By Command. Duchêne
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Extra info for Flight Without Formulae - Simple Discussions on the Mechanics of the Aeroplane, (2nd Ed.)
3. 5) componentwise (cf. 21)): 3 lT~q = 3 LLWpilTijWqj. 7) in the study of the stressed state of a medium at a point. 2. As is known from linear algebra, any symmetric matrix can be represented in the form (cf. 8)) o (UU* - 1) where U is an orthogonal matrix. 1) with an orthogonal matrix W, we see that the stress tensor IIlTijll takes the form 0] 0 oS2 o U*W*. 8) 0 • The passage from the stress tensor IIlTij II to the diagonal stress tensor (cf. 8)) by the above procedure is called the reduction of the stress tensor to the principal axes.
We know that the stress tensor at any point of an isotropic medium is determined by the Murnaghan formulas (cf. 1) where p = kl~~k3' if the stressed state of the medium in a neighborhood of a fixed point can be obtained from the standard state by adiabatic deformation (with constant entropy S) described by a linear transformation c= Ilc;;1I = U [~. 2) = det V = +1. 1), the decomposition of C contains the matrix V. 1). It turns out that for such a characteristic we can take the so-called metric deformation tensor .
E = E(kl' k2, k3, 5). We emphasize that for parameters describing the stressed state we take the extension coefficients kl' k2, k3, and the rotation matrix U but not the matrix V because, the standard state of an isotropic medium with the entropy 5 remains unchanged under any rotation V Indeed, a stressed stated is associated with two matrices which differ only by the right orthogonal factor V(l) or V(2), and are the same in an isotropic medium. That is why for parameters describing the stressed state we take only the orthogonal matrix U and the extension coefficients kl' k2, k3.
Flight Without Formulae - Simple Discussions on the Mechanics of the Aeroplane, (2nd Ed.) by Command. Duchêne