TMM4175 Polymer Composites
Tsai-Wu criterion¶
The Tsai-Wu failure criterion is a specialization of the general quadratic failure criterion and is expressed in the form
\begin{equation} \begin{split} f = & F_1\sigma_1 + F_2\sigma_2 + F_3\sigma_3 + F_{11}\sigma_1^2 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2+ \\ & F_{44}\tau_{23}^2 + F_{55}\tau_{13}^2 + F_{66}\tau_{12}^2 + 2F_{23}\sigma_2\sigma_3 + 2F_{13}\sigma_1\sigma_3 + 2F_{12}\sigma_1\sigma_2 \end{split} \tag{1} \end{equation}where failure is predicted when $f\geq1$
The coefficients are expressed by the basic strength parameters,
\begin{equation} \begin{aligned} &F_1=\frac{1}{X_T}-\frac{1}{X_C}, \quad F_2=\frac{1}{Y_T}-\frac{1}{Y_C}, \quad F_3=\frac{1}{Z_T}-\frac{1}{Z_C} \\ &F_{11}=\frac{1}{X_T X_C}, \quad F_{22}=\frac{1}{Y_TY_C}, \quad F_{33}=\frac{1}{Z_TZ_C} \\ &F_{44}=\frac{1}{S_{23}^2}, \quad F_{55}=\frac{1}{S_{13}^2}, \quad F_{66}=\frac{1}{S_{12}^2} \\ &F_{23}=f_{23}\sqrt{F_{22}F_{33}}, \quad F_{13}=f_{13}\sqrt{F_{11}F_{33}}, \quad F_{12}=f_{12}\sqrt{F_{11}F_{22}} \end{aligned} \tag{2} \end{equation}where $f_{12},f_{13},f_{23}$ are experimentally determined interaction coefficients.
Solving the quadratic equation:
\begin{equation} \begin{split} & (F_1\sigma_1 + F_2\sigma_2 + F_3\sigma_3)R + \\ &(F_{11}\sigma_1^2 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + F_{44}\tau_{23}^2 + F_{55}\tau_{13}^2 + F_{66}\tau_{12}^2 + 2F_{23}\sigma_2\sigma_3 + 2F_{13}\sigma_1\sigma_3 + 2F_{12}\sigma_1\sigma_2)R^2 = 1 \end{split} \tag{3} \end{equation}Such that
\begin{equation} \begin{aligned} &a=F_{11}\sigma_1^2 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + F_{44}\tau_{23}^2 + F_{55}\tau_{13}^2 + F_{66}\tau_{12}^2 + 2F_{23}\sigma_2\sigma_3 + 2F_{13}\sigma_1\sigma_3 + 2F_{12}\sigma_1\sigma_2 \\ &b= F_1\sigma_1 + F_2\sigma_2 + F_3\sigma_3 \\ &c=-1 \end{aligned} \tag{4} \end{equation}and
\begin{equation} R=\frac{-b+\sqrt{b^2 -4ac}}{2a} \tag{5} \end{equation}Observe that the parameter $a$ can be zero. In such a case the $R$ will be undefined. In those cases we just return zero as written in this code:
In [1]:
def fE_tsaiwu(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']
f12, f13, f23 = m['f12'], m['f13'], m['f23']
F1, F2, F3 = (1/XT)-(1/XC) , (1/YT)-(1/YC) , (1/ZT)-(1/ZC)
F11, F22, F33 = 1/(XT*XC) , 1/(YT*YC) , 1/(ZT*ZC)
F44, F55, F66 = 1/(S23**2) , 1/(S13**2) , 1/(S12**2)
F12 = f12*(F11*F22)**0.5
F13 = f13*(F11*F33)**0.5
F23 = f23*(F22*F33)**0.5
a=F11*(s1**2) + F22*(s2**2) + F33*(s3**2)+ 2*(F12*s1*s2 + F13*s1*s3 + F23*s2*s3)+\
F44*(s23**2) + F55*(s13**2) + F66*(s12**2)
if a==0:
return 0
b=F1*s1 + F2*s2 +F3*s3
c=-1
R=(-b+(b**2-4*a*c)**0.5)/(2*a)
fE=1/R
return fE
Example¶
In [2]:
import matlib
m1=matlib.get('Carbon/Epoxy(a)')
stressCases = (150, 0, 0, 0, 0, 0) , (-120,-60,0,0,0,0) , (1000,60,0,0,0,0) , (0,0,0,0,0,150)
for s in stressCases:
fE=fE_tsaiwu(s,m1)
print(fE)
Failure envelope, $\sigma_2$ versus $\sigma_1$:¶
In [3]:
import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline
fig,ax = plt.subplots(figsize=(8,4))
s1,s2=[],[]
from math import cos,sin,radians
for a in np.linspace(0,360,3600):
s1i=cos(radians(a))
s2i=sin(radians(a))
fe=fE_tsaiwu((s1i,s2i,0,0,0,0),m1)
s1.append(s1i/fe)
s2.append(s2i/fe)
ax.plot(s1,s2,'-',color='blue',label='Tsai-Wu',linewidth=1)
ax.plot((0,),(0,),'+',color='black',markersize=50)
ax.legend(loc='best')
ax.set_xlabel(r'$\sigma_1$',fontsize=14)
ax.set_ylabel(r'$\sigma_2$',fontsize=14)
ax.grid(True)
plt.tight_layout()
