Optimal orientation of anisotropic solids

Results are presented for finding the optimal orientation ofan anisotropic elastic material. The problem is formulated asminimizing the strain energy subject to rotation of the materialaxes, under a state of uniform stress. It is shown that a stationaryvalue of the strain energy requires the stress and strain tensorsto have a common set of principal axes. The new derivation ofthis well-known coaxiality condition uses the six-dimensionalexpression of the rotation tensor for the elastic moduli. Usingthis representation it is shown that the stationary conditionis a minimum or a maximum if an explicit set of conditions issatisfied. Specific results are given for materials of cubic,transversely isotropic (TI) and tetragonal symmetries. In eachcase the existence of a minimum or maximum depends on the signof 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 materialsymmetry is not aligned with the symmetry directions. Expressionsare given for the optimal orientation of TI and tetragonal materials,and are in agreement with results of Rovati and Taliercio obtainedby a different procedure. A new concept is introduced: the straindeviation angle, which defines the degree to which a state ofstress or strain is not optimal. The strain deviation angleis zero for coaxial stress and strain. An approximate formulais given for the strain deviation angle which is valid for materialsthat are weakly anisotropic.