic response of a structure. For this purpose, one may hold an impact hammer and strike a target by hand. However, it is almost impossible to maintain the right angle between the specimen and impact hammer. To solve this problem, the impact hammer was installed in the special loading apparatus as shown in Fig. 2. The impact hammer used (PCB Electronics, 086B03, 2.3 mV/ N) was ®xed to the rod attached to the origin of rotation. The impulse could be adjusted by changing the length of the spring as the four steps, which was attached to the impact loading apparatus The range of impact velocity was 1.40±2.40 m/s. The impact velocity was measured from the time of passage between the laser sensors connected to an electronic timer. The tip diameter and mass of impactor were 12.7 mm and 3.45 kg, respectively. In addition, on the opposite surface against the impact point, two strain gages were mounted to measure the dynamic strain responses. Two strain gages, as shown in Fig. 1, were attached at 50 mm and
21
i1
where the constraint means that the impact force must be compressive. 4. Experimental procedure 4.1. Materials and specimen
Fig. 1. Con®guration of specimen (unit:mm).
The materials used in this study were plain-weave Eglass/epoxy composite plates (volume fraction, 10.62). The mechanical properties of this material were obtained from unidirectional tensile test, according to ASTM D-3039-93 [12], and summarized in Table 1. The Poisson ratio, xy, and shear modulus, Gxy, were obtained by using the 0 /90 strain-rosette and the 45 strain-gage method, respectively. The specimen was Table 1 Mechanical properties of plain-weave glass/epoxy composite Exx (GPa)
Eyy (GPa)
Gxy (GPa)
19.96
19.96
3.1
0.136
Fig. 2. Schematic diagram of impact loading apparatus.
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
80 mm away from the left edge of the specimen, respectively. The impact force and dynamic strain were recorded on a personal computer. The sampling rate used was always 100 kHz. 5. Results and discussion 5.1. Impact force evaluation In the Rayleigh±Ritz method, a governing equation is obtained by transforming a spatially continuous system to a discrete system of modal amplitudes using the assumed displacement mode. Thus, the number of modes required for the Rayleigh±Ritz method strongly in¯uences the accuracy of solution. To know how many modes must be included, the response signals must be analyzed by using the fast-Fourier transform method to obtain a frequency response. The recorded impact force and dynamic strain for the gage at (50 mm, 20 mm) in a B=2.3 mm plate are analyzed using the FFT analysis and the results are shown in Fig. 3(a). Most of the amplitudes are distributed in the frequency range lower than 5 kHz. Thus, the 1010 vibration modes of the plate used in this study are appropriate. To verify the assumption of correspondence used in the object function Eq. (20), the frequency characteristics of the impact force must be examined together with that of the dynamic strain over the concerned frequency range. To this end, the two responses of Fig. 3(a) are examined over the low frequency band and the results are shown in Fig. 3(b). It can be seen that the impact force corresponds to the dynamic strain regardless of the frequency. Based on the above results, the impact forces are obtained by minimizing the object function, Eq. (20). The computer code and search algorithm are the ADS
139
(Automatic Design Synthesis), published in public domain, and the modi®ed method of feasible directions, respectively [13]. The analysis results for B=2.3 mm are shown in Fig. 4 and Table 2 together with the experimental results. The results obtained by the proposed method are well matched with the experimental results regardless of applied impulses. Additionally, it takes a very short CPU time of about 90 s to compute the impact forces. On the other hand, though Yen's method, Eq. (15), has several advantages as previously mentioned, his method has the shortcoming that the accuracy and the eciency of the solution depend signi®cantly on the used time interval because of discretizing the force function in time domain. The eciency of Yen's method can be improved by using a longer time interval but simultaneously it causes a very poor accuracy of the solution. However, the proposed method is found to have good accuracy and the independence on the time interval due to the adoption of the two assumptions already mentioned. There are various analytical techniques that have been used to evaluate the impact response in composite structures. These techniques including the proposed analytical method are based on the beam or plate theory, which describes the overall deformation of a structure. These techniques give an exact solution to the certain categories of impact problems, but is restricted to a narrow range of conditions. It has been particularly noted that the techniques cannot be applied to a thick structure. For thick structures, local contact deformation plays a major role in the dynamic response but the governing equation, based on the beam or plates theory, cannot describe it near the impact point. To overcome this problem, the nonlinear contact law considering the change of contact area with time, is generally incorporated in the governing equation. It is, unfortunately, impossible to incorporate the contact law in the present
Fig. 3. The frequency responses of impact force and dynamic strain. (a) Frequency response for wide-band. (b) Frequency response for narrowband.
