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1

Spatially varying fuzzy multi-scale uncertainty propagation in unidirectional

fibre reinforced composites

Susmita Naskara, Tanmoy Mukhopadhyayb,*, Srinivas 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 (T. Mukhopadhyay)

Abstract

This article presents a non-probabilistic fuzzy based multi-scale uncertainty propagation framework

for studying the dynamic and stability characteristics of composite laminates with spatially varying

system properties. Most of the studies concerning the uncertainty quantification of composites rely on

probabilistic analyses, where the prerequisite is to have the statistical distribution of stochastic input

parameters. In many engineering problems, these statistical distributions remain unavailable due to

the restriction of performing large number of experiments. In such situations, a fuzzy-based approach

could be appropriate to characterize the effect of uncertainty. A novel concept of fuzzy representative

volume element (FRVE) is developed here for accounting the spatially varying non-probabilistic

source-uncertainties at the input level. Such approach of uncertainty modelling is physically more

relevant than the prevalent way of modelling non-probabilistic uncertainty without considering the

ply-level spatial variability. An efficient radial basis function based stochastic algorithm coupled with

the fuzzy finite element model of composites is developed for the multi-scale uncertainty propagation

involving multi-synchronous triggering parameters. The concept of a fuzzy factor of safety (FFoS) is

discussed in this paper for evaluation of safety factor in the non-probabilistic regime. The results

reveal that the present physically relevant approach of modelling fuzzy uncertainty considering ply-

level spatial variability obtains significantly lower fuzzy bounds of the global responses compared to

the conventional approach of non-probabilistic modelling neglecting the spatially varying attributes.

Keywords: composite laminates; spatially varying fuzzy uncertainty; fuzzy representative volume

element; fuzzy multi-scale analysis; radial basis function

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1. Introduction

Composite materials are increasingly being utilised as various components of aerospace,

civil, automobile, mechanical and sports structures due to their high strength-to-weight ratio and cost

effectiveness. Composites materials can also be aero-elastically customized to meet different design

conditions. 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 [Dey et al. (2018a)]. The issue aggravates further due to uncertain

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

defects during the service life. The structural performances of composite materials are often subjected

to significant element of risks due to being susceptible to multiple uncertainties. Hence it is important

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

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

Deterministic analyses of composite structures have received considerable attention from the

scientific community [Reddy (2003), Chakrabarti et al. (2013), Liew (1996), Liew and Huang (2003),

Neves and Ferreira (2016), Tornabene et al. (2017, 2018)]. The aspect of considering uncertainty in

the analysis and design of composite structures are becoming increasingly recognised in last few

years. Probabilistic approaches are found to be predominantly used to quantify the effect of

uncertainty in composite structures following intrusive [Lal and Singh (2010), Scarth and Adhikari

(2017), Zhou et al. (2017)] as well as non-intrusive [Shaw et al. (2010), Dey et al. (2016a, 2018),

Umesh and Ganguli (2013), Naskar et al. (2017a, 2017b), Karsh et al. (2018), Sakata et al. (2008),

Naskar and Sriramula (2017a, 2017b, 2017c, 2018), Mukhopadhyay and Adkhikari (2016b);

Mukhopadhyay et al. (2016c)] methods. The prerequisite of carrying out a probabilistic analysis is to

3

Fig. 1 Fuzzy multi-scale analysis of composite laminates considering the coupled effect of material

and geometric uncertainty (a) Typical distribution of a typical material property y (such as E1) along a

cross-sectional view (X-Z plane) of two laminae for a particular realization in case of the

conventional approach where spatial variation of y is ignored (b) Typical distribution of a material

property y along a cross-sectional view (X-Z plane) of two laminae for a particular realization in case

of the present approach considering spatial variation of material properties (c) Typical representation

of a fuzzy based analysis

have the statistical distribution of stochastic input parameters available. Composites being a multi-

layerd complex and expensive material, statistical distributions of the stochastic input parameters

may not always be available due to restriction of performing large number of experimnents to

characterize the material properties. In such situation, a non-probabilistic approach (such as fuzzy

uncertainty) could be suitable to characterize the effect of uncertainty. The fuzzy based uncertainty

propagation approach is found to be adopted in very limited number of studies concerning

composites [Dey et al. (2016b), Pawar et al. (2012), Pawar and Ganguli (2007), Muc and Kedziora

(2001)]. However, these studies do not consider the inevitable spatial variability in material

properities of each of the laminas. This approach of modelling the non-probabilistic form of

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uncertainty without accounting spatial variability has limited practical relevance. Thus in this paper,

we aim to consider the ply-level spatial variability of the material properties of composites in a more

realistic non-probabilistic framework (refer to figure 1). The aspect of spatial variation of lamina

properties is illustrated in figure 1(b) for a particular 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).

In general, the fuzzy finite element analysis [Lei and Chen (2002), Mace et al. (2005)] can

couple the potential of finite element method and uncertainty modeling capability of the fuzzy

approach [Zadeh (1975)] in the presence of sparse input data. A way of viewing the fuzzy input-

output parameters is the universality of an interval variable. In the interval approach, the

uncertainties of input parameters are expressed by interval values where a statement about the

likelihood of the individual values within the interval is not given. In order to solve the

corresponding finite element system, the interval arithmetic technique may be used. In the fuzzy

finite element method, input variables are characterized by fuzzy sets to represent data and

information possessing non-statistical uncertainties. The underlying principle of fuzzy representation

is that the boundary values are generally less possible and therefore have a lower likelihood

compared to the expected value. In this way, the fuzzy finite element based method provides a

generalization of the interval finite element method using multiple α-cuts. It can be noted in this

context that the intervals do not represent the numerical values of a parameter, but a knowledge

regarding the range of possible values (lower and upper bounds) that the parameter can possess. In

fuzzy approach, a membership function is introduced to generalize the interval based approach.

The propagation of uncertainty in composite structures possesses significant computational

challenges as the evaluation of structural responses (i.e. output) for composites normally involves a

complicated and computationally expensive numerical process like finite element method. This paper

develops a radial basis function (RBF) [Beatson (1999), Fasshauer (1997), Hon and Mao (2001),

Kansa (1990a, 1990b), Kansa and Hon (2000)] based non-probabilistic uncertainty propagation

method coupled with the fuzzy finite element model of composites to investigate the dynamics and

5

stability characteristics of the structure in an efficient manner. In the fuzzy RBF method, the RBF

model acts as an emulator (surrogate) for the expensive finite element model at the zero level

membership function. The input-output domain corresponding to the zero level membership function

is mapped using the RBF based metamodel formed on the basis of Sobol’s sequence [Sobol (1967)].

Thereby, the RBF models can be used for prediction corresponding to the higher α-levels of fuzzy

membership function. To incorporate the RBF based approach in fuzzy analysis, the interval

variables corresponding to each alpha-cut can be treated as a random variable with uniformly

distributed probability distribution function. In the possibilistic interpretation of uncertainty [(Denga

et al. (2012), Möller and Beer, (2004)], the fuzzy variables can be thought as generalized interval

variables, allowing one to use the techniques intended for the interval analysis such as classical

interval arithmetic [Moore (1966)], affine analysis [Degrauwe et al. (2010)] or vertex theorems [Qiu

et al. (2005)]. Fuzzy based uncertainty quantification is found to be adopted in various engineering

analyses [Babuška and Silva (2014), Rao and Annamdas (2008), Díaz-Madroñero et al. (2004),

Chandrashekhar and Ganguli (2009), Chowdhury and Adhikari (2012), Altmann et al. (2012)].

