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The coupled effect of manufacturing uncertainty and a critical service-life damage condition (delamination) is investigated on the natural frequencies of laminated composite plates. In general, delamination is an unavoidable phenomenon in composite materials encountered often in real-life operating conditions. We have focused on the characterization of dynamic responses of composite plates considering source-uncertainty in the material and geometric properties along with various single and multiple delamination scenarios. A hybrid high dimensional model representation based uncertainty propagation algorithm coupled with layer-wise stochastic finite element model of composites is developed to achieve computational efficiency. The finite element formulation is based on Mindlin's theory considering transverse shear deformation. Numerical results are presented for the stochastic natural frequencies of delaminated composites along with a comprehensive deterministic analysis. Further, an inevitable effect of noise is induced in the surrogate based analysis to explore the effect of various errors and epistemic uncertainties involved with the system.
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1
Effect of delamination on the stochastic natural frequencies of
composite laminates
T. Mukhopadhyaya*, S. Naskarb, P.K. Karshc, S. Deyc, Z. Youa
aDepartment of Engineering Science, University of Oxford, Oxford, UK
bSchool of Engineering, University of Aberdeen, Aberdeen, UK
cMechanical Engineering Department, National Institute of Technology Silchar, India
*Corresponding author’s e-mail: tanmoy.mukhopadhyay@eng.ox.ac.uk
Abstract
The coupled effect of manufacturing uncertainty and a critical service-life damage condition (delamination) is
investigated on the natural frequencies of laminated composite plates. In general, delamination is an
unavoidable phenomenon in composite materials encountered often in real-life operating conditions. We have
focused on the characterization of dynamic responses of composite plates considering source-uncertainty in
the material and geometric properties along with various single and multiple delamination scenarios. A hybrid
high dimensional model representation based uncertainty propagation algorithm coupled with layer-wise
stochastic finite element model of composites is developed to achieve computational efficiency. The finite
 Numerical results
are presented for the stochastic natural frequencies of delaminated composites along with a comprehensive
deterministic analysis. Further, an inevitable effect of noise is induced in the surrogate based analysis to
explore the effect of various errors and epistemic uncertainties involved with the system.
Keywords: delamination; manufacturing uncertainty; laminated composite plate; surrogate based finite
element method
1. Introduction
Laminated composites have gained preference in various engineering applications such as aerospace,
naval, automobile, micro-electro-mechanical-systems (MEMS) and civil structures due to high strength and
stiffness with weight-sensitivity, increased toughness, mechanical damping, as well as tailoring of structural
properties. In the recent year, wide application of composite materials has drawn an increased attention to its
operational reliability and safety. The exhaustive usage of such structures has warranted the detail
understanding of damage modes and their consequences in global structural responses. One of the most
2
significant setbacks of fibre-reinforced polymer composites is the propensity to onset, growth and propagation
of delamination. In other words, one of the principle modes of failure in laminated composites is the
delamination or separation of layers along the interfaces. In general, delamination occurs due unaccounted
tension and shear developed at inter-laminar zones due to various factors such as free edge effect,
discontinuities in structural elements, localized disturbances during manufacturing and operating conditions.
The presence of inter-laminar debonding or delamination is often laid buried between the layers as it is hidden
from superficial visual inspection. Subsequently the growth and propagation of delamination in conjunction
with other modes of damage also remain unaccounted whereas it may reduce stiffness of the structure
drastically leading to failure and instability of the structure.
Besides delamination in composites, these advanced materials are susceptible to various forms of
source-uncertainties in material and structural attributes due to complex manufacturing process (such as intra-
laminate voids and excess matrix voids, excess resin between plies, incomplete curing of resin, porosity,
variations in lamina thickness, fibre orientation and fibre properties) and complicated design requirements.
Such uncertainties affect the global structural responses significantly. The coupled effect of delamination and
inevitable source-uncertainties can drastically influence the structural responses computed based on
deterministic assumptions.
Fundamental principles of stochastic mechanics are required to understand the probabilistic dynamic
behavior of delaminated composites. It involves significant challenges to apply the original concepts
developed for isotropic materials to laminated composites where material and geometric anisotropy prevails.
Hence it is essential to investigate the complexities arising from unknown sources initiating the debonding of
constitutive layers and its uncertain means of propagation. The free vibration characteristics of delaminated
composites can show drastic fluctuations from the computed deterministic values due to randomness in
material and geometric properties and damages incurred during the service-life depending on number, size or
shape and location of delamination. Moreover, due to the involvement of random system properties, the
vibration characteristics of such structures with delamination can behave differently in different modes.
A plenty of research work is reported on deterministic free vibration analysis of laminated composite
plates and shells [1-10]. The aspect of delamination in composites has also received adequate attention in the
deterministic domain [11-39]. Stochastic analysis of composite and sandwich structures considering source-
uncertainty is found to be studied by many researchers including the aspects of multi-scale analysis,
3
optimization and reliability assessment [40-59]. However, the compound effect of delamination and source-
uncertainty has not been investigated yet for the dynamic responses of composite structures. In this paper, we
aim to investigate this coupled effect on the natural frequencies of composite plates.
A careful review of literature concerning uncertainty quantification in composites shows two
prominent approaches: perturbation based approach and Monte Carlo simulation based approach. The major
drawback in a perturbation based approach can be identified as the requirement of intensive analytical
derivation and lack of the ability to obtain complete probabilistic description of the response quantities.
Moreover, this approach is valid only for a low degree of stochasticity in the input parameters. A Monte Carlo
simulation (MCS) approach for uncertainty quantification does not have these critical lacunas. But the MCS
approach is computationally very demanding because of the requirement of carrying out large number (~104)
of repetitive simulations corresponding to a random set of input parameters. For the analyses of composite
structures including the effect of delamination, even one such simulation is normally very computationally
intensive and time consuming. In such situation the panacea is a surrogate based Monte Carlo simulation [60-
64], which is adopted in this study.
In the uncertainty quantification of responses by employing surrogate based approach, the original
finite element model is replaced by an efficient pseudo simulation model, which is effective but economical.
The surrogate models get necessary information about the nature of the response outputs by algorithmically
selected design points drawn from the entire domain. An efficient hybrid high dimensional model
representation (HDMR) [65-66] based uncertainty propagation algorithm coupled with layer-wise stochastic
finite element (FE) model of the delaminated composites is developed in this paper for the stochastic free
vibration analysis. The optimal design points are drawn from a pseudo random Sobol sequence [67]. In this
context, another source of uncertainty needs be accounted in the analysis. The information acquired from the
selected design points (input-output dataset) for forming surrogate models is a second source of uncertainty
besides the conventional source-uncertainties in material and geometric parameters (refer to Figure 1). In the
present study, simulated noise is introduced to account for such second source of uncertainty that can be
tantamount to incorporating measurement error of responses, modelling and simulation error and other
       