140
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Fig. 4. Comparisons between predicted and experimental impact force histories. (a) Impact location 1 and 3. (b) Impact location 2 and 4.
5.2. Thickness eect
Table 2 The relative errors and CPU time (B=2.3 mm) Spring location
1
2
3
4
[(FEXPÿFPRE)/FEXP] CPU times
2.82 88.7
4.85 84.9
4.33 92.1
5.98 93.2
analytical method and hence the applicability of the method on a thick structure must be proved. To this end, the applicability of the number of modes used for B=2.3 mm on thick plates, is discussed ®rst. Fig. 5 shows the frequency responses for B=4.0 mm and B=6.6 mm. Fig. 5(a) indicates that though the frequency characteristics are somewhat dierent according to the thickness, most of the amplitudes are distributed in the frequency range lower than 5 kHz for all thickness. An identical trend could be found for B=2.3 mm. The assumption of correspondence between the impact force and the dynamic strain is also applicable despite of the inconsistence between the two responses in a very low frequency range for B=6.6 mm in Fig. 5(b). According to these results, the analyses and experiments were conducted for thick plates. Fig. 6 shows the typical results for B=4.0 mm and B=6.6 mm. The results predicted by the present analytical method are essentially consistent with the experimental results for the entire thickness and impulses considered herein. It can be said, therefore, that the present analysis method can be used to predict the impact force of composite plates regardless of its thickness. Therefore, it can be concluded that the present method can be used to predict the impact force from the measured dynamic strain, when plain-weave glass/epoxy composite plates with various thicknesses are impacted. Additionally, this method requires a relatively short CPU time for computing the impact force.
It has been reported that the dynamic response and impact resistance of a structure are strongly in¯uenced by thickness [7]. This behavior is because the ¯exural and contact stinesses vary according to thickness and cause the change of the impact behavior of a structure. Therefore, it is important to understand the dynamic response of plain-weave glass/epoxy composite plates according to the thickness. Fig. 7 shows the impact force histories at the impact locations 1 and 3 for the entire thickness. It can be seen that the contact time duration decreases with the increasing of thickness, but the magnitude of impact force increases inversely. This behavior is essentially the same for all the impact locations. For this plate, since the dynamic response is governed by the ¯exural stiness, the contact duration between the impact hammer and the composite plate lengthens and the maximum impact force decreases. On the other hand, the dynamic behavior of the thick plates aected primarily by the contact stiness is opposite to that of the thin plates. Let us de®ne the factors representing the changes of ¯exural and contact behavior as the maximum de¯ection ratio, (wmax,i/Wmax,B) and the maximum local indentation ratio, ( max,i/ max,B), respectively. Here the symbols wmax,i and wmax,B denote the maximum de¯ections for each thickness and B=2.3 mm, respectively. The de®nitions of max,i and max,B are the maximum local indentations for each thickness and B=6.6 mm, respectively, which are calculated by the Sun's equation [14]. Fig. 8 shows the results for the impact locations 1 and 3. The maximum de¯ection ratio has the proportional relationship with the thickness whereas the maximum local indentation ratio has the inverse relationship. It may be, therefore, said that the dynamic response of the plain-weave glass/epoxy composite plate
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
141
Fig. 5. The frequency responses of impact force and dynamic strain. (a) Wide band. (b) Narrow band.
Fig. 6. Comparisons between predicted and experimental maximum impact forces. (a) B=4.0 mm. (b) B=6.6 mm.
has been in¯uenced by both of the bending and contact according to the change of thickness. For the understanding of the thickness eect on the dynamic response, the absorbed impact energies for each impact location are obtained from the force histories (Fig. 7). The results for locations 1 and 3 are shown in Fig. 9. Here the absorbed impact energy means the energy absorbed by de¯ection and local indentation of the plate. It can be seen that though little marked dierence is observed between the energies with the thickness, this ®gure indicates the remarkable tendency regardless of the impact locations. The absorbed impact energy has a nonlinear relationship with the thickness. The energy reaches the maximum value at the speci®c thickness. In other words, the absorbed impact energy is dierent
according to the thickness and particularly shows the maximum at the speci®c thickness, though the applied impulses are identical for the entire thickness. This behavior is essentially the same for all the impact locations. The above result and Fig. 8 imply that the energy absorbed by the de¯ection matches with that absorbed by the local indentation. Consequently, the total absorbed energy becomes the maximum at about B=4.0 mm. We have investigated the impact damage mechanism with regard to the thickness change for the same material used in this study [15]. It was found that the impact damage behavior changes with the thickness. Consequently, the threshold and critical impact energy are at their maximum at about B=5.0 mm. In this, the threshold impact energy means the energy below that no
142
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Fig. 7. The impact force histories with plate thickness. (a) Impact location 1. (b) Impact location 3.
