Composites: Part B 40 (2009) 522–529
Contents lists available at ScienceDirect
Composites: Part B
journal homepage: www.elsevier.com/locate/compositesb
A multi-objective optimization approach for design of blast-resistant
composite laminates using carbon nanotubes
Mahmoud M. Reda Taha a,*, Arife B. Colak-Altunc a, Marwan Al-Haik b
a
b
Department of Civil Engineering, Univ. of New Mexico, Centennial Engineering Center, Albuquerque, NM 87131-0001, USA
Department of Mechanical Engineering, Univ. of New Mexico, Albuquerque, NM 87131-0001, USA
a r t i c l e
i n f o
Article history:
Received 28 January 2009
Received in revised form 22 April 2009
Accepted 23 April 2009
Available online 7 May 2009
Keywords:
A. Carbon-carbon composites (CCCs)
A. Nano-structures
B. Impact behavior
C. Laminate mechanics
Blast-resistant composites
a b s t r a c t
A reliable process for the design of blast-resistance composite laminates is needed. We consider here the
use of carbon nanotubes (CNTs) to enhance the mechanical properties of composite interface layers. The
use of CNTs not only enhances the strength of the interface but also significantly alters stress propagation
in composite laminates. A simplified wave propagation simulation is developed and the optimal CNT content in the interface layer is determined using multi-objective optimization paradigms. The optimization
process targets minimizing the ratio of the stress developed in the layers to the strength of that layer for
all the composite laminate layers. Two optimization methods are employed to identify the optimal CNT
content. A case study demonstrating the design of five-layer composite laminate subjected to a blast
event is used to demonstrate the concept. It is shown that the addition of 2% and 4% CNTs by weight
to the epoxy interfaces results in significant enhancement of the composite ability to resist blast.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Structural fiber reinforced plastic (FRPs) composites have been
used extensively in several high performance structural applications (e.g. vehicles, airplanes, etc.) for their significantly high
strength-to-weight ratio. Other plausible features of FRPs include
their manufacturing flexibility, which allows composites to
achieve material properties that are difficult to attain using single-phase materials. However, typical structural composites exhibit limited ductility, which limit their ability to absorb energy
under impact loading and ultimately leads to their failure [1]. This
limited energy absorption is related to the lack of plasticity mechanisms and dominant debonding at weak interfaces [2]. The dominant failure mechanism in composite laminates subjected to
impact-loading is a complex combination of delamination predominantly caused by mode II shear, matrix cracking caused by transverse shear, and translaminar fracture in terms of fiber fracture and
kinking [3]. There are several factors that dictate the above fracture
processes, such as the material variables, loading and environmental conditions and the source of impact. Amongst the material variables, the mechanical properties of fiber and matrix, particularly
the failure strains, interface properties, fiber configuration and
stacking sequence in angle-ply laminates play, important roles in
determining the impact damage resistance of composites [4].
* Corresponding author. Tel.: +1 505 277 1258; fax: +1 505 277 1988.
E-mail addresses: mrtaha@unm.edu (M.M. Reda Taha), burcu@unm.edu (A.B.
Colak-Altunc), alhaik@unm.edu (M. Al-Haik).
1359-8368/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compositesb.2009.04.020
Significant efforts for developing blast-resistant composites
have been conducted in the last three decades. Brennan and Prewo
[5] reported a composite made of a glass–ceramic matrix reinforced with silicon carbide fibers that exhibited exceptional
mechanical strength and toughness. Toughness measurements of
new silicon carbide fiber composites were reported to be 50 times
that of typical ceramic composites. Much interest was directed to
examine the impact strength of carbon/epoxy composites [6]. Kang
and Lee [7] reported a significant improvement in energy absorption of multi-laminate carbon composites by chain and plain
stitching. More recently, Tekalura et al. [8] investigated the advantages of using polyurea and E-glass vinyl ester layered composites
as a sandwich material for shock mitigation structures.
Research was also devoted to developing computational tools to
simulate composite behavior under impact loads and to quantify
damage in laminated composites (cf. [9]). Riccio and Tessitore
[10] demonstrated the strong dependence of composite behavior
on the loading conditions using nonlinear finite-element modeling.
