Impact response and damage in laminated composite cylindrical shells

Composite Structures 59 (2003) 15–36 www.elsevier.com/locate/compstruct

Impact response and damage in laminated composite cylindrical shells K.S. Krishnamurthy, P. Mahajan *, R.K. Mittal Department of Applied Mechanics, Indian Institute of Technology, New Delhi 110016, India

Abstract Impact response of a laminated composite cylindrical shell is determined both by the classical Fourier series and the finite element methods. Impact response determined by the finite element method also includes a prediction of the impact-induced damage deploying the semi-empirical damage prediction model of Choi–Chang. In the analytical method, the solution is obtained by means of an alternative numerical procedure incorporating the non-linear HertzÕs contact law which enables consideration of local indentation produced by the indentor on the impacted surface, instead of the Laplace Transforms technique used in an earlier work. The above procedure is also extended to the problem of impact on a cylindrical shell panel. The analytical procedure also provided information such as the natural frequencies of vibration of the impacted shell which helped in selecting appropriate mesh and time step sizes for the finite element method. A parametric study was carried out by the finite element method to determine the effect of varying the controlling parameters such as impactor mass, its approach velocity, curvature of the shell, on both the impact response and on the impact-induced damage. A reduction of the stiffnesses of the failed laminas on the impact response concurrently as the solution proceeded has also been incorporated. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fibre-reinforced composites; Impact behaviour; Finite element analysis; Laminates; Failure criterion

1. Introduction Understanding impact response and impact-induced damage in beams, plates and shells made of laminated composites under low velocity impact loading due to projectile hits or accidental dropping of heavy objects on their surface has been a topic of intense research. Such impacts (e.g. that of a blunt-nosed object) induce a region of high stress concentration in the vicinity of contact area leading generally to a localized damage which adversely affects the integrity of the structural member. Research in this area especially concerning laminated composite plates has been quite significant (exhaustive references are given by [1–3]); that concerning laminated composite shells, on the other hand, is however relatively small [4–12]. Closed form and semi-closed form solutions for impact on shells were given by [6,7] in recent times, and earlier by [4,5]. While analytical methods are based on firm mathematical and physical principles

*

Corresponding author. Tel.: +91-11-6853604x417; fax: +91-116858703. E-mail address: [email protected] (P. Mahajan).

and provide a better insight in understanding the factors controlling the impact response, etc., they are unsuitable for predicting impact-induced damage. Experimental studies, although indispensable for gaining first hand knowledge of the phenomenon and for the verification of the theoretical predictions, are too expensive when the experimental parameters have to be varied over a wide range. Finite element method, on the other hand, is best suited to fulfill these latter needs. While a number of studies concerning impact analysis using the finite element method exist, some improvements, as discussed in the following, are possible. In Ref. [8], e.g., the authors have carried out impact experiments on laminated composite cylindrical shell panels with various ply sequencing, using drop weight (¼3.09 kg) impact tests to study the impact response and the impact-induced damage. A comparison of the experimental contact force–time variation of the shell with that predicted by the finite element analysis was provided. In [9] also, the authors have carried out low velocity drop weight impact tests on flat plates and composite cylindrical shell panels and compared the response with solutions obtained using an analytical series method. The aim of the authors was to improve the analytical method that

0263-8223/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 2 3 8 - 6

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

incorporated large deflections with special axial mode shape functions. A finite element study to determine the impact response of a laminated composite shell panel using an impactor of small mass (modeling a projectile) for laminated composite shells considering variations of impactor mass, velocity, curvature of shell, etc., was made in [10]. Prediction of impact-induced damage prediction was however not considered in all the abovementioned works. While the determination of impact response and prediction of damage in a laminated composite shell was made by [11] using the finite element method, only the effect of shell curvature on the contact force and damage was shown in this work. The effect of failed laminas on the stiffness of the laminate was also not accounted for in determining the impact response. In [12], the authors carried out a numerical simulation of the quasi-static loading experiment on laminated composite shells by the finite element method, assuming that low velocity impact force and deformation can be simulated by static tests. The authors predicted matrix cracking failure in cylindrical and spherical composite shells by applying the general Tsai-Wu failure criterion [13] for composite materials. However, delamination failure was not considered. The authors have used a form of HertzÕs contact law given by [14] which does not account for the permanent indentation due to impact. Also, the study being essentially a static analysis, does not compute impact response as a function of time. From all this, it was observed that finite element studies which simultaneously addressed the influence of the impact problem parameters on both the impact response and impact-induced damage of laminated composite shells subject to impact by a small mass impactor (simulating the impact of a flying foreign object) were very few in number. A few other observations are can also be made as discussed in the following. 1. Prediction of impact-induced damage by the failure criteria (for matrix cracking and delaminations) given by Choi and Chang [15,16] based on impact experiments on laminated composite plates, and made suitable for predicting impact-induced damage, were more appropriate than a general failure criterion for composite materials such as the Tsai-Wu criterion as done by [14]. (In [16], a comparison of the damages predicted by the finite element analysis incorporating this failure criteria with the experimental observations are shown.) 2. Since the strength of a laminate is adversely affected due to impact-induced damage, it is desirable to incorporate the effect of such damage in impact response analysis by a proper modification of the terms of the stiffness matrices of the failed laminas, accounting for the loss of contribution of such laminas in resisting the internal stresses. Even though Choi and Chang [15] carried out such an analysis involving reduction of stiffness matrices for impact on a laminated composite plate, it is felt that the effect of such reductions (made

concurrently in each time step of the numerical solution procedure in which damage is predicted) on the contact force and shell deformation histories should also be brought out by a comparison of results with those obtained by using unchanged laminate stiffnesses, for the present case of impact on laminated composite shells. 3. A computation of the energy transferred to the shell and the part of the impact energy that causes indentation would provide a useful insight into the nature of apportioning of total energy during impact. Further, the influence of the parameters of the impact problem on these fractions of the impact energy can be determined. 4. The Fourier series method, which provided information regarding the natural frequencies of vibration of the impacted structure, helped in selecting appropriate mesh and time step sizes, etc. for the finite element method since the accuracy of finite element solutions are known to be significantly affected by these factors. In an earlier study by the present authors [17], some of the above discussed observations for an impactdamage analysis were incorporated by considering the impact on a laminated composite cylindrical shell panel. The present paper attempts to complete this task by including all the above-mentioned observations, for both a cylindrical shell panel as well as a full cylinder. The method of Fourier series for determining the impact response of a laminated composite cylindrical shell using an alternative numerical solution procedure is briefly discussed first.

2. Determination of impact response by the method of Fourier series An analytical solution was given for the problem of simply supported orthotropic cylindrical shell subject to impact loading by [5]. The method is based on an expansion of the load, the displacements and rotations in a double Fourier series having its spatial and temporal variables separated and satisfying the end boundary conditions of simple support. The objective of the analysis is to obtain a simultaneous solution for the impact force and the dynamic response of the impacted cylinder. A closed form solution was given by the authors for a uniformly distributed impulse loading on a rectangular area at the centre of the cylinder (Fig. 1). By assuming the displacements of the impactor and the impacted shell at any time to be equal (thus ignoring the local indentation), the authors solved for the contact force using Laplace transforms. However, an alternative numerical solution to Laplace transforms had been earlier used for determining the impact response of a composite plate by [18], which was originally used by Timoshenko [19] for the impact on a beam. The same procedure was recently used by the present authors for impact on a composite cylinder [20]. The procedure does

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36



F ðtÞ k

2=3

Z Z t 1 t dt F dt m 0 0 X X Pmn pRL mpx sin  R/ðl2  l1 Þm1 m n xmn L Z t  cos nh F ðsÞ sin xmn ðt  sÞ ds

17

¼ v0 t 

ð3Þ

0

Fig. 1. A composite cylinder being impacted by a spherical object. An enlarged view of the rectangular contact area and the definition of positive directions of coordinate lines and displacements are shown. The ends are assumed to be simply supported.