A careful review of the literature concerning the uncertainty analysis of composites reveals

that the effect of spatially varying non-probabilistic uncertainty has not been addressed so far.

Moreover, the consideration of multi-scale uncertainty propagation, where the uncertainty can be

modeled in the micromechanical properties in a physically more accurate way compared to the

conventional macromechanical analyses, is scarce to find in the scientific literature. The present

article aims to quantify the effect of spatially varying non-probabilistic uncertainty in composites

following an efficient multi-scale RBF based framework. Thus a multi-scale approach is followed,

wherein the effect of non-probabilistic uncertainty is included in the elementary micromechanical

level first and then the effects are propagated towards the global responses (dynamics and stability)

via an efficient surrogate of the actual finite element model. For this purpose, the concept of fuzzy

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

structures. This article hereafter is organized as, section 2: theoretical formulation for the fuzzy finite

element analysis of composites; section 3: description of the RBF based fuzzy multi-scale

6

uncertainty propagation algorithm; section 4: results and discussion demonstrating the influence of

spatially varying fuzzy uncertainty on the dynamic and stability characteristics of composite

laminates; section 5: summary and perspective of the present study in context to the available

scientific literature; section 6: conclusion.

2. Theoretical formulation for fuzzy finite element analysis of composites

In the fuzzy concept of uncertainty, a set of transitional states across the members and non-

members are described by a membership function

()

i

p

that specifies the degree to which the

elements in the domain belong to the fuzzy set [Zadeh (1975)]. Considering a triangular membership

function, the fuzzy number can be given as,

],,[)

~

(

~L

i

M

i

U

ii pppp

(1)

Here

M

i

p

,

U

i

p

and

L

i

p

denote the mean, upper and lower bounds, respectively. The parameter

~

denotes the fuzziness corresponding to the α-cut, in which α is known as the membership grade

ranging from 0 to 1.

The Gaussian probability distribution function can be approximated by a triangle for the

fuzzy analysis [Hanss and Willner (2000)]. The area under triangular membership function is

equated to the area under the normalised Gaussian distribution function to approximate the Gaussian

probability distribution function as a triangle (refer to figure 2(a)) [Adhikari and Khodaparast

(2014)], leading to a triangular fuzzy membership function as

i

j

i

ip

XX ˆ

1,0max

)(

)(

(2)

Here

X

2

,

i

X

ˆ

and

X

represent the mean and standard deviation respectively, for the

equivalent Gaussian distribution. The triangular shaped membership function (

)(ip

), as employed in

this work, can be described as

7

Otherwise

pppforpppp

pppforpppp

ip

U

ii

M

i

M

i

U

i

M

iiip

M

ii

L

i

L

i

M

ii

M

iip

0

),(/)(1

),(/)(1

)(

)(

)(

(3)

Thus, the fuzzy input number (

i

p

) can be defined into the set

i

P

of (m+1) intervals

)( j

i

p

using the α-

cut method as below

].............,,[)

~

()()()2()1()0( m

i

j

iiiii pppppP

(4)

Here m is the number of α-cuts. The interval of the j-th level of the i-th fuzzy number is expressed by

],[ ),(),()( Uj

i

Lj

i

j

ippp

(5)

where

),( Lj

i

p

and

),( Uj

i

p

denote the lower and upper bounds of the interval (the superscripts L and U

are used to denote the lower and upper bounds respectively) at the j-th level, where,

),( Lm

i

p

=

),( Um

i

p

=

M

i

p

at j = m. A triangular fuzzy membership function as discussed in this section can represent a

Gaussian distribution quite well. Besides that, it is logical to have equidistant bounds with respect to

the mean values corresponding to different α-cut levels when there is no experimental data available

for the input parameters. However, the area equivalence analogy as presented here can be adopted for

other distributions in future.

A numerical procedure of interval analysis at a number of α-levels [Moens and Hanss (2011)]

can be applied to propagate uncertainty in a system where the uncertain model parameters are

constituted by fuzzy input numbers. In this analysis, the range of the response components (fuzzy

output) on a specific level of membership function is searched within a particular α-level in the input

domain. Thus the analysis corresponding to each α-cut resembles to an interval analysis for the

system. Figure 2(b) [Moens and Vandepitte (2005, 2006); Adhikari and Khodaparast (2014)]

depicts the scheme of fuzzy uncertainty propagation for a typical case of two input parameters and

one output response with four α-cut levels, but the idea can be readily generalised for the case of

multiple input-output systems and any other number of α-cut levels. In the approach adopted in this

paper, we characterize a fuzzy variable with a set of interval variables by means of the membership

function. The lower and upper bounds of interval variables at different α-levels (i.e.,

),(),( ,U

i

L

ipp

)

8

(a)

(b)

Fig. 2 (a) Linear approximation of a Gaussian distribution by triangular fuzzy membership functions

(b) Scheme for fuzzy analysis containing transformation

)(

of a fuzzy variable

i

p

to ξ ∈ [0, 1] for

different α-cuts

can be converted into the normalized values in the range of 0 and 1 by means of a transformation

function

. The fuzzy uncertainty propagation begins with a deterministic solution at

1

and

continues through an interval analysis towards the lower

-cut levels. The interval variables ‘

i

’ are

characterized by a normalised random variable with uniform probability distribution as

101

() 2

0

i

i

pelsewhere

(6)

If the input-output relationship of a particular system is known to be of monotonic nature, the lower

and upper bounds of the fuzzy outputs corresponding to a particular α-cut level can be found by

carrying out very few number of simulations, otherwise a minimization and maximization

algorithm needs to be employed involving multiple simulations. In case of such complex input-

9

output relationships, an efficient surrogate based Monte Carlo simulation approach can be adopted

to carry out the search for minimum and maximum values of the output quantities of interest for a

particular α-cut level. In this analysis, radial basis function [Beatson (1999), Fasshauer (1997), Hon

and Mao (2001), Kansa (1990a, 1990b), Kansa and Hon (2000)] is employed as a surrogate

[Mukhopadhyay et al. (2016b), Karsh et al. (2018b), Maharshi et al. (2018), Metya et al. (2017),

Mahata et al. (2016), Dey et al. (2015, 2016d, 2016e, 2018b)] of the actual finite element model of

composites.

The radial basis function model is dependent upon the distance to a central point

ˆj

x

and a

shape parameter called

c

in some cases. For a set of nodes

12

ˆ ˆ ˆ

, ,..........., n

N

x x x R

, the RBF

centred at

ˆj

x

can be expressed as

where

ˆˆ

j

xx

represents the Euclidian norm. The RBF models can be expressed as

2

2ˆˆ

ˆ

( ) j

xxc

j

f x e

(Gaussian)

1

22

ˆ

( ) )( ˆˆ

jj xxx cf

(Multi-quadratic)

1

22

ˆˆ

ˆ

)( )( jj xxx cf

(Inverse multi-quadratic)

(8)

2

ˆˆ

(ˆ ˆ ˆ

log) jjjxxfx xx

(Thin plate splines)

ˆ

()

ˆ

()

ˆˆ , 1,3,5,......

