-
Composite Structures 28 (1994) 61-72 '~ ; ' "' : L
Impact response of laminated composite plates: Prediction and
verification
H. V. Lakshminarayana, R. Boukhili & R. Gauvin Department of
Mechanical Engineering, Ecole Polytechnique de Montreal, Canada
Methods and procedures for predicting the impact response of
laminated com- posite plates using a commercial finite element
system are described. Results of element evaluation, procedure
verification and a correlation study are presen- ted and discussed.
The need for a hybrid experimental-numerical approach and combined
geometric and material nonlinear finite element analysis is iden-
tified. A methodology for the prediction of delamination (onset and
growth) is outlined.
1 INTRODUCTION
Laminated composite plates are easily damaged by impacts,
especially those normal to the plane of the laminate. Systematic
study of such problems can be divided into three distinct areas:
structural mechanics, damage mechanics and residual strength
prediction. A study in that order will provide a mechanistic basis
for both design and assessment of damage tolerance.
Reliable and accurate prediction of the impact response of
multilayered anisotropic plates cover- ing a wide range of
parameters is the focus of this study. Impact response means
contact force his- tory, deformation history, surface strain
history, stress distribution across the laminate thickness
(including interlaminar stresses), nature and extent of damage and
stiffness and strength loss associated with that damage. Parameters
signifi- cantly influencing the impact response include impact
velocity/energy, material system, ply orientation and stacking
sequence, plate geometry (shape, size and thickness), wall
construction (solid laminate, integrally stiffened, and sand-
wich), support conditions, pre-stress state and initial curvature
(curved panels).
A review of previous work is available in a monograph L2 and two
review papers, 3,4 which provide a background to the present study.
A critical assessment of this vast literature, with particular
reference to the focus of the present study, can be summarized as
follows. (1) Finite element analysis of linear and transient
structural behaviour has been the focus of intense research,
61
while few studies have included nonlinear effects. Almost all of
them make use of special-purpose programs. Such programs, valuable
in their own right, are difficult to access, are not well docu-
mented and hence are difficult to use by the practitioners. (2)
Assessment of convergence and accuracy of the deflection history,
stress distribu- tions and damage zones calculated by the Finite
Element Method (FEM) have not been given explicit attention. (3)
Experimental investigations that document the response recorded
during the impact test are rather limited. The majority of them
provide a qualitative/quantitative descrip- tion of the accumulated
damage but not its growth. This information is vital to perform
corre- lation studies. (4) Few investigators have consid- ered
full-scale components. The majority of the reported studies use
generic structures such as beams, circular plates, square plates
and cylindri- cal panels. There is a real danger in extrapolating
conclusions drawn from studies on test specimens to real life
components.
It is more appropriate to use commercial FEM systems. They are
widely distributed, well docu- mented and user friendly. However,
there is a need to verify the accuracy of material models, finite
elements and analysis procedures in such systems before using them
for the intended appli- cation, namely, prediction of the impact
response of composite plates. The predictability issue itself
demands a correlation study. This, in fact, is the aim and scope of
the present study.
A brief description of the specific methods and procedures used
is given in the next section. In
Composite Structures 0263-8223/94/S07.00 1994 Elsevier Science
Limited, England. Printed in Great Britain
-
62 H. V. Lakshminarayana, R. Boukhili, R. Gauvin
succeeding sections, results of element evaluation, procedure
verification and correlation study are presented and discussed. The
presentation con- cludes by identifying directions for further
work.
2 METHODS AND PROCEDURES
Numerical results for this study were generated using ABAQUS --
a general-purpose finite ele- ment code with emphasis on nonlinear
applica- tions. ~ This program is capable of modelling multilayered
anisotropic materials. It provides elements suitable for dynamic
analysis of compo- site plates and shells taking into account
bending- membrane coupling and transverse shear deformation
effects. Among these, the S8R, an isoparametric quadrilateral
plate/shell element, is employed in the present study. This element
has eight nodes and six engineering degrees of free- dom at each
node. The user can specify within each element an arbitrary number
of layers, each with its own thickness, ply orientation and ortho-
tropic elastic properties. The formulation uses a value of 5/6 for
the shear correction factors as default. However, the user has the
option to use any other value through independent input of
transverse shear stiffness. The element output includes membrane
stress resultants, bending stress resultants and transverse
shear-stress resultants, either at the nodes or at the four inte-
gration points. The user can also request ply-by- ply stresses at
the integration points and at a maximum of 3 section points within
each ply. It should be noted that the formulation of S8R does not
ensure the continuity of interlaminar normal and shear stresses at
the interfaces between plies.
