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Research Papers

Design of Nonperiodic Microarchitectured Materials That Achieve Graded Thermal Expansions

[+] Author and Article Information
Jonathan B. Hopkins

Mechanical and Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095
e-mail: hopkins@seas.ucla.edu

Lucas A. Shaw

Mechanical and Aerospace Engineering,
University of California, Los Angeles,
Los Angeles, CA 90095
e-mail: lukeshaw@ucla.edu

Todd H. Weisgraber

Materials Engineering Division,
Lawrence Livermore National Laboratory,
Livermore, CA 94550
e-mail: weisgraber2@llnl.gov

George R. Farquar

Materials Engineering Division,
Lawrence Livermore National Laboratory,
Livermore, CA 94550
e-mail: gfarquar@dnatrek.com

Chris D. Harvey

Materials Engineering Division,
Lawrence Livermore National Laboratory,
Livermore, CA 94550
e-mail: harvey1@llnl.gov

Christopher M. Spadaccini

Materials Engineering Division,
Lawrence Livermore National Laboratory,
Livermore, CA 94550
e-mail: spadaccini2@llnl.gov

1Corresponding author.

Manuscript received September 14, 2015; final manuscript received December 8, 2015; published online May 4, 2016. Assoc. Editor: Venkat Krovi.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Mechanisms Robotics 8(5), 051010 (May 04, 2016) (11 pages) Paper No: JMR-15-1265; doi: 10.1115/1.4032248 History: Received September 14, 2015; Revised December 08, 2015

The aim of this paper is to introduce an approach for optimally organizing a variety of nonrepeating compliant-mechanism-like unit cells within a large deformable lattice such that the bulk behavior of the lattice exhibits a desired graded change in thermal expansion while achieving a desired uniform stiffness over its geometry. Such lattices with nonrepeating unit cells, called nonperiodic microarchitectured materials, could be sandwiched between two materials with different thermal expansion coefficients to accommodate their different expansions and/or contractions induced by changing ambient temperatures. This capability would reduce system-level failures within robots, mechanisms, electronic modules, or other layered coatings or structures made of different materials with mismatched thermal expansion coefficients. The closed-form analytical equations are provided, which are necessary to rapidly calculate the bulk thermal expansion coefficient and Young's modulus of general multimaterial lattices that consist first of repeating unit cells of the same design (i.e., periodic microarchitectured materials). Then, these equations are utilized in an iterative way to generate different rows of repeating unit cells of the same design that are layered together to achieve nonperiodic microarchitectured material lattices such that their top and bottom rows achieve the same desired thermal expansion coefficients as the two materials between which the lattice is sandwiched. A matlab tool is used to generate images of the undeformed and deformed lattices to verify their behavior and an example is provided as a case study. The theory provided is also verified and validated using finite-element analysis (FEA) and experimentation.