Batra and Hassan [11] also used a continuum damage mechanics
approach in their characterization of damage due to blast loading
of unidirectional fiber reinforced composites. Moreover, Donadona
et al. [12] successfully implemented and validated a 3D continuum
damage mechanics finite-element modeling approach for composite laminates subjected to low-velocity impact. Such investigations
provided a roadmap to identify the desired characteristics of composites to resist blast events. Here we suggest the addition of carbon nanotubes to enhance blast resistance of structural composites
and we discuss below a numerical approach to determine the
optimal carbon nanotubes content to enhance blast resistance.
M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
2. Carbon nanotubes
Carbon nanotubes (CNTs) have drawn significant attention from
the research community for their attractive mechanical properties.
CNTs represent a unique form of carbon that can be visualized by
considering a single grapheme sheet representing a lattice of carbon atoms distributed in a hexagonal pattern [13]. A single wall
carbon nanotube (SWCNT) is 1–3 nm in diameter [14]. The attractive properties of SWCNTs might be attributed to their unique
nanostructure. SWCNTs possess exceptional mechanical [15] properties and superior thermal and electric properties compared to
macroscale fibers such as graphite, Kevlar, SiC and alumina. The
strength, elastic modulus and the fracture properties of carbon
nanotubes are an order of magnitude higher than those of most
common composite materials. The properties of CNTs are presented in Table 1 after [16]. A schematic representation of a
SWCNT is shown in Fig. 1(a) and a transmission electronic microscope (TEM) image of several SWCNTs is shown in Fig. 1(b).
Numerous experiments have been reported on the enhanced
mechanical properties of SWCNTs. Of interest here is the significantly high stiffness (Young’s modulus of elasticity) of SWCNTs
which was measured using atomic force microscopy [17]. These
experiments suggested that the SWCNT Young’s modulus is usually around 1 TPa, and SWCNTs are slightly stiffer than multiwalled nanotubes. Moreover, the tensile strength of SWCNTs obtained from direct tension tests is found to vary between 50 and
150 GPa for different strain rates and temperatures [16,18]. In spite
of the fact that the fracture toughness of SWCNTs are not yet measured experimentally, researchers [19] found that absorbed energy
to failure of a composite based on polyvinyl-SWCNTs was 18 times
higher than Kevlar fibers and 48 times higher than graphitic fibers.
Various methods have been successfully utilized to synthesize
CNTs, such as arc discharge, laser ablation pyrolysis and chemical
vapor deposition [20]. Researchers reported successful inclusion
of CNTs in polymer matrices using various techniques to avoid
the difficulty of uniformly dispersing CNTs in polymer [21] including the use of magnetic fields [22]. We suggest here that CNTs can
be dispersed in epoxy forming the interface of the composite laminates. CNTs at concentrations of 2–5% in epoxy can significantly
enhance the tensile strength and tensile modulus of the composite
interface. We suggest that using classical Voigt’s mixture rule [23];
we can approximately predict the properties of CNT strengthened
epoxy. We hypothesize that such a change of strength and stiffness
of the interface will play a major role in blast resistance as it alters
the stress to strength distribution inside the composite laminate.
3. Stress waves propagation in composite laminates
To simulate this effect, we adopt a simplified approach for stress
wave propagation in bounded elastic media after Kolsky [24] and
Al-Haik and Almyadmeh [25]. The velocity of longitudinal waves
C 0 in a medium of density q and bulk modulus K can be computed
as
C0 ¼
sffiffiffiffi
K
ð1Þ
q
This method allows computing reflection and refraction of both
dilation and distortion waves at free boundaries and at an interface
between two media. We limit our discussion here to the propagation of longitudinal waves at the interface to predict the normal
stresses at both sides of an interface in the composite laminate
due to an incident pressure. However, the proposed approach can
be extended to include torsional and flexural stress waves [24].
Considering a limited width dx of the composite layer as shown in
Fig. 2 with the applied stress rxx and the transmitted wave produc@ rxx
dx, one can
ing the stress at the other side of the interface rxx þ
@x
write the following differential equation
q
Table 1
Mechanical properties of CNTs compared to typical structural materials [16].
