Axisymmetric problem

S. A. Ambartsumyan, The Axisymmetric Problem of a Circular Cylindrical Shell Made of Material with Different Stiffness in Tension etnd Compression, Izvesttya Akademii Nauk SSSR Mekhanika, No. 4, 1965, pp. 77-85 English translation N69-11070, STAR. 

T. W. Shield and D. M. Bogy, Some Axisymmetric Problems for Layered Elastic Media Part I — Multiple Region Contact Solutions for Simply-Connected Indenters, Thansac-tions of the ASME, vol. 56, pp. 798-806, Dec 1989. 

Using the spreadsheet representation of the CVD flow field (Fig. 2.23), evaluate the strain-rate tensor at the point z = 0.045789474 m and r = 0.026666667 m. Although this is a two-dimensional axisymmetric problem, be aware that there are 0 components in the tensor. 

For a two-dimensional axisymmetric problem, all other vorticity components vanish exactly. For the stagnation-flow problem, it has been established that du/dr = 0 thus cog = dv/dz =rdV/dz. It is apparent that 2 = cog/r is a function of z alone. Thus, like the radial velocity, the scaled vorticity also exhibits a radially independent boundary layer. 

J. M. Charrier, S. Shrivastava, and R. Wu, Free and Constrained Inflation of Elastic Membranes in Relation to Thermoforming - non-axisymmetric Problems, J. Strain Anal., 24, 55-74 (1989). 

The use of cylindrical coordinates is particularly suitable in the solution of axisymmetrical problems. It is worth noting that for a non-simply connected cross section, as occurs in the case of a hollow cylinder, the compatibility equations are not sufficient to guarantee single-valued displacements. In this situation, the displacements themselves must be considered. 

The residual thermal stresses were calculated by a finite element computer programme Super-Sap. The model is 10mm in diameter, 6mm in thickness and has 11 graded layers. Due to the axisymmetric problem, an axisymmetric finite element modle is used and 1/4 part of die material is considered. The FE model includes 1200 elements and 1275 nodal points. The thermal load is raised from the sintering temperature 1300 °C to room temperature. Moreover, an insulated heat condition is considered at the flank boundary. 

Chapter 7 Creeping Flows (Two-Dimensional and Axisymmetric Problems)10... 

Note that a common form for the second term in each of these equations for the 2D and axisymmetric problems is... 

Thus, for this class of axisymmetric problems, where Tr = 0,... 

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