A model is presented for estimating the dimensions of delaminations in continuous fiber reinforced composite plates subjected to non-penetrating (low velocity) impact of a solid object. The model is based on dimensional analysis, and results in two simple, closed form expressions, one providing the delamination length, the other the delamination width. Comparisons of delamination lengths and widths calculated by these expressions with experimental data suggest that the model predicts these dimensions with reasonable accuracy

Finn, Scott R.

Springer, George S.

English

NASA Technical Reports Server

. 
Department of AERONAUTlCS and ASTRONAUTICS 
STANFORD UNUVER SUTV 
f l / , / c  15 
Q/. /?/I 
Final Report to 
NASA Ames Research Center 
Grant No. NCA2-311 g ( r 6  
DELAMINATIONS IN COMPOSITE PLATES CAUSED 
BY NON-PENETRATING IMPACT 
? 
. 
Scott R. Finn and George S. Springer 
Department of Aeronautics and Astronautics 
Stanford University, Stanford, California 94305 
( N A S A - C R - 1 8 5 0 6 3 )  DELAMINATIONS I N  COMPOSITE N09-2949 3 
PLATES CAUSE0 BY NON-PENETRATTNG INPACT 
? Fina l  Report (Stanford Univ,) 24 pCSCL 110 
Unci as 
G 3 / 2 4  0 2 3 3 3 4 6  
October 1989 
https://ntrs.nasa.gov/search.jsp?R=19890020122 2020-03-20T00:32:37+00:00Z
Final Report to  
NASA Ames Research Center 
Grant No. NCA2-311 
DELAMINATIONS IN COMPOSITE PLATES CAUSED 
BY NON-PENETRATING IMPACT 
Scott R. Finn and George S. Springer 
Department of Aeronautics and Astronautics 
Stanford University, Stanford, California 94305 
October 1989 
ABSTR.ACT 
A model is presented for estimating the dimensions of delaminations in continuous 
fiber reinforced composite plates subjected to non-penetrating (“low velocity”) impact of 
a solid object. The model is based on dimensional analysis, and results in two simple, 
closed form expressions, one providing the delamination length, the other the delamination 
width. Comparisons of delamination lengths and widths calculated by these expressions 
with experimental data suggest that the model predicts these dimensions with reasonable 
accuracy. 
1 
I. 1NTR.ODUCTION 
Delaminations in composites caused by the impact of a solid object are of great practical 
concern. Delaminations may reduce significantly the strength and stiffness of the material, 
especially when the impacted laminate is under compression. Therefore, in order to utilize 
composites to their full advantage, we must be able to assess their susceptibility to impact. 
Specifically, it is important to know the locations and dimensions of delaminations which 
may occur due to impact. 
Recently, Wu and Springer [l] described a method for estimating the locations and sizes 
of delaminations inside fiber reinforced organic matrix plates impacted by a solid object. Wu 
and Springer’s analysis utilizes a three dimensional, transient, finite element stress analysis 
coupled with a failure criterion based on dimensional analysis. In addition to the delami- 
nation sizes and locations, this analysis also provides detailed behavior of the impactor and 
the impacted plate (e.g., displacement and velocity of the impactor, contact force as a func- 
tion of time, and displacements, velocities, stresses and strains in the plate as functions of 
position and time). The disadvantage of the Wu-Springer analysis is that it requires long 
computation times on mainframe computers. 
Frequently, in practice, the full history of the impactor and the plate is not needed; 
an estimate of the locations and sizes of the delaminations suffices. This investigation was 
undertaken to develop a method which provides this information simply and easily, without 
recourse to very long computer calculations. The present method is based on the failure 
criteria proposed by Wu and Springer, but does not require the time consuming finite element 
calculations. Instead, the method described in this paper results in closed form expressions, 
which readily yield the locations and sizes of delaminations caused by non-penetrating (“low 
velocity”) impact. 
2 
11. THE PR.OBLEk1 
We consider a rectangular plate (thickness h) made of a continuous fiber-reinforced, 
organic matrix composite (Figure 1). The plate consists of n plies or lamina. The ply 
orientations are arbitrary and need not be symmetric with respect to the mid-surface of the 
plate. Perfect bonding between each ply is assumed. 
