For the Kelvin state as considered in Example 15.4, explicitly justify the displacement and stress results given in relations (15.2.8) and (15.2.10).

Data from example 15.4

Equation 15.2.8

Equation 15.2.10

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Consider next the combined Kelvin problem with unit loads aa (a = 1, 2, 3) acting along each of three coordinate directions, as shown in Fig. 15.6. This singular state is denoted by S"(x) and is given by the potentials p = 0, y = -C- R 1 8 (1-v) The displacements and stresses corresponding to this case become +(3-4v) ai] = C Xaxi 2R R UR = TR$ = C(1-2v) of= C [3x a Xixj R R As a special case of this state, consider aa= [0, 0, 1], which would be state S*(x) with potentials - Jijxa) - -+ (1-2v)(daixj + dajxi dijxa, sino where C = 0==4=0, = (15.2.9) and in spherical coordinates (R0,0) (see Fig. 1.6) yields the following displacements and stresses 2C(1-v) cosp C(3 - 4v) sind R 2 R OR=-2C(2-v)- 9 coso R = up R TRO= Te = 0 de=0=C(12v)- (15.2.7) Ug = 0 coso R (15.2.8) (15.2.10)