first attempt for a surrogate based dynamic analysis of delaminated composite plates (refer to Figure 2)
coupled with random material and geometric properties including the effect of inevitable noise. After the
4
Fig. 1 Surrogate based stochastic analysis under the influence of noise
(a)
(b)
(c)
Fig. 2 (a, b) Composite plate showing forces and moments (c) Delamination in composite plates at crack tip
introduction section, this paper is organized as, section 2: brief description of the mathematical model for
stochastic dynamic analysis of delaminated composite plates; section 3: Hybrid HDMR based FE algorithm
for layer-wise stochastic modelling of delaminated composites including the effect of noise (detail
mathematical formulation of HDMR based surrogate modelling is provided as APPENDIX); section 4: results
and discussion; section 5: summary and perspective; section 6: conclusion.
2. Stochastic dynamics of delaminated composite plates
hLb
considered as shown in Figure 2. According to the first-order shear deformation theory, the displacement field
of the plates is described by
5
),(),(),,( yxzyxuzyxu x
),(),(),,( yxzyxvzyxv y
(1)
where,
u
,
v
and
w
denotes displacements corresponding to the reference planes, while
x
and
y
represents x
and y axes rotations respectively. θ’ about the x-axis.
For the shell, the constitutive equations is given by [68]
})]{([}{
DF
(2)
where
 
is a typical representation of stochasticity. Here the force resultant is given by
T
yxxyyxxyyx HHMMMNNNF })(,)(,)(,)(,)(,)(,)(,)({}{
/2
/2
{ } { , , , , , , , }
T
h
x y xy xz yz xy xz yz
h
F dz
 



(3a)
The strain vector can be expressed as
T
yzxzxyyxxyyx kkk },,,,,,,{}{
(3b)
The elements of elastic stiffness matrix
)]([
D
is given by
n
kk
z
zonijijijij jidzzzHDBA k
k
1
26,2,1,],,1[])}({[)](),(),([
1
n
k
z
zkijsij jidzHS k
k
15,4,])([)]([
1
(4)
The parameter
s
represents the shear correction factor (in the present study
s
=5/6) and
])([ ij
H
denotes
off-axis elastic constant matrix elements and expressed by
T
onijoffij THTH
)]([])([)]([])([ 1
1
1
for i, j = 1,2,6
T
onijoffij THTH
)]([])([)]([])([ 2
1
2
for i, j = 4,5
(5)
22
22
22
12
2
)]([
dccdcd
cdcd
cddc
T
and
cd
dc
T)]([ 2
(6)
in which
)(
Sinc
and
)(
Cosd
, wherein
)(
is random fibre orientation angle.
66
2212
1211
00
0
0
)]([
H
HH
HH
Honij
for i, j = 1,2,6
5545
4544
)]([ HH
HH
Honij
for i, j=4,5
(7)
Where
6
2112
1
11 1vv
E
H
,
2112
2
22 1vv
E
H
,
2112
212
12 1vvEv
H
,
135523441266 ,GHandGHGH
In the FE formulation, an isoparametric quadratic element with 64 elements and 225 nodes is considered in
this paper and each node has five degrees of freedom (DOF) (three translations and two rotations). For
composite plate the mass per unit area is given by
n
k
z
z
k
k
dzP11
)()(
(8)
The mass matrix is given by
Vol
voldNPNM )()]([)]([)]([)]([
(9)
The stiffness matrix can be expressed by
 
