def analytical_comparison():
'''for the case when tau is deterministic,
there is an analytical solution.
The differences are caused by the additional matrix
stiffness due to broken fibers, which are in the CB model
added to matrix stiffness. As the matrix E grows and the V_f
decreases, the solutions tend to get closer'''
tau, E_f, E_m, V_f = 0.2, 200e3, 25e3, 0.3
r, shape, scale = 0.00345, 5., 0.02
# analytical solution for damage controlled test
ctrl_damage = np.linspace(0.0, .99, 100)
def crackbridge(w, tau, E_f, E_m, V_f, r, omega):
Kf = E_f * V_f * (1 - omega)
Km = E_m * (1 - V_f) + E_f * V_f * omega
Kc = Kf + Km
T = 2. * tau * V_f * (1. - omega) / r
c = np.sqrt(Kc * T / Km / Kf)
return c * np.sqrt(w) * (1 - omega)
def w_omega(tau, E_f, E_m, V_f, r, omega, shape, scale):
Kf = E_f * V_f * (1 - omega)
Km = E_m * (1 - V_f) + E_f * V_f * omega
Kc = Kf + Km
T = 2. * tau * V_f * (1. - omega) / r
return (-np.log(1. - omega)) ** (2. / shape) \
* scale ** 2 * Km * Kf / Kc / T
w_lst = [w_omega(tau, E_f, E_m, V_f, r, omega, shape, scale)
for omega in ctrl_damage]
epsf = crackbridge(np.array(w_lst), tau, E_f, E_m, V_f, r, ctrl_damage)
plt.plot(np.array(w_lst), epsf * E_f, color='red',
lw=4, ls='dashed', label='analytical')
reinf = Reinforcement(r=r, tau=tau, V_f=V_f, E_f=E_f,
xi=RV('weibull_min', shape=shape, scale=scale),
n_int=20)
ccb = CompositeCrackBridge(E_m=E_m, reinforcement_lst=[reinf],
Ll=1000., Lr=1000.)
stress = []
w_arr = np.linspace(0.0, np.max(w_lst), 100)
for w in w_arr:
ccb.w = w
stress.append(ccb.max_norm_stress)
plt.plot(w_arr, stress, color='blue', lw=2, label='CB model')
plt.legend(loc='best')
def bundle_comparison(w_arr, L, shape, scale, E):
'''bundle (Weibull fibers) response for comparison with the CB model'''
from scipy.stats import weibull_min
eps = w_arr / L * (1. - weibull_min(shape, scale=scale).cdf(w_arr / L))
plt.plot(w_arr / L, eps * E, lw=4, color='red', ls='dashed', label='FB model')
bundle = Reinforcement(r=0.13, tau=0.00001, V_f=0.9999, E_f=E,
xi=RV('weibull_min', shape=shape, scale=scale),
n_int=50)
ccb = CompositeCrackBridge(E_m=25e3,
reinforcement_lst=[bundle],
Ll=L / 2.,
Lr=L / 2.)
eps = []
for w in w_arr:
ccb.w = w
eps.append(ccb.max_norm_stress / E)
plt.plot(w_arr / L, np.array(eps) * E, color='blue', lw=2, label='CB model')
plt.legend(loc='best')