not ignore the local indentation, which becomes important as the thickness of the shell increases and is also a measure of the energy consumed in causing damage. Details of the Fourier series procedure in determining the deflection of the shell as a function of time at the point of impact are given in [5]. The essential steps are as follows: During impact, the relative approach of the two colliding bodies would be Z Z t 1 t aðtÞ ¼ wi ðtÞ  ws ðtÞ ¼ v0 t  dt F dt  ws ðtÞ ð1Þ m 0 0 where wi ðtÞ and ws ðtÞ are the displacements of the center of the impactor and the point on the mid-surface of the shell, respectively. ÔmÕ is the mass of the striker and v0 its initial velocity, and F ðtÞ is the contact force. Considering a rectangular contact patch of area R/ðl2  l1 Þ shown in Fig. 1, over which the impact force is assumed act uniformly, the expression for the transverse deflection of the mid-surface of the cylinder for the transient loading F ðtÞ, is given to be [5]:

in which the contact force F ðtÞ is the unknown quantity. In the above equation, k ¼ HertzÕs contact stiffness, m1 is the mass of the shell, Pmn is the Fourier term for the load, and xmn is the circular frequency. By approximating the contact force versus time variation by means of a series of steps loads, the above equation is solved numerically as discussed in [18,26]. The final expression will be:  2=3 n Fn ðn DsÞ2 X ¼ v0 n Ds  Dnjþ1 Fj k m j¼1 X X Pmn pRL mpx  sin R/ðl2  l1 Þm1 m n x2mn L n X  cos nh Fj fcos xmn j Ds j¼1

 cos xmn ðj  1Þ Dsg

ð4Þ

in which Fn is the contact force at nth time step and the term n X Dnjþ1 Fj ¼ 2½ðn  1ÞF1 þ ðn  2ÞðF2  F1 Þ þ ðn  3Þ j¼1

 ðF3  F2 þ F1 Þ þ þ ðn  jÞðFj  Fj1 þ Fj2  Fj3 þ  F1 Þ þ þ ðFn1  Fn2 þ Fn3  F1 Þ þ ð1=3Þ  ðFn  Fn1 þ Fn2  F1 Þ being substituted for the double integration term on the right hand side of Eq. (3). In the same manner, solution for impact on a cylindrical shell panel (shown in Fig. 2) is also obtained using appropriate Fourier series expressions for displacements, rotations and the impact force, which satisfy the boundary conditions of the shell panel.

ws ðtÞ ¼ wðL=2; 0; tÞ ¼

X X Pmn pRL mpx sin R/ðl2  l1 Þm1 m n xmn L Z t  cos nh F ðsÞ sin xmn ðt  sÞ ds

ð2Þ

0

Substituting the above expression for ws ðtÞ, the displacement of the impactor wi ðtÞ using the initial conditions, and ÔaÕ from Hertz contact law (F ¼ ka3=2 ) into Eq. (1) leads to the final integral equation:

Fig. 2. Composite cylindrical shell panel and the definition of coordinate lines.

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

3. Impact response and prediction of impact-induced damage by the finite element method Solution of the impact problem by the finite element method including a prediction of impact-induced damage can be summarized in the following steps: (a) Determining the contact force between impacting mass and the shell as a function of time. (b) Applying the contact force to find the transient dynamic response of the impacted structure as a function of position and time (displacements, strains, stresses, etc.). (c) Predicting damage in the laminate using appropriate failure criteria. 3.1. Computation of contact force Contact force is determined by the simultaneous solution of the equation of motion for a rigid impactor and the equations of dynamic equilibrium of the impacted shell while also considering the local indentation produced by the impact force. The equations of dynamic equilibrium for the shell are obtained by the application of the principle of virtual work and the DÕAlembertÕs principle. The equations for the shell, after the discretization and elimination of the arbitrary virtual displacements are obtained as: Z M€a þ Ca_ þ BT r dv ¼ f ð5Þ v

in which the mass matrix M, and damping matrix C and the internal restoring force vector PðaÞ and external applied vector f are obtained after the summation of the element contributions. aT ¼ fu1 ; v1 ; w1 ; a1 ; b1 ; . . . ; uN ; vN ; wN ; aN ; bN g is the vector of nodal variables and N is the total number of nodes. The present formulation uses the well known isoparametric shell element [21] with five degrees-of-freedom at each node and satisfying the assumptions of the Mindlin–Reissner theory that the normal to the middle surface remains straight after deformation though no longer remaining normal to the middle surface. Lin and Lee [22] also used this shell element for a shell structure under impact loading. The element has been in common usage for the analysis of When R shells. R T the displacements are large, PðaÞ ¼ T B r dv ¼ ð B DB dvÞa ¼ KðaÞa, i.e., the stiffness v v matrix is dependent on the structural displacements. The bar suffix is used to denote the non-linear strain–displacement relationship. The computer program by Hwang [23] is presently updated to work for impact loading and prediction of impact-induced damage. The program is also upgraded to include geometric nonlinearity on the basis of [24,25]. Solution of the equations is obtained by means of NewmarkÕs time stepping algorithm combined with a predictor–corrector scheme.

With the isoparametric formulation of [21], transformation of the elastic properties of laminas can be made from lamina axes to each elementÕs local axes. At each time step of the direct integration procedure, the computed local stresses in each lamina, oriented with respect to the local axes, are transformed back to its principal directions for the application of failure criteria. The program assumes a constant shear strain across the thickness and uses shear correction factors which take into account the changing shear moduli (G13 and G23 ) from lamina to lamina [24,25]. The indentation made by the impactor is known to form an elliptical contact area having a parabolic contact pressure distribution on it [27,28]. However, since this contact area is so small in comparison with the dimensions of the plate or shell, it is convenient to assume a resultant point load acting at a node in the finite element method. Neglecting damping, the equations of dynamic equilibrium for the shell can be rewritten as ½ M f€ag þ ½ KðaÞ fag ¼ fF g

where fF g

¼ f0; 0; . . . ; Fc ; . . . ; 0; 0g

T

ð6Þ

in which Fc is the contact force. 3.1.1. Equation of motion for the impactor Closely following [10] the equation of motion for a rigid impactor by the application of NewtonÕs second law is given by: € i ¼ Fc mi w

ð7Þ

€ i are the impactor mass and acceleration where mi and w respectively (subscript ÔiÕ stands for impactor). The negative sign indicates that the force is acting in the direction opposite to the displacement of the impactor. The initial displacement and velocity of the impactor are wi ¼ 0 and w_ i ¼ v0 . In order to solve Eqs. (6) and (7) above, the contact force between the laminated shell and the impactor must be known. This is provided by the third equation in the form of the HertzÕs contact law. The HertzÕs law is assumed to be applicable to the loading phase, and the expression given below by Tan and Sun [29] for the unloading phase. These are repeated here for convenience: f ¼ ka3=2 for loading  2:5 a  a0 and f ¼ fm for unloading am  a0

ð8Þ

In the above, aðtÞ ¼ wi ðtÞ  ws ðtÞ is the depth of indentation. wi and ws are the displacements of the impactor and displacement of the point on the mid-surface of the shell below the point of contact at time t. am is the indentation when the contact force reaches its maximum value fm , and, a0 ¼ 0

when am < acr

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

" a0 ¼ am 1 



acr am

2=5 # for am P acr

acr known as the critical indentation. This has been taken to be 8:03  102 mm for Graphite/epoxy laminates by, e.g., in [10,30,31] and the same value is used here also. The expression for k, when bodies having surfaces of revolution come in contact with one another, has been taken in accordance with the one given in [10] and is given below for both the cylindrical and spherical shells as: !  1=2 , 2 4 1 1 ð1  mi Þ 1 kcyl ¼ þ þ ; 3 ri 2Rcyl E2 Ei  1=2 ,  4 1 1 1  m2i 1 þ þ ksph ¼ 3 ri rsph E2 Ei In the above, ri , mi , and Ei are the radius, the PoissonÕs ratio, and the YoungÕs modulus of the impactor. E2 is the modulus of elasticity transverse to the fibre direction of the lamina coming in contact with the impacting mass. The displacement of impactor at the time step (n þ 1) is determined by applying NewmarkÕs integration scheme for the differential equation (Eq. (7)) as:  2 Dt ðwi Þnþ1 ¼ ðwi Þn þ Dtðw_ i Þn þ ð€ w i Þn 4  2 Dt  ð9Þ ðFc Þnþ1 4mi Substituting the above expression for wi in the contact laws defined in Eq. (8), and denoting the displacement of the mid-surface of the shell at the point of contact as Ôws Õ, one obtains, for the time step n þ 1 the following implicit expressions for the contact force: For the loading phase: ðFc Þnþ1 ¼ k½q  ðws Þnþ1  b1 ðFc Þnþ1 1:5

