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Research Papers

Optimization of Composite Fracture Properties: Method, Validation, and Applications

[+] Author and Article Information
Grace X. Gu

Laboratory for Atomistic and
Molecular Mechanics (LAMM),
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: gracegu@mit.edu

Leon Dimas

Laboratory for Atomistic and
Molecular Mechanics (LAMM),
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: leon_dim@mit.edu

Zhao Qin

Laboratory for Atomistic and
Molecular Mechanics (LAMM),
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: qinzhao@mit.edu

Markus J. Buehler

Laboratory for Atomistic and
Molecular Mechanics (LAMM),
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: mbuehler@mit.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 22, 2016; final manuscript received April 7, 2016; published online May 5, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(7), 071006 (May 05, 2016) (7 pages) Paper No: JAM-16-1149; doi: 10.1115/1.4033381 History: Received March 22, 2016; Revised April 07, 2016

A paradigm in nature is to architect composites with excellent material properties compared to its constituents, which themselves often have contrasting mechanical behavior. Most engineering materials sacrifice strength for toughness, whereas natural materials do not face this tradeoff. However, biology's designs, adapted for organism survival, may have features not needed for some engineering applications. Here, we postulate that mimicking nature's elegant use of multimaterial phases can lead to better optimization of engineered materials. We employ an optimization algorithm to explore and design composites using soft and stiff building blocks to study the underlying mechanisms of nature's tough materials. For different applications, optimization parameters may vary. Validation of the algorithm is carried out using a test suite of cases without cracks to optimize for stiffness and compliance individually. A test case with a crack is also performed to optimize for toughness. The validation shows excellent agreement between geometries obtained from the optimization algorithm and the brute force method. This study uses different objective functions to optimize toughness, stiffness and toughness, and compliance and toughness. The algorithm presented here can provide researchers a way to tune material properties for a vast number of engineering problems by adjusting the distribution of soft and stiff materials.

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Figures

Grahic Jump Location
Fig. 1

Model formulation. Targeted material property optimization is performed by algorithmic assignment of stiff and soft elements in a multiphase building block. The prescribed binary distribution of element stiffness defines stiffness and volume ratios for each initial geometry. The material contains an edge crack and undergoes tensile loading under mode I failure with displacement controlled boundary conditions (“dx”). a is the length of the sample (square), and b is the length of the crack.

Grahic Jump Location
Fig. 2

Algorithm organization. The algorithm takes in as an input a random initial population of soft and stiff elements. At every iteration, an objective function value of the system is calculated for every element when only it is switched. Elements switch from soft to stiff and vice versa. The switch that generates the highest objective value increase is kept for the next iteration. This process is repeated until there is no switch that generates a higher objective value compared to the previous iteration at which point the algorithm exits outputting the final geometry.

Grahic Jump Location
Fig. 3

Optimized solutions for maximizing effective compliance and effective stiffness for 8 × 8 and 12 × 12 grid size systems. (a) The maximum compliance solution is a geometry in which the soft (black) and stiff (gray) materials are in series with the loading conditions. From different initial random geometries, the algorithm leads to the optimized design of in-series materials. (b) The maximum stiffness solution is a geometry in which the soft and stiff materials are in parallel. Similarly, the algorithm leads to the optimized design starting from random initial geometries. These final geometries show that there is no single optimum solution for these problems, but many optimal solutions solving the same objective function.

Grahic Jump Location
Fig. 4

Larger grid size solutions for minimizing stiffness (maximizing compliance) and maximizing stiffness. Convergence of solutions: (a) For a 16 × 16 system, the optimal solution obtained from the algorithm remains the same. The graph shows that the initial effective stiffness starts out high and decreases as iteration increases. (b) The optimal solution remains the same for maximizing stiffness and the graph shows an increase in effective stiffness with iteration. These two graphs show the solutions converging to the theoretical optimal values of compliance and stiffness.

Grahic Jump Location
Fig. 5

Solutions from brute force method to validate algorithm when optimizing for toughness. The chart describes the differences between systems A and B. System A has a grid size of 8 × 8 and an edge crack that is 50% of the system length, with a being the length of the sample. The algorithm begins with a random geometry and generates the solution that exactly matches the solution obtained from brute force (i.e., checking every possible solution and selecting the best). The algorithm, however, is orders of magnitude faster than the brute force method. System B has a grid size of 10 × 10 and an edge crack that is 20% of the system length. The algorithm also obtains the same solution as the brute force method. Tbrute is the toughness obtained from the brute force method and Talg is the toughness obtained from our algorithm, and the ratio shows unity. This performance confirms that our algorithm is effective and robust.

Grahic Jump Location
Fig. 6

Various case studies. (a) The geometry for the objective function that optimizes toughness modulus only. (b) The geometry for the objective function that optimizes toughness along with stiffness. More elements need to be in parallel to maintain high stiffness and also toughness. (c) The geometry for the objective function that optimizes toughness along with compliance. More elements are spread out to be in series with each other. Variables: f is the objective function, T is the toughness modulus, T0 is the initial geometry toughness modulus, Eeff is the effective stiffness of the system, and E0 is the effective stiffness of the initial population.

Grahic Jump Location
Fig. 7

Strain field for homogenous design compared to optimized design for 40 × 40 grid size. White space represents the stiff material and black space represents the soft material. Delocalization of strain seen in optimized design from crack tip area to soft material locations.

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