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1

Probabilistic micromechanical spatial variability quantification in

laminated composites

S. Naskara, T. Mukhopadhyayb, S. Sriramulaa

aSchool of Engineering, University of Aberdeen, Aberdeen, UK

bDepartment of Engineering Science, University of Oxford, Oxford, UK

Email address: tanmoy.mukhopadhyay@eng.ox.ac.uk (Tanmoy Mukhopadhyay)

Abstract

This article presents a probabilistic framework to characterize the dynamic and stability parameters of

composite laminates with spatially varying micro and macro-mechanical system properties. A novel

approach of stochastic representative volume element (SRVE) is developed in the context of two

dimensional plate-like structures for accounting the correlated spatially varying properties. The physically

relevant random field based uncertainty modelling approach with spatial correlation is adopted in this

paper on the basis of Karhunen-Loève expansion. An efficient coupled HDMR and DMORPH based

stochastic algorithm is developed for composite laminates to quantify the probabilistic characteristics in

global responses. Convergence of the algorithm for probabilistic dynamics and stability analysis of the

structure is verified and validated with respect to direct Monte Carlo simulation (MCS) based on finite

element method. The significance of considering higher buckling modes in a stochastic analysis is

highlighted. Sensitivity analysis is performed to ascertain the relative importance of different

macromechanical and micromechanical properties. The importance of incorporating source-uncertainty in

spatially varying micromechanical material properties is demonstrated numerically. The results reveal that

stochasticity (/ system irregularity) in material and structural attributes influences the system performance

significantly depending on the type of analysis and the adopted uncertainty modelling approach, affirming

the necessity to consider different forms of source-uncertainties during the analysis to ensure adequate

safety, sustainability and robustness of the structure.

Keywords: composite laminate; micromechanical random field; spatially correlated material properties;

stochastic natural frequency; stochastic buckling load; stochastic mode shape

2

Contents

1. Introduction ........................................................................................................................................................ 2

2. Stochastic dynamics and stability analysis of composite plates ...................................................................... 6

3. HDMR based surrogate modelling coupled with DMORPH algorithm ........................................................ 8

4. Stochastic representative volume element based framework for uncertainty quantification ................... 12

4.1. Concept of SRVE ....................................................................................................................................... 12

4.2. Characterization of correlated material properties based on Karhunen-Loève expansion ................... 14

4.3. Description of the uncertainty quantification framework ........................................................................ 17

4.3.1. Monte Carlo simulation ...................................................................................................................... 17

4.3.2. Modelling of source-uncertainty at the input level ............................................................................. 18

4.3.3. Propagation of uncertainty based on HDMR coupled with DMORPH algorithm ............................. 20

5. Results and discussion ...................................................................................................................................... 22

5.1. Stochastic dynamic analysis ...................................................................................................................... 24

5.1.1. Validation and convergence study ...................................................................................................... 24

5.1.2. Results for stochastic dynamic analysis.............................................................................................. 26

5.2. Stochastic stability analysis ....................................................................................................................... 38

5.2.1. Validation and convergence study ...................................................................................................... 38

5.2.2. Results of stochastic stability analysis ................................................................................................ 43

6. Summary and perspective ................................................................................................................................ 50

7. Conclusion ......................................................................................................................................................... 55

Acknowledgements ................................................................................................................................................... 56

References ................................................................................................................................................................. 60

1. Introduction

Composite structures are widely used in modern aerospace, construction, marine and automobile

applications because of high strength and stiffness with lightweight and tailorable properties. Even though

laminated composite structures have the advantage of modulating large number of design parameters to

achieve various application-specific requirements, this concurrently brings the challenge of manufacturing

the structure according to exact design specifications. Large-scale production of such structures according

to the requirements of industry is often subjected to large amount of variability arising from unavoidable

manufacturing imperfections (such as intra-laminate voids and excess matrix voids, excess resin between

plies, incomplete curing of resin, porosity, variations in lamina thickness and fibre properties), lack of

experiences and complexity of the structural configuration. The issue aggravates further due to uncertain

3

operational and environmental factors and the possibility of incurring different forms of damages and

defects during the service life. In general, uncertainties can be broadly classified into three categories,

aleatoric (due to variability in structural parameters), epistemic (due to lack of adequate information about

the system) and prejudicial (due to absence of stochasticity characterization of the structural system)

(Agarwal et al. (2004), Oberkampf et al. (2001), Sriramula and Chryssanthopoulos (2009), Dey et al.

(2018a)). Composite structures being susceptible to multiple forms of uncertainties, the structural

performances are often subjected to a significant element of risk. Thus it is of prime importance in case of

composite structures to quantify the effect of source-uncertainties so that an inclusive design paradigm

could be adopted to avoid any compromise in the aspects of safety and serviceability.

Composite structures have received immense attention from the engineers and scientists

concerning their static, dynamic and stability behaviour (Chakrabarti et al. (2011, 2013), Biswal et al.

(2016), Dey et al. (2016d), Mandal et al. (2017), Kumari and Behera (2017). Recent studies on the

vibration and buckling analysis of advanced lightweight structures (like composites and FGM) in the

deterministic regime include non-homogeneity, non-linear behaviour, shear deformation, rotary inertia

and effect of elastic foundation in the analysis (Sofiyev et al. (2012, 2017), Sofiyev and Kuruoglu

(2014), Haciyev et al. (2018)). Following several decades of deterministic studies, the aspect of

considering the effect of uncertainty in material and structural attributes have recently started receiving

due attention from the scientific community. Both probabilistic (Sakata et al. (2008), Goyal and Kapania,

(2008), Manan and Cooper (2009), Dey et al. (2016a, 2016e, 2018b, 2019), Naskar et al. (2017b), Naskar

and Sriramula (2017a, 2017b, 2017c), Naskar (2017)) as well as non-probabilistic (Dey et al. (2016b),

Pawar et al. (2012)) approaches have been investigated to analyse the influence of variability in the

material and structural attributes of composite structures. Plenty of researches have been reported based

on intrusive methods to quantify the uncertainty of composite structures (Lal and Singh (2010), Scarth

and Adhikari (2017)), wherein the major drawback can be identified as the requirement of intensive

analytical derivation and lack of the ability to obtain complete probabilistic description of the response

quantities for systems with spatially varying attributes. A non-intrusive method based on Monte Carlo

simulation, as adopted by many researchers (Dey et al. (2016a, 2016g, 2015d), Mukhopadhyay and

4

Fig. 1 (a) Typical distribution of a material property E1 along a cross-sectional view (X-Z plane) of two

laminae for a random realization in case of the layer-wise random variable approach (b) Typical

distribution of a material property E1 along a cross-sectional view (X-Z plane) of two laminae for a

random realization in case of the random field approach

Adhikari (2016c)), can obtain comprehensive probabilistic descriptions for the response quantities of

composite structures. Besides consideration of random variability in material and structural attributes,

recent studies related to uncertainty quantification of laminated composite structures include the effect of

environmental (Dey et al. (2015a)), operational (Dey et al. (2015b)) and service life conditions (Naskar et

al. (2017), Karsh et al. (2018a)) following the non-intrusive approach. A careful consideration of

available scientific literature unveils that most of the studies conducted so far to quantify the effect of

uncertainty in composite structures are based on a ply-level random variable based approach, where the

spatial variation of stochastic parameters in the laminae is neglected. In the previous studies, the material

and structural properties of a lamina are assumed constant spatially (i.e. along the x-y plane) for a

particular realization (refer to figure 1(a)). Modelling of uncertainty in composite structures based on such

random variable based approach is of limited practical resemblance. Therefore, it is essential to consider

5

the effect of spatial variability in the material properties (Kazimierz and Kirkner (2001)) of the laminae to

quantify the effect of uncertainty accurately.

