12: Mechanical Behaviour of Materials

Example number 12.1, Page number 12.115

In [40]:
#importing modules
import math
from __future__ import division

#Variable declaration
sigma0=8.55;
K=2.45;      
sigma=10**-3;      #steel size(mm)

#Calculation
sigma=sigma0+(K/math.sqrt(sigma));      #yield strength

#Result
print "yield strength is",round(sigma,3),"kg/mm**2"

yield strength is 86.026 kg/mm**2

Example number 12.2, Page number 12.115

In [41]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=70*10**9;     #young's modulus(Pa)
gama=1;     #surface energy(joule/m**2)
C=1*10**-6;    #depth(m)

#Calculation
sigma_f=math.sqrt(2*E*gama/(math.pi*C));      #fracture strength(GPa)

#Result
print "fracture strength is",round(sigma_f/10**9,3),"GPa"

fracture strength is 0.211 GPa

Example number 12.3, Page number 12.116

In [42]:
#importing modules
import math
from __future__ import division

#Variable declaration
ml=57800;     #load(N)
d=10*10**-3;    #diameter(m)
D=5;      #diameter after fracture(mm)
l=50;    #guage length(mm)
L=55;    #length after fracture(mm)

#Calculation
ts=ml/(math.pi*(d/2)**2);    #ultimate tensile strength(MPa)
de=(L-l)*100/l;          #ductility % of elongation(%)
dr=((2*D)**2-D**2)*100/(2*D)**2;      #ductility % of reduction(%)
t=(2/3)*ts*de/100;        #modulus of toughness(Pa)

#Result
print "ultimate tensile strength is",round(ts/10**6),"MPa"
print "ductility % of elongation is",de,"%"
print "ductility % of reduction is",dr,"%"
print "modulus of toughness is",int(t/10**6),"*10**6 Pa"

ultimate tensile strength is 736.0 MPa
ductility % of elongation is 10.0 %
ductility % of reduction is 75.0 %
modulus of toughness is 49 *10**6 Pa

Example number 12.4, Page number 12.116

In [43]:
#importing modules
import math
from __future__ import division

#Variable declaration
pl1=206850*10**3;     #proportional limit(Pa)
pl2=310275*10**3;     #proportional limit(Pa)
pl3=413700*10**3;     #proportional limit(Pa)
s2=0.0615;      #strain
s3=0.2020;      #strain
Y=2.0685*10**11;    #young's modulus(Pa)

#Calculation
e1=pl1/Y;     #elastic strain in 1st case
e2=pl2/Y;     #elastic strain in 2nd case
p2=s2-e2;     #plastic strain in 2nd case
r2=e2*100/p2;   #ratio of elastic and plastic strain in 2nd case
e3=pl3/Y;     #elastic strain in 2nd case   
p3=s3-e3;     #plastic strain in 2nd case 
r3=e3*100/p3;   #ratio of elastic and plastic strain in 3rd case

#Result
print "elastic strain in 1st case is",e1
print "ratio of elastic and plastic strain in 2nd case is",r2,"%"
print "ratio of elastic and plastic strain in 3rd case is",r3,"%"

elastic strain in 1st case is 0.001
ratio of elastic and plastic strain in 2nd case is 2.5 %
ratio of elastic and plastic strain in 3rd case is 1.0 %

Example number 12.5, Page number 12.117

In [44]:
#importing modules
import math
from __future__ import division

#Variable declaration
s=12411*10**3;      #stress(Pa)
t=0.0168;     #tension
e=0.127;     #elongation(cm)
l=15.24;     #length(cm)
g=9.8;
L=68.04;     #load(kg)

#Calculation
E_eff=s/t;     #effective modulus(Pa)
S=e/l;      
W=E_eff*S;
A=L*g/W;       #cross sectional area(m**2)

#Result
print "effective modulus is",E_eff/10**3,"*10**3 Pa"
print "cross sectional area is",round(A*10**4,4),"*10**-4 m**2"

effective modulus is 738750.0 *10**3 Pa
cross sectional area is 1.0831 *10**-4 m**2