The plate is supported along all four edges. Each edge may be either clamped or simply 
supported. A solid object of mass rn, speed V, and "nose" radius r impacts the center of 
the plate (Figure 1). It is assumed that during impact the impactor does not penetrate the 
surface. The objective is to develop a method which can be used to estimate the locations 
and sizes of delaminations caused by the non-penetrating impact. 
111. DELAMINATION 
The shape of the delamination is irregular. However, previous data have shown the 
major axis of the delamination is, in general, parallel to the fiber direction of the lower ply 
[2]. Therefore, we characterize the delamination between any two plies by the length and 
width WD which are taken to be the dimensions of the delamination in the directions parallel 
and perpendicular to the fiber direction in the lower ply, i.e., in the ply further from the 
impactor (Figure 2). 
By using dimensional analysis, Wu and Springer [l] obtained the following expressions 
for the delamination lengths and widths 
WD urnax& c7 Dee -="( e0 KI,  ) m Dee 
KI,  and l o  are ply properties which must be determined by tests. KI,  is the Mode 
I critical stress intensity factor (opening mode). eo is the critical crack length given by 
3 
transverse tensile strength measurements [l, 31 
where STT is the transverse ply tensile strength. 
Q* and Dee are laminate properties which can be calculated from laminated plate theory. 
Q* is proportional to the difference between the stiffnesses of the two plies adjacent to the 
delamination in the direction of the fibers of the “lower” ply (Figure 2). The parameter &* 
is defined as: 
Qzz and Qrar are the reduced stiffnesses parallel and perpendicular to the fibers. &e@ is the 
transformed reduced stiffness corresponding to the angle 8 between the fiber orientations of 
the two plies adjacent to the delamination (Figure 2). The superscripts l and u refer to the 
lower and upper plies. For convenience, an expression for &* in terms of the ply properties 
and the angle 8 is given in the Appendix. 
D& and DE are the flexural rigidities of the sections of the laminate above and below 
the delamination, respectively, in the direction of the fibers of the ply immediately below 
the delamination (Figure 2). These values can be found by considering the two sections as 
separate laminates and calculating their flexural rigidities according to [4] 
where Qee is the same as previously given, and’z is measured from the midplane of the 
individual section, not of the entire laminate. 
The remaining three parameters am,,, t m ,  and tf depend on the laminate as well as on 
the impact. amax is the maximum out-of-plane tensile stress under the point of impact. t m  is 
the time at which this stress occurs. Finally, tf is the duration of the stress. Wu and Springer 
[l] evaluated am=, t f ,  and t ,  using a three-dimensional, transient, finite element method. 
Here, our analysis differs from that of Wu and Springer. We propose to approximate these 
three parameters in a simple manner, as described below. 
4 
The maximum stress omax is taken to be proportional to the pressure P which arises at 
the flat interface of two elastic, semi-infinite cylinders when the two cylinders collide 
where k1 is a constant. The impact pressure P is given by [5] 
where V is the relative velocity between the two impacting surfaces, which, in our case, is 
the impact velocity, p is the density, and c is the dilatational wave speed in the material. 
The subscripts i and c refer to the impactor and the impacted cylinder, respectively. In our 
problem, the latter corresponds to the top ply of the composite plate. Thus, the wave speeds 
in the impactor and in the impacted composite may be calculated by the expressions 
where E; is the Young’s modulus of the impactor, and Evr is the transverse ply modulus of 
the composite. 
The time t ,  required to reach omax is taken to be proportional to the time it takes for 
the first wave (caused by the impact) to travel in the composite from the impact surface to 
the back surface of the laminate. Thus, we write 
t ,  = k2tw (9) 
and 
where k2 is a proportionality constant and h is the thickness of the laminate. For plates 
made of sublaminates of different materials, this expression can be generalized to 
where h(j)  is the thickness of the jth sublaminate, cp) is the wave speed in the material of 
that sublaminate, and n is the total number of sublaminates. 
5 
The duration of the stress t f  is approximated by the time during which the impactor 
is in contact with the impacted surface. 
t f  = t d  (12) 
This contact time is taken to be proportional to the contact time for the onedimensional 
problem of a rigid body of mass m impacting a massless, elastic cylinder of length h and 
diameter 2r (twice the nose radius of the actual impactor). 