We aim to quantify the effect of spatially varying lamina properties in this article to characterize

the probabilistic descriptions for the dynamics and stability characteristics of composite plates. The aspect

of spatial variation of lamina properties is illustrated in figure 1(b) for a random realization (i.e. a typical

sample of the Monte Carlo simulation), wherein it can be noticed that the stochastic attributes vary in the

x-y plane as well as along the z-axis (i.e. for different laminae). Most of the previous investigations in this

field have not considered the spatial variation of stochastic attributes as shown in figure 1 (Dey et al.

(2016a)). In practical situations, the stochastic attributes often being spatially correlated, it is essential to

account for the effect of such correlation to create a physically realistic model of uncertainty. We aim to

consider the spatially correlated material attributes in composite laminates based on Karhunen-Loeve

expansion (Karhunen (1947), Loève (1977)). However, even after ensuring a physically relevant

uncertainty model of composite laminates, as discussed above, the issue of propagation of uncertainty

following a computationally viable framework still remains to be addressed. The aspect of

computationally efficient uncertainty propagation in context to composite laminates is discussed in the

next paragraph.

Uncertainty quantification based on Monte Carlo simulation is a popular approach because of the

ability to obtain a comprehensive probabilistic description of the response quantities. However, the major

lacuna of this approach is that a Monte Carlo simulation requires thousands of expensive finite element

(FE) simulations to be carried out corresponding to the random realizations. Thus, direct Monte Carlo

simulation has limited practical use due to the computational intensiveness. To mitigate this lacuna we

have developed a surrogate modelling approach based on the high dimensional model representation

(HDMR) technique coupled with the diffeomorphic modulation under observable response preserving

homotopy (DMORPH) algorithm (Li and Rabitz (2012)) for accounting correlated spatially varying

attributes, wherein the uncertainty propagation can be realized following an efficient mathematical

medium.

6

In the present analysis, uncertainty of the system properties is considered in the elementary

micromechanical level to comprehensively analyse the dynamic and stability characteristics of composite

laminates. Thus a probabilistic approach is followed, wherein the effect of uncertainty is included in the

elementary micromechanical-level first and then the effects are propagated towards the global responses

via an efficient surrogate of the actual finite element model. For this purpose, the idea of stochastic

representative volume element (SRVE) is proposed in the context of two-dimensional plate-like

structures. This article hereafter is organized as, section 2: governing equations for analysing the

stochastic dynamics and stability of composite laminates; section 3: brief description of the surrogate

model based on HDMR coupled with the DMORPH algorithm; section 4: description of the SRVE

approach of uncertainty quantification considering spatially correlated material properties hinged upon the

Karhunen-Loève expansion; section 5: results and discussion demonstrating the influence of spatially

varying properties on the global responses of composite laminates; section 6: summary and perspective of

the present study in context to the available scientific literature; section 7: conclusion.

2. Stochastic dynamics and stability analysis of composite plates

In present article, a laminated composite plate with thickness h, length L and width b is analysed

as shown in figure 2 and 3. The governing equation for stochastic free vibration analysis of a composite plate

without damping can be expressed as (refer to the APPENDIX for detail formulation)

( ) ( ) 0MK

(1)

where

)()()(

ee KKK

. In the finite element formulation of this study, an eight noded element is

considered, wherein each node has five degrees of freedom (two rotations and three translations). The natural

frequencies

k

and mode shapes

()

f

k

S

of the composite plate are obtained by solving an eigenvalue

problem based on QR iteration algorithm (Bathe (1990), Rayleigh (1945))

2

( ) ( ) [ ( )] [ ( )] ( )

ff

k k k

K S M S

(2)

where

nk ,....,3,2,1

. The superscript f is used to denote the frequency analysis. Here the orthogonality

relationship is satisfied as

7

(a) (b)

Fig. 2 (a) Force acting on the plate (b) Moment acting on the plate

Fig. 3 Laminated composite plate with layer number

[ ( )] [ ( )] ( )

f T f

i k ik

S M S

and

2

[ ( )] [ ( )] ( ) [ ( )]

f T f

i k k ik

S K S

(3)

where

nki ,....3,2,1,

and the Kronecker delta functions

ik

=0 for

ki

;

ik

=1 for

ki

. The problem of

stability analysis is solved through another eigenvalue problem as:

( ) ( )[ ( )] ( )

b b b

e k e k

K S K S

(4)

where

()

b

is the stochastic buckling load factor and

b

k

S

gives the buckling modeshapes. The

superscript b is used to denote the buckling analysis.

8

3. HDMR based surrogate modelling coupled with DMORPH algorithm

In this section, a brief overview is given for the surrogate modelling approach on the basis of high

dimensional model representation (HDMR) coupled with the diffeomorphic modulation under observable

response preserving homotopy (DMORPH) algorithm. In general, the surrogate models (Dey et al.

(2016a, 2016f, 2018, 2015e), Mukhopadhyay (2019), Mukhopadhyay et al. (2016b, 2016c), Karsh et al.

(2018b), Maharshi et al. (2018), Mahata et al. (2016), Metya et al. (2017)) are employed to reduce the

number of function evaluations based on actual simulation/ experimental models in a Monte Carlo

simulation (refer to figure 4) or a process involving iterative simulations (such as optimization), which

need large number of realizations corresponding to random set of input parameters. The surrogate models

can encompass any prospective combination of all the input variables within the analysis domain.

Thousands of sets of the design input parameters can be generated and pseudo analyses for each set can be

efficiently executed by adopting the corresponding surrogate based prediction models. The development

of surrogate models is performed in three typical steps: selection of optimal sample points (which are able

to collect information of the whole design space) to construct surrogate model, evaluation of responses

(i.e. output) corresponding to each of the sample points and formulation of the mathematical/ statistical

prediction model to obtain an efficient input-output relationship based on the sample set (containing a set

of input parameters and corresponding output parameters).

The present HDMR (Dey et al. (2015c, 2016c, 2017) Mukhopadhyay et al. (2015, 2016a)) based

surrogate modelling algorithm is particularly suitable for high dimensional systems (i.e. large number of

input parameters) and correlated system properties. The HDMR can form an efficient model to predict the

random output responses (e.g. natural frequency and buckling load) in the stochastic analysis domain.

This approach is able to treat both independent as well as correlated input variables. The function of D-

MORPH here is to verify the component function orthogonality following a hierarchical approach. The

present formulation decomposes function

)(S

with the component functions by input parameters,

),...,,( 21 kk

SSSS

. As the input parameters are considered to be independent in nature, the component

functions can be projected by vanishing condition. In the present analysis, the component functions is

portrayed, wherein a unified framework for general HDMR dealing with both correlated as well as

9

Fig. 4 Surrogate based analysis of stochastic system (Here

()x

and

( ( ))yx

are the symbolic

representation of stochastic input parameters and output responses respectively.

denotes the

stochasticity of parameters.)

independent variables is developed. For various stochastic input parameters, the output quantity is

calculated as follows (Li and Rabitz (2012), Li et al. (2002a, 2002b))

).,....,,(.......),()()( 21.......12

1 1

0kkkk

kk

ikkji jiijii SSSSSSS

(5)

kkuuu SS )()(

(6)

where

0

is the zero-th order component function, which represents the mean value.

