0
Technical Brief

On the Sufficient Symmetry Conditions for Isotropy of Elastic Moduli

[+] Author and Article Information
C. Ayas

Structural Optimization and Mechanics,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands
e-mail: c.ayas@tudelft.nl

C. Tekog̃lu

Department of Mechanical Engineering,
TOBB University of Economics and Technology,
Sög̃ütözü,
Ankara 06560, Turkey

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 7, 2018; final manuscript received April 6, 2018; published online May 8, 2018. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 85(7), 074502 (May 08, 2018) (5 pages) Paper No: JAM-18-1079; doi: 10.1115/1.4039952 History: Received February 07, 2018; Revised April 06, 2018

The structural symmetry of a material can be manifested at a multitude of length scales such as spatial arrangement of atoms in a crystal structure, preferred orientation of grains in a polycrystalline material, alignment of reinforcing particles/fibers in composites or the micro-architecture of members in cellular solids. This paper proofs, in a simple yet rigorous manner, that six axes of fivefold structural symmetry is necessary and sufficient for isotropy of the elastic moduli tensor in the three-dimensional (3D) context.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Christensen, R. M. , 1987, “Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor,” ASME J. Appl. Mech., 54(4), pp. 772–777. [CrossRef]
Christensen, R. M. , and Waals, F. M. , 1972, “Effective Stiffness of Randomly Oriented Fibre Composites,” J. Compos. Mater., 6(3), pp. 518–535. [CrossRef]
Christensen, R. M. , 1986, “Mechanics of Low Density Materials,” J. Mech. Phys. Solids, 34(6), pp. 563–578. [CrossRef]
Gibson, L. J. , and Ashby, M. F. , 1997, Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, Cambridge, UK. [CrossRef]
Ashby, M. F. , Evans, A. G. , Fleck, N. A. , Gibson, L. J. , Hutchinson, J. W. , and Wadley, H. N. G. , 2000, Metal Foams: A Design Guide, Butterworth-Heinemann, Oxford, UK.
Deshpande, V. S. , Ashby, M. F. , and Fleck, N. A. , 2001, “Foam Topology: Bending Versus Stretching Dominated Architectures,” Acta Mater., 49(6), pp. 1035–1040. [CrossRef]
Dresselhaus, M. S. , and Dresselhaus, G. , 1991, “Note on Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor,” J. Mater. Res., 6(05), pp. 1114–1118. [CrossRef]
Kearsley, E. A. , and Fong, J. T. , 1975, “Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks Up to Eight,” J. Res. Natl. Bureau Standards B, 79B(1), pp. 49–58. https://nvlpubs.nist.gov/nistpubs/jres/79b/jresv79bn1-2p49_a1b.pdf
Malvern, L. E. , 1969, Introduction to the Mechanics of a Continuous Medium (Prentice Hall Series in Engineering of the Physical Sciences), Prentice Hall, Upper Saddle River, NJ.
Pronk, T. N. , Ayas, C. , and Tekglu, C. , 2017, “A Quest for 2D Lattice Materials for Actuation,” J. Mech. Phys. Solids, 105, pp. 199–216. [CrossRef]
Gurtner, G. , and Durand, M. , 2014, “Stiffest Elastic Networks,” Proc. R. Soc. London A: Math., Phys. Eng. Sci., 470(2164), p. 20130751.
Tancogne-Dejean, T. , and Mohr, D. , 2018, “Elastically-Isotropic Truss Lattice Materials of Reduced Plastic Anisotropy,” Int. J. Solids Struct., 138, pp. 24–39. [CrossRef]
Berger, J. B. , Wadley, H. N. G. , and McMeeking, R. M. , 2017, “Mechanical Metamaterials at the Theoretical Limit of Isotropic Elastic Stiffness,” Nature, 543(7646), pp. 533–537. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 1

A hexagonal cupola micro-architecture depicted in 3D. Note that, as the out-f-plane (x3 direction) thickness is much smaller than the in-plane dimensions; such a material can be modeled assuming plane stress conditions, and therefore, such lattice materials are referred to as 2D lattices; see also Ref. [10].

Grahic Jump Location
Fig. 2

Two-dimensional lattice materials. Solid (black) lines represent the struts while dashed lines represent (possible) unit cells for the lattices. Struts that coincide with an edge of a unit cell are removed from the figure to improve visibility: (a) rectangular, (b) square, (c) fully triangular, and (d) hexagonal lattice.

Grahic Jump Location
Fig. 3

Schematic illustration of a regular (pentagonal) dodecahedron with six independent surface normals each of which is an axis of fivefold symmetry

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In