TMM4175 Polymer Composites

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Maximum stress criterion¶

The Maximum stress failure criterion makes a simple asessment where each stress component is compared to the corresponding strength. The criterion predicts failure if one of the following six conditions is False:

\begin{equation} \begin{aligned} & -X_C < \sigma_1 < X_T && -Y_C < \sigma_2 < Y_T && -Z_C < \sigma_3 < Z_T && \mid\tau_{12}\mid <S_{12} && \mid\tau_{13}\mid < S_{13} && \mid\tau_{23}\mid < S_{23} \end{aligned} \tag{1} \end{equation}

The criterion can alternatively be formulated using a max function:

\begin{equation} f = \text{max}\Big( -\frac{\sigma_1}{X_C}, \frac{\sigma_1}{X_T}, -\frac{\sigma_2}{Y_C}, \frac{\sigma_2}{Y_T}, -\frac{\sigma_3}{Z_C}, \frac{\sigma_3}{Z_T}, \frac{|\tau_{12}|}{S_{12}}, \frac{|\tau_{13}|}{S_{13}}, \frac{|\tau_{23}|}{S_{23}}\Big) \tag{2} \end{equation}

where failure is predicted when $f \geq 1$.

The stress exposure factor is obtained by multiplying the load proportionality factor $R$ to each stress components and equating the result to unity:

\begin{equation}\text{max}\Big( -\frac{R\sigma_1}{X_C}, \frac{R\sigma_1}{X_T}, -\frac{R\sigma_2}{Y_C}, \frac{R\sigma_2}{Y_T}, -\frac{R\sigma_3}{Z_C}, \frac{R\sigma_3}{Z_T}, \frac{R|\tau_{12}|}{S_{12}}, \frac{R|\tau_{13}|}{S_{13}}, \frac{R|\tau_{23}|}{S_{23}}\Big) = 1 \tag{3} \end{equation}

such that

\begin{equation} f_E = \frac{1}{R} = \text{max}\Big( -\frac{\sigma_1}{X_C}, \frac{\sigma_1}{X_T}, -\frac{\sigma_2}{Y_C}, \frac{\sigma_2}{Y_T}, -\frac{\sigma_3}{Z_C}, \frac{\sigma_3}{Z_T}, \frac{|\tau_{12}|}{S_{12}}, \frac{|\tau_{13}|}{S_{13}}, \frac{|\tau_{23}|}{S_{23}}\Big) \tag{4} \end{equation}

Implementation:

The following function returns the stress exposure factor $f_E$ for a stress state s and material m:

def fE_maxstress(s,m):
    # Using local varibles for easier coding and readability..
    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']
    fE=max(s1/XT,-s1/XC,s2/YT,-s2/YC,s3/ZT,-s3/ZC,abs(s12/S12),abs(s13/S13),abs(s23/S23))
    return fE

Example:

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_maxstress(s,m1)
    print('fE =',fE,'when stress=',s)

fE = 0.08333333333333333 when stress= (150, 0, 0, 0, 0, 0)
fE = 0.3333333333333333 when stress= (-120, -60, 0, 0, 0, 0)
fE = 1.5 when stress= (1000, 60, 0, 0, 0, 0)
fE = 2.142857142857143 when stress= (0, 0, 0, 0, 0, 150)

Failure envelope in the $\sigma_1$ - $\sigma_2$ -plane:

import matplotlib.pyplot as plt
%matplotlib inline

#Data points for the rectangle in the s1-s2 plane
s1=(m1['XT'],-m1['XC'],-m1['XC'],m1['XT'],m1['XT'])
s2=(m1['YT'],m1['YT'],-m1['YC'],-m1['YC'],m1['YT'])
fig,ax = plt.subplots(figsize=(8,4))
ax.plot(s1,s2,'--',color='blue',linewidth=1)

#Origo
ax.plot((0,),(0,),'+',color='black',markersize=50)

#Adding some stress states

s1a, s2a = -600, -50
s1b, s2b = 180, 0
s1c, s2c = 1000, 60

ax.scatter((s1a,s1b,s1c),(s2a,s2b,s2c),color='red')
fE1=fE_maxstress((s1a,s2a,0,0,0,0),m1)
fE2=fE_maxstress((s1b,s2b,0,0,0,0),m1)
fE3=fE_maxstress((s1c,s2c,0,0,0,0),m1)
ax.text(s1a,s2a,'  '+r'$f_E=$'+str(fE1),fontsize=14)
ax.text(s1b,s2b,'  '+r'$f_E=$'+str(fE2),fontsize=14)
ax.text(s1c,s2c,'  '+r'$f_E=$'+str(fE3),fontsize=14)
ax.set_xlabel(r'$\sigma_1$',fontsize=14)
ax.set_ylabel(r'$\sigma_2$',fontsize=14)
ax.grid(True)
plt.tight_layout()

Maximum stress envelope in the $\sigma_1$ - $\sigma_2$ -plane plane can be constructed using a more generic method where we sweep through $\theta=0...360^{\circ}$ where $\sigma_1=\cos(\theta)$ and $\sigma_2=\sin(\theta)$. This method can be utilized for most failure criteria.

fig,ax = plt.subplots(figsize=(8,4))

# empty list of the stresses in the 1-2 plane:
s1,s2=[],[]

# sweeping 360 degrees using 10 points per degrees:

import numpy as np

for a in np.linspace(0,360,3600):
    s1i=np.cos(np.radians(a))
    s2i=np.sin(np.radians(a))
    fe=fE_maxstress((s1i,s2i,0,0,0,0),m1)
    # then scaling by the load-proportionality ratio (1/fE):
    s1.append(s1i/fe)
    s2.append(s2i/fe)

ax.plot(s1,s2,'--',color='blue',linewidth=1)

ax.plot((0,),(0,),'+',color='black',markersize=50)

ax.set_xlabel(r'$\sigma_1$',fontsize=14)
ax.set_ylabel(r'$\sigma_2$',fontsize=14)

ax.grid(True)

plt.tight_layout()

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