Abstract

The present study addresses the integration of an analytical uncertainty quantification approach to multi-scale modeling of single-walled carbon nanotube (SWNT)-epoxy nanocomposites. The main highlight is the investigation of the stochasticity of nanotube orientations, and its effects on the homogenized properties. Even though the properties of SWNT-epoxy nanocomposites are well-studied in the literature, the natural stochasticity that arises from the nanotube orientations has not been observed. To understand the effects of the variability in SWNT orientations to material properties of interest, an analytical uncertainty quantification algorithm is utilized. The analytical scheme computes the propagation of the orientational uncertainty to the volume-averaged properties with a linear solution and uses the transformation of random variables principle to obtain the variations in non-linear properties. The results indicate that the uncertainty propagation affects the macro-scale properties, including stiffness, thermal expansion, thermal conductivity, and natural frequencies.

References

References
1.
Wang
,
R.-M.
,
Zheng
,
S.-R.
, and
Zheng
,
Y.-P.
,
2011
,
Polymer Matrix Composites and Technology
,
Woodhead Publishing in Materials
,
Philadelphia, PA
, pp.
1
7
.
2.
Sugita
,
Y.
,
Winkelmann
,
C.
, and
La Sapanora
,
V.
,
2010
, “
Environmental and Chemical Degradation of Carbon/Epoxy Lap Joints for Aerospace Applications, and Effects on Their Mechanical Performance
,”
Compos. Sci. Technol.
,
70
(
5
), pp.
829
839
. 10.1016/j.compscitech.2010.01.021
3.
Treacy
,
M. J
,
Ebbesen
,
T.
, and
Gibson
,
J.
,
1996
, “
Exceptionally High Young’s Modulus Observed for Individual Carbon Nanotubes
,”
Nature
,
381
(
6584
), pp.
678
680
. 10.1038/381678a0
4.
Yakobson
,
B. I.
, and
Avouris
,
P.
,
2001
, “Mechanical Properties of Carbon Nanotubes,”
Carbon Nanotubes
,
Springer
,
New York
, pp.
287
327
.
5.
Walters
,
D.
,
Ericson
,
L
,
Casavant
,
M.
,
Liu
,
J.
,
Colbert
,
D.
,
Smith
,
K.
, and
Smalley
,
R.
,
1999
, “
Elastic Strain of Freely Suspended Single-Wall Carbon Nanotube Ropes
,”
Appl. Phys. Lett.
,
74
(
25
), pp.
3803
3805
. 10.1063/1.124185
6.
Thess
,
A.
,
Lee
,
R.
,
Nikolaev
,
P.
, and
Dai
,
H.
,
1996
, “
Crystalline Ropes of Metallic Carbon Nanotubes
,”
Science
,
273
(
5274
), pp.
483
487
. 10.1126/science.273.5274.483
7.
Wilder
,
J. W.
,
Venema
,
L. C.
,
Rinzler
,
A. G.
,
Smalley
,
R. E.
, and
Dekker
,
C.
,
1998
, “
Electronic Structure of Atomically Resolved Carbon Nanotubes
,”
Nature
,
391
(
6662
), pp.
59
62
. 10.1038/34139
8.
Odom
,
T. W.
,
Huang
,
J.-L.
,
Kim
,
P.
, and
Lieber
,
C. M.
,
1998
, “
Atomic Structure and Electronic Properties of Single-Walled Carbon Nanotubes
,”
Nature
,
391
(
6662
), pp.
62
64
. 10.1038/34145
9.
Dresselhaus
,
M.
, and
Eklund
,
P.
,
2000
, “
Phonons in Carbon Nanotubes
,”
Adv. Phys.
,
49
(
6
), pp.
705
814
. 10.1080/000187300413184
10.
Hone
,
J.
,
2001
, “Phonons and Thermal Properties of Carbon Nanotubes,”
Carbon Nanotubes
,
Springer
,
New York
, pp.
273
286
.
11.
Fasanella
,
N. A.
, and
Sundararaghavan
,
V.
,
2015
, “
Molecular Dynamics of SWNT/Epoxy Nanocomposites
,”
56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
,
Kissimmee, FL
,
Jan. 5–9
, pp. 1–11.
12.
Fasanella
,
N. A.
, and
Sundararaghavan
,
V.
,
2015
, “
Atomistic Modeling of Thermomechanical Properties of SWNT/Epoxy Nanocomposites
,”
Modell. Simul. Mater. Sci. Eng.