ABAQUS provides two procedures for calcu- lating the response of
structures subjected to impulsive loads. They are modal analysis
and dynamic analysis by direct implicit integration of the
equations of motion. The modal analysis is limited to linear
transient behaviour. Its converg- ence and accuracy is dependent on
the number of natural modes considered in the analysis. The pulse
shape and the duration (t~) for which the contact force acts (in
comparison with the period T~ of the fundamental mode of free
vibration of the structure) strongly influence the response cal-
culated by the modal method. For complex pulse shapes and for
(t,,/Ti)'~ 1, a very large number of modes is required for
convergence. Unfortu- nately, accurate determination of the
eigenvalues and eigenfunctions associated with higher modes
places very heavy demands on computational resources by the FEM,
Incidentally, eigenvalue extraction in ABAQUS is done using the
sub- space iteration method. The transient response calculation by
the direct integration of the equa- tions of motion is applicable
to linear as well as nonlinear structural behaviour. When applying
this step-by-step method, the time step At should be selected with
caution, because a system of non- linear algebraic equations must
be solved at each time increment. This is done in ABAQUS itera-
tively by using Newton's method or, if preferred, the
quasi-Newton's method. This time stepping, nonlinear
equation-solving procedure is computa- tionally expensive. The
principal advantage of this procedure is that it is unconditionally
stable, which means that there is no mathematical limit on the size
of the time increment that can be used. In practice, At should be
small enough to ade- quately define the history of excitation, its
value being chosen on the basis of the shortest period which
corresponds to the highest natural mode likely to contribute to the
response.
A recommended methodology (' for determin- ing the contact force
history is to measure the local contact stiffness of the composite
plate in static tests and use this in conjunction with a finite
element model. This approach, however, could not be considered
truly predictive since it requires fabrication of the plate and
static indentation tests for every impactor under consideration.
Alternat- ively, it can be measured during impact tests using
suitable instrumentation and data acquisition sys- tem. This
approach is applicable to the nonlinear response also.
The nature and extent of impact-induced damage is estimated by
first calculating the dyamic stresses and their spatial
distribution in the laminate and incorporating these in appropri-
ate failure criteria. Multiple matrix cracks, dela- minations and
fiber breaks are the failure modes observed after impact tests.
These failure modes and complex interactions between them compli-
cate the prediction of damage. The tensor poly- nomial failure
criterion proposed by Tsai and Wu 7 is employed in the present
study to calculate the failure index (FI) given by
FI=k' lo , + F2o~ + F~o~ + F11o~ + F2:o!
+ F(~,o~ + _F~2o 102
where k~j are the strength tensors; 7 o~, o 2 are the lamina
stresses in the fiber direction and trans- verse direction
respectively; and o~, is the in-plane
-
Impact response of laminated composite plates 63
shear stress. Loci of points at which FI = 1 define the damage
zone. The failure modes are identified using the maximum stress
criteria. 7
The stiffness and strength loss due to impact- induced damage
are not at present truly predict- able. A suggested approach
involves flexure tests. ~
Evaluation of the accuracy of the S8R element for the analysis
of composite plates in general is presented in the next section. We
identify three distinct procedures: (1) modal analysis (procedure #
1); (2) linear and transient response analysis (TRA) (procedure
#2); and (3) nonlinear and transient response analysis (NLTRA)
(procedure # 3). The convergence and accuracy of each one of these
is verified in a section entitled procedure verification. Finally,
a critical assessment of their predictability with particular
reference to the impact response of laminated composite plates is
presented in Section 5.
3 ELEMENT EVALUATION
Application of the full set of test problems pro- posed in Ref.
9 to the S8R element is not pre-
6 I I Ill Ill _ W(O,O) E,h~ 10 s W= Pa"
5111111 I t II IIIIII
CPT Solution
1
0 -1 -2 -3
10 10 10 h/a
Fig. 1. Effect of (h/a) on the central deflection predicted
using the S8R element (8 x 8 mesh, whole plate).
sented here. The results showed that a particular problem, a
homogeneous, anisotropic, clamped, square plate under uniform
pressure, provided an intensive measure of element performance.