523
@2u
@2u
¼E 2
2
@x
@t
ð2Þ
This equation can be solved by considering a linear displacement
time function uðtÞ. This leads us to realize the fundamental relationship of the stress with particle velocity as
Material
Modulus
(GPa)
Strength
(MPa)
Fracture toughness,
K IC ðMPa m1=2 Þ
Steel 4140
Ti–6Al–4V
C–C
composites
Kevlar 49
SWCNTS
190–210
113
125
1646
910
1200–1600
54–62
44–66
70
rxx ¼ qC 0
186
12,000
3400
150,000
60–80
100
The term qC 0 is known as the characteristic impedance. Defining the
particle velocity V ¼ @u=@t we can re-write Eq. (3) to relate the particle velocity to the stress as
@u
@t
ð3Þ
Fig. 1. (a) Schematic representation of single wall carbon nanotube (SWCNT). (b) TEM image showing SWCNTs produced by arc discharge (courtesy of Al-Haik M, 2008).
524
M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
form the step-by-step in-time analysis to obtain the stress pulse
time history.
4. Multi-objective optimization
Fig. 2. Assumptions for longitudinal wave propagation.
V¼
rxx
qC
ð4Þ
The above method can be extended to estimate the reflected and
transmitted waves and stresses at an interface between two media.
Assuming an incident stress to the interface rI , part of that stress
will be transmitted to the second media and part will be reflected.
The transmitted stress can be computed by considering the continuity of velocity across the interface plane V I ¼ V T þ V R where V I
is the incident particle velocity, V R is the reflected particle velocity
and V T is the transmitted particle velocity. This is shown schematically in Fig. 3. Relating the incident stress to the incident particle
velocity at the first layer before the interface V I can be computed as
VI ¼
rI
qI C I
ð5Þ
Equilibrium at the interface also requires that
ðrI þ rR ÞA1 ¼ rT A2
ð6Þ
where A1 and A2 are areas of Layers 1 and 2, respectively. By substituting Eq. (5) in Eq. (2) and solving Eq. (6) one can deduce the transmitted stress rT and the reflected stress rR as
2A1 q2 C 2
rI
A1 q1 C 1 þ A2 q2 C 2
A2 q2 C 2 A1 q1 C 1
¼
rI
A1 q1 C 1 þ A2 q2 C 2
rT ¼
ð7Þ
rR
ð8Þ
For the composite laminates considered in the current investigation, all the areas will be assumed identical; A1 ¼ An ; n ¼ 1; . . . ; N
where N is the total number of layers. It becomes obvious from
Eqs. (7) and (8) that direction of the reflected pressure rR is a function of the characteristic impedance of each layer. The stress distribution can therefore be altered by optimally distributing the
impedance (stiffness) of the composite layer. By adding CNTs to
the epoxy interface, one will increase the characteristic impedance
of the layer and alter the reflected and transmitted pressure. It is
important to realize that the final stress in each layer is the algebraic sum of both the reflected and transmitted stress waves occurring in the layer at different time steps. Therefore, a step-by-step intime analysis is required to account for all the stress propagation in
the laminate. A computer code is thus developed after [25] to per-
The above simulation method can predict the stresses at the different composite layers due to an incident pressure applied to the
composite laminate. We suggest that adding CNTs to the epoxy
interface can alter the stress distribution in the composite layers.
Moreover, adding CNTs to epoxy will also enhance the ultimate
tensile strength of the interface and thus might alter the failure
mode to occur in the composite fiber instead of the interface. However, CNTs are expensive materials and therefore one shall optimize the amount of CNTs to be added such that the composite
strength against blast is enhanced while limiting the cost. Furthermore, if more than one interface exists (which is the case in most
composite laminates), nonlinear behaviors can be observed with
alternating the impedance of the interface layers by using different
CNT content in each layer.
Classical optimization approaches have always considered a
single-objective function while dealing with all other objectives
as constraints. The fundamental difference between single-objective and multi-objective optimization is the ability of multi-objective optimization to avoid the artificial fixes needed for singleobjective optimization methods in order to address tradeoffs between the two different objectives. For instance, we might need
to limit failure of all interface layers, the optimal CNT contents of
the layers are those needed to distribute the stress in the composite laminate such that the maximum stress in each interface layer
does not exceed its strength, which is also a function of the CNT
content. We suggest that such a design can be achieved by getting
candidate designs to lie in the Pareto front, typically known as Pareto-optimal solutions after Pareto [26]. De Kruijf et al. [27] reported successful microstructural optimization to handle the
tradeoffs between alternative microstructures to achieve desired
conductivity and stiffness in composite materials.
