How to fill graph in empty window with matplotlib

You get an empty plot due to an error in your 'plt.xlabel' line. The special sign $ is causing the problem. If you read it as a raw string, it should be fixed:

import matplotlib.pyplot as plt
import numpy as np

alpha =np.deg2rad(np.linspace(0,180,1000))
sigma_1=190 #kPa
sigma_3=145 #kPa

sigma_delta=(sigma_1+sigma_3)/2 + ( (sigma_1-sigma_3)/2 ) * np.cos(2*alpha) #kPa

def sigma_alpha(s1,s3,a): 
    return (  (s1+s3)/2 + ( (s1-s3)/2 ) * np.cos(2*alpha)  )

def tau_alpha(s1,s3,a):
    return (( (s1-s3)/2 )*np.sin(2*alpha) )

plt.figure(figsize=(18,10))
plt.plot(alpha, sigma_alpha(sigma_1, sigma_3, alpha), label=r'$sigma_{$alpha$}$', linewidth=3, color='green') 
plt.plot(alpha, tau_alpha(sigma_1, sigma_3, alpha), label = r'$tau_{$alpha$}$', linewidth=3, color='red') 
plt.xlabel(r'Angle $\alpha$ [\circ]$')
plt.ylabel('Spenning[kPa]')
plt.title('Spenningsvariasjon for plan $alpha$ = 0-180')
plt.legend(loc=5, framealpha=1, prop={'size':18})
plt.grid()
plt.minorticks_on()

plt.grid(b=None, which='both', axis='x', linewidth=0.5, linestyle='-')
plt.grid(b=None, which='both', axis='y', linewidth=0.5, linestyle='-')

plt.show()

Note that I've changed N_DEGREES to \circ to obtain the angle character in your label.