ð10Þ

For the unloading phase: ðFc Þnþ1 ¼ k1 ½q  ðws Þnþ1  a0  bðFc Þnþ1

2:5

ð11Þ

where q ¼ ðwi Þn þ Dtðw_ i Þn þ ðDt2 =4Þð€ wi Þn ; b1 ¼ Dt2 =4mi k1 ¼ Fm =ðam  a0 Þ

2:5

Using the initial conditions, w_ i ¼ v0 , wi ¼ ws ¼ 0, an approximate value of Fc is found from the loading law (Eq. (10)) by means of a root finding algorithm, e.g., the Newton–Raphson method. This force is now applied at the selected node of shell and the nodal displacement ws is next found from a solution of shell equations (Eq. (6)). With this value of ws , Fc is recomputed using Eq. (10), as was done previously. The process is repeated till the required accuracy is achieved. The contact force is now used to calculate acceleration, velocity and displacement of the impactor for the next time step. A switch-over to the unloading phase (Eq. (11)) in the computation is

19

made after the contact force reaches its peak value. During impact on a flexible structure it has been observed that multiple contacts can occur sometimes between the impactor and the impacted structure. This is due to the fact that the two colliding bodies may be moving at velocities differing both in their magnitudes and directions, i.e., either both moving in the same direction or in opposite directions, at a given point of time. In the present computer code, a second contact is modelled as suggested in [30]. Validation of the current finite element code has been made with the same example as used by other authors [10,30] in which a simply supported composite plate [0=90=0=90=0]s having the dimensions 200 mm  200 mm  2:69 mm is impacted by a steel sphere of 12.7 mm diameter travelling at 3 m/s. Comparison of contact force and deflection histories was shown previously in [17]. 3.2. Prediction of impact-induced damage According to the damage prediction model for laminated composite plates under impact loading given by Choi and Chang [15], failure in the form of a matrix crack, due to transverse shear or bending stresses, is initiated in a ply group. Once a matrix crack is predicted Table 1 Material and structural property data for the composite laminates Cylindrical shell panel

Composite cylinder

Laminate characteristics: Geometry: Thickness ¼ 2:54 mm Chord length, a ¼ 254 mm Length ¼ arc length (variable) Radius ¼ 1:27, 2.54, 25.4 m (variable) [0=90=90=0] graphite/epoxy

Radius ¼ 0:10 m Length ¼ 0:42 m Thickness ¼ 2:5 mm [0=90=0=90]s graphite/epoxy

Material constants: E1 ¼ 144:8 GPa E2 ¼ 9:65 GPa G12 ¼ G13 ¼ 7:10 GPa, G23 ¼ 5:92 GPa m12 ¼ 0:30 q ¼ 1389:2 kg/m3

E1 ¼ 138:0 GPa E2 ¼ 9:00 GPa G12 ¼ G13 ¼ 7:10 GPa, G23 ¼ 3:24 GPa m12 ¼ 0:30 q ¼ 1540:0 kg/m3

Lamina strengthsa Yt ¼ 55:20 MPa Yc ¼ 294:0 MPa ST ¼ 32:94 MPa SL ¼ 101:1 MPa

Yt ¼ 55:20 MPa Yc ¼ 294:0 MPa ST ¼ 32:94 MPa SL ¼ 101:1 MPa

Indentation constant, acr ¼ 8:03  102 mm

Indentation constant, acr ¼ 8:03  102 mm

Impactor characteristics: Diameter ¼ 12:7 mm Material density ¼ 7870 kg=m3 E ¼ 200 GPa a

Diameter ¼ 12:0 mm Material density ¼ 7800:0 kg=m3 E ¼ 200 GPa

Strengths are as given in Ref. [11].

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

in a lamina at a particular point, it is assumed to traverse throughout the thickness of the ply group to which the lamina belongs. Delamination failure is next checked at the upper and lower interfaces of that cracked ply group. The two failure criteria given below: Matrix cracking criterion: n

ryy nY

2

n þ

ryz nS i

2

¼ e2M n

eM P 1 failure;

eM < 1 no failure

n

Y ¼ n Yt Y ¼ n Yc

if ryy P 0; if ryy < 0

Delamination criterion: " n Da

ryz nS i

2

 nþ1 þ

rxz nþ1 S i

 nþ1

2 þ

ryy nþ1 Y

2 #

¼ e2D

nþ1

e2D P 1 failure;

e2D < 1 no failure

nþ1

assumed as given in Table 1. In order to account for the loss of load carrying capacity of the failed laminas due to matrix cracking, the reduced the stiffness matrix is made to have the diagonal terms Ex , Gxz , Gxy as the only non-zero terms of matrix. The 6  6 stiffness matrix given in Ref. [16] is, in the present case, a 5  5 matrix as appropriate for the five degree-of-freedom shell element used. Complete solution process in pseudocode format is shown in Fig. 3. In the above failure criteria, x, y, and z denote the principal material directions of each lamina. Bars over the stress terms indicate average stresses in each lamina. 3.3. Energy transferred to the shell during impact

nþ1

Y ¼ Yt Y ¼ nþ1 Yc

if ryy P 0; if ryy < 0

ð12Þ are applied at all points where stresses have been computed in every time step. Lamina strengths are again

If Es is the energy absorbed by the structure during impact it can be calculated as Es ¼ Eð1  e2 Þ, where E is the initial kinetic energy of the impactor, and ÔeÕ, the coefficient of restitution is found from the expression R F ðtÞ dt ¼ mvð1 þ eÞ in which the impulse is equated to

Fig. 3. Algorithm for computation of contact force and prediction of impact-induced damage.

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

21

the change in momentum of the impactor [32]. The energy imparted by the striking object to the shell from the beginning of impact up to time ÔtÕ is the work done by the contact the displacement of the shell, R t force through Rt i.e., Es ¼ 0 F dw ¼ 0 F ðtÞw_ ðtÞ dt where ÔwðtÞÕ is the lateral displacement of the shell. A part of the impact energy is used up in causing indentation. Denoting this part Rof the energy as Ed , it may be computed as a Ed ¼ 0 max F da where amax is the maximum indentation. A part of this energy is dissipated in causing damage or permanent indentation, and the remaining part is recovered elastically during unloading [33]. The ratio Ed =Es can be plotted as a function of time over the contact period for different values of the parameter of the impact problem, viz., curvature of the shell, change of ply orientation, edge conditions etc. to examine the influence that these parameters have on the ratio Ed =Es .

4. The method of Fourier series expansions for the problem of impact on a composite cylindrical shell and a shell panel––example problems (a) Cylindrical shell: An eight layered [0=90=0=90]s laminated composite cylindrical shell with simply supported ends is impacted at its center by a sphere of steel of radius equal to 6 mm with an initial approach velocity of 3 m/s. The properties of this cylinder are as given in Table 1(b). For the analytical method based on Fourier series, the optimum number of terms for summation is determined by the required accuracy and computational efficiency. Accuracy obviously increases with the number of terms, though not very significantly beyond some finite number. Practical considerations like computation time limit it to an even smaller number. Fig. 4 shows the convergence of the solutions with increasing number of terms of the Fourier series and it was observed that the differences in variation of contact force with time for m; n > 24 are small and for m; n ¼ 252 and m; n ¼ 501, these are very close. It was further noted that computation times for m; n ¼ 12, 24, 51, 102, 252, and 501 were 3, 9, 65 s, 4.75, 45 min, 312 h respectively on a personal computer. Presently, the upper limit of m,n was kept equal to 252. The size of the time step (Dt) is generally chosen to be a fraction of the period corresponding to the highest frequency of the vibrating structure to ensure stability of the numerical solution process [34]. Dt is commonly taken to be equal to ð1=10ÞTk , where Tk ¼ 2p=xk is the period corresponding to the circular frequency xk , and k is the mode number of the highest frequency. Considering the fact that for impact analysis a time step size close to 1 ls is most frequently used, it was noted this corresponded approximately to 1/10 of 9:32376  106 s which was the period of the 50th vibration mode. A time step size of 9:5  107 s has been chosen currently. Impact response is determined for two