1
1
1
1
)]([)]([)]([)]([
ddBDBK T
(10)
69] are employed to define the equation of motion for the free
vibration system having n DOF and given by
0}{)]([][)]([
KM
(11)
where
)]([
M
represents mass matrix and
)]([
K
represents elastic stiffness matrix, while { represents the
             
transverse shear deformation. In the free vibration analysis QR iteration algorithm is applied to determine the
stochastic natural frequencies
)]([
n
by solving standard eigenvalue problem [70].
The cross-sectional view of delaminated composite crack tip is illustrated in Figure 2 (multipoint
constraint delamination model), in which a common node is formed by the nodes of three plate elements.
The plate element 1 with thickness h illustrates the undelaminated portion, while plate elements 2 and 3 show
the delaminated portion. Delamination occurs at the interface of the plate element 2 and 3, where h2 and h3 are
the thicknesses of the elements 2 and 3, respectively. Before application of constraints condition, the elements
1, 2 and 3 are free to deform. At the crack tip, the nodal displacements of elements 2 and 3 can be given by
[71]
xjjjj zzuu
)( ''
(12)
yjjjj zzvv
)( ''
'jj ww
(where, j = 2, 3)
7
where
'j
u
,
'j
v
and
'j
w
represents the mid-plane displacements in the x, y and z direction respectively.
'j
z
represents the z-coordinate of mid-plane of element j while
x
and
y
denotes rotations about x and y axes,
respectively. For the element 1 also the given equation is valid. The common node have relationship for
transverse displacements and rotations as
wwww 321
(13)
xxxx
321
yyyy
321
At the crack tip, in-plane displacements have same magnitude for all the three elements and they have
relationship as
x
zuu
'
2
'
1
'
2
(14)
y
zvv
'
2
'
1
'
2
x
zuu
'
2
'
1
'
3
y
zvv
'
3
'
1
'
3
where
'
1
u
denotes the displacement of the element 1 at the mid-plane. Equations (13) and (14) are the
multipoint constraint equations, which have relationship for the nodal displacements and rotations of elements
at the crack tip. These equations satisfy the compatibility equations of displacement and rotations and these
equations are employed for the finite element formulation in the present study. The mid-plane strains
between elements 2 and 3 have relationship as,
}{}{}{ '
1
'' kz jj
(where j = 2, 3)
(15)
Where
}{ '
denotes the normal strain vector at mid-plane and {k} denotes the curvature vector. For elements
1, 2 and 3 curvature vector have same value at the crack tip. This equation is the special case for element 1
1 is equal to zero. For the element 2 and 3, in-plane stress-resultants
)}({
N
and moment resultants
)}({
M
are expressed by
}{)]([)]([(}{)]([)}({ '
1
'kBAzAN jjjjj
(where j = 2, 3)
(16)
}{)]([)]([(}{)]([)}({ '
1
'kDBzBM jjjjj
(where j = 2, 3)
}{])([)}({ *yAH jj
(where j = 2, 3)
In other form,
8
6,2,1,...)([
)(
)(
)(
1 1 1
'
'
'
'
8
1
 
 
jidz
k
k
k
zzdz
k
k
k
dzH
N
N
Nk
k
k
k
k
k
z
z
z
z
z
zxy
y
x
j
xy
y
x
xy
y
x
n
kij
xy
y
x
(17)
6,2,1,...)([
)(
)(
)(
1 1 1
'2
'
'
'
8
1
 
 
jizdz
k
k
k
zdzz
k
k
k
zdzH
M
M
Mk
k
k
k
k
k
z
z
z
z
z
zxy
y
x
j
xy
y
x
xy
y
x
n
kij
xy
y
x
8
1'
'5,4,...])([
)(
)(
1
n
k
z
zyz
xz
ij
y
xjidzH
H
Hk
k
Thus in matrix form
'
'
'
'
'
5545
4544
66
'
6626
'
2616
'
16662616
26
'
2622
'
2212
'
12262212
16
'
1612
'
1211
'
11161211
66
'
6626
'
2616
'
16662616
26
'
2622
'
2212
'
12262212
16
'
1612
'
1211
'
11161211
000000
000000
00)()()()()()()()()(
00)()()()()()()()()(
00)()()()()()()()()(
00)()()()()()()()()(
00)()()()()()()()()(
00)()()()()()()()()(
)(
)(
)(
)(
)(
)(
)(
)(
yz
xz
xy
y
x
xy
y
x
jjj
jjj
jjj
jjj
jjj
jjj
xy
x
xy
y
x
xy
y
x
k
k
k
SS
SS
BzDBzDBzDBBB
BzDBzDBzDBBB
BzDBzDBzDBBB
AzBAzBAzBAAA
AzBAzBAzBAAA
AzBAzBAzBAAA
H
H
M
M
M
N
N
N
(18)
where
)]([
A
denotes the extension coefficients,
)]([
B
denotes the bending-extension coupling coefficient
and
)]([
D
denotes the bending stiffness coefficients. For the element 1 the relationship among these
coefficients can be given as
 