Fig. 8. The indentation and de¯ection ratio with plate thickness. (a) Impact location 1. (b) Impact location 3.
strength reduction occurs. In addition, the critical impact energy denotes the energy at which a plate completely fails. Those results are somewhat inconsistent with the results obtained in this study in that the absorbed impact energy is maximum at about B=4.0 mm. This may be because no impact damage occurs in this study. However, for the above quoted author's study, the impact damage such as transverse and longitudinal damage occurs. Consequently, the energy absorbing mechanism changes. 6. Conclusions
Fig. 9. The variation of absorbed impact energy with plate thickness.
In this study, a new analytical method was proposed to predict the impact force history of plain-weave glass/ epoxy composite plates with various thicknesses using
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
the measured dynamic strain. For the veri®cation of the proposed analysis method, the impact hammer test was conducted. Additionally, the thickness eect on the dynamic response of the composite was investigated. The following conclusions have been drawn. 1. The impact force and dynamic strain could be separated into frequency and amplitude. Additionally, their amplitudes at any frequency corresponded to each other regardless of the plate thickness. 2. On the basis of the above assumptions, the new analytical method was proposed to predict the impact force using the dynamic strain. The validity and eciency of the method were proved by the impact hammer test. The method was also applicable to various plate thicknesses. 3. The dynamic response of the plain-weave glass/ epoxy composite plates subjected to the transverse impact was signi®cantly in¯uenced by the thickness. As the thickness increased, the absorbed impact energy by the de¯ection decreased whereas that by the local indentation increased. 4. For the composite plate without any impact damage, the thickness having the maximum absorbed impact energy is somewhat inconsistent with that for the plate with impact damage. This can be explained by the change of the energy absorbing mechanism due to the occurrence of the impact damage. Acknowledgements This study is supported by Korea Science and Engineering Foundation through Research Fund No. 981-1004026-2. The authors gratefully appreciate the support.
143
References [1] Abrate S. Impact on laminated composite materials. Applied Mechanics Review 1991;44(4):155±90. [2] Doyle JF. An experimental method for determining the dynamic contact law. Experimental Mechanics 1984;24(1):10±16. [3] Doyle JF. Determining the contact force during the transverse impact of plates. Experimental Mechanics 1987;27(1):68±72. [4] Doyle JF. Experimentally determining the contact force during the transverse impact of an orthotropic plate. Journal of Sound and Vibration 1987;118:441±8. [5] Yen CS, Wu E. On the inverse problem of rectangular plates subjected to elastic impact, Part 1: method development and numerical veri®cation. Transaction of the ASME 1995;62:692± 7. [6] Kim MS, Kim NS, Lee HC. Impact analysis of laminated composite plates using higher-order shear deformation theory. Transaction of KSME(A). 1991;27:735±750 (in Korean) [7] Cantwell WJ, Morton J. Geometrical eects in the low velocity impact response of CFRP. Composite Structures 1989;12:39± 60. [8] Reissner E. The eect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945; 18:67±9. [9] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. Journal of Applied Mechanics 1970;37:1031± 6. [10] Cairns DS, Lagace PA. Transient response of graphite/epoxy and Kevlar/epoxy laminates subjected to impact. AIAA Journal 1989;27:1590±6. [11] Yen CS, Wu E. On the inverse problem of rectangular plates subjected to elastic impact, Part II: experimental veri®cation and further applications. Transaction of the ASME 1995;62:699±705. [12] ASTM D 3039-93. Standard test method for tensile properties of ®ber resin composite. [13] Vanderplaats GN. ADS user's manual. Engineering Design Optimization, Inc., 1985. [14] Tan TM, Sun CT. Use of statical indentation laws in the impact analysis of laminated composite plates. Journal of Applied Mechanics 1985;52:6±12. [15] Kim JK, Kang KW. Impact damage behavior and evaluation of residual strength in plain woven glass/epoxy composites. Transaction of KSME(A) 1999;23:294±303 (in Korean).