Fig. 4. Effect of varying the number of terms of the Fourier series included for summation in the solution for contact force. A laminated composite cylinder (lay-up: [0=90=0=90]s , R ¼ 0:1 m, h ¼ 2:5 mm, L ¼ 0:42 m) is impacted at 3 m/s by a steel sphere of 12 mm diameter.

cases, viz., (i) impact force acting as a point load, and (ii) when it is distributed over a contact patch. For the latter case, square contact areas with (i) side length (denoted as ÔaÕ) ¼ 0.0005 m and (ii) a ¼ 0:005 m were chosen. Contact force and deflection histories are shown in Fig. 5(a) and (b). The results of the finite element analysis for these two cases are also shown in these figures. (b) Cylindrical shell panel: An open cylindrical laminated composite shell panel of [0=90=90=0] (as shown in Fig. 1(b)) has the properties as given in Table 1(a). The panel is impacted by a steel sphere of 6 mm at a velocity of 10 m/s. Results from finite element analysis and from the present series solution are shown in Fig. 5(c).

5. Parametric study by the finite element method A number of case studies using two types of shells, namely, (A) an open cylindrical panel, and (B) a full circular cylinder were considered. The properties of the laminate in (A) and the impactor are taken to be the same as those given in [11] and typical values in the case of (B) are used. These are given in Table 1. In each case, the contact force and the deflection histories of the shell are followed by plots of the impact-induced damage (for the open cylindrical shell panel only the results not included in the previous study [17] are discussed). (A) Cylindrical shell panel: Impactor mass was varied by means of a multiplying factor (¼ 1, 2, or 3), keeping the impactor velocity and curvature fixed (impactor velocity ¼ 30 m/s and R=a ¼ 10). Impactor masses are

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

The results are given in Fig. 7(a)–(e). The effect of edge conditions with a [0=90=90=0] laminate was discussed previously in [17]. The same is presently examined with a change of ply orientation using [45=  45=  45=45]. Impact response and impact-induced damage for this are shown in Fig. 8(a) and (b). (B) Cylindrical shell: The effect of the impact problem parameters for the case of a simply supported laminated composite cylinder were made in a manner to similar to that for the shell panel. The effect of varying impactor mass on the contact force, deflection of shell under the point of impact and damage are shown in Fig. 9(a) and (b) respectively, and of the impactor velociy in Fig. 10(a)–(c) respectively. Impact-induced damage at an impactor velocity of 30 m/s was also obtained for the [45=  45=45=  45]s for comparing with the damage in the [0=90=0=90]s laminate (Fig. 10(d)). The effect of curvature of the shell on the impact response was determined using composite shells of diameters 0.05, 0.1, and 0.25 m, all subjected to impact at 10 m/s, while the impactor mass ¼ 1M are in Fig. 11(a) and (b). The length of the

denoted as 1M, 2M or 3M. To study the effect of impactor velocity, it was varied from 10–30 m/s (R=a ¼ 10 and impactor mass ¼ 1M). Contact force and deflection histories the corresponding damages for these case studies were presented in the previous study [17]. For the present, impact response of two impactors with masses 2M and 3M but both having the same initial approach velocity of 30 m/s, was compared with the impact response of an impactor of mass 1M and having initial approach velocities of 42.4 and 52.4 m/s, corresponding to the initial kinetic energies of the impactors of mass 2M and 3M traveling at 30 m/s, respectively. Contact force histories and the damages are for these two cases are shown in Fig. 6(a) and (b) (Fig. 6(c) is from the previous paper when only impactor mass is varied). To examine the effect of curvature in greater detail, it was varied over a wide range by using the ratio of radius (R) to chord length (a), e.g., R=a ¼ 0:5; 0:75; 1; 5; 10; 100. Of these, curvatures corresponding to R=a ¼ 0:5; 0:75; 1:0 may be regarded as curvatures of higher range, and the remaining, viz., R=a ¼ 5; 10; 100 to be of lower range. 300

(a)

2

0.15

Fourier series [ a x a = (0.005m) ] 2 Fourier series [ a x a = (0.0005m) ] Fourier series (Point load) FEM result

Deflection ( x 10 m )

-3

Contact force (N)

a x a = Area of contact patch

200

100

(b)

0.10

0.05 FEM result Fourier series ( Point load ) 2 Fourier series [ a x a = (0.005m) ] 2 Fourer series [ a x a =(0.0005m) ] a x a = Area of contact patch

0 0

50

100

150

200

0 0

250

50

100

150

200

250

Time ( µ sec.)

Time ( µ sec.) 1000

0.8

(c)

FEM result Fourier series ( Point load )

800 0.6 -3

Contact force (N)

Deflection ( x 10 m ) 600 0.4 400

Contact force

0.2

200

0 0

50

100

150

200

0 250

Time ( µ sec.)

Fig. 5. (a,b) Variations of contact force and deflection of the shell at the point of impact during impact on a composite cylinder. Results for the concentrated and distributed contact force are shown. Also plotted are the results from FEM analysis. Other conditions of the shell are as given in Fig. 4. (c) Variation of contact force and mid-point deflection of the shell with time during impact on a simply supported cylindrical shell panel. Comparison of solutions determined by the Fourier series and the finite element methods is shown. Shell panel is of [0=90=90=0] lay-up; R ¼ 2:54 m; chord length, a ¼ 0:254 m; and t ¼ 2:54 mm. Impactor diameter ¼ 12 mm; impactor velocity ¼ 10 m/s.

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

(a)

23

6000

Contact force ( N )

Impactor velocity = 52.4 m/s, KE = 11.4 J Impactor velocity = 42.4 m/s, KE = 7.6 J Impactor mass = 3M, KE = 11.4 J Impactor mass = 2M, KE = 7.6 J

4000

2000

0 0

200

400

600

800

1000

Time ( µ sec )

(b) Impactor mass = 1M Impactor velociy = 52.4 m/s K.E. of impactor = 11.4 J

Along the chord ( m )

Along the chord ( m )

Impactor mass = 1M Impactor velociy = 42.4 m/s K.E. of impactor = 7.6 J

0.05

-0.05

0

0.05

0.10

0.15

0.20

0.25

Parallel to axis ( m )

0.05

-0.05

0

0.05

0.10

0.15

0.20

0.25

Parallel to axis ( m )

Fig. 6. (a) Comparison of contact force histories for the impact cases with the same KE but with different masses and velocities. Impactor diameter ¼ 12:6 mm; M ¼ 8:44 gm; impactor velocity ¼ 30 m/s when impactor masses are 2M and 3M. Laminate is of [0=90=90=0] lay up; thickness ¼ 2:54 mm, R=a ¼ 2:54. (b) Impact induced overall damage by an impactor of mass 1M (¼8.44 gm) having equivalent higher approach velocities corresponding to kinetic energies of impactor with masses 2M and 3M, respectively. (c) Effect of impactor mass on the overall damage in the laminate. Top, middle, and bottom figures correspond to multiplying factors 1, 2 and 3, respectively, for the impactor mass (¼8.44 gm). Impactor velocity ¼ 30 m/s.

cylinder was kept equal to twice the diameter in all the cases. The effect of edge constraints on the impact response was obtained for fully clamped and simply supported conditions is shown in Fig. 12 and that of change in ply orientation in Fig. 13, with two ply arrangements, [0=90=0=90]s and [45=  45=45=  45]s , respectively. Fig. 14 shows the comparison of impact response for the two cases when the stiffness matrices of the failed laminas are modified or unchanged respectively. 5.1. Effect of impact problem parameters on contact force and damage (A) Cylindrical shell panel Impactor mass: It was observed that with the increase of impactor mass, the deflection of the shell panel increased proportionately, whereas the maximum contact force did not increase in the same manner. This was

because the relative approach of the two masses did not increase proportionately. The increases in the maximum contact force were 10.14% and 15.34% corresponding to M ¼ 2, and 3, respectively. The extent of damage also increased with the impactor mass, as larger energies were input. Damages mostly occurred on the side opposite to the impacted surface and the overall shape of the damaged area approached that of a peanut (Fig. 6(c)). Considering a rectangular area bounding the damaged area, the areas of the damage were determined to be 1.21, 3.31, and 5.6 cm2 respectively corresponding to the three masses. Impactor velocity: In this case the contact force and the shell displacements increased in equal measure corresponding to the increments in impactor velocity. Impact-induced damage also increased proportionately with the increase in contact force, and in turn with the impactor velocity.