A
A
z
h
h
z
dzzzHdzzzHDBA
2
222 ),,1)](([),,1)](([)]([),([),(([
(19)
where
)]([
H
represents the transformed reduced stiffness [59] while
o
t
z
denotes the z-co-ordinate of mid-
plane of t-th sub-laminate. The formulation of the multi-point constraint conditions leads to un-symmetric
stiffness matrix. The resultant forces,
)]({
N
, moments,
)]({
M
and transvers shear resultants,
)}({
H
at
the delamination point satisfy the given equilibrium conditions,
321 )]({)}({)}({)]({
NNNN
(20)
3322321 )}({)]({)]({)}({)}({)]({
NzNzMMMM
321 )]({)}({)}({)]({
HHHH
9
3. Hybrid HDMR based FE algorithm for layer-wise stochastic modelling of delaminated composites
Stochastic dynamic analysis of delaminated composites is carried out using a hybrid HDMR based FE
algorithm. Detail mathematical formulation for the HDMR based surrogate model is provided as APPENDIX.
This article focuses on the coupled effect of source-uncertainty (due to inherent randomness in material and
geometric parameters) and delamination with different degree of severity and location. The effect of noise is
investigated in the surrogate based uncertainty quantification algorithm. In this paper, the combined effect of
geometric and material uncertainties along with delamination in laminated composites is considered as
follows:
 
1 2 12 13 23
( ) ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ) ( , ) ( )E E G G Gg s
 
  
(21)
where
( , )
represents the effect of delamination. The parameters
and are used to denote the location
and severity of delamination. The quantity
()s
represents the effect of noise in the surrogate based analysis,
wherein s is used to denote the noise level. The compound effect of materials and geometric source-
uncertainties can be expressed as
 
1 2 12 13 23
1 2 3 4
1(1) 1( ) 2(1) 2( ) 12(1) 12( ) 13(1) 13( )
5 6 7 8
23(1) 23( ) (1) ( ) (1) ( ) (1) ( )
( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )
( .... ), ( .... ), ( .... ), ( .... ),
( .... ), ( .... ), ( .... ), ( .... )
l l l l
l l l l
E E G G G
E E E E G G G G
G
g
G
 


 
   





(22)
where θi denotes the ply-orientation angle, ρi is the mass density, G12(i) and G23(i) represent the shear moduli,
E1(i) and E2(i) l
the number of layer in the laminated composite plate. The parameter
is the stochasticity in input
parameters. It is considered that the randomness in input parameters is distributed within a certain band of
tolerance from central deterministic mean value which follows a uniform distribution. For presenting
numerical results, it is considered as 10% and
10º for material properties and ply orientation angle
respectively according to industry standard, unless otherwise indicated. Figure 3 shows the hybrid surrogate
(HDMR) based FE algorithm for uncertainty quantification followed in this paper. A description about Monte
Carlo simulation can be found in Naskar et al. (2018) [57].
Besides the source-uncertainties in material and geometric properties (
)(g
), another source of
uncertainty is considered in the present analysis in terms of noise (
()s
). The simulated noise is introduced to
account for the effect of measurement error of responses, modeling and simulation error and other epistemic
10
Fig. 3 Flowchart for surrogate based uncertainty quantification of composite laminates (to analyze the
coupled effect of source-uncertainty and delamination) including the influence of noise. Representative
figures      -random sampling, surrogate modelling and probability
distributions are shown corresponding to the respective steps.
uncertainties involved in the system [59]. While formation of the surrogate model, gaussian white noise with
a specific level (s) is induced in the set of output responses as
ijN ij ij
f f s
 
(23)
where, f represents the natural frequency corresponding to a particular mode of vibration, subscript i is the
frequency number, and subscript j is sample number in the design point set. The parameter
ij
denotes a
function that creates random numbers. Here the subscript N is used to represent the noisy frequency. Thus the
simulated noisy dataset is generated by considering a gaussian noise in the responses, while the input design
points remain constant. Thereby Monte Carlo Simulation is carried out for each noisy dataset following a non-
intrusive method as shown in Figure 3. Thus thousands of surrogate based Monte Carlo Simulations are
performed with the noisy design points to quantify the effect of noise corresponding to a particular level.
11
4. Results and discussion
In the present study, the first three natural frequencies of a cantilever laminated composite plate with
length 1m, breadth 1m and thickness 0.0004m are analysed corresponding to different ply orientation angle,
degree of orthotropy, degree of stochasticity, material and geometric properties, position of delamination, and
number of delamination. Both deterministic as well as stochastic results are presented for a comprehensive
analysis. The material properties of the composite plate are assumed as: E1 = 138.0 GPa, E2 = 8.96 GPa and µ
= 0.3, while values of G12, G13, and G23 are determined by relationship of E and µ as G12 = G13 = 7.1 GPa, G23
= 2.84 GPa, where the symbols have usual meanings. The first three natural frequencies are abbreviated as
FNF (first natural frequency), SNF (second natural frequency) and TNF (third natural frequency) respectively.
4.1. Validation and convergence study
In the surrogate assisted stochastic analysis of laminated composites, two different forms of validation
and convergence study are needed to be carried out. The first validation is for the finite element model of
delaminated composite plate along with mesh convergence study. A second type of validation is also needed
here concerning the performance (efficiency and accuracy) of the surrogate model in predicting the responses
along with a convergence study for minimizing the number of design points required for forming surrogate
models. Table 1 shows the mesh convergence study and comparative results with scientific literature for the
deterministic first natural frequency of an undelaminated composite plate. Keeping the computational aspect
in mind, a 6 x 6 mesh is adopted in this work. Further the deterministic first natural frequency is validated
with Krawczuk et al. [72] for delaminated composite plates considering various relative position of
delamination as shown in figure 4.
The optimum number of samples (drawn from Sobol sequence) to construct surrogate models are
decided based on sample-wise prediction performance (scatter plot) and a comparative assessment with direct
Monte Carlo simulation (probability density function plot). Based on the results presented in figure 5, a
sample size of 1024 is adopted for surrogate model formation. A good agreement between the probabilistic
descriptions of natural frequencies and minimal deviation from the diagonal lines of the scatter plots
corroborate the accurate prediction capability of the surrogate models corresponding to the chosen sample
size. It can be noted in this context that the computational time required is exorbitently high for evaluating the
probabilistic responses through full scale direct MCS because of the involvement of large number of finite
element simulations (~104). However, in case of the present surroagate based method, although a same sample
12
Table 1 Non-    n L2 1h2)] of thre   - 
graphite- 
Ply orientation