)( ii S

and

),( jiij SS

represent first and second order component functions, respectively. The expression

).,....,,( 21.......12 kkkk SSS

is the residual contribution by input parameters. Here

},....,2,1{ kku

indicates the subset wherein

kku

. Note that

u

includes

(

u

), which is an empty set. According

to Hooker’s definition, the correlated variables can be expressed as

dSSwSSgArgkkuS ku uu

kkuRLg

uu u

u)()()(min)}|({

2

}),({ 2

(7)

10

0)()(,, uiuu dSdSSwSuikku

(8)

0)(,)()()()(:, vvuuvvuuv SgSdSSwSgSguv

(9)

The parameter

)(S

is obtained from the design points. Assuming H’ to be a Hilbert space on the basis of

{h’1, h’2, . . . , h’kk}, the bigger subspace

'

H

(⊃

'

H

) can be expanded by the extended basis {h’1, h’2, . . . ,

h’kk, h’kk+1, . . . , h’m}. Then the subspace

'

H

is decomposed as

' ' '

H H H

(10)

where

'

H

represents the orthogonal complement subspace of

'

H

within the subspace

'

H

. Component

functions of a second order HDMR expansion can be obtained from basis functions

}{

(Li et al.

(2006)),

kk

ri

i

r

i

rii SS 1

)0( )()(

(11)

)()()]()([),( 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 SSSSSS

(12)

The HDMR based expansions for

samp

N

sample points of

S

is represented by a linear system of

algebraic equations

RJ ˆ

(13)

where

represents a matrix (

samp

N

×

t

~

), where the elements are basis functions at

samp

N

values of

S

; J

denotes a vector having

t

~

dimension of all the unknown combination coefficients;

R

ˆ

is a vector having

samp

N

dimension, where the

l

-th element is

0

)( )(

l

S

.

)(l

S

is the

l

-th sample of

S

, and

0

represents

the average value of all

)( )(l

S

. Regression equation for the least squares can be written as

R

N

J

NT

samp

T

samp

ˆ

11

(14)

Because of using extended bases, some of the rows of the above expression become identical and these

can be removed for obtaining an underdetermined algebraic system of equation

VJA ˆ

(15)

11

This has many solutions for

J

composing a manifold

t

Y~

. Thus a solution

J

from

Y

is found to force

the HDMR component functions that satisfy hierarchical orthogonal condition. DMORPH regression can

provide a solution for ensuring the additional condition

)()()(

)( lvAAIlv

dlldJ t

(16)

where

is an orthogonal projector having the following properties

2

and

T

(17)

T

2

(18)

The free function vector can be adopted for ensuring the wide domain for

)(lJ

and to reduce the cost

))(( lJ

simultaneously

JlJ

lv

))((

)(

(19)

Then,

0

))(())((

)(

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

JlJ

P

JlJ

P

lvP

JlJ

llJ

JlJ

llJ

T

TT

(20)

The cost function is obtained in a quadratic form

JBJT

2

1

(21)

where, B denotes a positive definite symmetric matrix and

J

is expressed as

VAUVUVJ Trtrt

Trt

tˆ

)( ~

1

~~

(22)

Here the last columns

)

~

(rt

in

U

and

V

are obtained as

rt

U

~

and

rt

V

~

which is obtained by

decomposition of

B

T

rV

S

UB

00

0

(23)

The solution

J

in Y , which is unique, indicates the minimized cost function. Here DMORPH regression

is adopted to obtain J ensuring the HDMR component functions’ orthogonality in a hierarchical manner.

12

4. Stochastic representative volume element based framework for uncertainty quantification

4.1. Concept of SRVE

In this paper a concept of stochastic representative volume element (SRVE) is proposed for two-

dimensional plate-like structures to account for the effect of spatial randomness of material properties.

According to this approach, each of the representative units (structural element) is considered to be

stochastic in nature, instead of considering the homogenized mechanical properties of a conventional

representative volume element (RVE) throughout the entire solid domain. As per the traditional approach,

one RVE is analysed typically and the assumption is that a single RVE can represent the entire analysis

domain (Sriramula and Chryssanthopoulos (2009), Mukhopadhyay et al. (2017a, 2017b, 2018a)).

However, this approach of analysis can lead toeroneous outcomes, specially in case of stochastic systems

with spatial randomness in material and other attributes. To analyse such systems, it is essential to account

for the effect of the distribution of stocahstic mechanical properties along the spatial location of different

zones of a plate-like structure.

According to the present approach, the entire plate is assumed to be consisted of a finite number of

SRVEs. Thus mechanical properties of a SRVE are dependent on its stochastic material and structural

properties. Following this framework, it becomes feasible to consider the spatial randomness in a

structural system more realistically. The global responses (such as natural frequencies and buckling loads)

of the plate are computed by propagating the mechanical information acquired in the elementary local

level (SRVEs) towards the global level by combing (/assembling) the SRVEs applying the principles of

solid mechanics (finite element approach in the present study). Recently, a similar concept has been

proposed for analyzing hexagonal honeycomb-like lattices having spatial irregularity (Mukhopadhyay and

Adhikari (2016a, 2016b, 2017a), Mukhopadhyay et al. (2018a), Mukhopadhyay (2017)), wherein multiple

representative unit cell elements (RUCE) are analysed instead of the conventional approach of considering

a single unit cell. The entire lattice structure is assumed to be consisted of several RUCEs and the global

13

Fig. 5 SRVE based approach for analyzing spatially random two dimensional systems

mechanical properties of the entire irregular lattice can be computed by assembling the RUCEs based on

equilibrium and compatibility conditions. The concept of SRVE for analyzing one dimensional beam-like

structures with random material properties and crack density is first adopted by Naskar et al. (2017). In

this paper, we have generalized the concept for stochastic analysis of two-dimensional plate-like

structures with randomly inhomogeneous form of uncertainty (Mukhopadhyay and Adhikari (2017b)). It

can be noted that spatially correlated material properties can be conveniently accounted in this approach

based on the Karhunen-Loève expansion. The adoption of SRVEs in a plate-like structure is shown in the

figure 5, wherein the two-dimensional space is divided into a finite number of stochastic elements

(SRVEs) having dimensions of l1 and l2 in two mutually perpendicular directions of the two-dimensional

structure. Each of the SRVEs possesses different material and structural properties. Here, a parameter

characteristic length (

r

) can be defined as:

12

12

1

d

ll

rL L N

, where

d

N

denotes the number of divisions

along the two dimensions of the plate. As per the proposed concept of SRVE, the size (/number) of SRVE

is independent of the discretization in a finite element based numerical solution that could be adopted for

dynamic/ stability analysis of the composite plate. The size (/number) of SRVE would normally be

govorned by the spatial distribution of structural and material attributes along with the correlation length.

Once the size of a SRVE is decided, they could be discretized following conventional finite element

analysis using a mesh convergence study. In the present analysis, we have only considered spatial

14

variation of material properties; but future studies could include random spatial variation of

microstructural properties (such as micro-scale damage) using the proposed SRVE based framework. In

such problems, appropriate finite element meshing schemes would need to be adopted for each of the

SRVEs.