This problem, shown as the insert in Fig. 1, was chosen to evaluate
the combined effect of material aniso- tropy and shear deformation
on the accuracy of the S8R element. Numerical results were obtained
for a unidirectional laminate, made of a high- modulus
graphite/epoxy composite (EI/E 2 = 40, E2 = 5.17 GPa (0.75 x 106
psi), Gi2--- 3.10 GPa (0"45 x 10 ~' psi), G23/GI2=0"8, vl2=0.25 ),
for a ply orientation of 45 and for various values of the thickness
ratio (h/a). Computed results are com- pared with the converged
solutions given in Ref. 9. The agreement is very good for
displacements as well as stress resultants provided (h/a)> 0.01.
The effect of h/a on the predicted central deflec- tion is shown in
Fig. 1, which also has the classical plate theory (CPT) solution
for comparison. For a given thickness ratio, the inaccuracy
associated with the omission of transverse shear deformation
effects is obtained from this figure. Obviously, the accuracy of
the S8R element deteriorates for (hi a)
-
64 H. V. Lakshminarayana, R. Boukhili, R. Gauvin
P(t)
P max
P(t)
\
I I I /
I t 1
',\
t 0
h
w(t)
Contact Force History
a
h = 5.1 mm a = 203.2 mm
Pr.~x = 27.4 Kg to = 0.2 ms
t 1 = 0.25 t o
t
i i
0
90 '
0
90
0
0
90
0
90
0
Ply Orientation/Stacking Sequence
/;
/ i /
j / - J
J
Finite Element Model (Quarter Plate)
Fig. 2. Test problem for procedure verification.
improvement is noticed using a finer mesh and a smaller time
step, indicating that a converged and accurate solution has been
obtained. Figure 3 also indicates that two pulse shapes (triangular
and half sine-wave) produce almost identical results. Since the
half sine-wave pulse can be represented by an analytical
expression, it is ideally suited to create benchmarks for the
impact response of composites.
For the same problem, Fig. 4 shows displace- ment-time histories
calculated using procedure # 1 (modal analysis). There is no sign
of converg- ence as the number of modes used is increased from 10
to 15. These results do not show any comparison with the reference
solution in Ref. 11. Not only is the displacement history
different, the maximum amplitudes differ by one order of
magnitude. This observation is substantiated by the findings of
a recent round robin study.~2 It is noted that whenever the
duration of impact is very small (in comparison with the period of
the funda- mental mode of free vibration of the target), the number
of modes to be considered for converg- ence may be so large as to
be computationally inefficient. It appears that reliable prediction
of the impact response is outside the domain of the modal
analysis.
For the problem specified in Fig. 2, a com- parison of the
displacement histories calculated using procedure #3 (nonlinear and
transient response by direct implicit integration) and proce- dure
# 2 is made in Fig. 5. The two solutions are in fact identical
because at this low load level the resulting deflections are very
small (in comparison
-
0.22 0.24
0.2
0.18
0.16
0.14
~ 0.12
d o.1
0.08
I~1 0.06
5 0.04
- 0.02 G) E o,
0 -0,02 GI (3. .~_ -o.04 0 -o.os
-0.08
-0.1
-0.12 0
Impact response of laminated composite plates 6 5
P(t
Pmax = 27.4 Kg to= 0.2 ms t, = 0.25 t o
\ , t
11 I 0 I
0.5 1 1.5 2
Time t(miliseconds) Fig. 3.
0.22
0.2
0.18
0.16
-0.06
"- 0.14
"*"= 0.12 0 0 0.1 v
0.08 II
I~: 006 c- 0.04
E 0.02 0 0
oo -0.02 =~
a -0.04
-0,08
-0.1
Number of modes =10
D
r O
; f / i \ fltl= I, t l -o ~ ' i !1 Prnax = 27.4 Kg ' 1 ~_o (;t~
1, to= 0"2,ms
0 0.5 1 1.5
Time t(miliseconds) Central displacement history calculated
using procedure # 2.
2
1.8
1.6
1.4
1.2
1
"~. o.e o
-
0.24 0 .24
0.22
0.2
0.18
0.16 e.-
0.14
C~ 0.12 0
0.1
II 0.08
15 006 .. E 0.04
0 0.02
~. 0 .oo E3 -o.o2
-0.04
-0.06
-0.08
-0.1 0
Fig. 5.