Present FE model
Qatu and
Leissa [73]
4 x 4
6 x 6
8 x 8
10 x 10
15°
0.8588
0.8618
0.8591
0.8543
0.8759
30°
0.6753
0.6790
0.6752
0.6722
0.6923
45°
0.4691
0.4732
0.4698
0.4578
0.4831
60°
0.3189
0.3234
0.3194
0.3114
0.3283
Fig. 4 Validation for the deterministic finite element code with respect to published results (Krawczuk et al.
[72]) considering the effect of relative position of delamination
size as the direct MCS is considered, the requirement of carrying out actual finite element simulations is much
lesser compared to the direct MCS approach. Here it is equal to the number of samples required to form the
HDMR based surrogate model (i.e. 1024). Hence, the computational intensiveness (time and effort) in terms
of FE analyses are decreased significantly in comparison to full-scale direct MCS.
4.2. Deterministic analysis
In this section, deterministic results are presented to portray the fundamental influences of the location
and severity of delamination in cantilever composite plates. Table 2 presents the effect of severity of damage
considering a case of single delamination. It can be noticed that the natural frequencies reduce with increasing
percentage of delamination due to the reduction in stiffness. Table 3 shows the effect of single delamination
in a composite plate considering different locations of delamination along the span (x-y plane). From the
table, it can be noted that all three natural frequencies decrease with changing the delamination location from
near fixed end to near the free end. The effect of locational variation of delamination across the thickness is
shown in figure 6, which reveals an interesting trend of reduction of the frequencies up to mid-zone of the
13
Scatter Plot
Statistical distributions
Fundamental natural frequency
(a)
(d)
Second natural frequency
(b)
(e)
Third natural frequency
(c)
(f)
Fig. 5 (a,b,c) Scatter plots and (d, e, f) probability density function (PDF) plots    
frequencies (rad/s) considering HDMR model with respect to original finite simulation model considering
different sample size
interfaces and then a rise following a symmetric pattern. In this figure the results are presented considering a
higher number of plies to portray the effect of locational variation of delamination across thickness more
clearly. For the case of single delamination, it can be noted that the effect of increasing severity of damage
(percentage of delamination) is rather less, although the natural frequencies decrease marginally with the
increase of damage level. In contrast, the effect of multiple delamination is observed to be more noteworthy
for the natural frequencies as shown in Table 4 considering different laminate configurations. With the
increase in number of delamination (nd), the natural frequencies are found to decrease significantly.
14
Table 2 The effect of severity of delamination (% of delamination) on first three deterministic natural
frequencies considering a stacking sequence of [45/-45/45]
Parameters
FNF
SNF
TNF
Undelaminated composite plate
8.851
51.053
164.702
% of delamination
(Mid-point delamination)
16.66%
8.836266
51.02038
164.5625
25%
8.835081
51.01272
164.5404
33.33%
8.830061
50.99974
164.4979
50%
8.738607
50.81708
163.891
Table 3 The effect of location of delamination (along the span) on first three deterministic natural frequencies
considering a stacking sequence of -
Parameters
FNF
SNF
TNF
Undelaminated composite plate
8.851
51.053
164.702
Location of delamination
(Along span 25%
delamination)
Near fixed end
8.842589
51.03411
164.6553
Mid-point
8.835081
51.01272
164.5404
Near Free end
8.398455
50.04205
161.7076
4.3. Stochastic analysis
Stochastic results are presented in this section for natural frequencies of delaminated composite plates
considering the compound variation of the source-uncertainties (refer to section 2) along with the effect of
noise. Figure 7-13 and figure 15 show stochastic results concerning different aspects of single delamination,
while figure 14 considers multiple delamination in composites. Figure 7 shows the effect of increasing
percentage of delamination (along with the case of no delamination) on the first three natural frequencies of
composite laminates, wherein it can be noticed that the natural frequencies reduce marginally with the
increase in severity of damage while the probabilistic descriptions vary considerably. The effect of ply-
orientation angle on the stochastic natural frequencies of delaminated composite plates (single delamination
25%) is shown in figure 8 considering cross-ply and angle-ply laminates. The figure illustrates that cross-ply
laminates have higher range of natural frequencies compared to angle-ply laminates, whereas the probabilistic
response bounds are more in case of angle ply laminates. Figure 9 shows the effect of variation in degree of
orthotropy (DOO) on the stochastic natural frequencies of delaminated composite plates (considering single
delamination 25%), wherein all the natural frequencies along with their probabilistic bounds are found to
increase with the increase in degree of orthotropy. The effect of degree of stochasticity (i.e. level of source-
15
(a)
(b)
(c)
(d)
Fig. 6 (a-c) Variation of first three natural frequencies of an     --S
delaminated composite plate (single delamination) with varying relative location of delamination across the
thickness (d) Relative location of delamination (at the interfaces of two laminas) across thickness
Table 4 Effect of multiple delamination (33.33% mid-point delamination) on the first three natural
frequencies for various laminate configurations - - S family of composite
Ply
orientation
angle 
No delamination
(nd = 0)
Single delamination
(nd = 1)
Multiple delamination
(nd = 4)
FNF
SNF
TNF
FNF
SNF
TNF
FNF
SNF
TNF