4.2. Characterization of correlated material properties based on Karhunen-Loève expansion

Random fields are useful for modelling variables which have correlated spatial randomness. When

spatial variation of structural and material properties are considered in a randomly inhomogeneous

structural system, the properties are often found to be spatially correlated. The conventional approach to

deal with such random fields is to discretize it into different finite number of random variables. Available

schemes for discretizing the random fields can be classified into three groups, point discretization (e.g.,

midpoint (Klintworth and Stronge (1988)), shape function (Liu et al. (1986a, b)), integration point

method (Matthies et al. (1997)), optimal linear estimate (Li and Der (1993))); series expansion (e.g.,

orthogonal series expansion (Zhang and Ellingwood (1994))), and average discretization (e.g., spatial

average (Vanmarcke (1983), Vanmarcke and Grigoriu (1983)), weighted integral (Deodatis (1991),

Deodatis and Shinozuka (1991))).

The beneficial alternative for discretizing the random field is representing it in a generalized

Fourier type of series such as Karhunen-Loève (KL) expansion (Karhunen (1947); Loève, (1977)). Let us

consider a random field

,

x

with covariance function

12

( , )xx

defined in the probability space

( , , )FP

. The KL expansion for

,

x

can be expressed in the following form

1

,i i i

i

x x x

(24)

where

i

represents a set of random variables with no correlation.

i

and

i

x

denote the

eigenvalues and eigenfunctions of the covariance kernel

12

( , )xx

, satisfying the integral equation

1 2 1 1 2

( , )

N

i i i

d

x x x x x

(25)

In practice, the infinite series in Equation (24) is truncated as

15

1

,N

i i i

if

x x x

(26)

The above expression approaches to

,

x

in a mean square sense for

N

. The finite element

method (Huang et al. (2001)) can be applied to compute the eigensolutions for any covariance function

and domain in the random field. In case of linear and exponential covariance functions along with a

simple domain, the eigen solutions are feasible to be obtained analytically (Huang et al (2001)). Once

()s

and its eigen solutions are computed, the parameterization of

,

x

is carried out by the KL

approximation of Gaussian image,

1

,N

i i i

i

Gf

x x x

(27)

As per Equation (27), the KL approximation gives a parametric representation of

,

x

with M

random variables. It can be noted that this is not the only available method for discretizing the random

field

,

x

. However, KL expansion has uniqueness and error minimization properties that make it a

superior choice over the other methods (Huang et al. (2001)).

In the present article, the stochastic material properties (micro/ macro-mechanical) are modelled as

random fields and these are discretized using the KL expansion. To be specific, lognormal random fields

are considered for modelling the correlated material properties. The covariance function is expressed as

1 2 1 22eyz

y y b z z b

zz

(28)

where

y

b

and

z

b

represent the two planar directions. These parameters control the rate of covariance decay.

The eigensolutions of the covariance function can be obtained by solving the integral analytically (refer to

Equation 25)

1

2

1

2

2 2 1 1 2 2 1 1 1 1

, ( , ; , ) ,

aa

i i i

a

a

y z y z y z y z dy dz

(29)

where,

11

a y a

and

22

a z a

. Assuming that the eigen-solution can be separated in y and z

directions and substituting the covariance function

16

2 2 2 2

,yz

i i i

y z y z

(30)

2 2 2 2

,yz

i i i

y z y z

(31)

, the solution of equation (29) can be reduced to the product of the solutions of two equations having the

form

1

12

1

()

1 2 2

y

ay y b

y y y

i i i

a

y e y dy

(32)

Solution of the above equation, which is the eigensolution of an exponential covariance kernel with a one-

dimensional random field, can be obtained as

2

22

*2

**2 2

*

*

2

cos for odd

sin 2

2

sin 2for even

sin 2

2

z

z

i

ii

i

i

i

i

ii

i

i

i

bi

b

a

bi

b

a

(33)

Where

1

y

bb

or

1

z

b

and

1

aa

or

2

a

. The parameter

is either y or z. Here

i

and

*

i

are computed

from the solutions of the following equations

tan( ) 0

ii

ba

and

tan( ) 0

ii

ba

, respectively. It

can be noted that the KL expansion was formulated for discretization following gaussian random fields. In

case of lognormal random fields, as considered in the present study, the KL expansion is formulated on its

gaussian image. In the current study of laminated composite plates, the spatially correlated properties are

parameterized by the respective mean values (considered to be same as the deterministic value of a

particular parameter), the coefficient of variation (COV) and two correlation parameters. The degree of

stochasticity is defined as the coefficient of variation (

) of a particular stochastic input parameter.

17

4.3. Description of the uncertainty quantification framework

4.3.1. Monte Carlo simulation

Uncertainty quantification is part of modern structural analysis problems. Practical structural

systems need to face uncertainty, variability and ambiguity on a constant basis. Even after having

unprecedented access to the information due to recent improvement in various technologies, it is

impossible to accurately predict future structural behaviour during its service life. Monte Carlo

simulation, a computerized mathematical technique, lets us realize all the possible outcomes of a

structural system leading to better and robust designs for the intended performances. This technique was

first used by the engineers and scientists developing the atom bomb and it was named after a Monaco

resort town Monte Carlo. Since the introduction during World War II, this technique has been applied to

model various physical and conceptual systems across different fields covering engineering,

manufacturing, energy, finance, insurance, project management, transportation and environment.

Monte Carlo simulation (MCS) furnishes a range of prospective outcomes along with their

respective probability of occurrence. This technique performs uncertainty quantification by forming

probabilistic simulations of all prospective results accounting a wide range of values from the probability

distributions of any factor having an inherent uncertainty. This method simulates the outputs multiple

times, using a different set of random values each time, drawn from the probability distribution of

stochastic input parameters. Depending upon the nature of stochasticity, a Monte Carlo simulation may

involve thousands or tens of thousands of realizations (/function evaluations) before it can provide a

converged result depicting the distributions of possible outcome values of the response quantities of

interest. Thus Monte Carlo simulation provides not only a comprehensive idea of what could happen, but

also how likely it is to happen i.e. the probability of occurrence.

The mean (/expected) value of a function

()fx

having an n dimensional vector of random variables

and a joint probability density function

()x

, is expressed as

()

fE f x f x x dx

(34)

18

The variance of the function

()fx

is given as,

2

2()

ff

Var f x f x x dx

(35)

The multidimensional integrals, as shown in equation (34) and (35) are difficult to compute analytically

various types of joint density functions. Moreover, the integrand function

()fx

may not always be

available in an analytical form for the problem under consideration. Thus the only alternative way is to

calculate it by numerical means. It can be computed using the MCS approach, wherein N number of

sample points is generated following a suitable sampling scheme in the random variable space of n-

dimensions. The N number of samples drawn from a dataset must follow the same distribution as

()x

.

The function

()fx

is computed at each of the sampling points of the sample set

1,............, N

xx

.

Thus, the integral for the expected value can be expressed in the form of averaging operator as

1

1N

i

fi

E f x f x

N

(36)

In a similar manner, using sampled values of MCS as above, the equation (35) leads to

2

2

1

11

N

i

ff

i

Var f x f x

N

(37)

Thus the statistical moments can be obtained using a brute force MCS based approach, which is often

computationally very intensive due to the evaluation of function

()

i

fx

corresponding to the N-sampling

points , where N ~ 104. The noteworthy fact in this context is the adoption of a surrogate based Monte

Carlo simulation approach in the present study that reduces the computational burden of traditional (i.e.

brute force) Monte Carlo simulation to a significant extent.