Failure envelope, $\sigma_2$ versus $\sigma_1$ for constant nonzero values of $\tau_{12}$:¶
Terms of a, b in equtions (4) containing $\tau_{12}$ are now constants and therefor terms of the constant c:
\begin{equation} \begin{aligned} & a=F_{11}\sigma_1^2 + F_{22}\sigma_2^2 + F_{33}\sigma_3^2 + F_{44}\tau_{23}^2 + F_{55}\tau_{13}^2 + 2F_{23}\sigma_2\sigma_3 + 2F_{13}\sigma_1\sigma_3 + 2F_{12}\sigma_1\sigma_2 \\ &b= F_1\sigma_1 + F_2\sigma_2 + F_3\sigma_3 \\ &c=-1 + F_{66}\tau_{12}^2 \end{aligned} \tag{6} \end{equation}Modified function for the specific case:
In [4]:
def fE_tsaiwu_const_s12(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']
f12, f13, f23 = m['f12'], m['f13'], m['f23']
F1, F2, F3 = (1/XT)-(1/XC) , (1/YT)-(1/YC) , (1/ZT)-(1/ZC)
F11, F22, F33 = 1/(XT*XC) , 1/(YT*YC) , 1/(ZT*ZC)
F44, F55, F66 = 1/(S23**2) , 1/(S13**2) , 1/(S12**2)
F12 = f12*(F11*F22)**0.5
F13 = f13*(F11*F33)**0.5
F23 = f23*(F22*F33)**0.5
a=F11*(s1**2) + F22*(s2**2) + F33*(s3**2)+ 2*(F12*s1*s2 + F13*s1*s3 + F23*s2*s3)+\
F44*(s23**2) + F55*(s13**2)
if a==0:
return 0
b=F1*s1 + F2*s2 +F3*s3
c=-1 + F66*(s12**2)
R=(-b+(b**2-4*a*c)**0.5)/(2*a)
fE=1/R
return fE
Multiple failure envelopes:
In [5]:
fig,ax = plt.subplots(figsize=(12,8))
s1a,s2a,s1b,s2b,s1c,s2c,s1d,s2d,s1e,s2e=[],[],[],[],[],[],[],[],[],[]
from math import cos,sin,radians
for a in np.linspace(0,360,3600):
s1i=cos(radians(a))
s2i=sin(radians(a))
fea=fE_tsaiwu_const_s12((s1i,s2i,0,0,0,0),m1)
feb=fE_tsaiwu_const_s12((s1i,s2i,0,0,0,20),m1)
fec=fE_tsaiwu_const_s12((s1i,s2i,0,0,0,40),m1)
fed=fE_tsaiwu_const_s12((s1i,s2i,0,0,0,50),m1)
fee=fE_tsaiwu_const_s12((s1i,s2i,0,0,0,60),m1)
s1a.append(s1i/fea)
s2a.append(s2i/fea)
s1b.append(s1i/feb)
s2b.append(s2i/feb)
s1c.append(s1i/fec)
s2c.append(s2i/fec)
s1d.append(s1i/fed)
s2d.append(s2i/fed)
s1e.append(s1i/fee)
s2e.append(s2i/fee)
ax.plot(s1a,s2a,'-',color='blue',label=r'Tsai-Wu, $\tau_{12}$=0',linewidth=1)
ax.plot(s1b,s2b,'-',color='red',label=r'Tsai-Wu, $\tau_{12}$=20',linewidth=1)
ax.plot(s1c,s2c,'-',color='green',label=r'Tsai-Wu, $\tau_{12}$=40',linewidth=1)
ax.plot(s1d,s2d,'-',color='orange',label=r'Tsai-Wu, $\tau_{12}$=50',linewidth=1)
ax.plot(s1e,s2e,'-',color='cyan',label=r'Tsai-Wu, $\tau_{12}$=60',linewidth=1)
ax.plot((0,),(0,),'+',color='black',markersize=50)
ax.legend(loc='best')
ax.set_xlabel(r'$\sigma_1$',fontsize=14)
ax.set_ylabel(r'$\sigma_2$',fontsize=14)
ax.grid(True)
plt.tight_layout()
Inconsistency of the criterion¶
Imagine now that we, for whatever reason, find it reasonable to use a lower value of the transverse tensile strength $X_T$:
In [6]:
m1['YT']=25
Then, construct the failure envelopes for both cases:
In [7]:
fig,ax = plt.subplots(figsize=(8,4))
s1b,s2b=[],[]
from math import cos,sin,radians
for a in np.linspace(0,360,3600):
s1i=cos(radians(a))
s2i=sin(radians(a))
fe=fE_tsaiwu((s1i,s2i,0,0,0,0),m1)
# then scaling by the load-proportionality ratio (1/fE):
s1b.append(s1i/fe)
s2b.append(s2i/fe)
ax.plot(s1,s2,'-',color='blue',label=r'Tsai-Wu, $Y_T$=50',linewidth=1)
ax.plot(s1b,s2b,'-',color='red',label=r'Tsai-Wu, $Y_T$=25',linewidth=1)
ax.plot((0,),(0,),'+',color='black',markersize=50)
ax.legend(loc='best')
ax.set_xlabel(r'$\sigma_1$',fontsize=14)
ax.set_ylabel(r'$\sigma_2$',fontsize=14)
#ax.set_xticks([0])
#ax.set_yticks([0])
ax.grid(True)
plt.tight_layout()
So, reduced transverse tensile strength implies improved capacity in compression-compression?
To be discussed in lectures
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