8.016
35.627
115.711
7.954
35.438
115.314
7.935
35.378
115.196

8.107
39.262
133.232
8.056
39.122
132.872
8.046
39.078
132.758

8.467
46.254
158.215
8.434
46.189
157.974
8.431
46.171
157.904

8.851
51.053
164.702
8.830
50.999
164.497
8.829
50.986
164.448

9.072
52.012
158.057
9.053
51.93
157.749
9.051
51.919
157.664

8.951
50.296
145.491
8.931
50.236
145.167
8.927
50.220
145.074

8.731
48.202
134.891
8.710
48.148
134.559
8.704
48.133
134.463
16
(a)
(b)
(c)
Fig. 7 Probabilistic description of      due to varying severity of
delamination ( ) considering the case of single delamination
(a)
(b)
(c)
Fig. 8 Effects of delamination    angle ply and cross ply
laminate configurations
17
(a)
(b)
(c)
Fig. 9 Effect of variation of degree of orthotropy (DOO) on the   natural frequencies (rad/s) of
delaminated composite plates
(a)
(b)
(c)
Fig. 10 Effect of variation in degree of stochasticity (DOS) on the rst three natural frequencies (rad/s) of
delaminated composite plates
18
(a)
(b)
(c)
Fig. 11 Effects of individual variation of all material properties and structural attributes () 
natural frequencies (rad/s)
uncertainty) on first three natural frequencies is presented in figure 10. As expected, the response bounds are
noticed to be increased with increasing degree of stochasticity. Figure 11 presents a comparative assessment
in two different cases of source uncertainty in delaminated composite plates (considering single delamination
25%), compound effect of stochasticity in all materials properties only and the effect of stochsticity in
structural property (ply orientation angle). Besides significant difference in the probability distribution
between the two cases, the natural frequencies are found to decrease marginally in case of the compound
variation of material properties. From the probabilistic response bounds it can be discerned that the sensitivity
of ply orientation angle is one of the most predominant in the free vibration responses of composite plates.
Figure 12 and 13 show the effect of the location of delamination (considering single delamination
25%) on stochastic natural frequencies of composite plates. Figure 12 shows the effect of span-wise location,
wherein it can be noticed that the influence of delamination becomes more severe as the location changes
from the fixed end to the free end in a cantilever composite plate along with an increase in probabilistic
response bounds. Figure 13 presents the effect of location of delamination across the thickness of a composite
plate (i.e. single delamination is considered at the interface of different layers) on the stochastic natural
frequencies. While obtaining these results, a different laminate configuration with increased number of layers
19
(a)
(b)
(c)
Fig. 12 Effect of delamination location        considering spatial
variation along the span (fixed, middle, and free end) with 25% single delamination
(a)
(b)
(c)
Fig. 13 Effects of delamination location across the thickness (refer to figure 6 for the pictorial representation
of h/h ratio) on the first three natural frequencies (rad/s) considering a 25% single delamination case
20
(a)
(b)
(c)
Fig. 14 Effects of number of delamination (nd) on the first three stochastic natural frequencies (rad/s) for the
case of multiple delamination in composite plates
is used to portray the effect of the location of delamination across the thickness clearly. From the figure it can
be observed that the range of natural frequencies reduce as the location of delamination varies from the two
free surfaces towards the middle of thickness of the laminate.
The results in figure 14 are presented to analyse the effect of multiple delamination on stochastic
natural frequencies of composite plates considering a laminate configuration of --S.
Comparative probability distribution plots are presented considering the case of no delamination, single
delamination and four delaminations (nd = 0, 1, 4). It can be noticed from the figures that the range of natural
frequencies reduce with the increase in number of delaminations. The influence of noise on the stochastic
natural frequencies of delaminated composite plates is presented in figure 15 considering a single
delamination (25% mid-point delamination). It can be noticed that the stochastic bound of the probability
distributions increase with the increase in level of simulated noise (s).
5. Summary and perspective
This paper presents an efiicient stochastic bottom-up framework for analyzing the coupled effect of
delamination and source-uncertainties (
()
) on natural frequencies of composite plates. The simulated noise
studied in this article can be considered equivalent to accounting for the effect of measurement error of
21
(a)
(b)
(c)
Fig. 15 Effect of noise on the first three stochastic natural frequencies (rad/s) of delaminated composite plates
responses, modeling and simulation error and other epistemic uncertainties involved in the system. A
stochastic simulation involving thousands of finite element simulations (~104) becomes exorbitantly
computationally expensive. To mitigate this lacuna, a surrogate based approach (HDMR coupled with
DMORPH) in conjunction with the stochastic finite element formulation is adopted to obtain computational
efficiency (without compromising the accuracy of results) in the present analysis. In case of the structural
mechanics problems, where efficient analytical solutions [74-80] are not available, a surrogate based approach
can be developed to carryout multiple iterative function evaluations.
In the stochastic characterization of delaminated composites, there exists three distinct stages of the
analysis: uncertainty modelling at the input level, propagation of uncertainty to the global level and
quantification of the global responses such as natural frequencies. A layer-wise model of source-uncertainty
along with the effect of noise and delamination is adopted in the present analysis, as discussed in section 2.