For the one-dimensional problem the contact time is [6] 
For plates with sublaminates of different materials, the corresponding equation is 
( j )  2 (i) (15) 
(cc 1 Pc 
where h( j ) ,  cp), p p )  are the thickness, wave speed, and density of the jth sublaminate, 
respectively. 
By substituting Eqs. (6)-(14) into Eqs. (1) and (2) we obtain 
A and B are constants 
The constants A,  B, C2-C5, and CT-C10 must be obtained by fitting Eqs. (16a) and 
(16b) to experimentally determined delamination lengths and widths. Therefore, we per- 
formed a least squares fit of these equations to the data given in Reference 2 and summarized 
6 
in Tables 1 and 2. This procedure yielded the following values 
A = 180 B = 20 
C2 = 314 
c3 = 112 
C4 = 114 
Cs = 215 Clo = O  
C7 = 413 
Cg = 413 
Cg = 113 
For these values, the expressions for the delamination length and width become 
215 
As previously discussed, the multipliers (180 and 20) as well as the exponents are constants 
independent of material property, layup, and impact velocity. 
IV. CALCULATION PR.OCEDUR,E 
By using Eqs. (20) and (21) the delamination length and width can be estimated between 
any two plies in the plate. To perform the calculations we must first select the location (or 
locations) at which we wish to determine the dimensions of the delaminations. At each 
location of interest the delamination length and width are calculated by Eqs. (20) and (21). 
The parameters KI ,  and t o  are material properties and their values must be determined 
by static tests (Eq. 3). The value of Q* depends on the material, on the layup, and on the 
location under consideration, and can be calculated by Eq. (4). The flexural rigidities DTe 
and DE depend on the material and on the layup and can be calculated by Eq. ( 5 ) .  Note 
that the location of the delamination is taken into account through D L  and DE.  The impact 
pressure P, the duration of the stress t l ~ ,  and the time t ,  required to reach the maximum 
stress depend on the material, the layup, and on the impact, and are to be calculated by 
Eqs. (7), (lo), and (14). 
7 
The calculations can be performed readily by a personal computer or even on a pro- 
grammable calculator. To perform the calculations we have written a computer code (des- 
ignated as “SIMPACT”). The input parameters required by the code are given in Table 3. 
V. DISCUSSION 
To illustrate the use of Eqs. (20) and (21), delamination lengths and widths were calcu- 
lated for the conditions which existed in the tests reported in Reference 2 and summarized 
in Tables 1 and 2. Essentially, in these tests, Fiberite T300/934 graphite epoxy plates (16 
plies thick, 4 in by 4 in) were impacted by 1/2 in diameter aluminum spheres [2]. The plates 
were inspected by an ultrasonic scanning technique (C-scan) before and after impact. After 
impact the plates were also cut along their axes of symmetry, and were examined by an 
optical microscope. The measured shapes of the delaminations are shown as solid lines in 
Figures 3-6. Delaminations lengths and widths were also calculated by the model (Eqs. 20 
and 21) using the material properties listed in Table 4. The results of the calculations are 
indicated by dotted lines. The comparisons in these figures show that the calculated and 
measured delamination lengths and widths agree closely. 
A more quantitative assessment of the accuracy of the model can be made by comparing 
directly calculated delamination lengths with measured delamination lengths and calculated 
delamination widths with measured delamination widths. Such comparisons are made in 
Figures 7 and 8 and in Tables 1 and 2. Ideally, all the data should fall on the 45 degree solid 
lines. Deviations from these lines may be due to errors either in measurements or in the 
model. It is encouraging that the measured and estimated delamination dimensions agree 
within 25 percent for both the length and the width. 
The foregoing results indicate that the present model can be used to estimate the lengths 
and widths of delaminations under a wide range of conditions. It is hoped that future tests 
will verify the model also to be valid under as yet untested conditions. 
8 
VI. R.EFER.ENCES 
1. Wu, H. T. and Springer, G. S. “Impact Induced Stresses, Strains, and Delaminations 
in Composite Plates,” Journal of Composite Materials, Vol. 22, No. 6, June 1988, pp. 
