Question: Consider a stress function formulation for the axisymmetric problem discussed in Section 14.2. The appropriate compatibility relation for this case has been previously developed in

Consider a stress function formulation for the axisymmetric problem discussed in Section 14.2. The appropriate compatibility relation for this case has been previously developed in Example 8.11; see (8.4.76). Using the plane stress Hooke’s law, express this compatibility relation in terms of stress and then in terms of the Airy function to get the result:

0= d dr2 2011) (2-[(-(-3) (4]

Data from section 14.2

We start our study by re-examining Example 8.6, a plane axisymmetric problem of a hollow cylin- drical domain

Dimensionless Modulus of Elasticity, E/E, 2 1.8- 1.6 1.4- 1.2 0.8 0.6 0.4 0.2 8 0 0.1 0.2 0.3 K=-0.5 K = 0

Dimensionless Axial Displacement, uE/T 3.5 2.5 1.5- 0.5- 0 0.1 K = 0 (Homogeneous Case) 0.2 0.3 0.4 0.5

Pi b a Po

Or = de E(r) [du 1- (r) dr E(r) 1-v(r) +v(r = du [+(r)]) dr Note that the corresponding plane strain

where E, and n are constants and a is the inner boundary radius. Note that Eo has the same units as E and as

Dimensionless Young's Modulus, E/E 15 10 n = 0 (Homogeneous Case) 1.5 2 n=2 2.5 3.5 Dimensionless Distance,

This particular gradation model reduces the Navier equation (14.2.2) to du (n+1) du dr + dr r The solution to

As with the homogeneous example, we choose the special case with only internal pressure (po = 0), which gives

which matches with the solution shown in Fig. 8.9 for bla = 2. A plot of the nondimensional stress

Dimensionless Stress, , p 0.1 of -0.1- -0.2 -0.3 -0.4 -0.5- -0.6- -0.7- -0.8 -0.9 n = 0 (Homogeneous Case)

Dimensionless Stress, P 0.8 0.6 0.4 0,2 0 n = 0 (Homogeneous Case) n = 1/2 1.5 n=1 L 2 n=2 2.5 3 Distance,

corresponding results for the tangential stress shown in Fig. 14.8 show much more marked differences. WhileData from example 8.11

As a final example in this section, consider the problem of a thin uniform circular disk subject to constant

y a (0) X

The solution can be efficiently handled by using a special stress function that automatically satisfies the

Recall that the more general polar coordinate case was given as Exercise 7.17. Using Hooke's law for plane

Equation 8.4.76

d dr (rea) er = 0 -

0= dr2 d 102 1) (2 -[(-(-3) (4]

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