Optimal orientation of anisotropic solids

TL;DR

This paper derives conditions for the optimal orientation of anisotropic elastic solids to minimize strain energy, providing explicit solutions for cubic, TI, and tetragonal materials, and introduces the strain deviation angle concept.

Contribution

It presents a new derivation of the coaxiality condition, explicit optimal orientation formulas for specific symmetries, and introduces the strain deviation angle for assessing non-optimal states.

Findings

Optimal orientation depends on elastic constants and symmetry.

A stationary strain energy state is a minimum or maximum under specific conditions.

The strain deviation angle measures deviation from optimal coaxiality.

Abstract

Results are presented for finding the optimal orientation of an anisotropic elastic material. The problem is formulated as minimizing the strain energy subject to rotation of the material axes, under a state of uniform stress. It is shown that a stationary value of the strain energy requires the stress and strain tensors to have a common set of principal axes. The new derivation of this well known coaxiality condition uses the 6-dimensional expression of the rotation tensor for the elastic moduli. Using this representation it is shown that the stationary condition is a minimum or a maximum if an explicit set of conditions is satisfied. Specific results are given for materials of cubic, transversely isotropic (TI) and tetragonal symmetries. In each case the existence of a minimum or maximum depends on the sign of a single elastic constant. The stationary (minimum or maximum) value of energy can always be achieved for cubic materials. Typically, the optimal orientation of a solid with cubic material symmetry is not aligned with the symmetry directions. Expressions are given for the optimal orientation of TI and tetragonal materials, and are in agreement with results of Rovati and Taliercio \cite{Rovati03} obtained by a different procedure. A new concept is introduced, the strain deviation angle, which defines the degree to which a state of stress or strain is not optimal. The strain deviation angle is zero for coaxial stress and strain. An approximate formula is given for the strain deviation angle which is valid for materials that are weakly anisotropic.