# Chapter 8 : Mechanical Properties of Materials and Mechanical Tests¶

## Example 8.1 Page No : 269¶

In [1]:
import math

# Variables
Y = 180*10**9;			#Young's modulus of a certain material(in N/m**2)
E = 1.8;			#true surface energy (in J/m**2)
c = (5./2)*10**-6;			#Crack (in meter)

# Calculation
F_strength = math.sqrt((2*Y*E/(math.pi*c)))
p = 1000*math.pi*c/(2*Y) - 1.8

# Results
print 'fracture strength = %.2f MN/m**2'%(F_strength*10**-6)
print "plastic work required to propogate the crack : %.1f "%p

# book answer is wrong

fracture strength = 287.24 MN/m**2
plastic work required to propogate the crack : -1.8


## Example 8.2 Page No : 270¶

In [9]:
import math

# Variables
d_o = 12.7;			#tensile test specimen diameter (in mm)
d = 12;			#tensile test specimen diameter after load (in mm)
P = 76*10**3;			#load(in N)
pi = 22./7;
A_o = (pi/4)*(d_o**2);			#Initial area of cross section(in mm**2)
A = (pi/4)*(d**2);			#area of cross section after load of 76 kN

# Calculation
E_stress = P/A_o;			#engineering stress
T_stress = P/A;			#true stress
T_strain = math.log(A_o/A);			#true strain
E_strain = math.exp(T_strain)-1;			#engineering strain

# Results
print 'engineering stress in = %.f N/mm**2'%E_stress
print 'true stress in = %.2f N/mm**2'%T_stress
print 'engineering strain = %.2f'%E_strain
print 'true strain = %.2f'%T_strain

# rounding off error

engineering stress in = 600 N/mm**2
true stress in = 671.72 N/mm**2
engineering strain = 0.12
true strain = 0.11


## Example 8.3 Page No : 271¶

In [4]:

import math

# Variables
Y = 210.*10**9;			#Young's modulus of a certain material(in N/m**2)
E = 10.;			#true surface energy (in J/m**2)
c = (100./2)*10**-6;			#Crack (in meter)
pi = 3.14;

# Calculation
F_strength = (2*Y*E/(pi*c))**(1/2.);

# Results
print 'fracture strength in %.1e Newton/m**2'%F_strength

fracture strength in 1.6e+08 Newton/m**2


## Example 8.4 Page No : 271¶

In [6]:

import math

# Variables
l_o = 305.*10**-3;			#length of copper piece(in meter)
E = 110.*10**9;			#surface energy
stress = 276.*10**6;			#in Pa

# Calculation
dl = stress*l_o/E;			#resultant elongation(in meter)

# Results
print 'resultant elongation in = %.2f mm'%(dl*10**3)

resultant elongation in = 0.77 mm


## Example 8.5 Page No : 271¶

In [7]:

import math

# Variables
T_stress = 415.;			#True stress (in Megapascal)
T_strain = 0.10;			#True strain
K = 1035.;			#(in Megapascal)

# Calculation
n = (math.log(T_stress)-math.log(K))/math.log(T_strain);			#

# Results
print 'Strain hardening exponent for an alloy = %.2f'%n

Strain hardening exponent for an alloy = 0.40