TMM4175 Polymer Composites

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Interlaminar fracture mechanical testing¶

Interlaminar crack propagation can occur in mode I, II and III corresponding to the fracture energies $G_{Ic}$, $G_{IIc}$ and $G_{IIIc}$.

Double cantilever beam¶

Figure-1: DCB testing

The classical way to determine the fracture energy $G_{Ic}$ in mode I require crack-length monitoring during the fracture tests, which is generally difficul to perform.

An alternative is The Compliance Based Beam Method (CBBM) using beam theory, where the fracture energy is related to the change of compliance $C$ during fracture. This is expressed by the Irwin-Kies relation:

\begin{equation} G_{Ic} = \frac{F^2}{2b}\frac{dC}{da} \tag{1} \end{equation}

where the compliance is $C=\delta / F$

The compliance based on Timoshenko beam theory is:

\begin{equation} C = \frac{8 a^3}{E b h^3} + \frac{12a}{5 G b h} \tag{2} \end{equation}

Combining (1) and (2) to obtain:

\begin{equation} G_{Ic} = \frac{F^2}{b^2}\ \big( \frac{12 a^2}{E h^3} + \frac{6}{5 G h} \big) \tag{3} \end{equation}

Now, we solve (3) for $F$ and compute $\delta = CF$

Example:

b,h,E,G,GIc = 10, 4, 125000, 5000, 0.1

F, d = [0,], [0,]
import numpy as np
na = np.linspace(40,55,100)
tota = na[-1]-na[0] # total crack length

for a in na:
    Ft = ( (GIc*b**2)/(  ((12*a**2)/(E*h**3)) + (6/(5*G*h))  ) )**0.5
    dt = Ft*(   (8*a**3)/(E*b*h**3)  + (12*a)/(5*G*b*h)      ) 
    F.append(Ft)
    d.append(dt)
F.append(0)
d.append(0)
    
import matplotlib.pyplot as plt
%matplotlib inline
fig,ax = plt.subplots(nrows=1,ncols=1, figsize = (10,8))
ax.plot(d,F,'-',color='green')
ax.grid(True)
ax.set_xlim(0,)
ax.set_ylim(0,)
plt.show()

Verify that the area of the curves divided by the fracture area is equalt to $G_{Ic}$ using numerical integration:

energy=np.trapz(y=F,x=d)
print('GIc=',energy/(b*tota))

GIc= 0.10000033197578503

End-notch flexure¶

Intentionally left empty

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