24

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 4000

(a )

R/a=100 R/a=10 R/a=5 R/a=1.0 R/a=0.75 R/a =0.5

3000

0.05

Contact force (N)

Along the chord, m

Impactor mass =1M

-0.05

2000

1000

0

0.1

0.2

0 0

200

400

600

800

1000

800

1000

Time (µ sec)

Impactor mass = 2M

R/a=100 R/a=10 R/a=5 R/a=1.0 R/a=0.75 R/a=0.5

(b) 0.003

Deflection (m)

Along the chord, m

0.004 0.05

-0.05

0

0.05

0.10

0.15

0.20

0.25

0.002

0.001

Impactor mass = 3M

Along the chord, m

0 0

400

600

Fig. 7. (a,b) Effect of curvature of the shell on the variation of (a) contact force, and (b) mid-point deflection of the shell with time. Shell panel is of [0=90=90=0] lay-up; impactor mass ðMÞ ¼ 8:44 gm; impactor velocity ¼ 30 m/s. (c) Effect of curvature of the shell on the overall damage in the shell panel. (d) Effect of curvature of the shell on the overall damage in the shell panel. (e) Effect of curvature of the shell on the overall damage in the shell panel.

-0.05

0

(c)

200

Time ( µ sec)

0.05

0.05

0.10

0.15

0.20

0.25

Parallel to the axis. m

Fig. 6 (continued )

For the case of impactors having the same initial kinetic energy but with different mass–velocity combinations, it was observed from Fig. 6(a) that for a given initial kinetic energy, an impactor having a higher approach velocity has a more severe effect, both on the maximum contact force induced as well as on the resulting damage (Fig. 6(b)), when compared to an impactor with an equivalent larger mass and a lower approach velocity. In impact experiments on laminated composite plates, Olsson [35] has shown that the extent of damage due to the high velocity impactor covered a much larger area compared to that produced by the large mass impactor (the damaged areas are in the ratio of 17:1), while the maximum contact force increased

by 60%. Similar observations may be made in the present problem also, although the change of the impactor velocity is not large (maximum contact force increased by 30% and 60% respectively for the two higher velocities considered). Also, at the impactor velocity of 52.4 m/s the damage is significantly larger. It has been suggested in [35] that the higher rate of straining involved in a high velocity impact may be additionally responsible for the larger damage, although presently the higher damage is only due to the higher impact force generated, since the failure criteria used do not incorporate rate effects. Curvature of the shell: It appears that curvature has relatively less pronounced effect on the maximum contact force when compared to the effects of varying either impactor mass or velocity. Higher curvatures produced only a marginal increase (<5%) in the maximum contact force. It was noticed that the shell with the highest curvature, viz., R=a ¼ 0:5 had the smallest time of con-

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 0.050

0.050

Cylindrical shell panel [0/90/90/0], R/a=0.5 Impactor velocity=30m/s.

Cylindrical shell panel [0/90/90/0], R/a=1.0 Impactor velocity=30m/s. 0.025

Along the chord, m

Along the chord, m

0.025

0

-0.025

-0.050 0.15

0

-0.025

0.17

0.19

0.21

0.23

-0.050 0.08

0.25

0.10

Parallel to the axis

0.12

0.18

Cylindrical shell panel [0/90/90/0], R/a=5.0 Impactor velocity=30m/s.

0.025

0.025

Along the chord, m

Along the chord, m

0.16

0.050

Cylindrical shell panel [0/90/90/0], R/a=0.75 Impactor velocity=30m/s.

0

-0.025

(c)

0.14

Parallel to the axis, m

0.050

-0.050 0.09

25

0

-0.025

0.11

0.13

0.15

0.17

-0.050 0.08

0.19

(d)

Parallel to the axis, m

0.10

0.12

0.14

0.16

0.18

Parallel to the axis, m

Fig. 7 (continued)

tact and the shells with R=a ¼ 0:75, and R=a ¼ 1:0 had slightly higher contact times and lost all contact within about 250 ls. Shells with lower range of curvatures, viz., R=a ¼ 5; 10; 100 had more prolonged contact periods Change of curvature which affected the panelÕs compliance produced significant changes in the deflections under the point of impact. Decrease of curvature increased the panelÕs compliance thereby resulting in higher central deflections. The panelÕs deflection histories determined how long or short the contact duration was. This also determined the number of contacts an impactor made. In the present combination of impactor mass, velocity and the range of curvatures, a third contact was made by the impactor only for the case of R=a ¼ 5. It is also observed that the deflection histories for the two curvature ranges are clearly separated. Results suggest that the shape of the damage changed with curvature changes, e.g., as the curvature reduced, the shape of the damage tended towards the peanut shape (Fig. 7(c)–(e)). The shape and size of the damage seems to stabilize at lower curvatures, signifying that curvature had less effect on damage at low curvatures. In the present problem, one may conclude that curvature had less influence on the maximum contact force and af-

fected the impact duration to a limited extent. These results compare well with those in [10], for the case of varying shell curvature. An examination of the time of occurrence of damage revealed that while at the highest curvature, all damage took place before the peak contact force was reached; it was 73% for all cases of curvature of the lower range. Rest of the damage too was completed soon after the peak was reached in the latter case. (B) Cylindrical shell Effect of impactor mass: It was observed that doubling or trebling the mass did not increase the maximum contact force to the same extent but in smaller increments, although deflections showed larger increases (Fig. 9(a)). The influence of the relative masses of an impactor and a composite cylindrical shell was discussed in [5], and it was observed that as the mass increased, the contact duration also increased and finally approached a quasi-static loading situation. Similar results are observed for the shell panel wherein an increase in impactor mass lead to a longer contact duration before the first separation of the impactor from the shell took place. While the deflection increased proportionately with the increase in mass, the maximum contact

26

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 0.050

Shell panel [45/-45/-45/45] Clamped edges (+) Damage in lamina 1 (x) Damage in other laminas

Cylindrical shell panel [0/90/90/0], R/a=10.0 Impactor velocity=30m/s. 0.025

Along the chord, m

Along the chord, m

0.05

0

-0.05 -0.025

-0.050 0.08

0.10

0.12

0.14

0.16

0.18

0

0.05

Parallel to the axis, m

0.10

0.15

0.20

0.25

Parallel to the axis, m

0.050

shell panel [45/-45/-45/45], Simply supported.

Cylindrical shell panel [0/90/90/0], R/a=100 Impactor velocity=30m/s.

(+) Damage in lamina 1 (x) Damage in other laminas

0.025

Along the chord, m

Along the chord, m

0.05

0

-0.05 -0.025

-0.050 0.08

0.10

0.12

(e)

0.14

0.16

0.18

0

0.05

(b)

Parallel to the axis, m

Fig. 7 (continued)

0.15

0.20

0.25

Fig. 8 (continued)

3000

0.004 1500

Simply supported Clamped

0.0020 Impactor mass = 3 M Impactor mass = 2 M Impactor mass = 1 M

C. force

Contact force 0.003

Deflection (m)

0.0015

2000

0.002

1000

Contact force ( N )

Contact force, N

0.10

Parallel to the axis, m

1000

0.0010

500

Deflection ( m )

0.001 0.0005

0 0

(a)

200

400

0 600

0 0

Time (∆t= 9.5x e-07 sec.)

