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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

hLb

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

),(),(),,( yxwyxwzyxw

(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

z

onijijijij jidzzzHDBA

k

k

1

26,2,1,],,1[])}({[)](),(),([

1

n

k

z

z

kijsij jidzHS

k

k

1

5,4,])([)]([

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 1vv

Ev

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

dzP

11

)()(

(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

z

xy

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

z

xy

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,

dwgArgkku

ku 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

N

T

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

dl

ldG

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

G

lG

lv

))((

)(

(15)

Then we obtain

0

))(())((

)(

))(()())(())((

G

lG

P

G

lG

P

lvP

G

lG

l

lG

G

lG

l

lG

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 T

rtrt

T

rt

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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