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References

Valdevit, L. , Jacobsen, J. A. , Greer, J. R. , and Carter, W. B. , 2011, “ Protocols for the Optimal Design of Multi-Functional Cellular Structures: From Hypersonics to Micro-Architectured Materials,” J. Am. Ceram. Soc., 94, pp. s15–s34. [CrossRef]
Hopkins, J. B. , Lange, K. J. , and Spadaccini, C. M. , 2013, “ Designing Microstructural Architectures With Thermally Actuated Properties Using Freedom, Actuation, and Constraint Topologies,” ASME J. Mech. Des., 135(6), p. 061004. [CrossRef]
Ashby, M. F. , 2005, Materials Selection in Mechanical Design, 3rd ed., Butterworth-Heinemann, Burlington, VT.
Ye, H. , Lin, M. , and Basaran, C. , 2002, “ Failure Modes and FEM Analysis of Power Electronic Packaging,” Finite Elem. Anal. Des., 38(7), pp. 601–612. [CrossRef]
Miller, R. A. , and Lowell, C. E. , 1982, “ Failure Mechanisms of Thermal Barrier Coatings Exposed to Elevated Temperatures,” Thin Solid Films, 95(3), pp. 265–273. [CrossRef]
Cribb, J. L. , 1963, “ Shrinkage and Thermal Expansion of a Two Phase Material,” Nature, 220, pp. 576–577. [CrossRef]
Lakes, R. S. , 1996, “ Dense Solid Microstructures With Unbounded Thermal Expansion,” J. Mech. Behav., 7(2), pp. 85–92.
Lakes, R. , 2007, “ Cellular Solids With Tunable Positive or Negative Thermal Expansion of Unbounded Magnitude,” Appl. Phys. Lett., 90(22), p. 221905. [CrossRef]
Shapery, R. A. , 1968, “ Thermal Expansion Coefficients of Composite Materials Based on Energy Principles,” J. Compos. Mater., 2(3), pp. 380–404. [CrossRef]
Rosen, B. W. , and Hashin, Z. , 1970, “ Effective Thermal Expansion Coefficients and Specific Heats of Composite Materials,” Int. J. Eng. Sci., 8(2), pp. 157–173. [CrossRef]
Gibiansky, L. V. , and Torquato, S. , 1997, “ Thermal Expansion of Isotropic Multiphase Composites and Polycrystals,” J. Mech. Phys. Solids, 45(7), pp. 1223–1252. [CrossRef]
Sigmund, O. , and Torquato, S. , 1996, “ Composites With Extremal Thermal Expansion Coefficients,” Appl. Phys. Lett., 69(21), pp. 3203–3205. [CrossRef]
Sigmund, O. , and Torquato, S. , 1997, “ Design of Materials With Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method,” J. Mech. Phys. Solids, 45(60), pp. 1037–1067. [CrossRef]
Sigmund, O. , and Torquato, S. , 1999, “ Design of Smart Composite Materials Using Topology Optimization,” Smart Mater. Struct., 8(3), pp. 365–379. [CrossRef]
Chen, B. C. , Silva, E. C. N. , and Kikuchi, N. , 2001, “ Advances in Computational Design and Optimization With Application to MEMS,” Int. J. Numer. Methods Eng., 52(1–2), pp. 23–62. [CrossRef]
Lakes, R. S. , 1996, “ Cellular Solid Structures With Unbounded Thermal Expansion,” J. Mater. Sci. Lett., 15(6), pp. 475–477.
Lehman, J. J. , and Lakes, R. S. , 2014, “ Stiff, Strong, Zero Thermal Expansion Lattices Via Material Hierarchy,” Compos. Struct., 107, pp. 654–663. [CrossRef]
Jefferson, G. , Parthasarathy, T. A. , and Kerans, R. J. , 2009, “ Tailorable Thermal Expansion Hybrid Structures,” Int. J. Solids Struct., 46(11–12), pp. 2372–2387. [CrossRef]
Steeves, C. A. , Lucato, S. L. , He, M. , Antinucci, E. , Hutchinson, J. W. , and Evans, A. G. , 2007, “ Concepts for Structurally Robust Materials That Combine Low Thermal Expansion With High Stiffness,” J. Mech. Phys. Solids, 55(9), pp. 1803–1822. [CrossRef]
Silva, M. J. , Hayes, W. C. , and Gibson, L. J. , 1995, “ The Effects of Non-Periodic Microstructure on the Elastic Properties of Two-Dimensional Cellular Solids,” Int. J. Mech. Sci., 37(11), pp. 1161–1177. [CrossRef]
Silva, M. J. , and Gibson, L. J. , 1997, “ The Effects of Non-Periodic Microstructure and Defects on the Compressive Strength of Two-Dimensional Cellular Solids,” Int. J. Mech. Sci., 39(5), pp. 549–563. [CrossRef]
Liu, K. , Khandelwal, K. , and Tovar, A. , 2013, “ Multiscale Topology Optimization of Structures and Non-Periodic Cellular Materials,” 10th World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, May 19–24.
Ajdari, A. , 2008, “ Mechanical Behavior of Cellular Structures: A Finite Element Study,” Master's thesis, Northeastern University, Boston, MA.
Ravichandran, K. S. , 1995, “ Thermal Residual Stresses in a Functionally Graded Material System,” Mater. Sci. Eng., A201(1–2), pp. 269–276. [CrossRef]
Nemat-Alla, M. , 2003, “ Reduction of Thermal Stresses by Developing Two-Dimensional Functionally Graded Materials,” Int. J. Solids Struct., 40(26), pp. 7339–7356. [CrossRef]
Cho, J. R. , and Oden, J. T. , 2000, “ Functionally Graded Material: A Parametric Study on Thermal-Stress Characteristics Using the Crank–Nicolson–Galerkin Scheme,” Comput. Methods Appl. Mech. Eng., 188, pp. 17–38. [CrossRef]
Cowin, S. C. , and Mehrabadi, M. M. , 1995, “ Anisotropic Symmetries of Linear Elasticity,” ASME Appl. Mech. Rev., 48(5), pp. 247–285. [CrossRef]
Ball, R. S. , 1900, A Treatise on the Theory of Screws, The University Press, Cambridge, UK.
Phillips, J. , 1984, Freedom in Machinery: Volume 1, Introducing Screw Theory, Cambridge University Press, New York.
Roark, R. J. , Young, W. C. , and Budynas, R. G. , 2002, Roark's Formulas for Stress & Strain, 7th ed., McGraw Hill Companies, New York.

Figures

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Fig. 1

(a) A tunable thermal expansion unit cell, (b) a lattice with periodic unit cells, and (c) a graded lattice with nonperiodic unit cells

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Fig. 2

Thermal expansion versus Young's modulus Ashby chart

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Fig. 3

(a) and (b) Nonperiodic lattices can accommodate materials with different thermal expansions. (c) A unit cell with different sector geometries.

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Fig. 4

Numbered elements and bodies within the cell (a) and other lattice parameters (b)

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Fig. 5

Positive (a) and negative (b) thermal expansion versus Young's modulus

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Fig. 6

Plot for the first graded thermal expansion design attempt (a) showing the initial nonperiodic lattice design before (b) and after heating (c)

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Fig. 7

Plot for the final graded thermal expansion design whose Actual Lattice values correctly begin and end with the desired expansion values (a), final lattice design (b), and deformed lattice (c)

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Fig. 8

Fabricated cell dimensions (a), thermal expansion (b), and Young's modulus (c) validation using experimental data and verification using FEA

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Fig. 9

2 × 2 nonperiodic lattice design generated (a), predicted deformations after heating the lattice by a 100 °C temperature increase with an exaggeration factor of 10 (b), and FEA results using the same temperature increase and exaggeration factor (c)

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