Example number 12.6, Page number 12.117

In [45]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=35*10**10;     #youngs modulus(Pa)
gama=2;     #specific surface energy(J/m**2)
C=2*10**-6;    #length(m)
x=17700;    
y=2.1;
z=31.25;

#Calculation
sigma_f=math.sqrt(2*E*gama/(math.pi*C));     #fracture stress(Pa)
T=x/((sigma_f/(9.8*10**6))-y+z);    #transition temperature(K)

#Result
print "transition temperature is",round(T),"K"

transition temperature is 229.0 K

Example number 12.7, Page number 12.118

In [46]:
#importing modules
import math
from __future__ import division

#Variable declaration
h1=1;
h2=1;
k1=1;
k2=1;
l1=1;
l2=1;
l3=0;
s=3.5*10**6;      #stress(Pa)

#Calculation
x=math.sqrt(h1**2+k1**2+l1**2);
y=math.sqrt(h2**2+k2**2+l2**2);
z=math.sqrt(h2**2+k2**2+l3**2);
cos_phi=((h1*h2)-(k1*k2)+(l1*l2))/(x*y);
sin_phi=math.sqrt(1-(cos_phi)**2);
cos_theta=((h1*h2)+(k1*k2)+(l1*l3))/(x*z);
ss=s*cos_theta*cos_phi*sin_phi;        #critical resolved shear stress(Pa)

#Result
print "critical resolved shear stress is",round(ss/10**6,3),"MPa"

critical resolved shear stress is 0.898 MPa

Example number 12.8, Page number 12.119

In [47]:
#importing modules
import math
from __future__ import division

#Variable declaration
dz1=4*10**-18;    #diffusivity(m**2/s)
dz2=5*10**-13;    #diffusivity(m**2/s)
T1=773;    #temperature(K)
T2=1273;   #temperature(K)
T3=573;   #temperature(K)
T4=973;   #temperature(K)

#Calculation
x1=round(math.log(dz1),2);
y1=round(math.log(dz2),3);
x2=round(-1/(8.314*T1),7);
y2=round(-1/(8.314*T2),7);
x=round((x1-y1),3);
y=round((x2-y2),6);
Q=x/y;      #activation energy(J/mol)
z=round(y1-(y2*Q),4);
D0=math.exp(z);          #diffusion coefficient(m**2/Vs)
D1=D0*math.exp(-Q/(8.314*T3));    #diffusivity at 300 C(m**2/s)
D2=D0*math.exp(-Q/(8.314*T4));    #diffusivity at 700 C(m**2/s)

#Result
print "activation energy is",round(Q/10**3,3),"kJ/mol"
print "answer varies due to rounding off errors"
print "diffusion coefficient is",round(D0*10**4,3),"*10**-4 m**2/s"
print "diffusivity at 300 C is",round(D1*10**23,2),"*10**-23 m**2/s"
print "diffusivity at 700 C is",round(D2*10**15,3),"*10**-15 m**2/s"
print "answer given in the book is wrong"

activation energy is 192.393 kJ/mol
answer varies due to rounding off errors
diffusion coefficient is 0.394 *10**-4 m**2/s
diffusivity at 300 C is 11.37 *10**-23 m**2/s
diffusivity at 700 C is 1.846 *10**-15 m**2/s
answer given in the book is wrong

Example number 12.9, Page number 12.119

In [49]:
#importing modules
import math
from __future__ import division

#Variable declaration
D0=0.73*10**-4;     #diffusion coefficient(m**2/s)
Q=170*10**3;     #activation energy(J/mol)
R=8.314; 
T=873;    #temperature(K)

#Calculation
D=D0*math.exp(-Q/(R*T));     #diffusion(m**2/s)

#Result
print "diffusion is",round(D*10**15,1),"*10**-15 m**2/s"

diffusion is 4.9 *10**-15 m**2/s