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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.
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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
4
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
,
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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... Thus, the higher-order zigzag theories (HOZT) are more accurate than the HDSTs, as this class of theory predicts the continuation of the transverse shear stresses at the interfaces without defining or using the shear correction factor as required in FSDT. Recently some of the works are published regarding the application of machine learning techniques for the analysis of laminated composite and sandwich structures (Garg et al., 2022(Garg et al., , 2023Mukhopadhyay, 2017;Naskar et al., 2019Naskar et al., , 2020. From the review, it can be seen that only a few works are available regarding the free vibration and buckling analysis of the bio-inspired helicoidal plates. ...
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Helicoidal laminates inspired by mantis shrimp crustacean can sustain higher loads compared to conventional laminated structures. However, there is still a lack of bending studies on the plates inspired by these helicoidal structures. The buckling and free vibration studies have been undertaken in the present work on the laminated composite and sandwich plates inspired by the helicoidal structures of the biological characters. Investigations are carried out using higher-order zigzag theory. For sandwich plates, the top and bottom face sheets are assumed to be made up of helicoidal layup schemes. Different kinds of helicoidal schemes, namely, recursive, exponential, semicircular, Fibonacci, and linear helicoidal are considered during the analysis. The buckling and free vibration behavior of helicoidal-based laminated sandwich plates are compared with the conventional laminated sandwich plates such as uniform distribution and cross-ply configurations. The influence of the number of layers, helicoidal schemes, geometric properties, and end conditions on the buckling and free vibration analysis of bio-inspired helicoidal laminated composite and sandwich plates is carried out in detail.
... The machine learningassisted surrogate models are faster in predicting the behavior of the structures as compared to the conventional physics-based methods. Mukhopadhyay et al. proposed a series of machine learning-based algorithms for optimization, sensitivity analysis, uncertainty quantification and system identification concerning laminated composite and sandwich structures [43][44][45][46], including a seminal book on this topic [47]. Khan et al. [48] presented a detailed review on the damage assessment of smart composites using machine learning algorithms. ...
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Available shear deformation theories (SDTs) in the literature have their own merits and demerits. Among SDTs, first-order shear deformation theory (FSDT) and higher-order shear deformation theories (HSDT) are most widely used for the analysis of laminated composite and sandwich (LCS) beams. However, these theories are not able to predict the continuation of transverse shear stresses at interfaces across the thickness of the LCS beams. Due to the assumption of the constant variation of the transverse displacement field across the thickness of the layer, the FSDT is not able to predict the values for the transverse normal stresses. The present work aims to transform the stress variations across the thickness of LCS beams obtained from FSDT to the 3D Elasticity solutions using Gaussian Process Regression (GPR) based surrogate model. Further, the surrogate model is exploited to predict the variation of transverse normal stresses σzz across the thickness. Without large computational efforts, the proposed methodology will be able to capture the through-thickness stress variations equivalent to 3D Elasticity solutions, leading to an accurate yet efficient prediction.
... In the recent years, machine learning (ML) has emerged as a promising tool for the predictive modeling of polymer composites with significant computational efficiency [25][26][27][28][29][30][31][60][61][62][63][64]. Out of different ML algorithms, Artificial Neural Network (ANN) is a well-known predictive technique used successfully by various researchers to model the mechanical behaviour of polymer composites [32][33][34][35][36]. ANN is a universal function approximator that has the capability of handling large covariate spaces with significant level of accuracy [37]. ...
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This paper presents an experimental investigation supported by data-driven approaches concerning the influence of critical stochastic effects on the dynamic fracture toughness of glass-filled epoxy composites using a computationally efficient framework of uncertainty quantification. Three different shapes of glass particles are considered including rod, spherical and flaky shapes with coupled stochastic variations in aspect ratio, dynamic elastic modulus and volume fraction. An artificial neural network based surrogate assisted Monte Carlo simulation is carried out here in conjunction with advanced experimental techniques like digital image correlation and scanning electron microscopy to quantify the uncertainty and sensitivity associated with the dynamic fracture toughness of composites in terms of stress intensity factor under dynamic impact. The study reveals that the pre-crack initiation time regime shows the most prominent effect of uncertainty. Additionally, rod shape and the aspect ratio are the most sensitive filler type and input parameter respectively for characterizing dynamic fracture toughness. Here the quantitative results based on large-scale data-driven approaches convincingly demonstrate using a computational mapping between the stochastic input and output parameter spaces that the effect of uncertainty gets pronounced significantly while propagating from the compound source level to the impact responses. Such outcomes based on experimental data essentially bring us to the realization that quantification of uncertainty is of utmost importance for developing a reliable and practically relevant inclusive analysis and design framework for the dynamic fracture of particulate composites. With limited literature available on the determination of fracture toughness considering inertial effects, the present work demonstrates a novel and insightful experimental approach for uncertainty quantification and sensitivity analysis of dynamic fracture toughness of particulate polymer composites based on surrogate modeling.
... Random spatial discretization methods have been applied to study uncertainties in composite materials in the past. For example, Naskar et al. [126,127] proposed stochastic or fuzzy representative volume elements to include uncertainties in the microscale level, which were then propagated to the macroscopic level for dynamic and stability analyses. ...