Normally efficeint function evaluations (in terms of the stochastic input parameters) by direct closed-form
formulae are not available for complex structural systems like laminated composites. In such situation, a
numerical analysis technique like finite element method is adopted to obtain the response quantities. Finite
element analyses are normally very expensive and time consuming. Inclusion of the effect of delamination in
22
the present study makes the simulations more computationally intensive. The situation becomes worse in case
of a stochastic analysis that requires thousands of such finite element simulations to be carried out. The
surrogate based uncertainty propagation strategy, as adopted in this study, can develop a representative and
predictive mathematical/ statistical metamodel relating the natural frequencies to a number of stochastic input
variables. Thereafter the metamodels (response surface) are used to compute the dynamic characteristics
corresponding to a given set of input variables, instead of having to simulate the time-consuming FE model
repeatedly. The response surface here represents the results (or outputs) of the structural analyses
encompassing (in theory) every prospective combination of the stochastic input variables. Hence, thousands
of combinations of the stochastic input variables can be created and a pseudo analysis (efficient, yet accurate)
for each variable set can be performed by adopting the corresponding surrogate model. The final step in the
stochastic analysis is uncertainty quantification in the output responses, which is effectively carried out by
deriving the probabilistic distributions and the statistical moments.
The results in this article capture the influence of inevitable source-uncertainties in material and
structural attributes (manufacturing uncertainties) along with service-life conditions such as damage in
composites (delamination). The effect of source-uncertainty in all material properties and structural properties
are analyzed separately to ascertain their relative influence on the stochastic natural frequencies. It is noted
that the ply orientation angle is more sensitive to natural frequencies compared to the individual effect of
variability in material properties of the delaminated composites. Along with the previous studies [81], where
the individual effect of variation in different material properties are analyzed for undelaminated composites,
the present results can provide a comprehensive idea about the senstivity of various input parameters towards
the global dynamic behaviour of the structure in the presence of delamination. Effect of single as well as
multiple delamination on the stochastic dynamic responses of composite plates is analyzed considering the
aspects of location and severity of delamination. The results reveal that the effect of increasing number of
delamination is more sensitive to the natural frequencies compared to the effect of increasing severity of
delamination corresponding to a particular case of single delamination. Besides the conventional sources of
manufacturing uncertainty in material and structural properties, another source of uncertainty is considered in
this study in the form of noise, which can be considered tantamount to incorporating various forms of errors
involved in the system and other epistemic uncertainties, which are not explicitly addressed in the analysis.
23
6. Conclusion
The compound influence of source-uncertainties in material and structural attributes (manufacturing
uncertainty) and delamination (a service-life condition) on the natural frequencies of composite laminates is
analyzed including the effect of inevitable noise. The propagation of source-uncertainty can cause a dramatic
shift in the dynamic behavior of composites when coupled with the effect of damage. A hybrid HDMR based
finite element code is developed for delaminated composite plates to study the effect of single and multiple
delamination along with their locational sensitivity and severity. In-depth results are presented in both
deterministic as well as stochastic regime for a comprehensive understanding. Various laminate
configurations along with different degree of orthotropy and degree of stochasticity are analyzed to provide a
thorough insight on the stochastic dynamic behavior of delaminated composite structures. It is found that the
coupled effect of stochasticity/ source-uncertainty and delamination has significant influence on the dynamic
behaviour of composite structures. Thus it is imperative to include these aspects in subsequent analyses/
design process to ensure the desired robust, safe and sustainable system performance. Future research would
follow the probabilistic analysis of delaminated shell structures and consideration of other forms of damages
concerning composites in the stochastic regime. The present paper dealing with the efficient hybrid HDMR
based finite element analysis for the stochastic dynamics of delaminated composite plates would serve as a
valuable reference in such future studies.
APPENDIX
In the case of large number of input variables (i.e. high dimensional systems), a system can be
effectively analysed by using the high dimensional model representation (HDMR) [65]. By using the HDMR,
deterministic as well as stochastic relationships can be handled. The HDMR is employed to create a model for
prediction of the response output in the stochastic region (in this paper natural frequency is taken as output
       -MORPH algorithm is employed in the
formulation. In the present approach, the function
)(
is decomposed with component functions by input
parameters
),...,,( 21 kk
. The nature of the input parameters is independent and the component
functions are projected by vanishing condition. So, this technique has limitation for general formulation. The
output is determined for the different input variables as [65]
24
).,....,,(.......),()()( 21.......12
1 1
0kkkk
kk
ikkji jiijii
 