ˆ ˆ ˆ ˆ

log , 2,4,6,....

k

j

k

jj

j

j

f x x k

x x x x k

x

fx

(Biharmonic)

RBFs are not sensitive to the spatial dimension making the implementation much easier for high

dimensional systems [Beatson (1999), Powell (1987), Krishnamurthy (2003)]. RBF models do not

need any grid; the only geometric property requirement is the pairwise distances between the points.

In this paper, we have dealt with the spatially varying material properties for each lamina of the

composite plate. Thus we need to deal with an exorbitantly high dimensional input parameter space

ˆˆ

( , ) ,

ˆ

( ) 1 , ,

n

jj f x x c jNRfx

(7)

10

depending on the number of FRVEs (refer to section 3). A surrogate modelling technique such as

HDMR [Dey et al. (2016c), Mukhopadhyay et al. (2016a)] that requires inversion of a matrix to

obtain the coefficients would be practically impossible to cope with the present problem. On the

other hand, dealing with high dimensional input parameter spaces using RBF is not difficult as the

distances are easy to evaluate in any number of space dimension. In this study, the RBF models are

built using the quasi-random Sobol sequence, which has been reported to show a faster convergence

than other sampling techniques [Mukhopadhyay et al. (2015), Mukhopadhyay (2018)]. In this

context, it can be noted that the RBF models are required to be formed only for α = 0 in case of a

fuzzy analysis. The same RBF models can be used for prediction in other α – cut levels. Thus the

number of expensive finite element simulations required for the present fuzzy analysis is same as the

number of sample points drawn from Sobol sequence to form the RBF models corresponding to α =

0 level.

In the present analysis of laminated composites, the spatially varying material properties and

ply orientation angles are considered as the source of fuzzy uncertainty, while the natural frequencies

and buckling loads of the composite plate along with the respective mode shapes are the fuzzy output

parameters of interest (denoted by y in figure 2(b)). The governing equation for stochastic free vibration

analysis without damping can be expressed as [Dey et al. (2016c)]

( ) ( ) 0MK

(9)

where

( ) ( ) ( )

ee

K K K

. Here

()

e

K

and

()

e

K

are the geometric stiffness

matrix and elastic stiffness matrix respectively, while

()M

represents the mass matrix. 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

(10)

11

where

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

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

orthogonality relationship is satisfied as

[ ( )] [ ( )] ( )

f T f

i k ik

S M S

and

2

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

f T f

i k k ik

S K S

(11)

where

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

and the Kroneker delta functions

ik

=0 for

ki

;

ik

=1 for

ki

. It can be

noted here that we have followed a Monte Carlo simulation based approach for the present fuzzy

analysis of composite structures. A Monte Carlo simulation involves multiple realizations of the

finite element model considering the input parameters drawn algorithmically from a certain random

sequence. Thus each of the realizations of a Monte Carlo simulation can be regarded as a separate

deterministic analysis in the present non-intrusive approach. This essentially means that even though

the current investigation is collectively a stochastic analysis, the orthogonality relationship is

satisfied.

The problem of stability analysis is solved through another eigenvalue problem obtained by

minimizing the total potential energy as:

( ) ( )[ ( )] ( )

b b b

e k e k

K S K S

(12)

where

()

b

is the fuzzy buckling load factor and

b

k

S

gives the buckling modeshapes. The

superscript b is used to denote frequency analysis. The detail fuzzy finite element formulation for

composites is provided as a supplementary material.

3. Fuzzy representative volume element based framework for non-probabilistic uncertainty

quantification

3.1. Concept of FRVE

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

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

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

stochastic in nature with different material properties, instead of considering the homogenized

mechanical properties of a conventional representative volume element (RVE) throughout the entire

12

Fig. 3 Conventional RVE for unidirectional fiber reinforced composites

Fig. 4 (a –b) FRVE based approach for analyzing spatially varying fuzzy uncertainty (c - d) Typical

illustrative representation of spatially varying material properties for longitudinal Young’s modulus

of fiber (a micromechanical property) and Young’s modulus (in GPa) of the composite at macroscale

solid domain (refer to figure 3). 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)]. However, this approach of analysis can lead to eroneous outcomes,

specially in case of stochastic systems with spatial variability in material and other attributes. To

analyse such systems, it is essential to account for the effect of the randomness of stocahstic

mechanical properties along the spatial location of different zones of a plate-like structure.

13

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

number of FRVEs. Thus mechanical properties of a FRVE are dependent on its stochastic material

and structural properties and they are different for each of the FRVEs in a particular realization of

Monte Carlo simulation. In this study, it is considered that all the FRVEs have same membership

function for a particular material property. Following this framework, it becomes feasible to consider

the spatial randomness (/ non-probabilistic uncertainty) 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 (FRVEs) towards the

global level by combining (/assembling) the FRVEs 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 probabilistic spatial irregularity [Mukhopadhyay and

Adhikari (2016a, 2017a), Mukhopadhyay et al. (2017, 2018a, 2018b), Mahata and Mukhopadhyay

(2018)], 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 mechanical properties of the entire irregular lattice can be

computed by assembling the RUCEs based on equilibrium and compatibility conditions. A concept of

SRVE (stochastic representative volume element) for probabilistic analysis with random material

properties and crack density is adopted by Naskar et al. (2017a, 2018). In this paper, we have

developed the FRVE based concept for non-probabilistic analysis of two-dimensional plate-like

structures with inhomogeneous [Mukhopadhyay and Adhikari (2017b)] form of fuzzy uncertainty.

The adoption of FRVEs in a plate-like structure is shown in the figure 4, wherein the entire domain is

divided into a finite number of fuzzy elements (FRVEs) having dimensions of l1 and l2 in two

mutually perpendicular directions. 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.

It can be noted that when

1r

, the fuzzy system would behave like the conventional form of non-

probabilistic analysis without considering any spatial variation, as presented by Dey et al. (2016b).

14

As per the proposed concept of FRVE, the size (/number) of FRVE is independent of the

discretization in a finite element based numerical solution that could be adopted for dynamic/

stability analysis of the composite plate. Once the size of a FRVE 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 variation of micro and macro scale material

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

(such as micro-scale damage) using the proposed FRVE based framework. In such problems,

appropriate finite element meshing schemes would need to be adopted for each of the FRVEs.

