By Eglit M.E., Hodges D.H. (eds.)
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Extra resources for Continuum Mechanics Via Problems and Exercises. Part I: Theory and Problems
31 4. Deformation. Deformation Rate. Vorticity. The rate of relative change in volume of a particle is equal to the first invariant of the strain rate tensor which in turn is equal to divergence of the velocity vector dv, en = ^— = divw . OXi The distribution of velocity in a small volume of a continuum is expressed in terms of the strain rate tensor and the vorticity vector w 1 . 1 dvk = - c u r l « = - e 0 * — et where e^* are the components of Levi-Civita tensor. Namely, if Vo is the velocity of the particle situated at a point r 0 , the velocity of the particle situated at a point r0 + p is expressed by the Cauchy-Helmholtz formula v = v0 + — ei + uxp dpi + O(p') where 3> = (l/2)eyPiP J .
As in any coordinate system, the components e^ are expressed in terms of the displacement field by the formula i^ = - (ViWj + V > , - Vi«>*VjU>fc) . The components of the Almansi tensor in the basis e* coincide with the components of the Green tensor in the basis e a , ia0 =£ap- Conditions of compatibility In some important cases, the velocity field v of a flow in the continuum is determined by one function
34 The covariant components of a velocity field in a spatial cylindrical coor dinate system i 1 = r, x2 = tp, x3 = z have the form V\ = 0 , v2 = k , v3 = 0 (k = const) everywhere except the point r = 0. a) Draw the particle paths of the medium, find the value of the velocity of a particle and the physical components of the velocity. b) Calculate the components of the strain rate tensor, c) Find the vorticity vector. d) Find the principal axes of the strain rate tensor. Do they rotate in time in an individual particle?