This screen capture is taken from the Chegg site. The question was previously an

Question

This screen capture is taken from the Chegg site. The question was previously answered. For the question "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." Why are there 3 zones? I thought for the J integral we needed a closed loop solution. How were the K factors and the final Jtotal values derived?

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)

I need to see how the stress intensity factors were calculated.

1. 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. P

Explanation / Answer

Solution:

As per the diagram to find total J-integral we need to divide it into 3 zones as per given and solved. It is easy and best way to find the solution. The value will also be very closest to the way you want to solve. And the closed loop solution will be more complicated than this one. K value is actually not derived. When calculating the principal stress the values were combined and named as K and the value was useful to find the stress intensity so named as stress intensity factor. Stress intensity factor has the direct formula for various modes.

Hope it helped and for J-integral you should refer some articles which will explain it very well.

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