3.2. RBF based FRVE approach for spatially varying fuzzy uncertainty propagation

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

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

for analyzing the dynamic and stability characteristics of laminated composite plates. In the 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 relative effect in non-

probabilistic uncertainty propagation on a comparative basis. For analysing the effect of various

source-uncertainties, the following five cases 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( , )

413(1,1) 13( , )

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

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

( .... ),

macro e l macro e l macro e l

ma

C

macro

cro e l macro

g E E G G G t

E E E E G G

GG

56

23(1,1) 23( , ) (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

GG

tt

(13)

(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)

15

12

1 2 3 4

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

C

micro E E E G G V t

E E E E E G

g

E

(1,1) ( , )

5 6 7 8

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

9 10 11

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

.... ),

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

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

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 m m e l micro f f e l micro

G

GG

VV

12

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

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

f f e l micro f f e l

tt

(14)

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

1(1,1) ( , )

( ) ( .... )

M macro M M em l

Iacro

g

(15)

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

1(1,1) ( , )

( ) ( .... )

m micro m m em l

Iicro

g

(16)

(v) Individual effect for the variation of a single geometric property

1(1,1) ( , )

( ) ( .... )

IG G G eG Gl

g

(17)

where,

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 orientation

angle and thickness of lamina respectively (with conventional notations) for the ith FRVE situated in

the jth layer, where i = 1, 2, 3,…, e and j = 1, 2, 3, …, l. The function g in the above expressions is

used to denote a particular type of stochastic analysis, as evident from the involved elements in the

expressions. The parameter

in the expressions is a symbolic representation of Monte Carlo

simulation. 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 FRVE 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, while

G

is any one of the geometric properties. The material properties

are considered to vary spatially for both the macromechanical and micromechanical analyses.

16

Fig. 5 Flowchart for RBF based fuzzy multi-scale analysis of composites (Representative figures of

multi-scale finite element analysis, surrogate modelling and fuzzy uncertainty quantification are

shown)

However, considering practical aspects of modelling uncertainty in the geometric parameters from

the manufacturing point of view, spatial variation is neglected for ply orientation angle and thickness

of lamina; rather a layer-wise 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). Thus, we have

adopted a mixed form of uncertainty modelling with spatial variability in the material properties and

random fibre orientation angle with no relative variation in the direction along the profile of the

fibres. For example, if the deterministic value of a certain ply angle is 45o, one particular stochastic

realization could assume the value of 45.1o throughout the fiber in a straight profile.

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, for which a realistic

spatially varying non-probabilistic modelling approach is adopted in this article. After the uncertainty

17

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 for quantifying

output responses following a bottom-up framework. As the input-output relationships of composites

are normally complicated (/non-monotonic) in nature, a Monte Carlo simulation based approach can

be adopted for finding the response bounds corresponding to each α-cuts. However, direct Monte

Carlo simulation involving finite element models being a computationally intensive approach, we

have adopted a surrogate based scheme (RBF) in conjunction with the fuzzy finite element model of

composite plates as presented in figure 5.

4. Results and Discussion

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

three layered graphite-epoxy cross-ply ([0o/90o/0o]) composite plate (clamped at the four edges) with

an aspect ratio of 2 and degree of fuzziness k =10, unless otherwise mentioned. It can be noted here

that the degree of fuzziness (k) indicates the bound of input parameters with respect to the respective

nominal value at

0

(i.e.

%k

). A practically relevant spatially varying non-probabilistic

approach of uncertainty modelling is considered for characterizing the first three modes of vibration

and buckling of composite plates. Results are presented for two distinct cases: fuzzy uncertainty in

micromechanical and macromechanical material properties (refer to equation 13 – 17). 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

attributes presented in Table 1 are assumed as the source of fuzzy uncertainty 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 fuzzy uncertainty in macromechanical material properties, the analysis commences one

step ahead in the hierarchy i.e. the fuzzy source-uncertainty is assumed in the macromechanical

18

Table 1 Micromechanical material properties (deterministic) of composites

Material property

Numerical value

Longitudinal Young’s modulus of fibre (

1f

E

)

80 x 103 (Unit: MPa)

Transverse Young’s modulus of fibre (

2f

E

)

80 x 103 (Unit: MPa)

Poisson's ratio of fibre (

f

)

0.2

Shear modulus of matrix (

f

G

)

33.33 x 103 (Unit: MPa)

Mass density of fibre (

f

)

2.55 (Unit: gm/cc)

Mass density of matrix (

m

)

1.26 (Unit: gm/cc)

Young’s modulus of matrix (

m

E

)

4.2 x 103 (Unit: MPa)

Shear modulus of matrix (

m

G

)

1.567 x 103 (Unit: MPa)

Poisson's ratio of matrix (

m

)

0.34

Fibre volume fraction (

f

V

)

0.61

Table 2 Macromechanical material properties (deterministic) of

composites (

f

V

= 0.61)

Material property

Numerical value

Longitudinal Young’s modulus (

1

E

)

50.438 x 103 (Unit: MPa)

Transverse Young’s modulus (

2

E

)

9.952 x 103 (Unit: MPa)

Poisson's ratio (

12

)

0.2546

In-plane shear modulus (

12

G

)

3.742 x 103 (Unit: MPa)

Mass density (

)

2.049 x 103 (Unit: gm/cc)

Shear modulus (

13

G

)

3.742 x 103 (Unit: MPa)

Transverse shear modulus (

23

G

)

2.094 x 103 (Unit: MPa)

19

(a) (b)

Fig. 6 Typical representation of membership functions for the geometric parameters ply orientation

angle in rad (with a nominal value of 0 rad) and thickness in m (with a nominal value of 0.33x10-2m)

properties (as shown in Table 2) along with uncertain geometric parameters (

C

macro

g

). Subsequently,

the results obtained from these two different types of non-probabilistic analyses are compared to

ascertain the relative effect. Typical plots for the membership functions considered in the geometric

properties are shown in figure 6. Figure 7(a) and 7(b) show the representative membership functions

for micromechanical properties and macromechanical properties in a micromechanical analysis,

while figure 7(c) shows the membership functions for macromechanical properties in a

macromechanical analysis. It can be noted that figure 7(b) is obtained from figure 7(a) based on the

Halpin - Tsai principle following a fuzzy interpretation. In the present study, ±k° variation for ply

orientation angle and ±k% variation for material properties from fuzzy crisp values are considered

corresponding to α = 0, where k is referred as the degree of fuzziness. Non-dimensional results for

natural frequencies are presented as

2

20

( ) ( )

h

bD

, where

3

2

012 21

12(1 )

Eh

D

, while

for buckling load the adopted non-dimensionalisation scheme is:

23

2

( ), 0, 0

xx xy yy

N N b E h N N

.

4.1. Fuzzy dynamic analysis

4.1.1. Validation and convergence study

In the RBF based fuzzy 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 the composite plate (in-house FE code written in MATLAB) along with mesh convergence

20

(a)

(b)

(c)

Fig. 7 (a) Typical representation of membership functions for the micromechanical properties in

micromechanical analysis (b) Typical representation of membership functions for the

macromechanical properties in micromechanical analysis (c) Typical representation of membership

functions for the macromechanical properties in macromechanical analysis (The units are in

accordance with the nominal values presented in Table 1 and 2)

21

t/b = 0.001

t/b = 0.20

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 8 The convergence study of frequency parameters for cross-ply

(0 /90 / 0 )

o o o

simply supported

(SSSS) rectangular laminates. The reference results from Liew (1996) are shown by the green lines.

The results are shown for two different t/b ratios.

