By B. M. Fraeijs de Veubeke (auth.)

This ebook relies on lecture notes of the past due Professor de Veubeke. the topic is gifted at a degree appropriate for graduate scholars in engineering, physics, or arithmetic. a few publicity to linear algebra, advanced research, variational calculus, or easy continuum mechanics will be useful. the 1st 3rd of the publication comprises the basics of the idea of elasticity. Kinematics of continuing media, the notions of pressure and equilibrium, conservation of power, 'and the elastic constitutive legislation are every one taken care of first in a nonlinear context, then really expert to the linear case. the rest of the ebook is given to 3 vintage functions of the idea, each one supplemented by way of unique re sults in accordance with using complicated variables. each of the 3 themes - Saint-Venant's conception of prismatic beams, airplane deformations, and the bending of plates - is first pre sented and analyzed regularly, then rounded out with quite a few particular and infrequently novel examples. the subsequent notational conventions are commonly in strength, other than the place famous on the contrary: decrease case boldface letters denote vectors or triples of Cartesian co ordinates, higher case boldface letters denote three x three matrices, repeated reduce case Latin subscripts are summed over (1,2,3), and non-repeated reduce case Latin subscripts are assumed to diversity over (1,2,3).

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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?

If the field w is considered in a simply connected domain, this is the sufficient condition for existence of a single-valued potential. Similarly to the relationship v = grad tp, the six components of a small strain tensor are expressed in terms of the three components of the displacement w, and, consequently, cannot be arbitrary. They satisfy the relationships d2ekl dxidxj | 6% dxkdxi d2eu dx^Oxj ^ _Q dxidxi (xi is a spatial Cartesian coordinate system); these relationships also are called the compatibility conditions for strain components.

17) in terms of bases e^ of the cylindrical and e'i of the Cartesian coordinate systems, b) A body rotates around an axis with the angular speed oj{t). Regarding the rotation axis as the coordinate line x3 of a cylindrical coor dinate system, find the physical components of the velocity and acceleration vectors of the body points in this system. 18a) is not the local basis of any coordinate system yk (x1, x2, x3). Tensor transformation rule. 20 a) Show that the elements of the matrix ||6y|| of a bilinear form are the covariant components of a tensor of second rank.