By A. Cemal Eringen
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Additional info for Continuum Mechanics of Single-Substance Bodies
2) arises from the motion of the material points of <€. Lemma 2. 20). 5) = [(dq/dt) + cuii(q x v) + v div q] • da From this there follows: Zorowski's criterion. 7) • da This may be seen by imagining that s(t) + c(t) consists of some fictitious material points having velocity v.
Thus the condition is necessary. It remains to show that the condition is also sufficient, namely, if / = 0, then the surface consists of the same material points. 11) (n) dt = dx'/v 1 = dx /v 2 2 = dx /v 3 3 These are the differential equations for the paths of material points. 12) x = x\X , t) k K v/here X are three arbitrary integration constants that identify the material points at an initial time / = 0. 7) is therefore given by K f=ils(X\X ,X ) 2 3 where \\i is an arbitrary function. This shows that whenever f = 0, a material point o n / = 0 remains on this surface throughout the motion.
Hence the ratios of the diameters of the ellipsoid at P to the corresponding diameters of the sphere are stretches. Thus the magnitudes of stretches in different directions at P vary as does the distance from p to the surface of the ellipsoid. The directions of the diameters of the ellipsoid with respect to their undeformed directions give the rotation. In an ellipsoid the lengths of the principal axes may be ordered as the largest, the middle-sized, and the smallest. Hence we have the following: Theorem 2 (Cauchy's Second Theorem).