,
23
(
6
), p.
065003
. 10.1088/0965-0393/23/6/065003
13.
Huyse
,
L.
, and
Maes
,
M. A.
,
2001
, “
Random Field Modeling of Elastic Properties Using Homogenization
,”
J. Eng. Mech.
,
127
(
1
), pp.
27
36
. 10.1061/(ASCE)0733-9399(2001)127:1(27)
14.
Sakata
,
S.
,
Ashida
,
F.
,
Kojima
,
T.
, and
Zako
,
M.
,
2008
, “
Three-Dimensional Stochastic Analysis Using a Perturbation-Based Homogenization Method for Elastic Properties of Composite Material Considering Microscopic Uncertainty
,”
Int. J. Solids Struct.
,
45
(
3–4
), pp.
894
907
. 10.1016/j.ijsolstr.2007.09.008
15.
Creuziger
,
A.
,
Syed
,
K.
, and
Gnaupel-Herold
,
T.
,
2011
, “
Measurement of Uncertainty in Orientation Distribution Function Calculations
,”
Scr. Mater.
,
72–73
, pp.
55
58
. 10.1016/j.scriptamat.2013.10.017
16.
Juan
,
L.
,
Liu
,
G.
,
Wang
,
H.
, and
Ullah
,
A.
,
2011
, “
On the Sampling of Three-Dimensional Polycrystalline Microstructures for Distribution Determination
,”
J. Microsc.
,
44
(
2
), pp.
214
222
. 10.1111/j.1365-2818.2011.03531.x
17.
Hiriyur
,
B.
,
Waisman
,
H.
, and
Deodatis
,
G.
,
2011
, “
Uncertainty Quantification in Homogenization of Heterogeneous Microstructures Modeled by XFEM
,”
Int. J. Numer. Methods Eng.
,
88
(
3
), pp.
257
278
. 10.1002/nme.3174
18.
Stefanou
,
G.
,
Savvas
,
D.
, and
Papadrakakis
,
M.
,
2015
, “
The Role of Microstructure Uncertainty in Stochastic Finite Element Analysis
,”
8th GRACM International Congress on Computational Mechanics
,
Volos, Thessaly, Greece
,
July 12–15
, Vol.
12
, pp.
1
10
.
19.
Kouchmeshky
,
B.
, and
Zabaras
,
N.
,
2009
, “
The Effect of Multiple Sources of Uncertainty on the Convex Hull of Material Properties of Polycrystals
,”
Comput. Mater. Sci.
,
47
(
2
), pp.
342
352
. 10.1016/j.commatsci.2009.08.010
20.
Madrid
,
P. J.
,
Sulsky
,
D.
, and
Lebensohn
,
R. A.
,
2014
, “
Uncertainty Quantification in Prediction of the In-Plane Young’s Modulus of Thin Films With Fiber Texture
,”
J. Microelectromech. Syst.
,
23
(
2
), pp.
380
390
. 10.1109/JMEMS.2013.2279500
21.
Niezgoda
,
S. R.
,
Yabansu
,
Y.
, and
Kalidindi
,
S. R.
,
2011
, “
Understanding and Visualizing Microstructure and Microstructure Variance as a Stochastic Process
,”
Acta Mater.
,
59
(
16
), pp.
6387
6400
. 10.1016/j.actamat.2011.06.051
22.
Stevens
,
G.
,
Atamturktur
,
S.
,
Lebensohn
,
R.
, and
Kaschner
,
G.
,
2016
, “
Experiment-Based Validation and Uncertainty Quantification of Coupled Multi-Scale Plasticity Models
,”
Multidiscip. Model. Mater. Struct.
,
12
(
1
), pp.
151
176
. 10.1108/MMMS-04-2015-0023
23.
Yin
,
X.
,
Lee
,
S.
,
Chen
,
W.
,
Liu
,
W. K.
, and
Horstemeter
,
M. F.
,
2009
, “
Efficient Random Field Uncertainty Propagation in Design Using Multiscale Analysis
,”
ASME J. Mech. Des.
,
131
(
2
), p.
021006
. 10.1115/1.3042159
24.
Sakata
,
S.
,
Ashida
,
F.
, and
Zako
,
M.
,
2008
, “
Kriging-Based Approximate Stochastic Homogenization Analysis for Composite Materials
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
21–24
), pp.
