Computational Mechanics by J.T. Oden

By J.T. Oden

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We present below the output of the corresponding program prog1-4. nb. 13: The trajectories of particles of a continuum. It is easy to see that the symbolic computation results by the above Notebook progl-4 . nb coincide with the results presented above by us, which we have obtained by hand. 0 a rapid visualization of the results. We show in Fig. 13 the trajectories of 27 particles of a continuum, which we have chosen in our Notebook in a sufficiently arbitrary way. 5. The displacement field is given in the Cartesian basis _ ei = ei = Ei in the form Determine the Lagrangian €ij and Eulerian Cij strain tensors.

1) because one can always determine some coordinate transform xilt=o = f(~j) such that xi = xi(f(~j), t) = Xi(~j,t) (see Fig. 10)] . To determine the invariant objects (vector, tensor) , one must define a coordinate system, which is specified with the aid of the basis vectors ~. For example, by specifying an orthonormal basis of a Cartesian coordinate system - - ei . 6) In this case, the coordinate system is frozen in the continuum and it deforms together with it and is generally curvilinear. 6) is local and determines the radius vector of particles located in a small neighborhood of a given particle whose position is taken as a coordinate origin ~i = O.

Same notations as in the above solution, which we have found by hand. It is easy to see that our calculations coincide with the output of the Mathematica program. 2. Find the components of a m etric tensor in the spherical coordinate system. Write the expression for the element of the length dB. Solution: Introduce the spherical coordinate system ~i shown in Fig. 1. 7. As has been shown in the foregoing problem, the metric tensor gij in an arbitrary coordinate system is determined by the formula gij = ax k ax k a~j a~j , where Xi = Xi(~j) determines the relation of arbitrary coordinates ~i to the Cartesian coordinates Xi.

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