1 Derive an expression for the J integral for an axisymmetri

Derive an expression for the J integral for an axisymmetrically notched bar in

tension shown below, where the notch depth is sufficient to confine plastic

deformation to the ligament.

GIVEN:-

For three zones 1,2 & 3 shown in the sketch in the problem, the three different diameters are D, d and again D.

SOLUTION:-

For Zone-1, Stress intensity factor K1 = 2P / (D)^1.5

For Zone-2 K2 = 4P / (d)^1.5

For Zone-3 K3 = 2P / (D)^1.5

So by Formula J-integral = K^2/E which gives the total J-integral Jtotal = J1 + J2 + J3

That is Jtotal = K1^2/E + K2^2/E + K3^2/E

Therefore, Jtotal = [2P / (D)^1.5]^2/E + [4P / (d)^1.5]^2/E + [2P / (D)^1.5]^2/E (ANSWER)