From Elasticity with Mathematica: An Introduction to Continuum Mechanics and Linear Elasticity


The change of coordinates from cartesian (x, y, z) to cylindrical polar (r, ? preserves all of the properties introduced in the previous sections. In fact, any coordinate transformation within the plane perpendicular to the z axis can be performed with the help of the TensorAnalysis package, provided the resulting coordinate system remains orthogonal.

For cylindrical polar coordinates the result has the form

Using MATHEMATICA, the derivation is performed in a few lines.

<b class="bold"><< Tensor2Analysis.m</b><b class="bold">SetCoordinates[Cylindrical[r, t, z]]</b><b class="bold">B = {{0, 0, 0}, {0, 0, 0}, {0, 0, Psi[r, t]}}</b><b class="bold">(Stress1 = Inc [B]) // MatrixForm</b>

Airy stress function

B := {{0, 0, 0}, {0, 0, 0}, {0, 0, A0 [x, y]}}

Airy stress function form of the Beltrami-Maxwell tensor potential B

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Topics of Interest

5.6 BIHARMONIC FUNCTIONS The above analysis of the plane problem demonstrates the important role played by biharmonic functions in the solution of elastic plane problems. The general form of the...

5.3 AIRY STRESS FUNCTION WITH A CORRECTIVE TERM: A 0(x, y) ?z 2A 1(x, y) We begin again with the Beltrami potential tensor given in (5.3). Our aim is to establish a form of the function A(x, y, z)...

5.1 PLANE STRESS The previously introduced expression for the stress tensor in terms of the Beltrami potential is Because div inc B = 0, the stress tensor defined in this way automatically satisfies...

5.7 THE DISCLINATION, DISLOCATIONS, AND ASSOCIATED SOLUTIONS In this section we consider a particular solution for the Airy stress function that allows some important properties of displacement...

3.7 Plotting Implicitly Defined Functions An implicitly defined function is given as an equation relating two variables, such as x 2 + y 2 = 1 (which describes a circle of radius one). Here the y...