Two categories of multi-objective optimization methods are
identified in the literature (cf. [28]). The first category utilizes classical single-objective optimization methods while reformulating
the problem to address multi-objectives considering preferences
between the different objective functions [29]. Methods in the first
category include techniques such as the weighted sum method, the
e-constraint method and the hierarchical optimization method
[29]. In the hierarchical optimization method the top priority
objective function is initially optimized and then the second priority function is optimized subject to the constraint that the top priority is not adversely affected. This procedure repeats itself for all
priority functions. In this paper, weighted sum method is used due
to its ability and relative simplicity in demonstrating different
optimization alternatives. The second category of multi-objective
optimization methods establishes an optimization method that is
multi-objective in nature by using evolutionary optimization such
as non-sorted genetic algorithms [30].
5. Case study
Fig. 3. Schematic representation of wave transmission and reflection at the
interface.
Here we present a case study to demonstrate the integration of
the above methods for designing a blast-resistant structural composite using carbon nanotubes. The composite of interest here is
a five-layer carbon fiber composite shown schematically in Fig. 4.
The properties of the materials used for the five composite layers
are presented in Table 2. We simulated a blast event, using scaling
laws after Smith and Hetherington [31]. This blast is aimed to produce an incident pressure of 140 MPa at the surface of the composite to assure the failure of the composite before the addition of
M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
525
noted by a1 and a2 , are defined as the design variables subject to
the following bounds
0 6 a1 6 amax
and 0 6 a2 6 amax
ð12Þ
amax is the maximum CNT ratio, which is set at 5% for practical
Fig. 4. Schematic representation of the composite laminate for the case study
showing five-layer laminate.
Table 2
Mechanical properties of the different layers of the structural composite.
Material
Modulus
(GPa)
Strength
(MPa)
Fracture toughness,
K IC ðMPa m1=2 Þ
Carbon fibers
Epoxy
SWCNTs
70
5.4
12,000
600
60
150,000
70
6
100
CNT. Using the stress wave propagation approach described above,
the stresses in the structural composite layers at different time
steps were simulated using the step-by-step in-time analysis.
The stress at each layer was then compared to the material
strength of that layer. Carbon nanotubes were then assumed to
be added to the epoxy interface at Layers 2 and 4. Addition of carbon nanotubes increased the interface strength and altered the
stiffness distribution in the composite and, therefore, the stress
propagation in the composite.
We implemented an optimization method to identify the Pareto-optimal design front. Two objective functions representing the
maximum tension and compression stress to strength ratios in
the composite denoted as f1 and f2 are defined as
f1 ¼ max16m6N
f2 ¼ max16m6N
maxðrTm Þ
rUT
m
maxðrCm Þ
rUC
m
reasons.
Two optimization methods (denoted here as OPT1 and OPT2)
were used to find the optimal values of a1 and a2 . The first optimization method denoted OPT1 implemented the Broyden–Fletcher–
Goldfarb–Shanno (BFGS) gradient based method which represents
a modified version of the quasi-Newton methods as described below [32]. The second optimization method denoted OPT2 is a nongradient based global optimization algorithm after Jones [33] that
balances between global and local search. The optimization process was developed under Design Analysis Kit for Optimization
and Terascale Applications (DAKOTA) environment (cf. [34]).
Fig. 5 provides a schematic representation of the optimization process and the optimization/simulation interface. Other optimization
methods can also be used to perform the required optimization.
We limit our discussion here to these two methods for space limitation. The choice of the two methods was to demonstrate the difference between gradient and non-gradient optimization
approaches.
¼ fa1 a2 g is
In this process, the vector of design variables a
passed by the optimization environment (DAKOTA) to the simulation algorithm to compute the maximum tensile and compressive
stresses in all the layers. The two objective functions f1 and f2 (Eqs.
(9) and (10)) are computed and then used to compute the combined objective function f j (Eq. (11)) using one set of weights wj1
and wj2 . Once the optimization process terminates, the j þ 1th set
of weights are used and the process is repeated until all assigned
weights are examined. Variation of weights enables forming the
Pareto front for multi-objective optimization. As the number of
weights increases, a fine Pareto front can be established. However,
increasing the number of weights results in increasing the computation time. Our preliminary experiments showed that 0.025 increments in weights are sufficient for obtaining a good Pareto front.
Fig. 6 shows the surface of the objective function for three different
sets of weights.