1953
1964
. 10.1016/j.cma.2007.12.011
25.
Chen
,
W.
,
Yin
,
X.
,
Lee
,
S.
, and
Liu
,
W. K.
,
2010
, “
A Multiscale Design Methodology for Hierarchical Systems With Random Field Uncertainty
,”
ASME J. Mech. Des.
,
132
(
4
), p.
041006
. 10.1115/1.4001210
26.
Yin
,
X.
,
Lee
,
S.
,
Chen
,
W.
,
Liu
,
W. K.
, and
Horstemeyer
,
M. F.
,
2008
, “
A Multiscale Design Approach with Random Field Representation of Material Uncertainty
,”
Proceedings of the ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2008
,
Brooklyn, NY
,
Aug. 3–6
.
27.
Clement
,
A.
,
Soize
,
C.
, and
Yvonnet
,
J.
,
2012
, “
Computational Nonlinear Stochastic Homogenization Using a Nonconcurrent Multiscale Approach for Hyperelastic Heterogenous Microstructure Analysis
,”
Int. J. Numer. Methods Eng.
,
91
, pp.
799
824
. 10.1002/nme.4293
28.
Clement
,
A.
,
Soize
,
C.
, and
Yvonnet
,
J.
,
2013
, “
Uncertainty Quantification in Computational Stochastic Multi-Scale Analysis of Nonlinear Elastic Materials
,”
Comput. Methods Appl. Mech. Eng.
,
254
, pp.
61
82
. 10.1016/j.cma.2012.10.016
29.
Yin
,
X.
,
Lee
,
S.
,
Chen
,
W.
,
Liu
,
W. K.
, and
Horstemeyer
,
M. F.
,
2010
, “
Enabling Integrated Material and Product Design Uncertainty Through Stochastic Constitutive Relations
,”
Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2010
,
Montreal, QC, Canada
,
Aug. 15–18
.
30.
Bessa
,
M. A.
,
Bostanabad
,
R.
,
Liu
,
Z.
,
Hu
,
A.
,
Apley
,
D. W.
,
Brinson
,
C.
,
Chen
,
W.
, and
Liu
,
W. K.
,
2017
, “
A Framework for Data-Driven Analysis of Materials Under Uncertainty: Countering the Curse of Dimensionality
,”
Comput. Methods Appl. Mech. Eng.
,
320
, pp.
633
667
. 10.1016/j.cma.2017.03.037
31.
Angelikopoulos
,
P.
,
Papadimitriou
,
C.
, and
Koumoutsakos
,
P.
,
2013
, “
Data Driven, Predictive Molecular Dynamics for Nanoscale Flow Simulations Under Uncertainty
,”
J. Phys. Chem. B
,
117
(
47
), pp.
14808
14816
. 10.1021/jp4084713
32.
Angelikopoulos
,
P.
,
Papadimitriou
,
C.
, and
Koumoutsakos
,
P.
,
2012
, “
Bayesian Uncertainty Quantification and Propagation in Molecular Dynamics Simulations: A High Performance Computing Framework
,”
J. Chem. Phys.
,
137
(
14
), p.
144103
. 10.1063/1.4757266
33.
Rizzi
,
F.
,
Jones
,
R. E.
,
Debusschere
,
B. J.
, and
Knio
,
O. M.
,
2013
, “
Uncertainty Quantification in MD Simulations of Concentration Driven Ionic Flow Through a Silica Nanopore. I. Sensitivity to Physical Parameters of the Pore
,”
J. Chem. Phys.
,
138
, pp.
194104
. 10.1063/1.4804666
34.
Rizzi
,
F.
,
Najm
,
H. N.
,
Debusschere
,
B. J.
,
Sargsyan
,
K.
,
Salloum
,
M.
,
Adalsteinsson
,
H.
, and
Knio
,
O. M.
,
2012
, “
Uncertainty Quantification in MD Simulations. Part I: Forward Propagation
,”
Multiscale Model. Simul.
,
10
(
4
), pp.
1428
1459
. 10.1137/110853169
35.
Rizzi
,
F.
,
Najm
,
H. N.
,
Debusschere
,
B. J.
,
Sargsyan
,
K.
,
Salloum
,
M.
,
Adalsteinsson
,
H.
, and
Knio
,
O. M.
,
2012
, “
Uncertainty Quantification in MD Simulations. Part II: Bayesian Inference of Force-Field Parameters
,”
Multiscale Model. Simul.