(a) Fig. 8. (a) Effect of boundary conditions on the variation of contact force and mid-point deflection of the shell with time during impact on a cylindrical shell panel. Shell panel is of [45=  45=  45=45] lay-up; R=a ¼ 10; impactor mass ¼ 1M; impactor velocity ¼ 30 m/s. (b) Effect of boundary conditions on the overall damage in the shell panel (a) with clamped edges (upper figure), and (b) with simply supported edges (lower figure). Other conditions are same as in (a). Enlarged view is a rectangular area enclosing damage is shown.

100

200

300

400

0 500

Time ( µ sec)

Fig. 9. (a) Effect of impactor mass on the variation of contact force and mid-point deflection of the shell with time during impact of a steel ball of 12 mm diameter on a composite cylindrical shell at mid-span (L ¼ 0:42 m; R ¼ 0:1 m and t ¼ 2:5 mm). The composite has a [0= 90=0=90]s lay-up. Impactor mass ð1MÞ ¼ 7:66 gm; impactor velocity ¼ 10 m/s. (b) Effect of impactor mass on damage. Other conditions are as in (a).

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 0.02

27

1500

0.0010 Contact force ( N ) Impactor velocity = 5m/s. Impactor velocity = 10m/s. Impactor velocity = 15m/s.

0.01

Contact force ( N )

Along diametrical direction, m

Impactor mass = 1M

0

0.0006

0.0004 Deflection ( m )

500

0.0002

0 0

0.20

0.21

0.22

(a)

0.01

0

-0.01

0.20

0.21

0.22

200

300

Time ( µ sec)

Fig. 10. (a) Effect of impactor velocity on the variation of contact force, and mid-point deflection of the shell with time during impact on the composite cylinder. Impactor velocity 5–15 m/s. Impactor mass ¼ 1M. Other conditions are as in Fig. 9(a). (b) Effect of impactor velocity on overall damage in the composite cylinder. Impactor velocity 10–15 m/s. An enlarged view of a rectangular area enclosing damage is shown. (c) Effect of impactor velocity on overall damage in the composite cylinder (Impactor velocity: 20–30 m/s). An enlarged view of a rectangular area enclosing damage is shown. (d) Enlarged view of the overall damage due to matrix cracking in the composite cylinder having a lay-up of [45=  45=45=  45]s (upper figure). Lower figure: Delamination failure at interface 1 on the laminateÕs inner side. Impactor mass ¼ 1M; impactor velocity ¼ 30 m/s. Other conditions are as given in (a).

Impactor mass = 2M

-0.02 0.19

0 100

0.23

0.02

Along diametrical direction, m

1000

-0.01

-0.02 0.19

0.0008

0.23

0.02

Along diametrical direction, m

Impactor mass = 3M 0.01

0

-0.01

-0.02 0.19

(b)

0.20

0.21

0.22

0.23

Parallel to axis, m Fig. 9 (continued)

force did not. The overall damage also changed in an incremental manner, but to a lesser degree as compared with a shell panel (Fig. 9(b)). Although the total number of points where failure was predicted increased when a physical count of these was made, the horizontal spread of the damage, however, did not increase due to an increase in impactor mass from 2 to 3 as much as it did when it was doubled. The increased damage was observed to be in the thickness direction. Impactor velocity: When the impactor velocity was increased in an incremental manner, say 5, 10, 15 m/s,

the peak contact force doubled and trebled (Fig. 10(a)). In comparison with the response obtained due to increase in impactor mass, such increases may be due to the fact that the kinetic energy of the impactor increased linearly with an increase in mass and quadratically with velocity. Impact at 5 m/s produced no damage, while at 10 m/s some damage could be observed. At 20 and 30 m/ s the damage is approaching a peanut shape (Fig. 10(b) and (c)). Effect of shell curvature: It seems that while the peak contact force increased with the increase in curvature of the cylinder, the magnitude of the maximum contact force is also affected by the size of the cylinder which changes significantly when the radius of curvature was increased from 0.05 to 0.25 m (Fig. 11(a)). With the increase in the radius of curvature, local compliance and hence deflections increased significantly. Damage increased with the increase in curvature; development of a pattern however is not clear (Fig. 11(b)). The total number of points where damage was predicted corresponding to the three curvatures, viz., 0.05, 0.1, and 0.25 m were noted to be 60, 32 and 0 respectively. Boundary conditions: No significant change in the contact force or displacement history was observable (Fig. 12) when change in boundary conditions from simply supported to fully clamped ends was made. This may be explained by the fact in the present impact problem the maximum contact force is attained before

28

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 0.02

0.02

Impactor velocity =20m/s

Along diametrical direction, m

Along diametrical direction, m

Impactor velocity = 10 m/s 0.01

0

-0.01

-0.02 0.19

0.20

0.21

0.22

0.01

0

-0.01

-0.02 0.19

0.23

0.20

0.21

Impactor velocity =30m/s

Along diametrical direction, m

Along diametrical direction, m

Impactor velocity =15m/s

(b)

0.23

0.02

0.02

0.01

0

-0.01

-0.02 0.19

0.22

Parallel to cylinder axis, m

Parallel to cylinder axis, m

0.20

0.21

0.22

0.01

0

-0.01

-0.02 0.19

0.23

(c)

Parallel to cylinder axis, m

0.20

0.21

0.22

0.23

Parallel to cylinder axis, m

Fig. 10 (continued)

it is affected by the conditions at the boundary. Displacement variation with time also tends to suggest this. Impact-induced damage was also not affected by the end conditions. Change of ply orientation: From Fig. 13, it seems that the shape of the contact force variation of the [0=90=0=90]s is sharper with a slightly smaller period of contact than that of the [45=  45=45=  45]s . Both the contact force and damage for the [0=90=0=90]s laminate were larger than those for the [45=  45=45=  45]s laminate. A computation of the ‘‘D’’ matrix of the laminate constitutive equations for above two ply stacking sequences revealed that the laminate with [45=  45= 45=  45]s ply sequence is more compliant than the [0=90=0=90]s laminate, an observation which conforms with the present results. Also, no damage was predicted in the [45=  45=45=  45]s case at 10 m/s (at which both the cylindrical shells were impacted) while other conditions remained the same. Effect of impact-induced damage: Fig. 14 clearly shows that the laminateÕs stiffness is affected. Since the extent of damage is quite small in this case, the reduction in the peak contact force is also small. However, it

is likely that a larger damage would affect the stiffness of the laminate more significantly. 5.2. Energy absorbed during impact Computation of impulse, I, energy absorbed by the shell, Es , the energy used for causing indentation/damage, Ed , and the ratio Ed =Es as functions of time was made for all the impact cases studied previously, i.e., with the composite cylinder and the composite cylindrical shell panel, by means of the finite element method (as discussed in Section 3.3). A comment may be made regarding Ed . Although impact-induced damage is caused by a combination of transverse shear and tensile stresses (due to bending), Ed is a useful measure for knowing the contribution made by it towards damage since part of the work done through a is non-recoverable, which represents the energy used in causing permanent indentation. Fig. 15 presents the results for the case of impact on a composite cylinder considering three different impact velocities, viz., 5, 10 and 15 m/s. In the upper figure, contact force variations are also shown for comparison (variation of impulse with time is shown

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 0.050

0.010

R=0.05m, L =0.2m Impactor velocity =10m/s

Along diametrical direction, m

Cylindrical shell [45/-45/45/-45]s Impactor velocity=30m/s.

Along diametrical direction, m

29

0.025

0

-0.025

0.005

0

-0.005

-0.010 0.100

-0.050 0.175

0.200

0.225

0.105

0.110

0.250

0.115

0.120

Parallel to axis, m

Along axial direction, m 0.010

0.050

Radius = 0.1m , L = 0.42m Impactor velocity = 10m/s

Along diametrical direction, m

Along diametrical direction, m

Cylindrical shell [45/-45/45/-45]s Impactor velocity=30m/s.

