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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

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23 For two second rank tensor a and b, consider the sums a,J + 6y of their components in every coordinate system. Show that they are not components of tensor unless a = 0 or b = 0. 24 Find the contravariant components of the sum of the tensors a = eiei and b = e 2 e 2 where e* is the basis of the coordinate system x*, x1 = x[ + x'2, x2 — x'2, x3 = x'-, and x[ are Cartesian coordinates. 26 Consider the components of tensors B = Bijkie'eieke' and e = e^e'e^ in every coordinate system and prove that a) sums Bijklekl are the components of a tensor, b) the following equalities are valid &iaeu = 5' J k 'e*, = B ^ , e t ' = Biiklekl.

Show the same for a cube and a regular octahedron. Use the properties of symmetry. 14A Such fields are referred to as axially symmetrical. 14 ? Write the results in the cylindrical coordinate system. 28 Determine the general form of scalar, vector and second-rank tensor fields which are invariant relative to the complete group of rotations and reflections. Such fields are referred to as spherically symmetrical. Write the results in a spherical coordinate system. , valid for all media) physical "conservation laws" are con­ sidered in this chapter.

38 Prove that, if the strain rate tensor is identical for all particles of a medium at an instant, then the vorticity vector is also identical for all the particles at this instant. 39 The strain rate tensor equals zero in all particles of a medium. Show that, in this case, the velocity field is described by the Euler formula for the velocity distribution in a rigid body v = v0 + fi x r where r is the radius-vector relative to a point O, Vo{t) is the velocity of this point, and f2(t) is a vector independent of r (the vector of instantaneous angular velocity).

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Continuum Mechanics Via Problems and Exercises. Part I: Theory and Problems by Eglit M.E., Hodges D.H. (eds.)


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