,
10
(
4
), pp.
1460
1492
. 10.1137/110853170
36.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2017
, “
Uncertainty Quantification of Microstructural Properties Due to Experimental Variations
,”
AIAA J.
,
55
(
8
), pp.
2824
2832
. 10.2514/1.J055689
37.
Acar
,
P.
,
Srivastava
,
S.
, and
Sundararaghavan
,
V.
,
2017
, “
Stochastic Design Optimization of Microstructures With Utilization of a Linear Solver
,”
AIAA J.
,
55
(
9
), pp.
3161
3168
. 10.2514/1.J056000
38.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2017
, “
Uncertainty Quantification of Microstructural Properties Due to Variability in Measured Pole Figures
,”
Acta Mater.
,
124
, pp.
100
108
. 10.1016/j.actamat.2016.10.070
39.
Acar
,
P.
,
Sundararaghavan
,
V.
, and
Fasanella
,
N.
,
2018
, “
Multi-Scale Optimization of Nanocomposites With Probabilistic Feature Descriptors
,”
AIAA J.
,
56
(
7
), pp.
2936
2941
. 10.2514/1.J056791
40.
Knox
,
C.
,
Andzelm
,
J.
,
Lenhart
,
J.
,
Browning
,
A.
, and
Christensen
,
S.
,
2010
, “
High Strain Rate Mechanical Behavior of Epoxy Networks From Molecular Dynamics Simulations
,”
Proceedings of the 27th Army Science Conference
,
Orlando, FL
, GP-09.
41.
Christensen
,
S.
,
2007
, “
Atomistically Explicit Molecular Dynamics Simulations of Thermosetting Polymers
,”
Proceedings of the 39th ISTC SAMPE Conference
,
Cincinnati, OH
, Vol.
39
, pp.
1
26
.
42.
Bunge
,
H.-J.
,
2013
,
Texture Analysis in Materials Science: Mathematical Methods
,
Elsevier
,
New York
.
43.
Ferrari
,
M.
, and
Johnson
,
G. C.
,
1989
, “
Effective Elasticities of Short-Fiber Composites With Arbitrary Orientation Distribution
,”
Mech. Mater.
,
8
(
1
), pp.
67
73
. 10.1016/0167-6636(89)90006-9
44.
Dunn
,
M. L.
,
Ledbetter
,
H.
,
Heyliger
,
P. R.
, and
Choi
,
C. S.
,
1996
, “
Elastic Constants of Textured Short-Fiber Composites
,”
J. Mech. Phys. Solids
,
44
(
9
), pp.
1509
1541
. 10.1016/0022-5096(96)00021-X
45.
Sundararaghavan
,
V.
, and
Zabaras
,
N.
,
2007
, “
Linear Analysis of Texture-Property Relationships Using Process-Based Representations of Rodrigues Space
,”
Acta Mater.
,
55
(
5
), pp.
1573
1587
. 10.1016/j.actamat.2006.10.019
46.
Sundararaghavan
,
V.
, and
Zabaras
,
N.
,
2005
, “
On the Synergy Between Texture Classification and Deformation Process Sequence Selection for the Control of Texture-Dependent Properties
,”
Acta Mater.
,
53
(
4
), pp.
1015
1027
. 10.1016/j.actamat.2004.11.001
47.
Kumar
,
A.
, and
Dawson
,
P.
,
2000
, “
Computational Modeling of FCC Deformation Textures Over Rodrigues’ Space
,”
Acta Mater.
,
48
(
10
), pp.
2719
2736
. 10.1016/S1359-6454(00)00044-6
48.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2016
, “
Utilization of a Linear Solver for Multiscale Design and Optimization of Microstructures in an Airframe Panel Buckling Problem
,”
57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
, AIAA SciTech Forum. AIAA Paper No. 2016-0156, pp.
1
16
.
49.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2016
, “
Utilization of a Linear Solver for Multiscale Design and Optimization of Microstructures
,”
AIAA J.
,
54
(
5
), pp.
1751
1759
. 10.2514/1.J054822
50.
Schapery
,
R. A.
,
1968
, “
Thermal Expansion Coefficients of Composite Materials Based on Energy Principles
,”
J. Compos. Mater.
,
2
(
3
), pp.
380
404
. 10.1177/002199836800200308
You do not currently have access to this content.