0.025

0

-0.025

0.005

0

-0.005

-0.010 0.200

-0.050 0.175

0.200

(d)

0.225

(b)

0.250

0.205

0.210

0.215

0.220

Parallel to cylinder axis, m

Along axial direction, m

Fig. 11 (continued) Fig. 10 (continued)

1000

0.0005 Simply supported ends Clamped ends

Contact force ( N )

1000

800

0.0010 Radius = 0.25m Radius = 0.1m Radius = 0.05m

Contact force ( N )

0.0008

750

0.0006

Contac t force (N)

1250

0.0004

600

0.0003 Deflection ( m )

400

0.0002

Deflection ( m ) 500

0.0004

250

0.0002

200

0 0

Contact force

100

200

0.0001

0 300

Time ( µ sec) 0 0

(a)

100

200

0 300

Time ( µ sec)

Fig. 11. (a) Effect of radius of curvature (R ¼ 0:05, 0.1, and 0.25 m) on the variation of contact force, and mid-point deflection of the shell with time in a composite cylindrical shell. Impactor mass ¼ 1M; impactor velocity ¼ 10 m/s; L ¼ 2R. Other conditions are as given in Fig. 9(a). (b) Effect of radius of curvature on the overall damage in the composite cylinder [0=90=0=90]s . No damage was observed in the cylinder of 0.25 m radius. Other conditions are same as in (a).

Fig. 12. Effect of edge conditions on the variation of contact force and mid-point deflection of the shell with time. Impactor mass ¼ 1M; impactor velocity ¼ 10 m/s. Other conditions are as given in Fig. 9(a).

only in some cases). Figs. 16–18 show plots of Es , Ed and Ed =Es versus time with parametric variations of impactor mass, curvature of the shell and ply orientation

30

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 2000

0.0010

[0/90/0/90]s [45/-45/45/-45]s 0.0008

Contact force ( N )

1500

Deflection ( m )

0.0006

1000 0.0004

500 0.0002

Contact force

0 0

100

0 300

200

Time ( µ sec)

Fig. 13. Effect of ply orientation on the variation of (a) contact force, and (b) mid-point deflection of the shell with time. The lay-ups are [0=90=0=90]s and [45=  45=45=  45]s respectively; impactor mass ¼ 1M; impactor velocity ¼ 20 m/s. Other conditions are as given in Fig. 9(a).

1000

0.0005 Stiffness matrix modified Stiffness matrix unchanged

Contact force ( N )

800

0.0004

600

0.0003

shape in all cases and has this peak. From Fig. 15, it seems that the fraction of the energy transferred to the shell and that used for indentation seems to decrease slightly with the increase in velocity. Increase of mass also shows greater absorption of indentation energy. Increase in curvature (and decrease of compliance) lead to a greater absorption of the energy for indentation as can be seen in Fig. 17. Change of ply orientation shows little affect on these curves (Fig. 18). Similar inferences could be drawn from the results for the shell panel. To obtain further insight into the overall transfer of energy during impact, the coefficient of restitution (considering the period of contact only up to the first loss of contact) for the several of impact cases are shown in Table 2. From an observation of the last column, it seems that energy transferred by the impactor to the shell is significantly different for the two types of cylindrical shells considered. Whereas the shell panels absorbed almost the whole of the initial kinetic energy of the impactor, energies absorbed by the cylindrical shell varied depending on the impact problem parameters. For the cylindrical shell, contact force histories and the values of the coefficient of restitution suggest that rebound of the impactor occurred during impact.

6. Conclusions

Deflection ( m ) 400

0.0002

200

0 0

0.0001

Contact force

100

200

300

0 400

Time ( µ sec)

Fig. 14. Effect modifying stiffness matrices of matrix-cracked laminas during step-by-step numerical solution of the equation of motion on the variation of contact force and mid-point deflection of the shell with time. Comparison is made with impact response obtained using unchanged or modified stiffness matrix. Impactor mass ¼ 1M; impactor velocity ¼ 10 m/s. Other conditions are as given in Fig. 9(a).

respectively. Results for the cylindrical shell panel are shown in Figs. 19–21. In all cases, it appears that the energy used for indentation is a minor fraction of the energy transferred to the shell, although not insignificant. The major part is obviously used in deforming the shell. The energy and impulse curves have a definite shape in all cases of impact on the cylindrical composite shell. Mittal and Jafri [36], while discussing the impact on an infinite orthotropic plate, have shown that the energy curve has a hump, which is also observed with impact on cylindrical shells. However, this peak is absent in the case of impact on the shell panel. However, Ed curve has the same

1. The intensity of impact, measured by the magnitude of the contact force developed as well as the damages produced for a given laminate, is mainly determined by the impactor mass and impactor velocity. Of the two, impactor velocity had a greater effect on the contact force developed as compared to the effect of the impactor mass. This is clearly due to the fact that higher kinetic energy is imparted when the impactor velocity is increased. Impact-induced damage also tends to increase with a higher velocity of impact for a given kinetic energy. In this regard, the response of a shell is comparable to that of a plate under impact. 2. The effect of curvature is more significant when the curvature is very high. When the curvatures are low, the effect of change of curvature had a relatively less influence on the maximum contact force developed and on the impact-induced damage in the case of shell panel. The maximum contact force increased only marginally with the increase of curvature. Deflections and time of contact increased with the decrease of curvature. The shape and size of the damage tended to stabilize towards the peanut shape (approximately) when the curvatures were low. However, the shape and size of the damage changed at higher curvatures. When complete cylindrical shell was the target, the effect of curvature on the impact response could not be correctly studied due to the fact that change of curvature significantly affected the size of the shell which can influence the impact re-

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 1600

0.8

31 0.16

Es (15m/s) 15m/s

400

0.12

I ( 15m/s)

10m/s

I (10m/s) 0.08

0.4

Es (10m/s) 0.04

I (5m/s)

5m/s

0.2

Impulse ( I ), N-sec.

800

F(t)

0.6

Energies, E s, Ed ( N-m)

Contact force, N

1200

Es (5m/s) Ed ( 15m/s) Ed (10m/s)

Ed (5m/s)

0

0 0

1 00

200

0 300

Time ( µ sec) 1

0.8

____ Impactor velocity = 5m/s __ __ Impactor velocity = 10m/s ------- Impactor velocity = 15m/s

Ed / Es

0.6

0.4

0.2

0 0

100

200

300

Time ( µ sec) Fig. 15. Upper figure: effect of impactor velocity on contact force (F ðtÞ), impulse (I), energy absorbed by the shell (Es ), and energy absorbed for indentation (Ed ) during impact on a composite cylinder. Results shown are for three different impactor velocities, viz., 5, 10, and 15 m/s respectively. (+) marked lines denote impulse (I), () marked lines denote Es , and unmarked lines denote Ed . Lower figure: the ratio (Ed =Es ) versus time.

sponse. This effect is also observed in the case of the shell panel at higher curvatures but to a lesser degree. 3. Response of a cylindrical shell panel is significantly different from that of a cylindrical shell both in the contact force variation with time and in the extent of damage developed. Damage suffered by the shell panel is significantly higher than that of the cylindrical shell.

4. The laminateÕs characteristic which had a major influence on the impact response was ply orientation. This had a significant influence on the damage resisting characteristics of the laminate especially in the case of a cylindrical shell panel. Cross-plied laminate suffered less damage especially in the case of a shell panel. However, the same was not true in the case of a full cylinder for which the extent of damage in a cross-plied laminate was

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 1.2

0.4

(a)

(a)

0.3

Es , Ed, N-m

Es , Ed , N-m

0.8

_____ R = 0.05m __ __ R = 0.1m -------- R = 0.25m

0.2

Es

0.4 0.1

_____ I. mass = 1M __ __ I. mass = 2M -------- I. mass = 3M

Es

Ed

Ed 0

0 0

100

200

300

0

400

50

100

150

200

250

200

250

Time (µ sec)

Time (µ sec) 1 1

(b)

(b) 0.8 0.8

_____ R = 0.05 __ __ R = 0.1m - - - - R = 0.25m

_____ Imp. mass =1M __ __ Imp. mass =2M - - - - - Imp. mass=3M

0.6

Ed / Es

Ed / Es

0.6

0.4 0.4

0.2 0.2

0 0

0 0

100

200

300

400

Time (µ sec) Fig. 16. Effect of varying impactor mass on the energy absorbed by the shell (Es ) and the energy used for causing indentation (Ed ) during impact on a composite cylinder ([0=90=0=90]s ) and the variation of the ratio (Ed =Es ) with time. Impactor velocity ¼ 10 m/s.