www.elsevier.com/locate/compscitech
An analysis of impact force in plain-weave glass/epoxy composite plates subjected to transverse impact Jung-Kyu Kim a,*, Ki-Weon Kang b a School of Mechanical Engineering, Hanyang University, Seoul 133-791, South Korea Department of Mechanical Design and Production Engineering, Hanyang University, Seoul 133-791, South Korea
b
Received 15 September 1999; received in revised form 29 June 2000; accepted 24 August 2000
Abstract In the present study, a new analytical method is developed for predicting the impact force from the dynamic strain of composite plates subjected to transverse impact. For this, two assumptions are introduced in this study. Firstly, the impact force and dynamic strain can be separated into frequency and amplitude. Secondly, the amplitude of the impact force corresponds to that of the dynamic strain at any frequency. By applying the Rayleigh±Ritz energy method and Lagrange's principle to a rectangular plate, the governing equation is derived, and the equation is solved by an optimal design technique. To verify this analytical method, impact tests are performed on plain-weave glass/epoxy composite plates having various thicknesses. The impact forces obtained by this analytical method agree well with the experimental results for all plate thicknesses studied. Additionally, the dynamic response, such as the impact force, is greatly in¯uenced by the thickness. This behavior is attributed to the dierent energy absorbing mechanisms: the absorbed energy is governed by the plate de¯ection in a thin plate, whereas the energy is governed by the local indention in a thick plate. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Plain-weave glass/epoxy composite plate; Impact force; Dynamic strain; Green's function; Thickness eect
1. Introduction Since composite materials are susceptible to damage resulting from the impact of foreign objects, an analysis of the dynamic response of composite materials subjected to impact is necessary for the assessment of damage resistance and the design of structures [1]. To understand the dynamic response of composite materials, the accurate evaluation of impact loading history applying to composites is important, and the thickness eect is well worth investigating because the impact force and the impact resistance of composite materials are strongly in¯uenced by thickness. To analyze the dynamic response of composite materials, several researchers have proposed techniques that can determine the impact force by taking the responses of an impacted structure itself. One is to solve the problem in a frequency domain by using the fast-Fourier transform method. Doyle [2±4] evaluated the impact forces on isotropic and orthotropic plates with this * Corresponding author. Fax: +82-2-22291-6707. E-mail address: [email protected] (J.-K. Kim).
technique. The measured dynamic strain, however, needed to be properly windowed and subsequently ®ltered to eliminate the boundary eect. The other method relies on the inverse method in a time domain. In this method, when some of the structural responses are known, the impact force can be determined. This method has been successfully applied to an aluminum alloy by Yen et al. [5]. However, it is not easy to identify the adequate time interval for discretizing the convolution integral, and consequently this causes the accuracy and the eciency of a solution to depend on the used time interval signi®cantly. Moreover, it is not clear whether this method is applicable to composite materials and to thick plates for which the dynamic response is governed by the local contact deformation. The thickness of composite materials is a dominant parameter that governs the dynamic response and damage mode of impacted ones. Thus, many pieces of research have been carried out on the dynamic response and damage behavior with the thickness. For example, Kim et al. [6] investigated the eect of thickness on the impact energy by ®nite-element analysis, and Cantwell et al. [7] experimentally demonstrated that the impact
0266-3538/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00203-7
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J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
resistance is the highest at the speci®c thickness for the laminated composite. However, since the dynamic response of composite materials depends not only on the thickness but also on the type of constituent material, these studies are not likely to completely explain the eect of parameters on it. Therefore, it can be said that for various composite materials a reliable mechanism for describing the dynamic response with the thickness is not yet available and more extensive research is necessary to characterize the impact behavior of composite materials. The purpose of this study is to develop a new analytical method to predict the impact force history of plainweave glass/epoxy composite plates and to evaluate the eect of thickness on the dynamic response. For these purposes, two assumptions are introduced. Firstly, the impact force and dynamic strain can be separated into frequency and amplitude. Secondly, the two responses correspond to each other regardless of frequency. Under these assumptions, applying the Rayleigh±Ritz energy method and Lagrange's principle to the rectangular plate with thickness B, the governing equation is derived, and the equation is solved by an optimal design technique. The proposed analytical method is veri®ed by impact hammer tests and the eect of thickness on the dynamic response is examined.