4.3.2. Modelling of source-uncertainty at the input level

The stochasticity in material properties (micro/ macro- mechanical properties) and geometric

properties (like ply-orientation angle and thickness of plate) are considered as stochastic input parameters

for analyzing the probabilistic dynamic and buckling characteristics of laminated composite plates. In the

i

x

i

x

19

present article, two separate forms of analysis have been performed considering the stochasticity in micro-

mechanical and macro-mechanical properties to understand and ascertain the cascading effect in

uncertainty propagation on a comparative basis. For analysing the effect of various source-uncertainties,

the following four cases of stochasticity are considered

(i) Compound effect for the simultaneous variation of macro-mechanical material properties (such as

Young’s moduli, shear moduli, mass density and Poisson’s ratio) and geometric properties (such as ply-

orientation angle and thickness of laminae)

1 2 12 13 23

1 2 3

1(1,1) 1( , ) 2(1,1) 2( , ) 12(1,1) 12( , )

45

13(1,1) 13( , ) 23(1,1) 23(

( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )

( .... ), ( .... ), ( .... ),

( .... ), ( ....

macro e l macro e l macro e l

macro e

C

macro

l macro

E E G G G t

E E E E G G

G G G

g

G

6

, ) (1,1) ( , )

7 8 9

(1,1) ( , ) (1,1) ( , ) (1,1) ( , )

), ( .... ),

( .... ), ( .... ), ( .... )

e l macro e l

macro e l macro e l macro e l

tt

(38)

(ii) Compound effect for the simultaneous variation of micro-mechanical material properties such as

Young’s moduli of fibre and matrix, shear moduli of fibre and matrix, Poisson ratios of fibre and matrix,

mass densities of fibre and matrix and volume fraction along with geometric properties (ply orientation

angle and thickness of laminae)

12

1 2 3 4

1 (1,1) 1 ( , ) 2 (1,1) 2 ( , ) (1,1) ( , ) (1,1) ( , )

( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )

( .... ), ( .... ), ( .... ), ( .... ),

f f m f m f m f m f

micro f f e l micro f f e l micro m m e l micr

C

m

o f f e l

mi

icr

c

o

r

E E E G G V t

E E E E E E G G

g

5 6 7 8

(1,1) ( , ) (1,1) ( , ) (1,1) ( , ) (1,1) ( , )

9 10 11

(1,1) ( , ) (1,1) ( , ) (1,1) ( , )

( .... ), ( ...., ), ( ...., ), ( .... ),

( .... ), ( .... ), ( .... ),

o m m e l micro f f e l micro m m e l micro f f e l

micro m m e l micro f f e l micro f f e l micr

GG

VV

12 (1,1) ( , )

( .... )

o f f e l

tt

(39)

(iii) Individual effect for the variation of a single macro-mechanical property

1(1,1) ( , )

( ) ( .... )

I

macro M macro M M e l

g

(40)

(iv) Individual effect for the variation of a single micro-mechanical property

1(1,1) ( , )

( ) ( .... )

I

micro m micro m m e l

g

(41)

Here

is a symbolic operator that generates a set of input parameters for carrying out the Monte Carlo

simulation. The parameters

1 , 2 , 12( , ) 13 , 23 , , , ,

( ) ( ) ( ) ,

, , , , ,,,,

i j i j i j i j i j i j i j i j i j

E E G G G t

denote the longitudinal

Young’s modulus, transverse Young’s modulus, shear moduli, Possoin’s ratio, mass density, ply

20

orientation angle and thickness of lamina respectively (with conventional notations) for the ith SRVE

situated in the jth layer, where i = 1, 2, 3,…, e and j = 1, 2, 3, …, l. For the stochasticity in micro-

mechanical properties,

1 ( , ) 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

, , , , , , , , , , ,

f i j f i j m i j f i j m i j f i j m i j f i j m i j f i j i j i j

E E E G G V t

denote (following conventional notations) Young’s moduli of fibre in longitudinal and transverse

directions, Young’s modulus of matrix, shear modulus of fibre and matrix, Poisson’s ratio of fibre and

matrix, mass density of fibre and matrix and volume fraction, ply orientation angle and thickness

respectively corresponding to

th

i

layer, respectively for the ith SRVE situated in the jth layer, where i = 1, 2,

3,…, e and j = 1, 2, 3, …, l. The quantities

M

and

m

denote any one of the macromechanical and

micromechanical properties. The material properties are considered to vary spatially (correlated variation

following the KL expansion) for both the macromechanical and micromechanical analyses. However,

considering practical aspects of modelling uncertainty in the geometric parameters, spatial variation is not

considered for ply orientation angle and thickness of lamina; rather a layer-wise uncorrelated random

variation is considered for the two geometric parameters (i.e.

(1, ) (2, ) (3, ) ( , )

...

j j j e j

and

(1, ) (2, ) (3, ) ( , )

...

j j j e j

t t t t

, for the jth lamina).

4.3.3. Propagation of uncertainty based on HDMR coupled with DMORPH algorithm

In a typical problem of uncertainty analysis, there are normally three aspects that need to be dealt

with. The first aspect is source-uncertainty modelling at the input level, which is discussed in the

preceding subsection. After the uncertainty in material and structural attributes is modelled in a

practically relevant way, the next concern is propagating the effect of uncertainty from the local input-

level to the global level of quantifying output responses following a bottom-up framework. As discussed

in section 1, direct Monte Carlo simulation being a computationally intensive approach, we have adopted

a surrogate based uncertainty propagation scheme for the present analysis. To achieve computational

efficiency, a HDMR based surrogate modelling framework coupled with the DMORPH algorithm is

developed in conjunction with the probabilistic finite element model of composite plates as presented in

figure 6.

21

Fig. 6 Flowchart for stochastic macro and micro mechanical analysis of laminated composites based on

surrogate models (Representative figures of finite element analysis, sobol’s quasi-random sampling,

surrogate modelling and uncertainty quantification are shown corresponding to the respective steps)

The surrogate models are constructed by choosing the sample points optimally from a domain Rn.

All input variables are rescaled in the range of 0 ≤ xi ≤ 1, where xi denotes the ith input parameter.

Generation of the random sample points is an important aspect to form the HDMR model because the

quality of the random sample points governs the convergence rate and prediction accuracy. Quasi-random

22

sequences (Niederreiter (1992)) (such as Halton (Halton (1960)), Sobol’ (Sobol (1967)), Faure (Faure

(1992))) having low discrepancy are often used for generating random sample points that ensures a

uniform distribution of input sample points in the design domain. It essentially results in a faster

convergence rate compared to pseudo-random sample points. In this work, Sobol’ sequence is used to

generate the input sample set as it shows a better convergence rate than Faure and Halton sequences

(Galanti and Jung (1997)). It can be noted in this context that in a surrogate based approach, first the

surrogate model is constructed using few optimally chosen design points. The same number (number of

design points) of finite element simulations (\function evaluations) is required to be performed for the

surrogate model formation. Here the HDMR model replaces the original finite element model (expensive)

effectively by an efficient mathematical/statistical model. After the HDMR based surrogate model is

constructed, thousands of virtual simulations can be conducted for various random combinations of the

input parameters using the efficient HDMR model.