Procedure #2 6t = 0.05 ms
7
P(t) to= 0.2 ms _
/ Pmax . . . . . . . . . .
o?,,J,,2t 0 f
I I "o I
0.5 1 1.5
T ime t(miliseconds)
Procedure #3 6t = 0.05 ms
0.22
0.2
0.18
0.16
~ 0.14 d o.12
0.1
0.08
g 0.04
0.02
~. o -0.02
-0.04 --
-0.06 I -0.08
-0.1 J 0
66 H. V. Lakshminarayana, R. Boukhili, R. Gauvin
I
0.5
P(t) = Pmeeqin (rrt/t o)
Pm~ = 27.4 Kg
VO 3 m/s O= 0.2 ms
l 4
1 1.5
T ime t(mi l iseconds)
Comparison of displacement-time histories calculated by
procedures # 2 and # 3 (impact velocity = 3 m/s).
with the thickness h) and hence nonlinear effects do not show
up.
Impact velocity is a very important parameter controlling the
response. Figures 6 and 7 provide a comparison of displacement time
histories calcu- lated using procedure # 2 and procedure # 3 at
V~=10 m/s and V0=30 m/s, respectively. Obviously, the responses
predicted by the two procedures do not agree with one another. Both
the displacement-time history as well as the maxi- mum amplitudes
predicted differ rather signifi- cantly. Basically, the load levels
are such that resulting deflections are of the order of the plate
thickness and therefore procedure # 3, which includes nonlinear
effects, is more appropriate. Unfortunately, the accuracy of the
numerical solu- tions presented in Figs 6 and 7 could not be veri-
fied due to the nonavailability of reference solutions.
Procedure # 3 is therefore more appropriate for numerical
solutions of impact tests. In fact, for complex structures, this
procedure may provide more cost-effective information than
experimen- tation. However, there is a need to further validate the
predictability of this procedure for the impact
response. This aspect is addressed in the next section.
5 CORRELATION STUDY
The predictability aspect of the procedures veri- fied in the
previous section, with particular refer- ence to the impact
response of composite plates is assessed using a bench mark. Bench
marks are fully specified and standard problems, which resemble
instances found in practical applica- tions, and for which
reference solutions have been obtained using both
analytical/numerical and experimental methods.
The bench mark chosen for this study was created by Aggour and
Sun. j3 It consists of a cross-ply laminated, E-glass/epoxy
composite (Ej=38"6 GPa (5"6x106 psi), E2=I0.34 GPa (1.5106 psi),
G2=4"14 GPa (0.6x106 psi), v,2 =0.25) square plate with all edges
clamped. The geometric parameters, ply orientation/stack- ing
sequence, and the finite element discretization used in the
computations are shown in Fig. 8. The contact force history
corresponds to an impact by
-
1
0.9
0.8
J::
o 0
II
,4.. I
(9 E
o
0.7
0.6
0.5
~, 0.4 d ~ 0.3
0.2
t 0.1 0
~ -0.1
~ -0.2
~. -0.3
i'~ -(I.4
-0.5
-0.6
-0.7
-0.8
Procedure #2 6t = 0.05 ms
o to
0 0.5 1 1.5 2
0.9
0.8
0.7
0.6
0.5 t-
0.4
C) 0.3 O
0.2
II 0.1
e- (9 E -0.1 (9 -0.2
~. -0.3 u) ~,~ -0.4
-0.5
-0.6
-0.7
-0.8 0
f Procedure #3 5t = 0.05 ms
~-~ ~W(t) ~ I h
I
0.5
P(t)= PmaxSin (~t/t 0) V 0 = 10 m/s Pm~ = 91.2 Kg
t 0= 0.2 ms I I
1 1.5 2
Time t(miliseconds) Time t(miliseconds) Fig. 6. Comparison of
displacement-time histories calculated by procedures # 2 and # 3
(impact velocity = 10 m/s).
0 [
4 2
-2 Vo= 30 m/s
Pro= = 364.8 Kg
to= 0.2 ms
-3 0
Procedure #2 : LTRA
P(t) 1
P(t) = Pm=Sin (nl/t o )
Prr,= = 364.8 Kg
Vo= 30 m/s t -- 0.2 ms
0
1.8
1.6
1.4
1.2
1
0.5 1 1.5
Time t(miliseconds)
- 0.8
c:) 0.6 0
O.4
II 0.2
t5 o, c (9 -0.2 E (9 -0.4 o 0 -0,6
-~ -0.8 a -1
-1.2
-1.4
-1.6
Impact response of laminated composite plates 6 7
Procedure #3 : NLTRA
6t = 0.05 ms
0.5 1 1.5 2
Time t(miliseconds) Fig. 7. Comparison of displacement-time
histories calculated using procedures # 2 and # 3 (impact velocity
= 30 m/s).
