Optimal orientation of anisotropic solids

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 [15] 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.

A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions

“…The eigentensors are diads of the form ω ⊗ ω, where ω is a symmetric tensor of order 2. The theorems on this spectral representation are provided in the articles by Rychlewski (1984), Blinowski et al (1996), and Sutcliffe (1992), Norris (2005Norris ( , 2006, Moakher (2008), Mehrabadi and Cowin (1990), Jemioło and Telega (1997). This representation makes it possible to find a new interpretation of the results by Bendsøe et al (1994), cf.…”

Euler-Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors

“…Thus, the last line in (20) vanishes for n 1 3, the last two for n 1 2, and all but the first line for n 1 1, while the terms that remain can be simplified using (21). The formulae for P 2n , n 1 2 and 3, are These identities apply for n 2 and n 3, respectively, since they reduce to, for example, the n 1 1 formula using (21).…”

“…C i jkl C i jkl 1 tr C t C. The six-dimensional version of the fourth-order tensor Q i jkl is Q 6 SO364, introduced by Mehrabadi et al [8], see also [20]. It may be defined in the same manner as Q 1 S[Q]S where [Q] is the matrix of Voigt elements.…”

Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

“…11], Yang et al [25], Chapman [6, pp. 92-93] and Norris [16]. We consider an orthonormal basis of {e 1 , e 2 , e 3 } in R 3 .…”

“…If it does, we want to find a basis with respect to which the symmetry group has form (15) and the tensor can be represented by matrix (16). To achieve these results, we formulate the following theorem.…”