5. Results and discussion

In this article, numerical results for stochastic dynamic and stability analyses are presented for a

three layered graphite-epoxy angle-ply ([45o/-45o/45o]) composite square plate, unless otherwise

mentioned. A practically relavent randomly inhomogeneous (Mukhopadhyay and Adhikari (2017b))

model of stochasticity with spatially correlated system parameters are considered for characterizing the

first three modes of vibration and buckling of composite plates. Results are presented for two distinct

cases: stochasticity in micromechanical and macromechanical material properties (refer to equation 38 –

41). The deterministic micromechanical properties (E-glass

21 43xK

Gevetex/

3501 6

epoxy) of

composite material are shown in Table 1 (Soden et al. (1998)). Applying Halpin-Tsai principle (Jones

(1999)) the deterministic macromechanical material properties are obtained with a volume fraction (

f

V

)

of 0.61 (refer to Table 2). Thus, for the case of stochasticity in micromechanical properties, the material

atributes presented in Table 1 are assumed as the source of stochasticity along with the uncertain

geometric parameters (

C

micro

g

) and thereby the macromechanical material properties are obtained based on

Halpin- Tsai principle to perform further analysis for quantifying uncertainty. For the case of stochasticity

23

Table 1 Deterministic micromechanical material properties of composites

Property

Value

Longitudinal Young’s modulus of fibre (

1f

E

)

80 GPa

Transverse Young’s modulus of fibre (

2f

E

)

80 GPa

Poisson's ratio of fibre (

f

)

0.2

Shear modulus of matrix (

f

G

)

33.33 GPa

Mass density of fibre (

f

)

2.55 gm/cc

Mass density of matrix (

m

)

1.265 gm/cc

Young’s modulus of matrix (

m

E

)

4.2 GPa

Shear modulus of matrix (

m

G

)

1.567 GPa

Poisson's ratio of matrix (

m

)

0.34

Fibre volume fraction (

f

V

)

0.61

Table 2 Deterministic macro-mechanical material properties of composites (

f

V

= 0.61)

Property

Value

Longitudinal Young’s modulus (

1

E

)

50.438 GPa

Transverse Young’s modulus (

2

E

)

9.952 GPa

Poisson's ratio (

12

)

0.2546

In-plane shear modulus (

12

G

)

3.742 GPa

Mass density (

)

2.049 gm/cc

Shear modulus (

13

G

)

3.742 GPa

Transverse shear modulus (

23

G

)

2.094 GPa

24

in macromechanical material properties, the analysis commences one step ahead in the hierarchy i.e. the

source-uncertainty is assumed in the macromechanical properties (as shown in Table 2) along with

uncertain geometric parameters (

C

macro

g

). Subsequently, the results obtained from these two different types

of analyses are compared to ascertain the cascading effect in stochasticity. Non-dimensional results are

presented following the scheme mentioned in the caption of Table 3 and 4.

5.1. Stochastic dynamic analysis

5.1.1. 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 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.

The results for validation and convergence study of the finite element code of a composite plate are

shown in Table 3, wherein non-dimensional natural frequencies are validated with the results available in

scientific articles (Liew and Huang (2003)). Based on the results presented in Table 3, a mesh size of 9

9

is adopted for the finite element analysis hereafter. The optimum number of samples (drawn from Sobol

sequence) to construct the surrogate models are decided based on the comparative performance (four

different statistical parameters: minimum value, maximum value, mean value and standard deviation)

with respect to direct Monte Carlo simulation. The convergence results for macro and micro mechanical

analyses showing the values of absolute error with different sample size are presented in figure 7 and 8.

From the figures it is evident that a sample size of 1024 provides reasonably accurate results for the

natural frequencies. To further examine the prediction capabilities of the surrogate models, scatter plots

are presented for the macro and micro mechanical analyses in figures 9(a-c) and 10(a-c), respectively.

Negligible deviation of the sample points from the diagonal line corresponding to the sample size of 1024

indicates the accuracy of prediction. The comparative probability density function plots on the basis of

25

Table 3 The convergence study of frequency parameters [

2

20

( ) ( )

h

bD

, where

3

2

012 21

12(1 )

Eh

D

] for cross-ply

(0 / 90 /0 )

o o o

simply supported (SSSS) rectangular laminates

t/b

Mesh

Mode 1

Mode 2

Mode 3

Present

FEM

Liew (1996)

Present

FEM

Liew (1996)

Present

FEM

Liew (1996)

0.001

5×5

6.8817

6.6253

9.7192

9.4653

24.4234

22.4607

6×6

6.7864

9.6212

24.0928

7×7

6.7056

9.5312

23.4168

8×8

6.6258

9.4684

22.4802

9×9

6.6251

9.4644

22.4520

10×10

6.6210

9.4401

22.2811

0.20

5×5

3.6867

3.5940

6.1626

5.7784

8.1347

7.3972

6×6

3.6314

6.0189

7.9167

7×7

3.6145

5.9180

7.6512

8×8

3.5978

5.7337

7.3368

9×9

3.5913

5.7331

7.3345

10×10

3.5839

5.7036

7.2366

surrogate based MCS corresponding to the sample size of 1024 and direct MCS are presented for the

macro and micro mechanical analyses in figures 9(d-f) and 10(d-f), respectively. A good agreement

between the probabilistic descriptions of natural frequencies corroborates the accurate prediction

capability of the surrogate models for further analyses. It can be noted in this context that computational

time required is exorbitently high for evaluating the probabilistic responses through full scale 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 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 finite element analyses

26

(a) Maximum

(b) Minimum

(c) Mean

(d) Standard Deviation

Fig. 7 Absolute percentage error in minimum value, maximum value, mean value and standard deviation

with respect to direct Monte Carlo simulation for macromechanical material properties for a coefficient of

variation of 0.6 in the stochastic input parameters (

C

macro

g

).

are decreased significantly in comparison to full-scale direct MCS.

5.1.2. Results for stochastic dynamic analysis

Having the FE model and the surrogate model validated, as shown in the preceding subsection,

stochastic results are presented in this subsection for the first three modes of vibration for a composite

plate with correlated spatially varying material properies (

C

macro

g

and

C

micro

g

). Figure 11 shows the

probabilistic descriptions for first three natural frequencies considering different degrees of stochasticity

for macro-mechanical and micro-mechanical material properties. For both the micro and macro

mechanical analyses, response bounds are noticed to substantially increase with the increasing degree of

stochasticity along with a marginal change in mean values. The probabilistic descriptions of the natural

frequencies differ from each other on the basis of the adopted type of analysis. A micromechanical

analysis, which is more accurate for considering the source uncertainty at a more elementary level, shows

higher degree of variability in the global responses due to the cascading effect in stochasticity.

27

(a) Maximum

(b) Minimum

(c) Mean

(d) Standard Deviation

Fig. 8 Absolute percentage error in maximum, minimum, mean and standard deviation with respect to

original MCS considering a coefficient of variation of 0.6 in the stochastic input parameters (

C

micro

g

).