22

Fig. 9 Convergence study and validation for the sample size required to form the RBF model in case

of first natural frequency (

C

micro

g

and

C

macro

g

). A typical membership function is shown considering

the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

study. A second type of validation is also needed here concerning the performance (efficiency and

accuracy) of the RBF model in predicting the responses along with a convergence study for

minimizing the number of design points (i.e. number of expensive finite element simulations)

required for forming the surrogate models.

The results for validation and convergence study of the deterministic finite element code of a

composite plate are shown in figure 8, wherein non-dimensional natural frequencies are validated

with the results available in scientific literature [Liew and Huang (2003)]. Based on the results

presented in figure 8, 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 RBF model are decided

based on the comparative performance (absolute error in predicting the lower and upper bounds for

23

Fig. 10 Convergence study and validation for the sample size required to form the RBF model in

case of second natural frequency (

C

micro

g

and

C

macro

g

). A typical membership function is shown

considering the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

each of the α-cuts) with respect to the direct Monte Carlo simulation approach. The results for macro

and micro mechanical analyses showing the values of absolute error for different sample sizes

corresponding to different α-cuts are presented in figure 9 - 11. From the figures it is evident that a

sample size of 4096 provides reasonably accurate results for the natural frequencies. It can be noted

in this context that the RBF model is formed only for α = 0, while the same RBF model can be used

for prediction corresponding to the other α-cuts. The lower and upper bounds are found based on a

Monte Carlo simulation based approach that requires a large number of finite element simulations

(~104) for each of the α-cuts. Hence, the computational time and cost in terms of finite element

analyses are reduced significantly compared to the direct Monte Carlo simulation based approach of

24

Fig. 11 Convergence study and validation for the sample size required to form the RBF model in

case of third natural frequency (

C

micro

g

and

C

macro

g

). A typical membership function is shown

considering the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

finding the lower and upper bounds corresponding to each α-cut. In this paper, the lower and upper

bounds corresponding to each α-cut are obtained based on 10,000 samples resulting in a total of 50,

000 samples. Thus it yields to a level of computational efficiency of more than 1/12 (~4096/50000)

times in terms of finite element simulations. It can be noted that the time taken for surrogate model

formation and prediction is negligible compared to a single finite element simulation. Thus we have

ignored the time taken for surrogate model formation and prediction, and reported the efficiency in

terms of the number of finite element simulations required in this study.

4.1.2. Numerical results for fuzzy dynamic analysis

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

subsection, results for fuzzy dynamic analysis are presented in this subsection for the first three

25

(a) (b)

(c)

Fig. 12 Effect of variation in degree of fuzziness (k) on the first three natural frequencies (

C

micro

g

)

modes of vibration considering a composite plate with spatially varying material properies (

C

macro

g

and

C

micro

g

). Figure 12 shows the variation of first three fuzzy natural frequencies considering different

degrees of fuzziness corresponding to α = 0 with the fuzzy uncertainty at the micro-mechanical level.

The response bounds are noticed to increase with the increasing degree of fuzziness. The fuzzy

uncertainty for first three natural frequencies is investigated for various laminate configurations

considering the

//

family of angle-ply composites and a cross-ply composite of configuration

[0o/90o/0o] accounting spatially varying fuzzy source-uncertainty in the micromechanical properties.

The fuzzy membership functions presented in figure 13 shows that the results corresponding to α = 1

varies depending on the effective stiffness of the structure, while the fuzzy bounds show different

behaviour corresponding to different laminate configurations. Figure 14 shows the influence of

boundary conditions on the fuzzy natural frequencies of composite plates for simply supported

(SSSS) and fixed (CCCC) boundaries at all the four edges. The results corresponding to α = 1 are

found to vary depending on the stiffness of the system, following a similar trend as deterministic

26

(a) (b)

(c)

Fig. 13 Effect of ply-orientation angle on the fuzzy natural frequencies of composite plates (

C

micro

g

)

analysis. Figure 15 shows the effect of aspect ratio of the composite laminated plates on the fuzzy

natural frequencies considering stochastic micromechanical material properties. The fuzzy natural

frequencies are noticed to reduce with the increase in aspect ratio for all the three natural frequencies

along with a reduction in fuzzy-bounds. Figure 16 shows the effect of change in degree of orthotropy

on the fuzzy natural frequencies considering source-uncertainties in the macromechanical properties.

The source-uncertainty in the macromechanical properties are considered in this case because of the

direct definition of degree of orthotropy as a ratio of longitudinal and transverse Young’s modulus at

macro-scale. The natural frequencies are found to increase with increasing degree of orthotropy for

all the three considered vibration modes. Comparative results considering the fuzzy source-

uncertainty at the micro and macro mechanical levels are presented in figure 17 considering two

different boundary conditions. To understand the relative effect of fuzzy uncertainty, the results in

this figure are normalized with respect to their respective deterministic values.

27

(a) (b)

(c)

Fig. 14 Effect of boundary condition on the fuzzy natural frequencies of cross-ply composite plates

with [0°/90°/0°] configuration (

C

micro

g

)

(a) (b)

(c)

Fig. 15 Effect of aspect ratio (AR) on the fuzzy natural frequencies of composite plates (

C

micro

g

) with

fixed boundary condition

28

(a) (b)

(c)

Fig. 16 Effect of degree of orthotropy (DO) on the fuzzy natural frequencies of composite plates (

C

macro

g

)

(a) (b)

(c)

Fig. 17 Comparative results for natural frequencies considering micromechanical (

C

micro

g

) and

macromechanical analyses (

C

macro

g

) of cross-ply composite plates

29

(a) (b)

(c)

Fig. 18 Individual effect of fuzzy uncertainty in the geometric properties (

I

G

g

) of composite plates

(Normalized results are presented with respect to the respective deterministic natural frequencies)

(a) (b)

(c)

Fig. 19 Individual effect of fuzzy uncertainty in the macromechanical (

I

macro

g

) properties

(Normalized results are presented with respect to the respective deterministic natural frequencies)

30

(a) (b)

(c)

Fig. 20 Individual effect of fuzzy uncertainty in the micromechanical properties (

I

micro

g

)

(a) (b)

(c)

Fig. 21 Effect of characteristic length (r) on the fuzzy natural frequencies (

C

micro

g

)

31

The results in this subsection so far are presented considering the compound effect of fuzzy

uncertainty at micro and macro-mechanical level. Figure 18 – 20 show the individual effect of fuzzy

source-uncertainties in micro and macro mechanical properties along with the geometric attributes.

Depending on the respective fuzzy response bounds, these figures can provide a clear idea regarding

the sensitivity of various individual material and geometric parameters to the fuzzy natural

frequencies of the composite plate. The nature of fuzzy random field of the stochastic micro and

macro mechanical material properties depend on the characteristic length (r) considered in the

analysis. A higher value of r indicates less rapid variation of the material property concerned and

vice versa. We have investigated the effect of characteristic length on the first three modes of

vibration. It can be noted that the system becomes a randomly homogenous system (analogous to

considering no spatial variation [Dey et al. (2016b)] when the characteristic length

1r

. The effect

of characteristic length on the first three fuzzy natural frequencies is presented in figure 21, wherein

a clear difference is noticed between the case of considering no spatial variation and the present

analysis considering spatially varying attributes. The results reveal that the conventional practise

[Dey et al. (2016b)] of neglecting the inevitable spatial variation in material properties over-

estimates the fuzzy response bounds significantly.