In these, p(x, y) is the lateral loading on the plate, w(x, y) the lateral displacement, and N0x, N0y, and N0xy the in-plane loadings. The kinetic energy of a plate is expressed as the surface integrals of the related mass and velocities ph T 2
a b 0 0
: wdydx
5
: where {} denotes the derivative with respect to time and is the density of the plate. In the Rayleigh±Ritz energy method, the assumed displacement mode shapes are used to transform the spatially continuous system to the discrete system of modal amplitudes [10]. Therefore, the governing equation of the plate, which is expressed in terms of the transformed system of modal amplitudes, can be derived by the minimization of the plate potential and kinetic energy. If the displacements are assumed to be separable into x, y and z, the planar and lateral displacements can be expressed as u x; y; z; t z x x; y; t z
N1 X N2 X Aij tXi x; x Yj y
v x; y; z; t z y x; y; t z
i
2. Dynamic analysis model
w x; y; t
The governing equation of equilibrium is deduced by the minimization of potential and kinetic energy. For the rectangular plate with the thickness B and in-plane size ab, the bending and shearing strain energies are given as the surface integral forms Ub
1 2
a b
ks Us 2
0 0
fgT Dfgdydx
a b 0 0
1
f gT Af gdydx
2
where {} and { } denote the plate curvature and shear strain in the Reissner±Mindlin plate theory [8], respectively and s is the shear correction factor [9]. Moreover, the external work on the plate is the sum of the lateral loading work W1 on the plate and the inplane loading work W2 at the mid-plane as follows: a b p x; y; tw x; y; tdydx W1 ÿ
3
0 0
1 W2 2
a b 0 0
w; w;
x y
"
N0x N0xy
N0xy N0y
#
w; w;
x y
j
i
N1 X N2 X j
Bij tXi xYj y;
N1 X N2 X Cij tXi xYj y
y
6
j
i
where Xi(x) and Yj(y) are the beam functions determined from the appropriate boundary conditions, and Aij(t), Bij(t) and Cij(t) are the time-varying modal amplitudes which must be determined. N1 and N2 are the number of beam functions in the x and y directions, respectively. The governing equation expressed in terms of modal amplitudes can be obtained by applying the following Lagrange's equation. d @T @T @V 0 : ÿ dt @rj @rj @rj
7
where V denotes the potential energy, which is de®ned as Ub+Usÿ(W1+W2) and rj is the modal amplitude. Therefore, the following equation can be developed by substituting Eqs. (1)±(6) into Eq. (7)
0 0 0 M22
r1 r2
K11 K21
K12 K22
r1 r2
0 P
8
dydx:
4
where {r1} denotes the column vector containing Aij(t) and Bij(t), and {r2} is the column vector composed of
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Cij(t). The mass matrix [M22] is a diagonal matrix due to the orthogonality characteristics of the beam functions. In addition, the components of the stiness matrix, [Kij], are obtained by integrating the assumed beam functions and their derivatives over the plate surface. The ®nal governing equation of the composite plate can be obtained by applying the static condensation scheme to Eq. (8) as follows: M22 fr2 g bK22 cfr2 g fPg
9
fr1 g ÿK11 ÿ1 K12 fr2 g
10
where K22 K22 ÿ K12 T K11 ÿ1 K12 :
3. Numerical analysis The eigenmode expansion method is introduced to solve Eq. (9). Let Eq. (9) be of Lank K, and !1, !2,. . .,!k and {e1}, {e2},. . .,{ek} be the eigenvalues and eigenvectors of the homogeneous equation, respectively. The unknown vector {r2}, therefore, can be expressed by linear superposition of the eigernvectors as follows: fr2 g
K X aj fej g
11
j1
where each aj is the function of time and can be obtained by solving each of the uncoupled equations as follows: mj a j kj aj p t fej gT fHg ;
j 1; 2; . . . ; K
12
where mj={ej}T [M22] {ej}, kj=mj!j2 and K is equal to N1N2. Thus, the modal amplitude Cij(t), contained in vector {r2}, can be obtained by solving Eq. (12) and the values of Aij(t) and Bij(t) can be determined by substituting the results into Eq. (10). Consequently, the dynamic response of the plate at a certain point (x1,y1) can be expressed in terms of these modal amplitudes as follows: " x1 ; y1 ; z; t ÿ x1 ; y1 ; z; t ÿz bP T K11 ÿ1 K12 fej gcbfej gT fHgc mj
t 0
K X
gj t
sin !j t=!j ; !j 6 0 !j 0 !j ;
137
14