The OPT1 method implements the BFGS algorithm. This algo 0 using an
rithm is based on updating an initial variable vector a
approximate Hessian matrix B0 . The approximate Hessian matrix
can be initiated as the identity matrix similar to the gradient des-
ð9Þ
ð10Þ
In this formulation, f1 is the maximum tensile stress to strength ratio in all individual layers and f2 is the maximum compressive stress
to strength ratio in all individual layers. N is the total number of layers in the composite; maxðrTm Þ is the maximum tensile stress observed in the mth layer due to the applied stress wave; rUT
m is the
ultimate tensile strength of the mth layer; maxðrCm Þ is the maximum
compressive stress observed in the mth layer due to the applied
stress wave; rUC
m is the ultimate compressive strength of the mth
layer.
The optimization problem is formulated as a multi-objective
nonlinear unconstrained optimization that targets minimizing
the combined objective function f j defined as
min f j ¼ wj1 f1 þ wj2 f2
ð11Þ
where wj1 and wj2 are the jth set of weights of the objective functions
f1 and f2 such that wj2 ¼ 1 wj1 and j ¼ 1; 2; 3; . . . ; J where J depends
on the weight increment. The CNT content in Layers 2 and 4, de-
Fig. 5. Schematic representation of the optimization process combining gradient
and non-gradient optimization methods. The figure shows the initialization steps,
opt
represents the vector of
the kth optimization iteration and the update process. a
j
optimal solution for the jth set of weights.
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M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
Fig. 6. Surface of the combined weighted objective function f j versus CNT content in Layer 2 and 4 represented by the variables a1 and a2 for three sets of weights: (a)
w1 ¼ 0:1; w2 ¼ 0:9, (b) w1 ¼ 0:5; w2 ¼ 0:5 and (c) w1 ¼ 0:9; w2 ¼ 0:1.
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M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
Fig. 7. Schematic representation of the search method used for non-gradient
optimization OPT2.
Fig. 8. Tensile stress to strength time history for all five layers of the original
composite without CNTs showing failure at Layer 4 as the stress to strength ratio
exceeds unity.
cent method. The step size sk for the kth iteration can be computed
k Þ as
using the gradient of the objective function rf ða
kÞ
Bk sk ¼ rf ða
ð13Þ
A maximum step size smax is typically used to limit the trust region
size. The maximum step size in the case study was set to 1000. The
kþ1 can be identified as
updated solutions a
a kþ1 ¼ a k þ bk sk
ð14Þ
k
where b is the optimal search direction satisfying Wolfe conditions
[35]. The approximate Hessian matrix of the k þ 1 iteration is updated by computing the change in gradient Dk as
kþ1
B
k
¼B þ
Dk DkT
DkT sk
Bk sk ðBk sk ÞT
skT Bk sk
ð15Þ
Convergence is checked by observing the change in the objective
k Þj. A convergence limit was set to
function gradient norm jrf ða
1e4 in the case study. Termination of the OPT1 method for each
set of weights was achieved by meeting the convergence limit or
performing the maximum number of iterations (function evaluations) set to 1000.
The second optimization method denoted OPT2 known as Dividing Rectangles (DIRECT) starts by scaling the feasible region into an
n-dimensional unit hypercube. The search begins at the center
point of the hypercube to find the optimal solution for the combined weighted objective function f. Sampling is used to select feasible points that will be the center of the potentially optimal
rectangles and these rectangles are further divided into smaller
rectangles until the termination criterion is met [36]. One can decide to divide a sub-region or a bigger sub-region by setting a
threshold to the ratio of the sub-region of interest to the largest
sub-region. Fig. 7 shows the design space partitioning with search
method. Termination is based on meeting the maximum number of
iterations (function evaluations) or reaching the minimum box size
k
Þ
ðb ¼ bmin . The minimum box size bmin was set here to 1e8 and the
maximum number of iterations was set to 1000.
Fig. 9. Pareto front and results for multi-objective optimization using two
optimization methods OPT1 and OPT2.
Table 3
Results of multi-objective optimization showing the significance of changing CNT
contents in interface Layers 2 and 4 represented by a1 % and a2 % on the objective
functions f1 and f2 . Solution 3 is selected as the optimal design of composite
laminates.
