Respuesta :
Answer:
0.024 m = 24.07 mm
Explanation:
1) Notation
[tex]\sigma_c[/tex] = tensile stress = 200 Mpa
[tex]K[/tex] = plane strain fracture toughness= 55 Mpa[tex]\sqrt{m}[/tex]
[tex]\lambda[/tex]= length of a surface crack (Variable of interest)
2) Definition and Formulas
The Tensile strength is the ability of a material to withstand a pulling force. It is customarily measured in units (F/A), like the pressure. Is an important concept in engineering, especially in the fields of materials and structural engineering.
By definition we have the following formula for the tensile stress:
[tex]\sigma_c=\frac{K}{Y\sqrt{\pi\lambda}} [/tex] (1)
We are interested on the minimum length of a surface that will lead to a fracture, so we need to solve for [tex]\lambda[/tex]
Multiplying both sides of equation (1) by [tex]Y\sqrt{\pi\lambda}[/tex]
[tex]\sigma_c Y\sqrt{\pi\lambda}=K[/tex] (2)
Sequaring both sides of equation (2):
[tex](\sigma_c Y\sqrt{\pi\lambda})^2=(K)^2[/tex]
[tex]\sigma^2_c Y^2 \pi\lambda=K^2[/tex] (3)
Dividing both sides by [tex]\sigma^2_c Y^2 \pi[/tex] we got:
[tex]\lambda=\frac{1}{\pi}[\frac{K}{Y\sigma_c}]^2[/tex] (4)
Replacing the values into equation (4) we got:
[tex]\lambda=\frac{1}{\pi}[\frac{55 Mpa\sqrt{m}}{1.0(200Mpa)}]^2 =0.02407m[/tex]
3) Final solution
So the minimum length of a surface crack that will lead to fracture, would be 24.07 mm or more.