where represents x or y. {} is the column vector, which can be obtained from the relationship between displacements and beam functions, Eq. (6). Thus, if the dynamic strain is given in Eq. (13), one can ®nd the force function at any position of the plate. Since an analytical solution is still not available for Eq. (13), Yen et al. [5] discretized the convolution integral in the time domain to develop an approximate numerical solution. By separating the concerned time period into n of the same intervals and by using the linear interpolation technique, Eq. (13) is transformed into the following algebraic equation: 8 9 2 3 8p 9 G1 ÿ1 > > 1> > > > > > > = < ÿ2 > 6 G2 G1 7 < p2 = 6 7 ÿz4 .. 15 .. .. .. 5 > .. > > > . . . > > ; > : . > > . > ; : Gn Gnÿ1 G1 l pn ÿn where Green's function, G means the transient strain response at (x1,y1) by a unit impulse force applied at (x0,y0) and l indicates the number of used strain gage. Yen et al. [11] reported Eq. (15) is very eective for determining the impact force of an aluminum plate and can be used to predict the impact force by using an incomplete strain response, which often occurs in the service condition of a structure. However, it is not easy to identify the adequate time interval, n for discretizing the convolution integral. Consequently the accuracy and eciency of the solution depend greatly on the used time interval. To overcome this shortcoming, it is assumed that the impact force can be separated into frequency and amplitude. Thus, the impact force can be approximately expressed as the following Mth-order Fourier series. p t
M X
pi t sin fi t
16
i1
where fi and pi are the frequency and the amplitude of impact force, respectively. By substituting Eq. (16) into Eq. (13), we can obtain a new equation with regard to the impact force and strain response as follows: " x1 ; y1 ; z; t ÿz
M X K X Gj x1 ; y1 ; t; x0 ; y0 pi i1 j1
j1
17
p t ÿ gj d 13
where G is Green's function, which is expressed as follows:
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J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Gj x1 ; y1 ; t; x0 ; y0 T K11 ÿ1 K12 fej g fej gT fHg sin wj tfi ÿ sin fi twj : mj wj f2i ÿ w2j 18 Obviously, Eq. (17) can be used to determine the impact force from the dynamic strain response. However, due to the complicated form of Green's function in Eq. (18), the impact force cannot be determined by an analytical method. Therefore, the optimization technique is adopted to approximately obtain the solution in this study. To establish an object function used in an optimization technique, the following assumptions are made. Firstly, the dynamic strain can be separated into frequency and amplitude such as in Eq. (19). Secondly, the frequency characteristics of impact force and dynamic strain are identical and the amplitudes of the two responses at any frequency correspond to each other. Based on these assumptions, the object function and constraint can be de®ned as Eqs. (20) and (21) " t
M X
"i t sin fi t
19
i1
Objective
( L X M X l1 i1
subjected to p t
K X "i z Gj pi
) 20
j1
M X pi t sin fi t40
250 mm in length and 40 mm in width, as shown in Fig. 1. Various thicknesses (B=2.3, 3.0, 4.0, 5.0 and 6.6 mm) were prepared to identify the thickness eect on the dynamic response. The specimens were clamped on two opposite edges and were left free on the other two edges. 4.2. Impact hammer test An impact hammer is generally used for evaluating the dynamic response of a structure. For this purpose, one may hold an impact hammer and strike a target by hand. However, it is almost impossible to maintain the right angle between the specimen and impact hammer. To solve this problem, the impact hammer was installed in the special loading apparatus as shown in Fig. 2. The impact hammer used (PCB Electronics, 086B03, 2.3 mV/ N) was ®xed to the rod attached to the origin of rotation. The impulse could be adjusted by changing the length of the spring as the four steps, which was attached to the impact loading apparatus The range of impact velocity was 1.40±2.40 m/s. The impact velocity was measured from the time of passage between the laser sensors connected to an electronic timer. The tip diameter and mass of impactor were 12.7 mm and 3.45 kg, respectively. In addition, on the opposite surface against the impact point, two strain gages were mounted to measure the dynamic strain responses. Two strain gages, as shown in Fig. 1, were attached at 50 mm and
21
i1
where the constraint means that the impact force must be compressive. 4. Experimental procedure 4.1. Materials and specimen
Fig. 1. Con®guration of specimen (unit:mm).
The materials used in this study were plain-weave Eglass/epoxy composite plates (volume fraction, 10.62). The mechanical properties of this material were obtained from unidirectional tensile test, according to ASTM D-3039-93 [12], and summarized in Table 1. The Poisson ratio, xy, and shear modulus, Gxy, were obtained by using the 0 /90 strain-rosette and the 45 strain-gage method, respectively. The specimen was Table 1 Mechanical properties of plain-weave glass/epoxy composite Exx (GPa)
Eyy (GPa)
Gxy (GPa)
19.96
19.96
3.1
0.136
Fig. 2. Schematic diagram of impact loading apparatus.