Wu, H. T. and Springer, G. S. “Measurements of Matrix Cracking and Delamination 
Caused by Impact on Composite Plates,” Journal of Composite Materials, Vol. 22, No. 
6, June 1988, pp. 518-532. 
3. Broek, D. Elementary Engineering fiacture Mechanics, Martinus Nijhoff Publishers, 
the Hague, the Netherlands, 1986. 
4. Tsai, S. W. Composites Design, Think Composites, Dayton, Ohio, 1987. 
5. Springer, G. S. Erosion by Liquid Impact, Scripta Publishing Co., Washington, DC, 
1976. 
6. Timoshenko, S. P. and Goodier, J. N. Theory of Elasticity, McGraw-Hill, New York, 
1970. 
7. Jones, R. M. Mechanics of Composite Materials, Scripta Publishing Co., Washington, 
DC, 1975. 
Johannesson, T. R. and Blikstad, M. “Mixed Mode Fracture of Grapphite/Epoxy Lami- 
nates,” Proceedings of the International Symposium of Composite Materials and Struc- 
tures, Technomic, Lancaster, PA, 1986. 
533-560. 
2. 
8. 
9 
APPENDIX 
EXPR.ESSION OF THE R,EDUCED STIFFNESS MATRIX Q 
The components of the reduced stiffness matrix Q are [7] 
Qss = Gzy (A.4) 
Qee = Q,, cos4 8 + 2(Q,, + 2QSs) sin2 8 cos2 8 + Qyy sin4 6 (A.5) 
where E,, is Young’s modulus in fiber direction, Eyr is Young’s modulus transverse to fiber 
direction, vzy and vyz are Poisson’s ratios, and G,, is the shear modulus. 
By substituting the appropriate expressions for Q,,, Qyy, and Qee, we obtain 
Q;, - [Qi, cos: +2 (Q& + 2&!3 sin2 8 cos2 8 + Q;y sin4 61 
Q!, - Q& Q* = ( A 4  
The superscripts I and u refer to the lower and upper plies, respectively. 
10 
c-l 
p! t 
8 
000 
000 
9 9 9  
000 
000 
9 9 9  
4-0 
N C a @ a  
3 e l M  
444 
3 
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n 
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4 4  
n 
0" 
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\ 
Y 0"
2 
=#k 
000 z z z  
2 
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4 a  
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m o o  b m m  P - o m  
n 
000 
000 
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3 
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44 
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31 
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000 
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Table 3 
SUMMARY OF T H E  INPUT A N D  OUTPUT PARAMETERS 
FROM T H E  “SIMPACT” CODE 
Input Parameters 
1) Impactor 
a) Mass, m 
b) Nose radius, 
c) Density, pj 
d) Young’s modulus, Ei 
e) Velocity, v 
2) Composite Plate 
a) Ply layup 
b) Material properties (see Table 4) 
Output Parameters 
1) Final delamination length 
2) Final delamination width 
Table 4 
MATERIAL PROPERTIES OF FIBERITE T300/934 
GRAPHITE/EPOXY (from Reference 2). 
Ply thickness, h 
Density, p 
Longitudinal Young's modulus, E,, 
Transverse Young's modulus, E,, 
Shear modulus in z-y direction, G,, 
Poisson's ratio in the s-y direction,' v,, 
Transverse tensile strength, STT 
Critical stress intensity factor of mode I,* KI, 
6.250 x in 
5.548 x lbm/in3 
2.109 x lo7 psi 
1.450 x lo6 psi 
8.251 x lo5 psi 
8.006 x lo3 psi 
1.183 x lo3 lbf/in'-' 
0.3 
'From Reference 4. 
*From Reference 8. 
I 
RADIUS: r 
VELOCITY: v IN-PLANE 
PRELOAD 
OR PRESTRAIN 
- .  
BOUNDARIES: 
SIMPLYSUPPORTED 
OR CLAMPED 
Figurc 1. Description of the problem 
Impactor 0 
1 
T I Upper layer 
Lower A 
Figure 2. Illustration of the delamination 
.i 
r - - - -  1 
I 
r--- 1 
VI 0 m 
I 
VI . - 
0 
VI - 
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VI 
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I 
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Delaminations in composite plates caused by non-penetrating impact

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