Solution #
a1 %
a2 %
Tensile stress/
strength ratio ðf1 Þ
Compressive stress/
strength ratio ðf2 Þ
1
2
3
4
5
1.64
1.80
2.01
2.74
3.66
4.88
4.15
3.97
3.44
4.20
0.332
0.297
0.277
0.226
0.218
0.411
0.416
0.421
0.437
0.449
6. Results and discussion
Numerical experiments for the case study were run under
Ubuntu 8.04 with 4 GB RAM. Results of the stress to strength ratios
at the different composite layers of the original composite (no carbon nanotubes) are presented in Fig. 8. It is obvious that the composite would have failed at the interface Layers 2 and 4, as the
stress/strength ratio exceeded unity. The results of the optimization methods are shown in Fig. 9. Two Pareto-optimal curves are
Fig. 10. Tensile stress to strength time history for all five layers of the composite
laminate including CNTs (2% in Layer 2 and 4% in Layer 4).
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M.M. Reda Taha et al. / Composites: Part B 40 (2009) 522–529
ber of iterations was never reached by either of them. In average,
OPT1 required 18 iterations and OPT2 required 819 iterations. Termination of OPT1 was due to the convergence limit and termination of OPT2 was due to the minimum box size limit. These
might be attributed to the existence of a number of local minima
in the surface of the combined weighted function f associated with
some weight combinations. Such local minima might influence the
performance of the gradient based method OPT1 but would not affect the non-gradient search method OPT2. This comparison is limited to the termination and convergence conditions imposed on
each method as described above.
7. Conclusions
Fig. 11. Comparison of maximum tensile stress to strength time-history of interface
layers with and without CNTs for (a) Layer 2 and (b) Layer 4.
shown in Fig. 9 using OPT1 and OPT2 optimization techniques. Each
point on the Pareto-optimal curve represents a viable solution with
a different combination of CNT contents a1 and a2 . It can be observed from Fig. 9 that the non-gradient search method (OPT2)
was more successful than the gradient based optimization (OPT1)
in finding solutions that resulted in a lower stress to strength ratio.
The optimal CNTs contents based on OPT2 is presented in Table 3.
While all the points on the Pareto front represent possible optimal
solutions, we select the solution corresponding to a relatively minimized value of both functions f1 and f2 , with weights 0.225 and
0.775, respectively. This means that the addition of 2% and 4% CNTs
to the two epoxy interface layers can significantly enhance the
blast resistance of the structural composite. Fig. 10 shows the
stress to strength time-history in the five layers of the composite
laminates due to the blast with the optimal CNT contents. Fig. 11
compares the stress to strength time history of the interface
(epoxy) layers with and without CNTs.
It can be observed that the use of CNTs with the polymer interface (here epoxy) can alternate the stress wave propagation in
structural composites and can significantly enhance the structural
composite resistance to blast events. Optimization for OPT1 required 0.28 s of CPU time whereas optimization for OPT2 required
5.18 s of CPU time. That indicates that OPT2 method will typically
need much longer computation time (about 19-folds) compared
with OPT1 method. Computation time was not a significant influence on this case study because of its relatively inexpensive simulation. However, this issue might be of interest when
computationally expensive simulation is considered.
Finally, both optimization methods were capable of identifying
the Pareto-optimal solution. However, the non-gradient search
method OPT2 was capable of identifying better solutions than
those found by the modified quasi-Newton method OPT1. By setting the maximum number of iterations to 1000, we gave sufficient
tolerance for both methods to converge since the maximum num-
A numerical approach for design of composite laminates for enhanced blast resistance using carbon nanotubes (CNTs) is presented. The hypothesis is based on the fact that mixing CNTs
with the polymer interface can result in changing the interface
mechanical characteristics including strength, stiffness and impedance. A simplified simulation for wave propagation in composite
laminates is developed. An optimization approach based on determining the optimal CNT content in the composite interface is
developed. A case study for enhancing the blast resistance of a
five-layer composite laminate is presented. Optimal CNT contents
of 2% and 4% are identified. Gradient and non-gradient based optimization methods were shown to be capable of establishing the
Pareto front. The non-gradient method proved to be more capable
of identifying the optimal solutions. Experimental investigations
are underway to prove the proposed hypothesis.
Acknowledgements
This research is mainly funded by the Army Research Office
(ARO) Grant # W911NF-08-1-0421. Moreover, Grants by Defense
Threat Reduction Agency (DTRA) (Grant # HDTRA1-08-1-0017
P00001) and National Science Foundation (NSF) Award # CMMI0800249 shall also be greatly acknowledged.
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