-
68 H.V. Lakshminarayana, R. Boukhili, R. Gauvin
P ,x
z [ P(t) r
- - . ~ X
a
P(t) .... \
/ '\ / '\
/ ',,
\ \ \
~_t t
Contact Force Histor~ P(t) = Pm~xSin (nUt o ) P~x = 310.1 Kg to=
0.25 ms
I ho o
/iiiiiii iiiiiii ii 9o! I!iiii!i!~iiiiiii!iiiiiii!ilili
iiiiiiliiiiiiiiiii!iiiiii!i
~i!i!i!ii!~!i!iiiiiiiiii~iii~iiiililiiiiiilililiiii!iiiiiiiiiJ 9o
o
t 0
Ply Orientation/Stacking Sequence
a = 139.7 mm
h=4.1 mm
J I
A B E H
Finite Element Model (Quater Plate).
Fig. 8. Test problem for correlation study.
a steel cylinder (diameter 9.5 mm and length 25.4 mm) at a
velocity E, = 22.6 m/s.
The calculated central displacement history using procedure # 3
is presented in Fig. 9. Results using procedure # 2 and test data
taken from Ref. 13 are also included in the same figure to enable a
three-way correlation. Results obtained using linear finite element
analysis by Aggour and Sun ~3 are in close agreement with those
obtained in the present study using procedure # 2. Furthermore, the
responses calculated using procedure # 3 and procedure # 2 are
identical, indicating that non- linear effects are negligible. It
is gratifying to note that computed results closely follow the
central deflection measured during an impact test for t-
-
Impact response of laminated composite plates 6 9
Procedure #2
6t = 0.025 ms
Present Study
Testdata
0.8
0.7
0.6
0.5
0.4
c~ 0.3 0.2
0.1
[ oc ~ -0.1
~ -0.2
~ -0.3
~ -o.g
-0.5
-0.6
-0.7
-0.8 0.5
0.9
0.8
0.7
0.6
0.5
~, 0.4
~ 0.3
0.2
I o.1 0{ g
~ -0.1
~ -0.2
~ -0.3
i~ -0.4
-0.5
-0.6
-0.7
-0.8 0
Procedure #3 6t = 0.05 ms
Prese~ Study l Testdata ]
l 0.5
Time t(miliseconds) Time t(miliseconds)
Fig. 9. Comparison of measured and predicted central
displacement histories.
1.8
1.6
1.4 l'k 0.8
0.6
c5 0.4
\
! -0.2
(1) -0.4
-0.6 ~ \' "t~r ~ 0 \ \ ~ -o8
-1
-1.2
-1.4 ~N -1.6
-1.8
Fig. 10.
0.5
Time t(miliseconds)
Procedure #2
VO= 40.0 m/s []
Vo= 30.0 m/s
Vo-- 2226 m/s
1
0.9
1.8
1.6- 1.4- 1.2
1 0.8 ...
0,6 o- o - 0.4
0.2 II i .,.., -0.2 1 e.-
E -o.4 ~ (:~ -0.6
~. -0.8
i~ -1 -1.2
-1.4
-1.6
-1.8 0.5
Procedure #3
Vo= 40.0 m/s
Vo= 30.0 m/s
Vo= 226 m/s
ji / ' / /
Time t(miliseconds)
Comparison of predicted displacement-time histories by linear
and nonlinear analysis procedures.
-
70 H. V. Lakshminarayana, R. Boukhili, R. Gauvin
1.4
.. .6
0,4
O" I
-02
1.6 Procedure #2 V = 40 m/s 14
0 t = 0~25 ms ,2 [ t= o2o ms i I t= 0z15 ms , i x:: ~f
"~'I~ 0.8!
0.6 I
0.4
0.2
0.1 -02 0.2 0.3 0.4 05 0 0.1 0.2 0.3 0.4
x/a x/a
Comparison of predicted displacement distributions by linear and
nonlinear analysis procedures. Fig. 11.
Procedure #3 V = 40 m/s
0
[t = o ms/ /t = 0.10 msl
0.5
the former cannot be truly calculated while infor- mation on the
latter is not available. The progres- sive failure finite element
analysis outlined is in fact a topic for further research.~4
Reliable prediction of displacement history as a function of
impact velocity (Fig. 10) constitutes only a fraction of the impact
response story. Spa- tial distribution of displacement (Fig. 11 ),
stress distribution across the laminate thickness (Fig. 12) and
damage growth (Fig. 13) are in fact even more important. The FEM in
general and ABAQUS in particular is able to provide such results.