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In this paper, a finite element-based framework is presented to model the probabilistic progressive failure of fiber-reinforced composite laminates with high fidelity and efficiency. The framework is based on the semidiscrete modeling approach that can be seen as a good compromise between continuum and discrete methods. The enhanced semidiscrete damage model (ESD2M) tool set comprises a smart meshing strategy with failure mode separation, a new version of the enhanced Schapery theory with a novel generalized mixed-mode law, and a novel probabilistic modeling strategy. These three joined components make the model efficient in capturing failure modes such as matrix cracks, fiber tensile failure, and delamination, as well as their interactions with high fidelity, while taking material nonuniformities into account. The model capabilities are demonstrated using single-edge notched tensile cross-ply laminates as an example. The ESD2M was not only capable of capturing the complex damage progression but also provided insights and explanations for some of the failure events observed in the laboratory. The presented framework efficiently integrates failure mode predictions with probabilistic modeling and enables Monte Carlo simulations to predict the ultimate failure strength with good accuracy, as well as its scatter.
... In order to give full play to the potential of composites and obtain an efficient design of composite structures under the premise that high structural reliability and safety are guaranteed, probabilistic design methodologies have been developed and gradually applied to the design of engineering structures [4,5]. Multiscale methodologies have been developed to simulate the probabilistic mechanical properties of composite structures because of the uncertainty in the mechanical properties' transfers from the micro-level to the structural level [6][7][8], as presented in Figure 1. Therefore, in probabilistic analyses of complex composite structures, the determination of the probabilistic mechanical properties of composite laminae is of great importance [9][10][11][12]. ...
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Considerable uncertainties in the mechanical properties of composites not only prevent them from having efficient applications but also threaten the safety and reliability of structures. In order to determine the uncertainty in the elastic properties of unidirectional CFRP composites, this paper develops a probabilistic analysis method based on a micromechanics theoretical model and the Monte Carlo simulation. Firstly, four commonly used theoretical models are investigated by calculating the deterministic elastic parameters of three unidirectional CFRP composites, which are compared with experimental outcomes. According to error analyses, the bridging model is the most brilliant one, with errors lower than 6%, which suggests that it can be used in probabilistic analyses. Furthermore, constituent parameters are regarded as normally distributed random variables, and the Monte Carlo simulation was used to obtain samplings based on the statistics of constituent parameters. The predicted probabilistic elastic parameters of the T800/X850 composite coincide with those from experiments, which verified the effectiveness of the developed probabilistic analysis method. According to the probabilistic analysis results, the statistics of the elastic parameters, the correlations between the elastic parameters, and their sensitivity to the constituent’s properties are determined. The moduli E11, E22, and G12 of the T800/X850 composite follow the lognormal distribution, namely, ln(E11)~N[5.15, 0.0282], ln(E22)~N[2.15, 0.0242], and ln(G12)~N[1.48, 0.0382], whereas its Poisson’s ratio, v12, obeys the normal distribution, namely, v12~N(0.33, 0.0122). Additionally, the correlation coefficients between v12 and E11/E22/G12 are small and thus can be ignored, whereas the correlation coefficients between any two of E11, E22, and G12 are larger than 0.5 and should be considered in the reliability analyses of composite structures. The developed probabilistic analysis method based on the bridging model and the Monte Carlo simulation is fast and reliable and can be used to efficiently evaluate the probabilistic properties of the elastic parameters of any unidirectional composite in the reliability design of structures in engineering practice.
Chapter
Every engineering application requires a comprehensive investigation of different parameters in different exposure conditions to come up with an optimal yet feasible product. Modeling the complex relationships between the various governing factors is extremely strenuous and generally requires the development of a mathematical tool. This has motivated researchers to look for time saving and less expensive computational techniques. Machine learning is perceived as the next big wave of innovation. It has revolutionized the field of Material Science by showing promising results when it comes to target oriented research. In the past few years, machine learning has been used as an efficient tool for predicting the material behavior but still there are inhibitions to use the various algorithms for large scale implementation. In an attempt to provide perspective on the usage of machine learning algorithms in polymer composites, this chapter summarizes the recent studies conducted on these composite materials using machine learning along with a general overview of its multifaceted applications like prediction, optimization, uncertainty quantification and sensitivity analysis.
Research
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First-order shear deformation theory (FSDT) is less accurate compared to higher-order theories like higher-order zigzag theory (HOZT).In case of large-scale simulation-based analyses like uncertainty quantification and optimization using FSDT, such errors propagate and accumulate over multiple realizations, leading to significantly erroneous results. Consideration of higher-order theories results in significantly increased computational expenses, even though these theories are more accurate. The aspect of computational efficiency becomes more critical when thousands of realizations are necessary for the analyses. Here we propose to exploit Gaussian process-based machine learning for creating a computational bridging between FSDT and HOZT, wherein the accuracy of HOZT can be achieved while having the low computational expenses of FSDT. The machine learning augmented FSDT algorithm is referred to here as modified FSDT (mFSDT), based on which extensive deterministic results and Monte Carlo simulation-assisted probabilistic results are presented for the free vibration analysis of shear deformation sensitive structures like laminated composite and sandwich plates considering various configurations. The proposed algorithm of bridging different laminate theories is generic in nature and it can be utilized further in a range of other static and dynamic analyses concerning composite plates and shells for accurate, yet efficient results.
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