The effect of fuzzy source-uncertainty 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 22 for first three modes of vibration. Fuzzy mode shapes

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 22(a – c)), a simply

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

C

macro

g

) properties

(refer to figure 22(d – f)), a clamped (CCCC) composite plate with stochasticity in the

micromechanical (

C

micro

g

) properties (refer to figure 22(g – i)) and a clamped (CCCC) composite

plate with stochasticity in the micromechanical (

C

micro

g

) properties (refer to figure 22(j – l)). From the

mode shapes presented in figure 22(a – l), it can be observed that the basic global pattern of the

32

Fig. 22 Fuzzy modeshapes and representative membership functions of the normalized eigenvectors (

C

micro

g

and

C

macro

g

)

33

stochastic mode shapes remains similar to the corresponding deterministic case. However, the value

of normalized eigenvectors becomes fuzzy in nature for each of the elements in the composite plate.

Fuzzy membership functions of the normalized eigenvectors of first three vibration modes for the

elements indicated in figure 22(o) are shown considering a clamped (CCCC) (refer to figure 22(m))

and a simply supported (SSSS) (refer to figure 22(n)) boundary conditions. The results for micro and

macro mechanical analyses are shown using lighter and darker shades of respective colours indicated

in figure 12(o). It can be noticed that the fuzzy membership functions depend significantly on the

type of analysis (micro and macro mechanical) and location of the element under consideration.

4.2. Fuzzy stability analysis

4.2.1. Validation and convergence study

The finite element code and the RBF model are validated first for analysing the buckling

loads similar to the case of dynamic analysis. The results of convergence study and validation of the

finite element code of a composite plate are furnished in figure 23, wherein the non-dimensional first

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

presented in the figure, 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 a comparative performance similar to the case of dynamic analysis

presented in the preceding subsection. The results for macro and micro mechanical analyses showing

the values of absolute error for different sample sizes corresponding to different α-cuts are presented

in figure 24 - 26. From the figures it is evident that a sample size of 4096 provides reasonably

accurate results for the first three buckling loads. It can be noted in this context that the RBF model

for a buckling load is formed only for α = 0, while the same RBF model can be used for prediction

corresponding to the other α-cuts. In this subsection, we have presented the results of stochastic

stability analysis considering the first three buckling modes for the sake of completeness. Besides

that, in case of fuzzy stability analysis, the buckling loads are found to have a fuzzy response bound

with respect to the corresponding deterministic values. Thus there exists a possibility of overlap in

the fuzzy responses of the buckling loads corresponding to different modes of buckling resulting in a

34

Fig. 23 The convergence study of uniaxial buckling load of four layer

(0 /90 /90 /0 )

o o o o

simply

supported (SSSS) rectangular laminates with respect to Liew and Huang (2003) and Neves and

Ferreira (2016)

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 fuzzy-uncertainty in the system parameters.

4.2.2. Numerical results for fuzzy stability analysis

Numerical results are presented in this subsection for the first three modes of buckling. Figure

27 shows the variation of first three fuzzy buckling loads considering different degrees of fuzziness

corresponding to α = 0 with the fuzzy uncertainty at the micro-mechanical level. The response

bounds are noticed to increase with the increasing degree of fuzziness. The effect of fuzzy uncertainty

for first three buckling loads is investigated for various laminate configurations considering the

//

family of angle-ply composites and a cross-ply composite of configuration [0o/90o/0o]

accounting spatially varying fuzzy source-uncertainty in the micromechanical properties. The fuzzy

membership functions presented in figure 28 indicate that the results corresponding to α = 1 varies

depending on the effective stiffness of the structure, while the fuzzy bounds show different behaviour

corresponding to different laminate configurations. Figure 29 shows the influence of boundary

conditions on the fuzzy buckling loads of composite plates for simply supported (SSSS) and fixed

(CCCC) boundaries at all the four edges. The results corresponding to α = 1 are found to vary

depending on the stiffness of the system, following a similar trend as deterministic analysis. Figure

35

Fig. 24 Convergence study and validation for the sample size required to form the RBF model in

case of first buckling load (

C

micro

g

and

C

macro

g

). A typical membership function is shown considering

the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

30 shows the effect of aspect ratio of the composite laminated plates on the fuzzy buckling loads

considering stochastic micromechanical material properties. The fuzzy buckling loads are noticed to

reduce with the increase in aspect ratio for all the three modes of buckling along with a reduction in

fuzzy-bounds. Figure 31 shows the effect of change in degree of orthotropy on the fuzzy buckling

loads considering source-uncertainties in the macromechanical properties. The first three buckling

loads are found to increase with increasing degree of orthotropy. Comparative results considering the

fuzzy source-uncertainty at the micro and macro mechanical levels are presented in figure 32

considering two different boundary conditions. To understand the relative effect of fuzzy uncertainty,

the results in this figure are normalized with respect to their respective deterministic values.

36

Fig. 25 Convergence study and validation for the sample size required to form the RBF model in

case of second buckling load (

C

micro

g

and

C

macro

g

). A typical membership function is shown

considering the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

Figure 33 – 35 show the individual effect of fuzzy source-uncertainties in micro and macro

mechanical properties along with the geometric attributes. Depending on the respective fuzzy

response bounds, these figures can provide a clear idea regarding the sensitivity of various individual

material and geometric parameters to the fuzzy buckling loads of the composite plate. The nature of

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

characteristic length (r) considered in the analysis. Similar to the case of dynamic analysis, we have

investigated the effect of characteristic length on the first three modes of buckling. It can be noted

that the system becomes a randomly homogenous system (analogous to considering no spatial

variation [Dey et al. (2016b)] when the characteristic length

1r

. The effect of characteristic length

on the first three fuzzy buckling loads is presented in figure 36, wherein a clear difference is noticed

37

Fig. 26 Convergence study and validation for the sample size required to form the RBF model in

case of third buckling load (

C

micro

g

and

C

macro

g

). A typical membership function is shown considering

the compound effect of stochasticity in micromechanical parameters (

C

micro

g

).

between the case of considering no spatial variation and the present analysis considering spatially

varying attributes. The results reveal that the conventional practise of neglecting the inevitable

spatial variation in material properties over-estimates the fuzzy response bounds of the buckling

loads. The effect of fuzzy uncertainty in micro and macro mechanical attributes is studied on the

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

are presented in figure 37 for first three modes of buckling. Fuzzy mode shapes 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 37(a – c)), a simply supported (SSSS)

composite plate with stochasticity in the macromechanical (

C

macro

g

) properties (refer to figure 37(d –

f)), a clamped (CCCC) composite plate with stochasticity in the micromechanical (

C

micro

g

)properties

38

(a) (b)

(c)

Fig. 27 Effect of variation in degree of fuzziness on the first three buckling loads (

C

micro

g

)

(a) (b)

(c)