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
80 mm away from the left edge of the specimen, respectively. The impact force and dynamic strain were recorded on a personal computer. The sampling rate used was always 100 kHz. 5. Results and discussion 5.1. Impact force evaluation In the Rayleigh±Ritz method, a governing equation is obtained by transforming a spatially continuous system to a discrete system of modal amplitudes using the assumed displacement mode. Thus, the number of modes required for the Rayleigh±Ritz method strongly in¯uences the accuracy of solution. To know how many modes must be included, the response signals must be analyzed by using the fast-Fourier transform method to obtain a frequency response. The recorded impact force and dynamic strain for the gage at (50 mm, 20 mm) in a B=2.3 mm plate are analyzed using the FFT analysis and the results are shown in Fig. 3(a). Most of the amplitudes are distributed in the frequency range lower than 5 kHz. Thus, the 1010 vibration modes of the plate used in this study are appropriate. To verify the assumption of correspondence used in the object function Eq. (20), the frequency characteristics of the impact force must be examined together with that of the dynamic strain over the concerned frequency range. To this end, the two responses of Fig. 3(a) are examined over the low frequency band and the results are shown in Fig. 3(b). It can be seen that the impact force corresponds to the dynamic strain regardless of the frequency. Based on the above results, the impact forces are obtained by minimizing the object function, Eq. (20). The computer code and search algorithm are the ADS
139
(Automatic Design Synthesis), published in public domain, and the modi®ed method of feasible directions, respectively [13]. The analysis results for B=2.3 mm are shown in Fig. 4 and Table 2 together with the experimental results. The results obtained by the proposed method are well matched with the experimental results regardless of applied impulses. Additionally, it takes a very short CPU time of about 90 s to compute the impact forces. On the other hand, though Yen's method, Eq. (15), has several advantages as previously mentioned, his method has the shortcoming that the accuracy and the eciency of the solution depend signi®cantly on the used time interval because of discretizing the force function in time domain. The eciency of Yen's method can be improved by using a longer time interval but simultaneously it causes a very poor accuracy of the solution. However, the proposed method is found to have good accuracy and the independence on the time interval due to the adoption of the two assumptions already mentioned. There are various analytical techniques that have been used to evaluate the impact response in composite structures. These techniques including the proposed analytical method are based on the beam or plate theory, which describes the overall deformation of a structure. These techniques give an exact solution to the certain categories of impact problems, but is restricted to a narrow range of conditions. It has been particularly noted that the techniques cannot be applied to a thick structure. For thick structures, local contact deformation plays a major role in the dynamic response but the governing equation, based on the beam or plates theory, cannot describe it near the impact point. To overcome this problem, the nonlinear contact law considering the change of contact area with time, is generally incorporated in the governing equation. It is, unfortunately, impossible to incorporate the contact law in the present
Fig. 3. The frequency responses of impact force and dynamic strain. (a) Frequency response for wide-band. (b) Frequency response for narrowband.
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J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Fig. 4. Comparisons between predicted and experimental impact force histories. (a) Impact location 1 and 3. (b) Impact location 2 and 4.
5.2. Thickness eect
Table 2 The relative errors and CPU time (B=2.3 mm) Spring location
1
2
3
4
[(FEXPÿFPRE)/FEXP] CPU times
2.82 88.7
4.85 84.9
4.33 92.1
5.98 93.2
analytical method and hence the applicability of the method on a thick structure must be proved. To this end, the applicability of the number of modes used for B=2.3 mm on thick plates, is discussed ®rst. Fig. 5 shows the frequency responses for B=4.0 mm and B=6.6 mm. Fig. 5(a) indicates that though the frequency characteristics are somewhat dierent according to the thickness, most of the amplitudes are distributed in the frequency range lower than 5 kHz for all thickness. An identical trend could be found for B=2.3 mm. The assumption of correspondence between the impact force and the dynamic strain is also applicable despite of the inconsistence between the two responses in a very low frequency range for B=6.6 mm in Fig. 5(b). According to these results, the analyses and experiments were conducted for thick plates. Fig. 6 shows the typical results for B=4.0 mm and B=6.6 mm. The results predicted by the present analytical method are essentially consistent with the experimental results for the entire thickness and impulses considered herein. It can be said, therefore, that the present analysis method can be used to predict the impact force of composite plates regardless of its thickness. Therefore, it can be concluded that the present method can be used to predict the impact force from the measured dynamic strain, when plain-weave glass/epoxy composite plates with various thicknesses are impacted. Additionally, this method requires a relatively short CPU time for computing the impact force.
