Chapter 7, Problem 7.43P

Interpretation Introduction

Interpretation:

The integration of $da/dn=C{\left(\Delta K\right)}^n$ gives the equation $N=\left[\frac{2\left[{\left(a_c\right)}^{\frac{\left(2-n\right)}{2}}-{\left(a_i\right)}^{\frac{\left(2-n\right)}{2}}\right]}{\left(2-n\right)C{f}^n\Delta {\sigma }^n{\pi }^{\frac{n}{2}}}\right]$ should be calculated.

Concept introduction:

Crack growth rate is defined as the rate at which propagation of crack is considered before failure with respect to stress intensity factor $\left(\Delta K\right)$.

The variation of crack growth rate with stress intensity factor is shown in the graph below:

The relationship for power law model is given by,

  $da/dn=C{\left(\Delta K\right)}^n$

Where,

C and n are empirical constants depending upon the material.

  $\left(\Delta K\right)$ is stress intensity factor.

Explanation of Solution

The given equation for basis is:

  $da/dn=C{\left(\Delta K\right)}^n$ ..................... (1)

The value of stress intensity factor has a dependency on the size of the material.

The value of growth rate increases in unstable manner till fracture occurred,

  $\Delta K=K_{\max }-K_{\min }$ ....................... (2)

Defining the relation based on geometrical factor and stress, the value of maximum and minimum stress intensity factor is,

  $\begin{array}{l}{K}_{\max }=f{\sigma }_{\max }\sqrt{\pi a}\\ K_{\min }=f{\sigma }_{\min }\sqrt{\pi a}\end{array}$

Putting the values in the formula of stress intensity factor,

  $\Delta K=\left(f{\sigma }_{\max }\sqrt{\pi a}\right)-\left(f{\sigma }_{\min }\sqrt{\pi a}\right)$ ..................... (3)

The difference between the maximum and minimum stress,

  ${\sigma }_{\max }-{\sigma }_{\min }=\Delta \sigma$

Replacing minimum and maximum stress with $\Delta \sigma$, and substituting the values in the formula below:

  $\Delta K=\left(f{\sigma }_{\max }\sqrt{\pi a}\right)-\left(f{\sigma }_{\min }\sqrt{\pi a}\right)$

  $\Delta K=f\Delta \sigma \sqrt{\pi a}$

Considering the value of stress intensity factor in terms of constant variable n,

  ${\left(\Delta K\right)}^n=f^n\Delta {\sigma }^n{\left(\sqrt{\pi a}\right)}^n$

On rearranging equation (1),

  $\begin{array}{l}\frac{da}{dN}=C{\left(\Delta K\right)}^n\\ dN=\frac{da}{C{\left(\Delta K\right)}^n}\end{array}$

Putting the value of $\left(\Delta K\right)$ in equation (1),

  $\begin{array}{l}dN=\frac{da}{C{\left(\Delta K\right)}^n}\\ {\left(\Delta K\right)}^n=f^n\Delta {\sigma }^n{\left(\sqrt{\pi a}\right)}^n\\ dN=\frac{da}{C^n{f}^n\Delta {\sigma }^n{\left(\sqrt{\pi a}\right)}^n}\end{array}$

Integrating the above equation with respect to initial and final flaw size. By Applying limits,

  ${\displaystyle \underset{0}{\overset{N}{\int }}dN}={\displaystyle \underset{a_i}{\overset{a_c}{\int }}\frac{da}{C^n{f}^n\Delta {\sigma }^n{\left(\sqrt{\pi a}\right)}^n}}$

Thus, the required value of equation is $N=\left[\frac{2\left[{\left(a_c\right)}^{\frac{\left(2-n\right)}{2}}-{\left(a_i\right)}^{\frac{\left(2-n\right)}{2}}\right]}{\left(2-n\right)C{f}^n\Delta {\sigma }^n{\pi }^{\frac{n}{2}}}\right]$.