Fig. 28 Effect of ply-orientation angle on the fuzzy buckling loads of composite plates (

C

micro

g

)

39

(a) (b)

(c)

Fig. 29 Effect of boundary condition on the fuzzy buckling loads of cross-ply composite plates with

[0°/90°/0°] configuration (

C

micro

g

)

(a) (b)

(c)

Fig. 30 Effect of aspect ratio (AR) on the fuzzy buckling loads of composite plates (

C

micro

g

)

40

(a) (b)

(c)

Fig. 31 Effect of degree of orthotropy (DO) on the fuzzy buckling loads of composite plates (

C

micro

g

)

(a) (b)

(c)

Fig. 32 Comparative results for buckling loads considering micromechanical (

C

micro

g

) and

macromechanical analyses (

C

macro

g

)

41

(a) (b)

(c)

Fig. 33 Individual effect of fuzzy uncertainty in the geometric properties (

I

G

g

) of composite plates

(a) (b)

(c)

Fig. 34 Individual effect of fuzzy uncertainty in the micromechanical properties (

I

micro

g

)

42

(a) (b)

(c)

Fig. 35 Individual effect of fuzzy uncertainty in the macromechanical properties (

I

macro

g

)

(a) (b)

(c)

Fig. 36 Effect of characteristic length (r) on the fuzzy buckling loads (

C

micro

g

)

43

Fig. 37 Fuzzy modeshapes and representative membership functions of the normalized eigenvectors (

C

micro

g

and

C

macro

g

)

44

(refer to figure 37(g – i)) and a clamped (CCCC) composite plate with stochasticity in the

micromechanical (

C

micro

g

) properties (refer to figure 37(j – l)). From the mode shapes presented in

figure 37(a – l), it can be observed that the basic global pattern of the fuzzy mode shapes remains

similar to the deterministic case. However, the value of normalized eigenvectors becomes fuzzy in

nature for each of the elements in the composite plate. Fuzzy membership functions of the

normalized eigenvectors of first three buckling modes for the elements indicated in figure 37(o) are

shown considering a clamped (CCCC) boundary condition (refer to figure 30(m)) and a simply

supported (SSSS) boundary condition (refer to figure 37(n)). The results for micro and macro

mechanical analyses are shown using lighter and darker shades of respective colours indicated in

figure 37(o). It can be noticed that the membership function plots depend significantly on the type of

analysis (micro and macro mechanical) and location of the element under consideration.

5. Summary and perspective

This paper presents an efficient non-probabilistic bottom-up framework for analyzing the

fuzzy dynamics and stabilty of composite laminates with spatially varying system properties. A

novel idea of FRVE is proposed in the context of two dimensional plate-like structures to incorporate

the spatially varying fuzzy material properties in conjunction with the FE analysis. It can be noted

here that the conventional methods of representative volume element (RVE) based analyses and

other available analytical solutions can not consider the effect of spatial variability in the system

properties. In the proposed FRVE based approach, various other spatially varying system properties

(such as varying intensity of matrix cracking, varying fibre properties, fibre breakage etc.) can also

be easily accounted following a non-probabilistic framework in future. However, an investigation

involving Monte Carlo simulation and finite element analysis becomes exorbitantly computationally

expensive. To mitigate this lacuna, a RBF based approach is adopted to achieve computational

efficiency (without compromising the accuracy of results) in the present analysis.

In the fuzzy multiscale characterization of composite laminates, there exists three distinct

stages of the analysis: fuzzy uncertainty modelling at the input level, propagation of uncertainty to

45

the global level and quantification of the global responses such as dynamics and stability

characteristics. In most of the studies concerning the fuzzy uncertainty quantification of composite

structures, due to simplicity of the approach in implementation, the inevitable spatial variation of the

material properties are ignored for any particular realization of Monte Carlo simulation [Dey et al.

(2016b)]. However, in reality, the material properties are not constant spatially in a plate-like

structure; rather they are spatially varying (as shown in figure 4(c-d)). In the present study, we have

adopted a practically relavant spatially vatying non-probabilistic uncertainty modelling strategy. Two

separate analyses have been performed considering the source-uncertainty in micro and macro

mechanical properties to present a comparative perspective. After modelling the uncertainty in a

practically relavant way, the effect of uncertainty needs to be propagated towards the global

responses from the elementary input level. The propagation of uncertainty is carried out throgh a

stochastic computational model of the structure using a Monte Carlo simulation based minimization-

maximization algorithm. In general, for complex composite laminated structures with spatially

varying system properties, the performance functions are not available as explicit functions of the

fuzzy input variables. Thus the performance functions or responses (such as natural frequencies and

buckling loads) of the structure can only be computed numerically at the end of an intensive

structural analysis procedure (such as the FE method), which is often exorbitantly time-consuming

and computationally expensive. The surrogate based uncertainty propagation strategy, as adopted in

this study, can develop a representative and predictive mathematical/ statistical metamodel relating

the natural frequencies and buckling loads to a number of fuzzy input variables. Thereafter the

metamodels (response surface) are used to compute the dynamic and stability characteristics

corresponding to a given set of fuzzy input variables, instead of having to simulate repeatedly the

time-consuming FE analysis. 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 fuzzy input variables can be created and a pseudo

analysis (efficient, yet accurate) for each variable set can be performed by adopting the

corresponding RBF model. It can be noted in this context that the RBF model is formed only for α =

46

0, while the same RBF model can be used for prediction corresponding to the other α-cuts. The final

step in the fuzzy analysis is uncertainty quantification in the output responses, which is effectively

carried out by deriving membership functions for the output quantities of interest.

In case of the fuzzy micro-mechanical analysis, the source-uncertainty is accounted in the

micro-mechanical material properties (spatially varying) of composites. Thereafter, the equivalent

macro-scale material properties are computed based on the proposed FRVE approach. In the next

stage, the equivalent macro-scale material attributes are fed into the finite element code to compute

the global responses (like natural frequencies and buckling loads) of the composite plate. Thus, in the

fuzzy micromechanical analysis, stochasticity is essentially accounted at the micro-scale level first,

and then the effect is propagated to the macro-scale level to characterize the global responses of the

structural system. It is common in scientific literature to refer this genre of analysis for composite

materials and structures considering micro-mechanical properties based on representative volume

element (RVE) as multi-scale analysis [May et al. (2014)]. As fuzzy material and structural

parameters are considered in the present analysis leading to a non-probabilistic characterization of

the structural responses, the study is referred as fuzzy multi-scale analysis in this paper. It can be

noted in this context that there is a possibility of modelling spatial variability of the fibre orientation

angle as well. However, we have restricted ourselves with modelling the orientation of the fibres in a

straight profile within a random bound (i.e. the orientation of the fibres are stochastic in nature, but it

does not vary along the profile in a relative sense). In case of unidirectional fibre reinforced

composites, form a manufacturing point of view, the present representation of uncertainty in fiber

orientation could be viewed as more practically relevant as the possibility of occurring high degree of

spatial randomness in the profile of the fibres is relatively negligible compared to variation in the

fiber angle with a straight profile (because the fibres would naturally have a tendency to maintain a

straight profile during the manufacturing process due to the inherent stiffness). For this reason, we

have adopted a mixed form of uncertainty modelling with spatial variability in the material properties

and random fibre orientation angle (meso-scale property) with no relative variation in the direction

along the profile of the fibres.