The probabilistic variations for first three natural frequencies are investigated for various laminate

configurations considering the

//

family of composites for stochasticity in micro-mechanical

properties. The probabilistic descriptions presented in figure 12 show that the mean values, depending on

the effective stiffness of the structure, decrease up to a critical value of

45o

, and after this point they

increase again for higher values of

. Figure 13 shows the influence of different boundary conditions on

the stochastic natural frequencies of composite plates. Results are presented for simply supported (SSSS)

and fixed (CCCC) boundaries at all the four edges considering stochasticity in micro-mechanical

properties. The range of natural frequencies is found to vary depending on the stiffness of the system,

following a similar trend as deterministic analysis. Figure 14 shows the effect of aspect ratio of the

composite laminated plates on the probabilistic variation of natural frequencies considering stochasticity

in micro-mechanical material properties. The stochastic natural frequencies are noticed to reduce with the

increase in aspect ratio.

28

Fundamental natural frequency

(a)

(d)

Second natural frequency

(b)

(e)

Third natural frequency

(c)

(f)

Fig. 9 (a-c) Scatter plots for HDMR based analysis (normalized on the basis of respective deterministic

values) for various sample sizes of Sobol sequence considering the stochasticity in macro-mechanical

attributes (

C

macro

g

) with respect to direct MCS (with coefficient of variation of 0.6 in the stochastic input

parameters) ; (d-f) Probability density function (pdf) of natural frequencies (normalized with respect to

the corresponding deterministic values) for macro-mechanical attributes (

C

macro

g

) obtained by direct MCS

and HDMR model based on a sample size of 1024 (with coefficient of variation of 0.6 in the stochastic

input parameters)

29

Fundamental natural frequency

(a)

(d)

Second natural frequency

(b)

(e)

Third natural frequency

(c)

(f)

Fig. 10 (a-c) Scatter plots for HDMR based analysis (normalized on the basis of respective deterministic

values) for various sample sizes of Sobol sequence considering the stochasticity in micro-mechanical

attributes (

C

micro

g

) with respect to direct MCS (with coefficient of variation of 0.6 in the stochastic input

parameters) ; (d-f) Probability density function (pdf) of natural frequencies (normalized with respect to

the corresponding deterministic values) for micro-mechanical attributes (

C

micro

g

) obtained by direct MCS

and HDMR model based on a sample size of 1024 (with coefficient of variation of 0.6 in the stochastic

input parameters)

30

(a)

(b)

(c)

Fig. 11 Probability density function (pdf) plots for macro (

C

macro

g

) and micro (

C

micro

g

) mechanical analyses

of natural frequencies for different degree of stochasticity (

). Normalized results are used with respect to

the corresponding deterministic values for plotting the pdfs. The bar plots in the inset indicate the

percentage increase of stochastic bounds for a micromechanical analysis with respect to the

macromechanical analysis.

Figure 15 presents the relative coefficient of variation (RCOV) for individual stochastic effect of

different uncertain input parameters considering both the macro (

I

macro

g

) and micro (

I

macro

g

) mechanical

analyses. To obtain these figures, MCS are carried out for the variation of each of the micro and macro

mechanical material parameters individually. The figures can provide a clear understanding regarding the

relative sensitivity of various stochastic system parameters (input) to the output natural frequencies in the

macro and micro mechanical analyses. For stochasticity in macro-mechanical material properties, it is

observed that mass density, longitudinal Young’s modulus and transverse Young’s modulus (in

decreasing order of sensitivity) are most sensitive to the first three natural frequencies, while the shear

31

(a)

(b)

(c)

Fig. 12 Probability density function (pdf) plots for natural frequencies considering different ply

orientation angle (

) for stochasticity in micro-mechanical properties (

C

micro

g

)

moduli and Poisson’s ratio are comparatively less sensitive. For the micro-mechanical analysis, it is found

that the most sensitive system parameters (stochastic input) according to decreasing order of sensitivity

are mass density of fibre, longitudinal Young’s modulus of fibre and volume fraction, while transverse

Young’s modulus of fibre, Poisson’s ratio of fibre and shear modulus of fibre are the least sensitive

parameters. Two different forms of analyses, as carried out here considering macro and micro mechanical

material properties, render an in-depth understanding regarding the relative influence of various stochastic

input parameters (source-uncertainty). For example, the macromechanical analysis (refer to figure 15(a))

shows that mass density of composites is the most sensitive parameter for low frequency vibration modes;

however, the micromechanical analysis (refer to figure 15(b)) provides information in more depth

showing that mass density of fibre is the most sensitive parameter. Outcomes of such sensitivity analyses

serve as an important guideline for efficient uncertainty quantification (including dimensionality

reduction) and subsequent analysis/ design and quality control of input parameters.

32

CCCC

SSSS

Fundamental natural frequency

(a)

(d)

Second natural frequency

(b)

(e)

Third natural frequency

(c)

(f)

Fig. 13 (a-c) Probability density function (pdf) plots of natural frequencies for micro mechanical analysis

(

C

micro

g

) considering clamped boundary condition (CCCC) (d-f) Probability density function (pdf) plots of

natural frequencies for micro mechanical analysis (

C

micro

g

) considering simply supported boundary

condition (SSSS)

33

(a)

(b)

(c)

Fig. 14 Probability density function (pdf) plots of natural frequencies for different values of aspect ratios

(AR) considering micro-mechanical properties (

C

micro

g

)

The random field of the stochastic micro and macro mechanical material properties depend on the

correlation length considered in the analysis. We have investigated the effect of correlation length on the

first three modes of vibration. Depending on the value of correlation length, the random field of all the

micro and macro mechanical properties would vary; representative plots are presented in figure 16

showing the spatial distribution of micro and macro mechanical material properties concerning the

longitudinal Young’s modulus for a single random realization considering different values of correlation

length. It can be noted that two extreme cases can be realized when the correlation length tends to the

upper and lower limits. The system becomes a randomly homogenous system (analogous to the random

variable based approach) when the correlation length is very high, while it becomes an uncorrelated

randomly inhomogeneous system when the correlation length is very low. The typical probability

distribution of a representative micromechanical property (E1f) and a macromechanical property (E1) for a

randomly chosen SRVE having three different laminae is shown in figure 17 considering a correlation

34

(a)

(b)

Fig. 15 (a) Relative coefficient of variation (RCOV) for the natural frequencies considering macro-

mechanical material properties (b) Relative coefficient of variation (RCOV) for the natural frequencies

considering micro-mechanical material properties

length of 1/50. The effect of correlation length on the probability density function plots of the first three

natural frequencies is presented in figure 18, wherein a clear difference is noticed between the lower and

higher values of correlation lengths. This, in turn indicates the difference in the probabilistic

characteristics of the natural frequencies corresponding to random field (lower values of correlation

length) and random variable (higher value of correlation length) approach. The figure shows that the

mean and response bound decrease in case of the random field based modelling of source-uncertainty

compared to the random variable based modelling.

35

CoL = 1

CoL = 1/20

CoL = 1/50

CoL = 1/100

CoL = 1/200

E1f

(a)

(b)

(c)

(d)

(e)

Em

(f)

(g)

(h)

(i)

(j)

Vf

(k)

(l)

(m)

(n)

(o)

E1

(p)

(q)

(r)

(s)

(t)

Fig. 16 Representative plots showing the spatial distribution of micro and macro mechanical properties concerning the longitudinal Young’s modulus for a random

realization considering different values of correlation length (CoL). Here X and Y axes in the figures are the two spatial directions of a lamina, while the Z axis shows

the value of corresponding material property.