 
(1)
kkuuu )()(
(2)
where
0
denotes the mean value, which is a zeroth order component function.
)( ii
represents the first
order component function,
),( jiij
resents the second order component functions, and
).,....,,( 21.......12 kkkk
denotes the residual contribution by input parameters, while
},....,2,1{ kku
represents the subset wherein
kku
for simplicity and empty set,
u
. The correlated
variables are defined as,
dwgArgkkuku uu
kkuRLg
uu u
u)()()(min)}|({
2
}),({ 2
(3)
0)()(,, uiuu ddwuikku
(4)
0)(,)()()()(:, vvuuvvuuv gdwgguv
(5)
The function
)(
is determined from the sample data by experiments or by modelling. The squared error can
be reduced to minimise the computational cost. Considering Q in Hilbert space expanded on the
basis
},....,,{ 21 kk
qqq
, the bigger subspace
Q
(
Q
) is expanded by extended
basis
},....,,,....,,{ 121 mkkkk qqqqq
. Then
Q
can be decomposed as
QQQ
(6)
where
Q
represents the complement subspace of
Q
[82] within
Q
. In the past works [83-85], basis
functions are employed to determine the component functions. The basis functions
}{
are used to estimate
the component functions of second order HDMR expansion as [85]
kk
ri
i
r
i
rii 1
)0( )()(
(7)
)()()]()([),( 1 1
)0(
1
)()( j
j
q
l
p
l
qi
i
p
ji
pq
kk
rj
j
r
jji
ri
i
r
iji
rjiij
 
(8)
The basis functions of
),( jiij
includes all the basis functions used in
)( ii
and
)( jj
. At
samp
N
sample
points of
, the HDMR expansions are denoted by the linear algebraic equation system
RG
(9)
where
is a matrix with size
samp
N
×
t
~
and all elements of this matrix are basis functions at the
samp
N
values
of
; G represents a vector with
t
~
dimension of all unknown combination coefficients;
R
denotes a vector
25
with
samp
N
-dimension wherein
l
-th element is
0
)( )(
l
.
)(l
represents the
l
-th sample of
, and
0
denotes the average value of all
)( )(l
. The regression equation for least squares of the above equation can
be given as
R
N
G
NT
samp
T
samp
11
(10)
Some rows of the above equation are identical due to the use of extended bases, and an underdetermined
algebraic equation system is obtained by removing these rows as
VGA
(11)
This has many of solutions for
G
consisting a manifold
t
Y~
. The main task is to determine a solution
G
from
Y
to force the HDMR component functions satisfying the hierarchical orthogonal condition. A solution
provided by D-MORPH regression ensures additional condition of exploration path which is denoted by
differential equation
)()()(
)( lvAAIlv
dlldG t
(12)
wherein
represents orthogonal projector ensuring
2
and
T
(13)
T
2
(14)
To ensure the wide domain for
)(lG
and reduce the cost
))(( lG
, the free function vector is selected and
expressed as
GlG
lv
))((
)(
(15)
Then we obtain
0
))(())((
)(
))(()())(())((
GlG
P
GlG
P
lvP
GlG
llG
GlG
llG
T
TT
(16)
The cost function is represented in quadratic form as
GBGT
2
1
(17)
where B is the positive definite symmetric matrix and
G
can be given as
VAUVUVG Trtrt
Trt
t
)( ~
1
~~
(18)
where the last columns
)
~
(rt
of
U
and
V
are denoted as
rt
U
~
and
rt
V
~
, found by decomposition of
B
26
T
rVUB
00
0
(19)
This unique solution
G
in Y 
D-       thogonality in hierarchical manner. In previous
literature, construction of the corresponding cost function
can be found [65].
Acknowledgement
PKK would like to acknowledge the financial support received from MHRD, India during the period of this
research work.
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Over the last few decades, uncertainty quantification in composite materials and structures has gained a lot of attention from the research community as a result of industrial requirements. This book presents computationally efficient uncertainty quantification schemes following meta-model-based approaches for stochasticity in material and geometric parameters of laminated composite structures. Several metamodels have been studied and comparative results have been presented for different static and dynamic responses. Results for sensitivity analyses are provided for a comprehensive coverage of the relative importance of different material and geometric parameters in the global structural responses.
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