Anisotropic Elasticity for HCP Crystal Structures

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The elastic behavior of a hexagonal close-packed (HCP) single crystal for a metal is anisotropic, meaning that the stiffness of the material depends on the direction of loading. This Demonstration illustrates the elastic anisotropy of 24 HCP metals when undergoing uniaxial stress, as shown in the 3D plot (left). This shows the variation of the elastic modulus (in GPa) as a function of direction. For reference, the axis of the plot is parallel to the axis of the HCP unit cell, and the axis is parallel to the direction of the HCP unit cell. The 2D plot (right) shows the elastic modulus on the plane. The anisotropy factor is determined by , where and are compliance constants of the selected material.

Contributed by: Matthew W. Priddy (March 2015)
Based on a program by: Megan Frary
Open content licensed under CC BY-NC-SA



The general relationship between stresses and strains in crystals can be represented using the fourth-rank tensors for stiffness and for compliance: and . Each involves 81 coefficients, but the number is drastically reduced for crystals of high symmetry, such as HCP.

The five independent elastic constants (, , , , and ) for the 24 HCP metals were taken from [1].


[1] D. Tromans, "Elastic Anisotropy of HCP Metal Crystals and Polycrystals," International Journal of Research and Reviews in Applied Science, 6(4), 2011 pp. 462–483. 4_ 14.pdf.

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