47

(a)

(b)

Fig. 38 A comparative perspective on probabilistic and non-probabilistic analyses considering

stochasticity in micromechanical properties (

C

micro

g

). Here the values of first natural frequency and

first buckling load are normalized with respect to the corresponding mean values. The densities for

plotting the probability density function plots are normalised with respect to the maximum value of

density so that the maximum value of the transformed density becomes 1 for the purpose of

providing a comparative perspective with respect to the fuzzy membership function.

No correlation between the material and structural properties are considered in this paper.

Normally, establishing the parameters for correlation between the input features (including spatial

correlation) requires substantial amount of experimental investigations. The whole point of carrying

out a fuzzy analysis being unavailability of such experimental data, we have avoided considering

correlation among the input parameters in this study. Moreover, consideration of a typical random

field based approach like gaussian or lognormal random fields, may make the current analysis

pseudo-probabilistic in nature. These reasons led to our decision of not considering any spatial

correlation in this paper on non-probabilistic analysis of composites. The present form of analysis

may slightly overestimate the fuzzy response bounds, but that is in the safer side from a design point

48

of view. However, it is interesting to notice that the present approach of spatially varying fuzzy

uncertainty modelling leads to a significantly reduced fuzzy response bound compared to the

conventional approach of modelling without such spatial variation (such as Dey et al. 2016b).

Oftentimes designers like to create a design factor or factor of safety out of the probability

analysis of structures. A fuzzy analysis can lead to evaluating the safety factor of design in the non-

probabilistic regime. In fact, a fuzzy based analysis could give us the opportunity to define the fuzzy

factor of safety (FFoS) corresponding to different alpha-cut levels. Normally, factor of safety (FoS)

for a design parameter is defined as: FoS = (Capacity of the system in terms of the design parameter /

Maximum allowable value of the design parameter). In case of fuzzy analysis, we have a bound

(defined by the upper and lower values) corresponding to each of the alpha-cuts. Thus a bound of the

factor of safety can be defined in case of the fuzzy analysis instead of a single value. Let us consider,

),( Lj

i

p

and

),( Uj

i

p

denote the lower and upper bounds of a fuzzy interval (the superscripts L and U are

used to denote the lower and upper bounds respectively) at the j-th level of α-cut for a certain

parameter i. The design value of that parameter is given by

D

i

p

. The bound of FFoS can be defined

as [

( , ) ( , )

,

j L D j U D

i i i i

p p p p

] for the j-th level of α-cut. It could be noted that the concept of FFoS

brings the notion of defining the level of safety corresponding to a particular degree of fuzziness.

Thus it would provide a clear perspective on how much the factor of safety could vary (in terms of a

bound) in case of a stochastic design for various degree of uncertainty (i.e. corresponding to different

α-cuts).

From an engineering perspective, the idea of a non-probabilistic fuzzy based analysis and a

probabilistic analysis is quite straight-forward. In a probabilistic analysis, the complete nature of

statistical distributions for the input parameters is known and this leads to obtaining the complete

probability distribution of the output quantities including the upper and lower bounds. In case of a

fuzzy based analysis, the statistical distributions of the input parameters are not known; rather an

idea about the mean value of the input parameters exists. The membership functions are defined for

the input parameters to establish lower and upper bounds corresponding to different degree of

49

fuzziness. Subsequently, the uncertainty in the output parameters is quantified in terms of lower and

upper bounds corresponding to different degree of fuzziness. The degree of fuzziness is analogous to

the level of uncertainty associated with a parameter. As a fuzzy analysis provides only the bounds of

output parameters instead of a complete probabilistic description between the bounds, the output of a

fuzzy analysis can be regarded as a subset of probabilistic analysis. It can be noted that the lower and

upper bounds of a parameter in the probability distribution coincide with the bounds of a

membership function corresponding to α = 0. The fascinating idea of combining the interval analyses

for different degree of fuzziness to form a membership function gives more information than the just

two bounds of the output parameters. Engineers, often being more interested in the range of

variability, a fuzzy analysis caters to this requirement quite well along with an additional factor of

the level of uncertainty (i.e. range of variability of the parameter corresponding to degree of

fuzziness). To provide a comparative perspective, figure 38 shows transformed probability density

function plots and membership function plots for a micromechanical analysis involving the first

natural frequency and first buckling load, wherein all the above-mentioned features become quite

evident. As the debate on ‘fuzziness is probability in disguise’ will continue; in essence, a fuzzy

based approach is capable of providing the engineers a way to comprehensively quantify uncertainty

in the absence of statistical distributions of the input parameters.

6. Conclusion

A bottom-up non-probabilistic analysis is presented in this article for multi-scale

quantification of the effect of fuzzy source-uncertainty in the dynamics and stability behaviour of

composite plates. A practically relevant non-probabilistic uncertainty modelling approach is proposed

considering spatially varying system attributes, for which a novel concept of fuzzy representative

volume element (FRVE) is developed. The non-probabilistic approach (such as fuzzy uncertainty), as

presented in this paper, can allow us to characterize the effect of uncertainty in a system when the

statistical distributions of the input parameters remain unavailable due to the restriction of performing

a large number of experiments. A non-linear and non-monotonic relationship between the output

quantities and the input parameters is evident in the fuzzy analysis in case of multiple numerical

50

results. The paper reveals that the conventional practise of neglecting the inevitable spatial variation

in material properties over-estimates the fuzzy response bounds of the natural frequencies and

buckling loads. Such outcome will have an influential impact on the current design procedures in the

non-probabilistic approach. Besides an in-depth analysis of fuzzy natural frequencies and buckling

loads, the effect of fuzzy uncertainty is investigated on the mode shapes of composite plates. The

fuzzy response bounds also provide a relative sense of sensitivity to the uncertainty in different

results presented in this paper. To understand the relative influence of different input parameters, the

individual effects of fuzzy uncertainty are analysed considering micro and macro mechanical

properties. For achieving computational efficiency, a RBF based stochastic analysis algorithm is

developed in conjunction with the fuzzy finite element model, wherein it is noted that the required

number of original FE simulations can be remarkably reduced without compromising accuracy of

results.

Even in the absence of adequate statistical information of the stochastic input parameters,

the fuzzy stochasticity in material/ structural attributes is found to influence the system performance

significantly depending on the degree of fuzziness, affirming the necessity to consider such forms of

source-uncertainties during the analysis to ensure adequate safety, sustainability and robustness of

the structure. Novelty of this paper includes the consideration of spatial variation in material

properties (micro/ macro) following the FRVE approach coupled with the RBF algorithm in context

to composite laminates. The idea of FRVE is quite generic in nature; this concept can be extended to

other structures and various other non-probabilistic systems with spatial variability in two/ three

dimensions.

Acknowledgements

SN and SS are grateful for the support provided through the Lloyd’s Register Foundation Centre.

The Foundation helps to protect life and property by supporting engineering-related education, public

engagement and the application of research.

51

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