Indeed a major problem is to find experimental techniques that can
provide data that are equally detailed to validate the results
presented here.
6 CONCLUDING REMARKS
A hybrid experimental-numerical approach is necessary to predict
the impact response of lami- nated composite plates covering a wide
range of the parameters involved. Experimental determi- nation of
contact force history is essential if the structural behaviour is
nonlinear. Nondestructive test methods are indispensable for
characterisa- tion of damage. Post-impact tests are needed to
determine residual stiffness and strength. The combined geometric
and material nonlinear finite element analysis capability required
for this pur- pose is available in commercial finite element
systems. However, accurate constitutive models
for composite laminates with multiple matrix cracks, fiber
breaks and delaminations are not yet available.
Delaminations are often the primary, life-limit- ing failure
modes. The task of developing methods and data for predicting the
onset and growth of delamination due to impact was not considered
in the present study. Accurate evalua- tion of interlaminar normal
and shear stresses a verified interface criteria and test methods
to measure material properties associated with such a criterion are
the prerequisites for prediction of delamination onset. It is to be
noted that an appropriate finite element model for this purpose
should be based on a lamination theory that enforces the continuity
of interlaminar normal and shear stresses at ply interfaces,~5 and
these are not yet implemented in commercial FEM systems such as
ABAQUS. For the prediction of delami- nation growth, the fracture
mechanics concept ~6 is indispensable. This procedure involves
calcula- tion of energy release rates associated with delam-
ination growth, the use of a mixed-mode fracture criterion and test
methods to measure interlami- nar fracture toughness. A drastically
different finite element modelling approach is necessary for the
numerical determination of energy release rates. These are topics
deserving further research.
This study sets the stage for confident applica- tion of a
commercial FEM system ABAQUS for numerical simulation of impact
tests conducted on composite plate and shell structures.
-
Impact response of laminated composite plates 71
at t = 0.2 ms, Vo= 40 m/s
__ 0
0'
ii:iii:!ii!:iiiii:iiiiii:iii:ii:i!ii:!iiii:iiiiii:iiiii!i:!:iiiiiiiii
90' ?::::..:.v....:.v..::.v.~..` ..v:.......v.....v.v...v.`
....:..:......` .:.v..::...
I1 ~ : ~ : ~ : ~ : ~ : ~ : ~ 90 '
~:~:~:~:~:~: :~:~:~:~ 90 ' I
......................................................... ~::..`
.:......` .:.v .`::~.:~.....:.v:.....v.....v .`` .` .` .:~.~.`
.:::......::.v..` .` .` .. i
:::~::~::~:~:~:~:~::~:~:~:~::~:~:~::~::~:~:~::~:~:~:~:~ 90 '
..:.v..` .:....::.` ....:::......` .:............:.v...v....:..`
.........v.....v:...v::.v
O'
-- O*
O
0
iiiii:!:!!ii!iiii:i:iiiii:iiiiiiiiii:i:iiii:iiiiiiiiiiiii:iiiiiii!
9o' (.:.~.....~.~.~.~.~::...~...~....~.....~...~.....~............`
....~..~..::...` .~.` ..` .~`~....~.` ....~ h
i!!!!:!i!i!!!!:ii!!i!ii!!:i!!i!!!!i!!i:!ii:ii!!!:!!!:ii!:!:!!ii!ii
9O'
~.~.):.~.~.~.)):~)~.)))~.~.~` ~.~.~.33)&:.:.:.~.))3~`
~.~.)~.)~.))~.:.:.:.~.~.333~:.~ 90' [ " 0' 0
h
!!!!!!!!!!i:!:i:!i:iiiiii!ii:i:i:iii!!i!!!!!!:iiiii:iiiiiiiiii:iiii
9o' :......:.....v:.v...v.......v~..`
:..:.v...~`v.......v.v......:..::..:.v....` ....` ....: ~0'
0
Fig. 12.
Procedure #3 ]
-1 i I I I
o
01/X
q
-10 ~/y
z/h
.1 o
otis Stress distribution across laminate thickness.
z/h
X = 1022.3 MPa
I
1
Y = 27.3 MPa
lO
S = 40.9 MPa
t =0.05 ms
~] t = 0.15 ms 900
~ t=O.2Oms i 0
V0= 40 m/s Procedure #3
Fig. 13. Growth of impact damage.
ACKNOWLEDGEMENTS
This research was supported by the Natural Sciences and
Engineering Research Council of Canada and la Direction de la
Recherche de l'Ecole Polytechnique de Montr6al.
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