higher than that in the laminate with angularly oriented laminas. The effect of ply orientation change on the maximum contact force was however small. 5. Contact force histories for a cylindrical shell panel suggest that multiple contacts occurred during impact, whereas the impact on a cylindrical shell did not show this behaviour. This may be attributed to the fact that cylindrical shell is a much larger structure and is also stiffer. The basic flexural behaviour of a cylindrical shell

50

100

150

Time ( µ sec ) Fig. 17. Effect cylinder radius on Es , Ed and the ratio (Ed =Es ) during impact on a composite cylinder ([0=90=0=90]s ). Impactor velocity ¼ 10 m/s.

is also different from that of a cylindrical shell panel. The coefficient of restitution for the cylindrical shell suggests rebound of the impactor. 6. It appears that the variations of the contact force and the deflection of the shell are not affected by the boundary conditions until the impact generated waves, which after being modified by the boundary conditions during their reflections reach back the point of impact. Boundary conditions had almost no influence on the maximum contact force, but however affected the contact force variations after that. Deflections however

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 1.6

0.25

33

4

(a)

Es

(a)

Es

____ Impactor velocity = 10m/s __ __ Impactor velocity = 20m/s ------- Impactor velocity = 30m/s

0.15

0.8

____ [0/90/0/90]s ------- [45/-45/45/-45]s

0.1

Impulse

Es , Ed (N -m)

3

Impu ls e, I ( N-sec .)

Ener gi es: Es , Ed ( N-m)

0.2 1.2

2

Es

0.4 0.05

1

Ed 0

0 0

50

100

150

200

Es

250

Time ( µ sec )

Ed

0 0.8

0

50

100

150

200

250

Time ( µ sec)

(b) 1 0.6

(b)

___ [0/90/0/90]s

Ed / Es

- - - - [45/-45/45/-45]s

0.8

____ Impactor velocity=10m/s.

0.4

__ __ Impactor velocity=20m/s. - - - - Impactor velocity=30m/s.

Ed / Es

0.6 0.2

0.4

0 0

50

100

150

200

250

Time ( µ sec )

Fig. 18. Effect of ply orientation on impulse (I), energy absorbed by shell (Es ), and energy absorbed for causing indentation (Ed ) during impact on a composite cylinder and the variation of the ratio (Ed =Es ) time. Impactor velocity ¼ 20 m/s.

0.2

0 0

50

100

150

200

250

Time ( µ sec)

changed quite significantly in the case of the shell panel, but to a lower degree in the case of the composite cylinder. Edge conditions in the case of a [0=90=90=0] shell panel had little influence on the extent of damage induced, but differences could be observed in the case of laminates having 45° plies. Clamped edges produced greater damage in this case. Boundary conditions also did not influence damage produced in the case of a cylinder. 7. Incorporating the influence of damaged laminas into the computation impact response concurrently has a significant effect on the computed impact response even when damages are small as in the present studies. 8. Damage predicted by the failure criteria adopted is in conformity with the basic postulates made regarding the initiation and growth of damage. Further, damage

Fig. 19. Effect of impactor velocity on energy absorbed by the shell (Es ), energy absorbed for causing indentation (Ed ), and the ratio (Ed =Es ) during impact on a shell panel (R=a ¼ 10; ply lay-up: [0=90=90=0]).

initiation and growth are similar to those in thin laminated composite plates, i.e., with a reverse pine-tree pattern. However these observations need to be examined by actual impact experiments. Thicker laminates have not been investigated in this study. 9. Plots of impact energy absorbed by the shell and that for forming the indentation, and the knowledge of the influence of the impact problem parameters on these can be useful for further understanding of the impact phenomenon and provide both qualitative and quantitative measures of the various energies. Shell panels

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K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36 12

4

(a) (a) ____ Impactor mass =1M __ __ Impactor mass =2M ------- Impactor mass =3M

Es

Es

3

Es , Ed ( N-m)

Es ,Ed ( N-m)

8

Es

___ [45/-45/-45/45] ----- [0/90/90/0] 2

4

Es

1 Ed 0 0

50

100

150

200

Ed

250

Time (µ sec) 1

0 0

(b) _____ Impactor mass =1M

0.8

50

100

150

200

250

Time ( µ sec)

1

__ __ Impactor mass =2M

(b)

- - - - - Impactor mass=3M

Ed / E s

0.6

0.8

____ [45/-45/-45/45] 0.4

- - - - [0/90/90/0]

Ed / Es

0.6 0.2

0.4 0 0

10

20

30

Time ( µ sec ) 0.2

Fig. 20. Effect of impactor mass on the energy absorbed by the shell (Es ), energy absorbed for causing indentation (Ed ), and the ratio (Ed =Es ) during impact on a shell panel (R=a ¼ 10; ply lay-up: [0=90=90=0]; impactor velocity ¼ 30 m/s).

0 0

50

100

150

200

250

Time ( µ sec)

absorbed almost the entire energy of impact whereas it depended on the conditions of impact in the case of a cylindrical shell. 10. The ÔdegeneratedÕ shell element of Ahmad et al. provided sufficiently accurate results. Close match of the solutions obtained by the finite element method and the analytical method existed. The series method, with information regarding natural frequencies of vibration of the impacted structure, provided a proper basis for adopting the size of the time step. The series solution also lent confidence in accepting the accuracy of finite element solutions which are known to be significantly affected by the choice of mesh used, element type, grading, etc. The finite element method, however, proved to be a powerful numerical tool as it enabled us to incorporate the effect of damage produced in the shell on its impact response, simultaneously as the solution progressed.

Fig. 21. Effect of ply orientation on energy absorbed by the shell (Es ), energy absorbed for causing indentation (Ed ), and the ratio (Ed =Es ) during impact on a shell panel (R=a ¼ 10; impactor mass ¼ 1M; impactor velocity ¼ 30 m/s).

This helped us in obtaining a quantitative measure of the extent to which the structural member has been weakened due to a given impact situation. 11. Present study limited itself to the problem of impact on laminated composite cylindrical shells. However, the software developed is useful for considering shells of other shapes. The general applicability of the Choi–Chang damage prediction model for composite shells and rate effects which may be important when the impactor velocities are in the higher range need to be examined through experiments.

K.S. Krishnamurthy et al. / Composite Structures 59 (2003) 15–36

35

Table 2 Effect of various problem parameters on coefficient of restitution and energy transfer Impact case

Area under the contact force–time curve (first contact), N s

Coefficient of restitution, e

Kinetic energy of the impactor, kg m2 /s2

Percent of energy transferred to the shell

Cylindrical panel (linear analysis) R=a ¼ 5 0.261 R=a ¼ 10 0.256 R=a ¼ 100 0.254

0.033 0.012 0.005

3.798 3.798 3.798

99.89 99.98 99.99

Cylindrical panel (non-linear analysis) R=a ¼ 5 0.261 R=a ¼ 10 0.257 R=a ¼ 100 0.255

0.033 0.017 0.009

3.798 3.798 3.798

99.89 99.96 99.99

Effect of impactor’s mass mass ¼ 1 m 0.256 mass ¼ 2 m 0.521 mass ¼ 3 m 0.748

0.012 0.028 0

3.798 7.596 11.391

99.9 99.92 99.95

Effect of impactor velocity V ¼ 10 m/s 0.086 V ¼ 20 m/s 0.171 V ¼ 30 m/s 0.256

0.021 0.015 0.012

0.422 1.688 3.798

99.95 99.97 99.98

Full cylinder impactor mass 1M 0.093 2M 0.191 3M 0.289

0.33 0.35 0.365

0.353 0.706 1.058

89.34 87.57 86.6

Radius R ¼ 0:05 m R ¼ 0:1 m R ¼ 0:25 m

0.101 0.094 0.085

0.427 0.326 0.208

0.353 0.353 0.353

81.75 89.34 95.65

Impactor velocity 5 m/s 10 m/s 15 m/s

0.047 0.094 0.138

0.351 0.326 0.312

0.088 0.353 0.794

87.68 89.34 90.26

Change of ply orientation [0=90=0=90]s 0.184 [45=45=45=45]s 0.200

0.303 0.419

1.411 1.411

90.8 82.4

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