Question

A large plate of some unknown material is known to have an scratches along the edge having a depth of 0.5mm, and a radius of curvature of 5μm. What is the ratio of maximum stress to applied stress (σ_m /σ_o) in the regions adjacent to these defects?

Answer 1

Due to the scratch there is the case of Stress Concentration in the material.

What is Stress Concentration?

Stress concentration is basically the region where the stress value in the material is typically greater than the surrounding regions. Stress concentration occurs due to fillets, notches, scratches or grooves.

One important thing to note is, for ductile material the redistribution of stress occurs in the material and the material could continue to carry load. Where as for brittle materials, it typically fails in the region of stress concentration.

Assumption Taken

1. Now given in the question is a hole(scratch) with a depth of 0.5mm. One assumption is taken here that the hole is considered to be elliptical in shape. So the half-length(a) will be equal to the depth of the scratch 0.5mm.

2. The radius of curvature given is considered to be the lower radius of curvature present on the material. So the max stress will occur near this lowest radius of curvature.

Formula Used

E. Kirsch derived the following equation known as the Inglis' Equation. Using it to find the value of maximum stress to the nominal stress(stress without any scratches).

Where,

a is the half crack length

is the radius of curvature at the end of crack

is the stress value at the end of crack

is the stress value applied on the material

Substituting the values

So the answer is 21