TMM4175 Polymer Composites
Hashin criterion¶
The 3D Hashin Criterion (1980) divides the mechanisms of failure of UD fiber composite materials intwo two groups: Fiber Failure (FF) and Inter Fiber Failure (IFF). Both of these are divided into tensile mode and compressive mode:
Fiber failure (FF) when $\sigma_1 > 0:$
\begin{equation} \Big( \frac{\sigma_1}{X_T} \Big)^2 + \frac{1}{S_{12}^2}\Big( \tau_{12}^2 + \tau_{13}^2 \Big) = 1 \tag{1} \end{equation}Fiber failure (FF) when $\sigma_1 \le 0:$
\begin{equation} -\frac{\sigma_1}{X_C} = 1 \tag{2} \end{equation}Inter fiber failure (IFF) when $(\sigma_2 + \sigma_3) \ge 0:$
\begin{equation} \frac{1}{Y_T^2}(\sigma_2+\sigma_3)^2 + \frac{1}{S_{23}^2}(\tau_{23}^2-\sigma_2 \sigma_3 ) + \frac{1}{S_{12}^2}(\tau_{12}^2+\tau_{13}^2 \ ) = 1 \tag{3} \end{equation}Inter fiber failure (IFF) when $(\sigma_2 + \sigma_3) < 0:$
\begin{equation} \frac{1}{4S_{23}^2} (\sigma_2+\sigma_3)^2 + \frac{1}{S_{23}^2}(\tau_{23}^2-\sigma_2 \sigma_3 ) + \frac{1}{S_{12}^2}(\tau_{12}^2+\tau_{13}^2 \ ) + \frac{1}{Y_C}\Big( \frac{Y_C^2}{4S_{23}^2} - 1 \Big)(\sigma_2+\sigma_3) = 1 \tag{4} \end{equation}Implementation
def fE_hashin(s,m):
s1,s2,s3,s23,s13,s12=s[0],s[1],s[2],s[3],s[4],s[5]
XT,YT,ZT,XC,YC,ZC,S12,S13,S23 = m['XT'], m['YT'], m['ZT'], m['XC'], m['YC'], m['ZC'], m['S12'], m['S13'], m['S23']
if s1>0:
R = ( 1/( (s1/XT)**2 + (1/S12**2)*(s12**2 + s13**2) ) )**0.5
fE_FF=1/R
if s1<=0:
fE_FF=-s1/XC
if (s2+s3)>=0:
temp=( (1/YT**2)*(s2+s3)**2+(1/S23**2)*(s23**2-s2*s3)+(1/S12**2)*(s12**2+s13**2) )
if temp==0:
fE_IFF = 0
else:
R = (1/temp)**0.5
fE_IFF = 1/R
if (s2+s3)<0:
b = (1/YC)*((YC/(2*S23))**2-1)*(s2+s3)
a = (1/(4*S23**2))*(s2+s3)**2+(1/S23**2)*(s23**2-s2*s3)+(1/S12**2)*(s12**2+s13**2)
if a==0:
fE_IFF = 0.0
else:
c=-1
R=(-b+(b**2-4*a*c)**0.5)/(2*a)
fE_IFF = 1/R
return fE_FF, fE_IFF
Example
import matlib
m1=matlib.get('Carbon/Epoxy(a)')
s = (1200, -20, 10, 30, 0, 0)
ff,iff = fE_hashin(s,m1)
print('Exposure factors for FF and IFF:')
print(ff,'and', iff)
Failure envelopes, $\sigma_2$ versus $\sigma_1$
Fiber failure (FF) is reduced to a simple maximum stress criterion when only $\sigma_2$ and $\sigma_1$ are present:
When $\sigma_1 > 0:$
\begin{equation} \Big( \frac{\sigma_1}{X_T} \Big)^2 = 1 \Rightarrow \frac{\sigma_1}{X_T}=1 \end{equation}When $\sigma_1 \le 0:$
\begin{equation} -\frac{\sigma_1}{X_C} = 1 \end{equation}Inter fiber failure (IFF) when $(\sigma_2 + \sigma_3) \ge 0$ is also just a maximum stress criterion;
\begin{equation} \Big(\frac{\sigma_2}{Y_T}\Big)^2 = 1 \Rightarrow \frac{\sigma_2}{Y_T}=1 \end{equation}Inter fiber failure (IFF) when $(\sigma_2 + \sigma_3) < 0:$
\begin{equation} \frac{\sigma_2^2}{4S_{23}^2} + \frac{1}{Y_C}\Big( \frac{Y_C^2}{4S_{23}^2} - 1 \Big)\sigma_2 = 1 \Rightarrow \end{equation}\begin{equation} \frac{Y_C^2}{4S_{23}^2} - \frac{1}{Y_C}\Big( \frac{Y_C^2}{4S_{23}^2} - 1 \Big)Y_C = 1 \quad \text{when} \quad \sigma_2 = -Y_C \end{equation}Failure envelope for the material Carbon/Epoxy(a):
A non-zero shear stress $\tau_{12}$ will reduce the envelope for all modes but the compressive fiber failure:
Failure envelopes, $\tau_{12}$ versus $\sigma_1$:
Failure envelopes, $\tau_{12}$ versus $\sigma_2$
Failure envelopes, $\sigma_3$ versus $\sigma_2$
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