# Chapter 6: Strengthening Mechanisms¶

### Example 6.1, Grain Size Measurement, Page No. 193¶

In [1]:
from math import sqrt

#variable declaration
sigma_i=150;
k=0.7;
n=6;

#calculation
N_x=2**(n-1);
N=N_x/(0.01)**2;             #in grains/in^2
N=N*10**6/25.4**2;             # in grains/m^2
D=sqrt(1/N);
sigma0=sigma_i+k/sqrt(D);

#result
print ('\nYield Stress = %g MPa')%(sigma0);

Yield Stress = 254.464 MPa


### Example 6.2, Strengthing Mechanism, Page No. 219¶

In [2]:
#variable declaration
sigma0=600;
G=27.6;
G=G*10**9           #conversion to Pa
b=2.5*10**-8;
b=b*10**-2;            #conversion to m
T0=sigma0/2;
T0=T0*10**6;             #conversion to Pa

#calculation
lambda1=G*b/T0;
Cu_max=54;
Cu_eq=4;
Cu_min=0.5;
rho_al=2.7;
rho_theta=4.43;
wt_a=(Cu_max-Cu_eq)/(Cu_max-Cu_min);
wt_theta=(Cu_eq-Cu_min)/(Cu_max-Cu_min);
V_a=wt_a/rho_al;
V_theta=wt_theta/rho_theta;
f=V_theta/(V_a+V_theta);
r=(3*f*lambda1)/(4*(1-f));

#result
print('\nParticle Spacing = %g m\nParticle Size = %g m')%(lambda1,r);

Particle Spacing = 2.3e-08 m
Particle Size = 7.35948e-10 m


### Example 6.3, Fiber Strengthing, Page No. 222¶

In [3]:
#variable declaration
Ef=380;
Em=60;

#calculation
#Case 1
f_f1=0.1;
Ec1=Ef*f_f1+(1-f_f1)*Em;

#Case 2
f_f2=0.6;
Ec2=Ef*f_f2+(1-f_f2)*Em;

#result
print('\nEc for 10 vol%% = %g GPa\n')%(Ec1);
print('\nEc for 60 vol%% = %g GPa\n')%(Ec2);

Ec for 10 vol% = 92 GPa

Ec for 60 vol% = 252 GPa



### Example 6.4, Load Transfer, Page No. 225¶

In [4]:
#variable declaration
sigma_fu=5;
sigma_fu=sigma_fu*10**9;         #Conversion to Pa
sigma_m=100;
sigma_m=sigma_m*10**6;        #Conversion to Pa
T0=80;
T0=T0*10**6;                   #Conversion to Pa
f_f=0.5;
d=100;
d=d*10**-6;
B=0.5;
L1=10;
L1=L1*10**2;                  #conversion to m
Lc=sigma_fu*d/(2*T0);
sigma_cu1=sigma_fu*f_f*(1-Lc/(2*L1))+sigma_m*(1-f_f);
sigma_cu1=sigma_cu1*10**-9;
print('\nsigma_cu = %g GPa for L=100um\n')%(sigma_cu1);

L2=2;
L2=L2*10**-3;          #conversion to m
sigma_cu2=sigma_fu*f_f*(1-Lc/(2*L2))+sigma_m*(1-f_f);
sigma_cu2=sigma_cu2*10**-9;
print('sigma_cu = %g GPa for L=2mm')%(sigma_cu2);

sigma_cu = 2.55 GPa for L=100um

sigma_cu = 0.596875 GPa for L=2mm