On the best position of elastic symmetry planes in an orthotropic body

Seregin, Gregory; Troitskii, V.A.

doi:10.1016/0021-8928(81)90022-8

Cited by 16 publications

(7 citation statements)

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“…A different approach is taken in Section 4 below, where the strain energy will be minimized directly. While Result 1 is not new but has been derived by several authors [16,14,5,19,2,15], the present derivation is novel and explicit. In particular, it allows us to go further and obtain the condition necessary for a minimum or maximum.…”

Section: Condition For Stationary Strain Energymentioning

confidence: 78%

“…The general problem of determining optimal orientations in anisotropic elasticity has been the subject of several studies in the last two decades, beginning with the work of Seregin and Troitskii [16] in the context of orthotropic solids. They determined the important coaxiality condition: a minimum or maximum of strain energy requires that the stress and strain share common principal axes.…”

Section: Introductionmentioning

confidence: 99%

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Optimal orientation of anisotropic solids

Norris

2005

The Quarterly Journal of Mechanics and Applied Mathematics

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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.

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Section: Condition For Stationary Strain Energymentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Optimal orientation of anisotropic solids

Norris

2005

The Quarterly Journal of Mechanics and Applied Mathematics

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“…However, the optimality conditions can be rewritten in terms of strains as [7] FISle=DI823831, F28~=D2812831, Note that due to (5.7) there are only three independent equations in (5.8) for determination of Euler angles | @ 2, and | 3-It appears that the system includes 32 solutions in terms of Euler angles. However.…”

Section: Numerical Examplementioning

confidence: 98%

Nonlinear constitutive behavior of orthotropic materials

Lellep¹,

Majak²

2000

Mech Compos Mater

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“…Indeed, from the optimal design of anisotropic elastic solids (see e.g. [18,20,21]), it is well known that the symmetry axes characterizing a maximum (or a minimum) stiffness layout are aligned to the principal stress directions. Unlike the approach followed in [9], where this alignment was iteratively enforced within the solution procedure, here the orientation field hðz 1 ; z 2 Þ is added to the unknown variables, and the expected alignment of symmetry axes and principal stress directions spontaneously arises from the minimization procedure.…”

Section: Problem Formulationmentioning

confidence: 99%