Chapter12-Fracture

Ex1-pg302

In [12]:
## Estimate fracture strength
import math
c = 2. ## crack of half length in micro meter
Y = 70. ## young’s modulus in GN m**-2
Gamma = 1. ## specific surface energy 
print("\n Example 12.1")
sigma_f = math.sqrt(2.*Gamma*Y*1e9/(math.pi*c*1e-6))/1e6
r = Y*1e3/sigma_f
print'%s %.2f %s %.2f %s  '%("\n Fracture strength of",math.ceil(sigma_f)," MN m^-2 is"   " ",(math.ceil(r/100.)*100.),"th of young’s modulus. ")
print("\n Thus Griffiths criterion bridges the gap between the \n observed and ideal strengths of brittle material")

 Example 12.1

 Fracture strength of 150.00  MN m^-2 is  500.00 th of young’s modulus.   

 Thus Griffiths criterion bridges the gap between the 
 observed and ideal strengths of brittle material

Ex2-pg304

In [13]:
## Estimate the brittle fracture strength at low temperatures
import math
Gamma = 1.5## specific surface energy in J/m**2
Y = 200 ## Young’s modulus in GN/m**2
c = 2. ## half length of crack

print("\n Example 12.2")
sigma_f = math.sqrt(2.*Gamma*Y*(1*10**9)/(math.pi*c*1e-6))

print'%s %.2f %s '%("\n Brittle fracture strength at low temperatures is ",sigma_f/1e6," MNm^-2 ")## answer in book is 310MNm**-2

 Example 12.2

 Brittle fracture strength at low temperatures is  309.02  MNm^-2

Ex3-pg306

In [17]:
## Estimate the temperature at which the ductility of brittle transition occurs at given strain rates
import math

Gamma = 2.## specific surface energy in J/m**2
Y = 350. ## Young’s modulus in GN/m**2
c = 2. ## half length of crack
de_dt1 = 1e-2 ## strain rate
de_dt2 = 1e-5 ## strain rate
print("\n Example 12.3")
print("\n Part A:")
sigma_f = math.sqrt(2.*Gamma*Y*1e9/(math.pi*c*1e-6))
sigma_y = sigma_f/1e6
T = 173600./(sigma_y-20.6-61.3*math.log10(de_dt1))## temperature calculation

print'%s %.2e %s %.2f %s '%("\n Transition temperature for strain rate",de_dt1," s^-1 is ",T," K")## answer in book is 300 K
print("\n\n Part B:")

T = 173600./(sigma_y-20.6-61.3*math.log10(de_dt2))## temperature calculation

print'%s %.2e %s %.2f %s  '%("\n Transition temperature for strain rate ",de_dt2," s^-1 is ",T," K")## answer in book is 230 K
## Solution in book for two parts is divided into three parts

 Example 12.3

 Part A:

 Transition temperature for strain rate 1.00e-02  s^-1 is  302.42  K 


 Part B:

 Transition temperature for strain rate  1.00e-05  s^-1 is  229.04  K