It has been reported that the dynamic response and impact resistance of a structure are strongly in¯uenced by thickness [7]. This behavior is because the ¯exural and contact stinesses vary according to thickness and cause the change of the impact behavior of a structure. Therefore, it is important to understand the dynamic response of plain-weave glass/epoxy composite plates according to the thickness. Fig. 7 shows the impact force histories at the impact locations 1 and 3 for the entire thickness. It can be seen that the contact time duration decreases with the increasing of thickness, but the magnitude of impact force increases inversely. This behavior is essentially the same for all the impact locations. For this plate, since the dynamic response is governed by the ¯exural stiness, the contact duration between the impact hammer and the composite plate lengthens and the maximum impact force decreases. On the other hand, the dynamic behavior of the thick plates aected primarily by the contact stiness is opposite to that of the thin plates. Let us de®ne the factors representing the changes of ¯exural and contact behavior as the maximum de¯ection ratio, (wmax,i/Wmax,B) and the maximum local indentation ratio, ( max,i/ max,B), respectively. Here the symbols wmax,i and wmax,B denote the maximum de¯ections for each thickness and B=2.3 mm, respectively. The de®nitions of max,i and max,B are the maximum local indentations for each thickness and B=6.6 mm, respectively, which are calculated by the Sun's equation [14]. Fig. 8 shows the results for the impact locations 1 and 3. The maximum de¯ection ratio has the proportional relationship with the thickness whereas the maximum local indentation ratio has the inverse relationship. It may be, therefore, said that the dynamic response of the plain-weave glass/epoxy composite plate
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
141
Fig. 5. The frequency responses of impact force and dynamic strain. (a) Wide band. (b) Narrow band.
Fig. 6. Comparisons between predicted and experimental maximum impact forces. (a) B=4.0 mm. (b) B=6.6 mm.
has been in¯uenced by both of the bending and contact according to the change of thickness. For the understanding of the thickness eect on the dynamic response, the absorbed impact energies for each impact location are obtained from the force histories (Fig. 7). The results for locations 1 and 3 are shown in Fig. 9. Here the absorbed impact energy means the energy absorbed by de¯ection and local indentation of the plate. It can be seen that though little marked dierence is observed between the energies with the thickness, this ®gure indicates the remarkable tendency regardless of the impact locations. The absorbed impact energy has a nonlinear relationship with the thickness. The energy reaches the maximum value at the speci®c thickness. In other words, the absorbed impact energy is dierent
according to the thickness and particularly shows the maximum at the speci®c thickness, though the applied impulses are identical for the entire thickness. This behavior is essentially the same for all the impact locations. The above result and Fig. 8 imply that the energy absorbed by the de¯ection matches with that absorbed by the local indentation. Consequently, the total absorbed energy becomes the maximum at about B=4.0 mm. We have investigated the impact damage mechanism with regard to the thickness change for the same material used in this study [15]. It was found that the impact damage behavior changes with the thickness. Consequently, the threshold and critical impact energy are at their maximum at about B=5.0 mm. In this, the threshold impact energy means the energy below that no
142
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
Fig. 7. The impact force histories with plate thickness. (a) Impact location 1. (b) Impact location 3.
Fig. 8. The indentation and de¯ection ratio with plate thickness. (a) Impact location 1. (b) Impact location 3.
strength reduction occurs. In addition, the critical impact energy denotes the energy at which a plate completely fails. Those results are somewhat inconsistent with the results obtained in this study in that the absorbed impact energy is maximum at about B=4.0 mm. This may be because no impact damage occurs in this study. However, for the above quoted author's study, the impact damage such as transverse and longitudinal damage occurs. Consequently, the energy absorbing mechanism changes. 6. Conclusions
Fig. 9. The variation of absorbed impact energy with plate thickness.
In this study, a new analytical method was proposed to predict the impact force history of plain-weave glass/ epoxy composite plates with various thicknesses using
J.-K. Kim, K.-W. Kang / Composites Science and Technology 61 (2001) 135±143
the measured dynamic strain. For the veri®cation of the proposed analysis method, the impact hammer test was conducted. Additionally, the thickness eect on the dynamic response of the composite was investigated. The following conclusions have been drawn. 1. The impact force and dynamic strain could be separated into frequency and amplitude. Additionally, their amplitudes at any frequency corresponded to each other regardless of the plate thickness. 2. On the basis of the above assumptions, the new analytical method was proposed to predict the impact force using the dynamic strain. The validity and eciency of the method were proved by the impact hammer test. The method was also applicable to various plate thicknesses. 3. The dynamic response of the plain-weave glass/ epoxy composite plates subjected to the transverse impact was signi®cantly in¯uenced by the thickness. As the thickness increased, the absorbed impact energy by the de¯ection decreased whereas that by the local indentation increased. 4. For the composite plate without any impact damage, the thickness having the maximum absorbed impact energy is somewhat inconsistent with that for the plate with impact damage. This can be explained by the change of the energy absorbing mechanism due to the occurrence of the impact damage. Acknowledgements This study is supported by Korea Science and Engineering Foundation through Research Fund No. 981-1004026-2. The authors gratefully appreciate the support.
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