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Research Papers

Optimization of Carbon Black Polymer Composite Microstructure for Rupture Resistance

[+] Author and Article Information
Bingbing San

Associate Professor
College of Civil and Transportation Engineering,
Hohai University,
Nanjing 210098, China;
Department of Civil Engineering and
Engineering Mechanics,
Columbia University,
New York, NY 10027
e-mail: bs2975@columbia.edu

Haim Waisman

Associate Professor
Department of Civil Engineering and
Engineering Mechanics,
Columbia University,
New York, NY 10027
e-mail: waisman@civil.columbia.edu

Manuscript received September 20, 2016; final manuscript received October 22, 2016; published online November 17, 2016. Assoc. Editor: Harold S. Park.

J. Appl. Mech 84(2), 021005 (Nov 17, 2016) (13 pages) Paper No: JAM-16-1464; doi: 10.1115/1.4035050 History: Received September 20, 2016; Revised October 22, 2016

Optimization of material microstructure is strongly tied with the performance of composite materials at the macroscale and can be used to control desired macroscopic properties. In this paper, we study the optimal location of carbon black (CB) particle inclusions in a natural rubber (NR) matrix with the objective to maximize the rupture resistance of such polymer composites. Hyperelasticity is used to model the rubber matrix and stiff inclusions, and the phase field method is used to model the fracture accounting for large deformation kinematics. A genetic algorithm is employed to solve the inverse problem in which three parameters are proposed as optimization objective, including maximum peak force, maximum deformation at failure-point, and maximum fracture energy at failure-point. Two kinds of optimization variables, continuous and discrete variables, are adopted to describe the location of particles, and several numerical examples are carried out to provide insight into the optimal locations for different objectives.

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Figures

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Fig. 1

Schematic representation of the phase field method applied to a rubber composite. The hatched circular region indicates reinforcing carbon black particles. Phase field models treat cracks as continuous entities where the extent of the damage to the material is characterized by the phase field parameter c. The parameter is 1 in the fully fractured phase and 0 in the undamaged phase. The width of the diffusive crack is determined by the parameter l0.

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Fig. 2

The GA method for optimal locations of particles

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Fig. 3

(a) Model and (b) mesh example for case 1

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Fig. 4

Mesh sensitivity study

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Fig. 5

Parametric studies for case 1: (a) force–angle curves, (b) stretch–angle curves, and (c) fracture energy–angle curves

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Fig. 6

(a) Force–displacement and (b) fracture energy–displacement curves in case 1

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Fig. 7

Snapshots showing the crack propagation in pure rubber matrix: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.64 mm, λ = 1.320 (peak force state); (c) Δu = 0.86 mm, λ = 1.430 (failure state); and (d) Δu = 0.92 mm, λ = 1.460 (after failure). For illustration purposes, elements with a phase field value of c > 0.8 have been removed.

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Fig. 8

Snapshots showing the crack propagation in the case of rubber with one particle at θ = 45 deg: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.64 mm, λ = 1.320 (peak force state); (c) Δu = 0.86 mm, λ = 1.430 (failure state); and (d) Δu = 0.88 mm, λ = 1.440 (after failure)

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Fig. 9

Force–displacement curves in the case of particle radius = 0.310 mm

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Fig. 10

(a) Model and (b) mesh example for case 2

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Fig. 11

Evolution of GA for case 2—(a) objective: maximum peak force, (b) objective: maximum stretch at failure-point, and (c) objective: maximum fracture energy at failure-point

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Fig. 12

(a) Force–displacement and (b) fracture energy–displacement curves in case 2

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Fig. 13

Snapshots showing the crack propagation of Loc.1 in case 2: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.64 mm, λ = 1.320 (peak force state); (c) Δu = 0.87 mm, λ = 1.433 (failure state); and (d) Δu = 0.88 mm, λ = 1.440 (after failure)

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Fig. 14

Snapshots showing the crack propagation of Loc.2 in case 2: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.64 mm, λ = 1.320 (peak force state); (c) Δu = 0.91 mm, λ = 1.457 (failure state); and (d) Δu = 0.93 mm, λ = 1.465 (after failure)

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Fig. 15

(a) Model and (b) mesh example for case 3

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Fig. 16

Evolution of GA for case 3—(a) objective: maximum peak force and (b) objective: maximum fracture energy at failure-point

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Fig. 17

Optimal locations in case 3

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Fig. 18

(a) Force–displacement and (b) fracture energy–displacement curves in case 3

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Fig. 19

Snapshots showing the crack propagation of Loc.1 in case 3: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.62 mm, λ = 1.310 (peak force state); (c) Δu = 0.87 mm, λ = 1.435 (failure state); and (d) Δu = 0.88 mm, λ = 1.440 (after failure)

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Fig. 20

Snapshots showing the crack propagation of Loc.2 in case 3: (a) Δu = 0.20 mm, λ = 1.100; (b) Δu = 0.62 mm, λ = 1.310 (peak force state); (c) Δu = 0.90 mm, λ = 1.450 (failure state); and (d) Δu = 0.95 mm, λ = 1.475 (after failure)

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