Using the general stress relations (15.4.25) for the stress concentration problem of Example 15.13, show that the

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Using the general stress relations (15.4.25) for the stress concentration problem of Example 15.13, show that the circumferential stress on the boundary of the hole is given by:

oo(a,0) = T (1 2 cos 20 1 + F

Verify that this expression gives a maximum at θ =±π/2, and explicitly show that this value will reduce to the classical case of 3T by choosing l₁ = l₂ = l = 0.

Equation 15.4.25

Example 15.13

The constitutive equations in polar coordinates read as or = 2(er+ee) + (2n + ker ge=2(er+ee) + (2n + kee =

For the polar coordinate case, the stress-stress function relations become 1 ap 182 18 1 ay + 00 r arde r30

where a 2 (w 1vw) = 2(1 v)h/ - r 1 a - a (Y 1w) = 2(1 v)1 (10) - r 4= - D 2 (Vp) r 30 Y( + k) k(2 + k)'

The appropriate solutions to equations (15.4.22) for the problem under study are given by T r(1 cos 20)+A log

T 0,- (1 + cos20) +- (+413 624) Cos 28 or = 08 +245 [314 A6(0/4) + (1+65) K (1/41)] 20020 -Ko(r/l)+1+ cos20 A

T Trd = - (+642 +24 64+) sin26 Ter A5 + 4 [+ Kor/h) + (1 + 125) Ki(r/h)]sin20 hr mez = - (7 + 642 +24-64+)

For boundary conditions we use the usual forms for the nonpolar variables, while the couple stress m, is

Using these conditions, sufficient relations can be developed to determine the arbitrary constants A, giving

F = 8(1-v): a 2a Ko (a/l) 4+ + R 11 K (a/11) 44 This then completes the solution to the problem. Let us now

Notice that for micropolar theory, the stress concentration depends on the material parameters and on the

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Question Posted: Jan 15, 2024 05:05 AM

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