A New Proof That the Number of Linear Elastic Symmetries in Two Dimensions Is Four

Journal Article · Tue Jun 14 00:00:00 EDT 2022 · Journal of Elasticity

DOI:https://doi.org/10.1007/s10659-022-09902-7· OSTI ID:1872862

Trageser, Jeremy ^[1]; Seleson, Pablo ^[2]

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1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

In this work, we present an elementary and self-contained proof that there are exactly four symmetry classes of the elasticity tensor in two dimensions: oblique, rectangular, square, and isotropic. In two dimensions, orthogonal transformations are either reflections or rotations. The proof is based on identification of constraints imposed by reflections and rotations on the elasticity tensor, and it simply employs elementary tools from trigonometry, making the proof accessible to a broad audience. For completeness, we identify the sets of transformations (rotations and reflections) for each symmetry class and report the corresponding equations of motions in classical linear elasticity.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Sandia National Laboratories (SNL-CA), Livermore, CA (United States); Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC05-00OR22725; NA0003525
• OSTI ID: 1872862
• Alternate ID(s): OSTI ID: 1877114
• Report Number(s): SAND2022-7981J
• Journal Information: Journal of Elasticity, Journal Name: Journal of Elasticity Vol. 150; ISSN 0374-3535
• Publisher: Springer
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
anisotropy
elasticity tensor
linear elasticity
symmetry transformations
two dimensions