Multiscale Optimization of Nanocomposites with Probabilistic Feature Descriptors

P OLYMER-MATRIX composite materials are widely used in high-performance applications due to their high specific strength, high specific stiffness, fatigue resistance, and ease of manufacturing. Thermoset polymers are the most predominant type of matrix system, and epoxies specifically are preferred for aerospace-grade components due to their superior mechanical properties and resistance to environmental degradation due tomoisture.Carbon nanotubes (CNTs) have been researched extensively due to their outstanding mechanical [1–3], electrical [4–6], and thermal properties [7,8]. Because of these exceptional properties and very high aspect ratios, forming CNT-polymer nanocomposites has become an attractive option to improve the properties of the polymer. This work particularly focuses on single-walled carbon nanotubes (SWNTs). SWNTs are an atomthick single layer of graphenewith a cylindrical structure and an elastic modulus around 1 TPa [9]. We have addressed a multiscale modeling and optimization methodology to further improve the structural and thermal performance of the SWNTs. The interface between theCNTand epoxy plays an important role in the thermomechanical properties of nanocomposites. Experimental characterization of this interface is difficult, and thus molecular modeling becomes an essential tool for relating molecular interfacial structure to bulk thermomechanical properties. Molecular dynamics (MD)was used previously to explore the effects on themechanical and dilatometric properties by adding pristine and covalently functionalized SWNTs to cross-linked polymers [10,11]. MD allows for the effects ofmechanical and thermal loading to be isolated and visualized in regions of interest where it may not be possible with experiments. MD, however, has well-known limitations regarding time and length scales; for example, the time step of an MD simulation is typically on the order of femtoseconds. It is defined by the fastest motion in the system, and vibrational frequencies are up to 3000 cm−1, corresponding to the period on the order of 10 fs [12]. The limiting factor of the length scale is the number of atoms that can be included in the simulation, on the order of 10–10 atoms, leading to a simulation on the order of 10–100 nm [12]. To address these issues, a number of different multiscale approaches have been implemented by various researchers tomodel SWNT-polymer nanocomposites.Liu et al. useda multiscale approach, via MD and micromechanics, to study the effect functionalization of CNTs on the damping characteristics of SWNTpolymer composites [13]. Seidel and Lagoudas calculated the effective elastic properties for CNT-reinforced composites through a variety of micromechanics techniques [14]. To accurately model the macroscopic behavior, many authors employ numerical techniques such as the finite element method (FEM). Li and Chou examined the effect of interfacial load transfer on the stress distribution, as well as the compressive behavior in CNT-polymer composites via a molecular structural mechanics model combined with FEM [15,16]. Namilae and Chandra also studied interfacial properties and developed a hierarchical multiscale methodology to link MD and FEM through atomically informed cohesive zonemodel parameters [17]. Spanos and Kontos developed a multiscaleMonte Carlo and FEM for the effective elastic properties of SWNT-polymer nanocomposites [18]. Tserpes et al. used amultiscale representative volume element formodeling the tensile behavior of CNT-reinforced composites to integrate nanomechanics andcontinuummechanics. [19]. Ionita usedmultiscale modeling via atomistic and mesoscale simulations to study SWNTepoxy composites [20]. Yang et al. developed a sequential multiscale bridging model to characterize the CNT size effects and weakened bonding effects at the interface of CNT-polyproplyene composites using MD and continuum micromechanics [21,22]. In many of the previous works, effects of preferential nanotube alignment are often modeled using expensive numerical (e.g., FEM) techniques. In this work, we introduce a multiscale modeling technique that allows efficient control over the volume fraction and alignment of SWNTs. SWNT alignment in a composite is modeled using a compact representation of the orientation space while accounting for symmetry. The nanocomposite structure is represented using an orientation distribution function (ODF) that captures the volume density of different SWNT orientations. The averaged material properties, such as the stiffness elements, thermal conductivity, and thermal expansion, are calculated using the volume averaging equations that are linear in terms of the ODF values. Several studies focusing on the improvement/optimization of SWNT chemical and physical properties such as position, shape, linkage, morphology, flame synthesis, and spectroscopic properties are addressed in the literature [23–26]. However, the present study introduces a fully automated multiscale design optimization framework for the first time to identify the optimum nanotube alignments of SWNTs in different macroscale engineering problems. We first develop an optimization methodology using linear programming (LP) to calculate the optimal values of the ODFs to enhance the mechanical and thermal properties of the SWNT.The objective of the LP problem is defined as the maximization of the stiffness parameter, C11. The design constraints are defined to ensure that the optimum design Presented as Paper 2018-1157 at the 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, 8–12 January 2018; received 6 October 2017; revision received 25 January 2018; accepted for publication 26 January 2018; published online 12 March 2018.Copyright© 2018by PinarAcar. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC atwww.copyright. com; employ the ISSN 0001-1452 (print) or 1533-385X (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa. org/randp. *Assistant Professor, Department of Mechanical Engineering. Member AIAA. Associate Professor, Department of Aerospace Engineering. Member AIAA. Department of Aerospace Engineering.