36

(a)

(b)

Fig. 17 Probability distribution of a representative micromechanical property (E1f) and a

macromechanical property (E1) for a randomly chosen SRVE having three different laminae considering a

correlation length of 1/50

(a)

(b)

(c)

Fig. 18 Effect of correlation length (CoL) on the probability distribution of the first three natural

frequencies of a composite laminate

The effect of stochasticity in micro and macro mechanical material properties are studied on the

vibration mode shapes considering two different boundary conditions (SSSS and CCCC). The results are

presented in figure 19 for first three modes of vibration. Stochastic mode shapes for first three modes of

37

Fig. 19 Stochastic modeshapes and representative probability distribution of the normalized eigenvectors

38

vibration considering a single random realization are presented in case of: a clamped (CCCC) composite

plate with stochasticity in the macromechanical (

C

macro

g

) properties (refer to figure 19(a – c)), a simply

supported (SSSS) composite plate with stochasticity in the macromechanical (

C

macro

g

) properties (refer to

figure 19(d – f)), a clamped (CCCC) composite plate with stochasticity in the micromechanical (

C

micro

g

)

properties (refer to figure 19(g – i)) and a clamped (CCCC) composite plate with stochasticity in the

micromechanical (

C

micro

g

) properties (refer to figure 19(j – l)). From the mode shapes presented in figure

19(a – l), it can be observed that the basic global pattern of the stochastic mode shapes remains similar to

the corresponding deterministic case. However, the value of normalized eigenvectors becomes stochastic

in nature for each of the elements in the composite plate. Probability distribution of the normalized

eigenvectors of first three vibration modes for the elements indicated in figure 19(o) are shown

considering a clamped (CCCC) boundary condition (refer to figure 19(m)) and a simply supported (SSSS)

boundary condition (refer to figure 19(n)). The results for micro and macro mechanical analyses are

shown using lighter and darker shades of respective colours indicated in figure 19(o). It can be noticed

that the probability density function plots depend significantly on the type of analysis (micro and macro

mechanical) and location of the element under consideration.

5.2. Stochastic stability analysis

5.2.1. Validation and convergence study

The FE code and the surrogate model are validated first for analysing the buckling loads similar to

the case of dynamic analysis as discussed in section 5.1.1. The results of convergence study and

validation of the finite element code of a composite plate is furnished in Table 4, wherein the non-

dimensional first buckling load is validated with the results available in scientific literature. Based on the

results presented in Table 4, a mesh size of 9

9 is found to be adequate for the finite element model. The

optimum number of samples (drawn from Sobol sequence) to form the surrogate models of buckling

loads are decided based on the comparative performance (four different statistical parameters: minimum

value, maximum value, mean value and standard deviation) with respect to direct MCS. The results of

convergence study for macro and micro mechanical analyses showing the values of absolute error with

39

Table 4 The convergence study of uniaxial buckling load of four layer

(0 /90 /90 /0 )

o o o o

simply

supported SSSS rectangular laminates

23

2

( ( ), 0, 0)

xx xy yy

N N b E h N N

t/b

Mesh

Buckling load

Present FEM

Reference results

0.001

5×5

23.3511

23.2928 (Neves and Ferreira (2016))

23.463 (Liew and Huang (2003))

6×6

23.3217

7×7

23.3190

8×8

23.2940

9×9

23.2927

10×10

23.2885

(a) Maximum

(b) Minimum

(c) Mean

(d) Standard Deviation

Fig. 20 Absolute percentage error in minimum value, maximum value, mean value and standard deviation

with respect to direct MCS for macro-mechanical analysis considering a coefficient of variation of 0.6 in

the stochastic input parameters (

C

macro

g

)

40

(a) Maximum

(b) Minimum

(c) Mean

(d) Standard Deviation

Fig. 21 Absolute percentage error in minimum value, maximum value, mean value and standard deviation

with respect to direct MCS for micro-mechanical analysis considering a coefficient of variation of 0.6 in

the stochastic input parameters (

C

micro

g

)

different sample size are presented in figure 20 and 21. From the figures it is evident that a sample size of

1024 provides reasonably accurate results for predicting the buckling loads. To further examine the

prediction capabilities of the surrogate models, scatter plots are presented for the macro and micro

mechanical analyses in figures 22(a-c) and 23(a-c), respectively. Negligible deviation of the sample

points from the diagonal line corresponding to the sample size of 1024 indicates the accuracy of

prediction. The comparative probability density function plots on the basis of surrogate based MCS

corresponding to the sample size of 1024 and direct MCS are presented for the macro and micro

mechanical analyses in figures 22(d-f) and 23(d-f), respectively. A good agreement between the

probabilistic descriptions of natural frequencies corroborates the accurate prediction capability of the

surrogate models for further analyses.

In this section, we have presented the results of stochastic stability analysis considering the first

three buckling modes for the sake of completeness (as presented in previous scientific literatures like Patel

41

First buckling load

(a)

(d)

Second buckling load

(b)

(e)

Third buckling load

(c)

(f)

Fig. 22 (a-c) Scatter plots for HDMR based analysis (normalized on the basis of respective deterministic

values) for various sample sizes of Sobol sequence considering the stochasticity in macro-mechanical

attributes (

C

macro

g

) with respect to direct MCS (with coefficient of variation of 0.6 in the stochastic input

parameters) ; (d-f) Probability density function (pdf) of buckling loads (normalized with respect to the

corresponding deterministic values) for macro-mechanical attributes (

C

macro

g

) obtained by direct MCS and

HDMR model based on a sample size of 1024 (with coefficient of variation of 0.6 in the stochastic input

parameters)

42

First buckling load

(a)

(d)

Second buckling load

(b)

(e)

Third buckling load

(c)

(f)

Fig. 23 (a-c) Scatter plots for HDMR based analysis (normalized on the basis of respective deterministic

values) for various sample sizes of Sobol sequence considering the stochasticity in micro-mechanical

attributes (

C

micro

g

) with respect to direct MCS (with coefficient of variation of 0.6 in the stochastic input

parameters) ; (d-f) Probability density function (pdf) of buckling loads (normalized with respect to the

corresponding deterministic values) for micro-mechanical attributes (

C

micro

g

) obtained by direct MCS and

HDMR model based on a sample size of 1024 (with coefficient of variation of 0.6 in the stochastic input

parameters)

43

(a)

(b)

(c)

Fig. 24 Probability density function (pdf) plots for macro (

C

macro

g

) and micro (

C

micro

g

) mechanical analyses

of first three buckling loads considering different degree of stochasticity (

). Normalized results are used

with respect to the corresponding deterministic values for plotting the pdfs. The bar plots in the inset

indicate the percentage increase of stochastic bounds for a micromechanical analysis with respect to the

macromechanical analysis.

and Sheikh (2016)). Besides that, in case of stochastic stability analysis, the buckling loads are found to

have a response bound (depending on the degree of stochasticity) with respect to the corresponding

deterministic values. Thus there exists a possibility of overlap in the stochastic responses of the buckling

loads corresponding to different modes of buckling resulting in a non-unique critical buckling mode with

the minimum value of buckling load. For this reason, it is essential to consider higher buckling modes in

case of stochasticity in the system parameters.

5.2.2. Results of stochastic stability analysis

Having the finite element model and the surrogate model validated, as shown in the preceding

subsection, stochastic results are presented in this subsection for the first three modes of buckling for a