TMM4175 Polymer Composites
Plot gallery¶
In [1]:
def plotMatIndex(names,xdata,ydata,xlabel,ylabel):
import numpy as np
import matplotlib.pyplot as plt
fig,ax=plt.subplots(figsize=(10,6))
area = 50
for k in range(0,len(names)):
ax.text(xdata[k]*1.01,ydata[k]*1.01,names[k])
ax.grid(True)
ax.scatter(xdata, ydata, s=area, c='green', alpha=0.5)
ax.set_xlabel(xlabel)
ax.set_ylabel(ylabel)
plt.show()
#EXAMPLE:
names=['E-glass', 'T1100', 'M60', 'Steel','Cryptonite']
xdata=[2.55, 1.79, 1.93, 7.8, 5.0]
ydata=[76, 324, 588, 201,400]
%matplotlib inline
plotMatIndex(names,xdata,ydata,'Density','Modulus')
In [2]:
def plotSquareArrayOfFibers(Vf):
import matplotlib.pyplot as plt
import numpy as np
fig,ax=plt.subplots(figsize=(3,3))
a=1.0
ax.set_xlim(-2.1*a,2.1*a)
ax.set_ylim(-2.1*a,2.1*a)
ax.set_axis_off()
d= ( (4*Vf*a**2)/(np.pi) )**0.5
circle1 = plt.Circle((-a, -a), d, color='black',fc='silver')
circle2 = plt.Circle(( a, -a), d, color='black',fc='silver')
circle3 = plt.Circle(( a, a), d, color='black',fc='silver')
circle4 = plt.Circle((-a, a), d, color='black',fc='silver')
ax.add_artist(circle1)
ax.add_artist(circle2)
ax.add_artist(circle3)
ax.add_artist(circle4)
ax.plot((-a,a,a,-a,-a),(-a,-a,a,a,-a), '--',color='black',linewidth=1)
plt.show()
#EXAMPLE:
%matplotlib inline
plotSquareArrayOfFibers(Vf=0.55)
In [3]:
def plotRandomDistributionOfFibers(dx,dy,d,n):
# dx and dy are dimensions
# d is a list or 1D array of diameters
# n is the number of attempts
import numpy as np
rx=dx*np.random.rand(n)
ry=dy*np.random.rand(n)
x, y = [], [] #lists of coordinates verified to fit
k=0 # counter: no. of fibers already added
for j in range(0,n): # all randomly generated coordinates
fits=True # initially assuming that the fiber fits
for p in range(0,k): # all previously verified coordinates
distance= ( (rx[j]-x[p])**2 + (ry[j]-y[p])**2 )**0.5 # vector length
if distance<(d[k]+d[p])/2: # does not fit if the distance is less than average diameter of the two
fits=False
if fits==True: # append only the ones that fit
x.append(rx[j])
y.append(ry[j])
k=k+1
print('Number of fibers:',k)
import matplotlib.pyplot as plt
fig,ax=plt.subplots(figsize=(5,5))
ax.set_xlim(0,dx)
ax.set_ylim(0,dy)
ax.set_axis_off()
for i in range(0,len(x)):
circle1 = plt.Circle( (x[i], y[i]), d[i]/2, color='black',fc='silver')
ax.add_artist(circle1)
#EXAMPLE
import numpy as np
fibd = np.random.normal(10, 1, 1000) # (mean, std, count)
%matplotlib inline
plotRandomDistributionOfFibers(dx=200,dy=200,d=fibd,n=20000)
In [4]:
def plotDistribution(d):
import matplotlib.pyplot as plt
fig,ax = plt.subplots(figsize=(8,6))
n,bins,patches=plt.hist(fibd, 20, density=True,facecolor='teal', alpha=0.6 )
ax.set_xlabel('Fiber diameter(um)')
ax.set_ylabel('Normalized number of fibers')
ax.set_title('Fiber diameter distribution')
ax.set_xlim(0,)
ax.grid(True)
plt.show()
#EXAMPLE:
import numpy as np
fibd = np.random.normal(10, 1, 1000)
%matplotlib inline
plotDistribution(fibd)
In [5]:
def illustrateStrains(ex,ey,exy,scaleFactor):
import numpy
import matplotlib.pyplot as plt
x=numpy.array([-0.5,0.5,0.5,-0.5,-0.5])
y=numpy.array([-0.5,-0.5,0.5,0.5,-0.5])
ux=(ex*x+0.5*exy*y)*scaleFactor
uy=(ey*y+0.5*exy*x)*scaleFactor
xd=x+ux
yd=y+uy
fig,ax=plt.subplots(figsize=(4,4))
ax.set_xlim(-1,1)
ax.set_ylim(-1,1)
ax.set_axis_off()
ax.set_title('Scale factor='+str(scaleFactor))
text=r'$\varepsilon_x=$'+str(ex)+'\n'+r'$\varepsilon_y=$'+str(ey)+'\n'+r'$\gamma_{xy}=$'+str(exy)
ax.text(0,0,text, ha='center',va='center')
ax.plot(x,y,'--',color='lightgreen')
ax.plot(xd,yd,'-',color='darkgreen')
ax.arrow(-0.9, -0.9, 1.0, 0.0, head_width=0.05, head_length=0.05, fc='black', ec='black')
ax.arrow(-0.9, -0.9, 0.0, 1.0, head_width=0.05, head_length=0.05, fc='black', ec='black')
ax.text(0.2,-0.9,'x')
ax.text(-0.9,0.2,'y')
# EXAMPLE:
%matplotlib inline
illustrateStrains( ex=0.003, ey=-0.001, exy=-0.003, scaleFactor=100)
In [6]:
def illustrateLayup(layup,size=(4,4)):
import matplotlib.pyplot as plt
from matplotlib.patches import Rectangle
fig,ax=plt.subplots(figsize=size)
tot=0
for layer in layup:
tot=tot+layer['thi']
hb=-tot/2.0
for layer in layup:
ht=hb+layer['thi']
if layer['ori']>0:
fco='lightskyblue'
if layer['ori']<0:
fco='pink'
if layer['ori']==0:
fco='linen'
if layer['ori']==90:
fco='silver'
p = Rectangle( (-0.6, hb), 1.2, layer['thi'], fill=True,
clip_on=False, ec='black',fc=fco)
ax.add_patch(p)
mid=(ht+hb)/2.0
ax.text(0.62,mid,str(layer['ori']),va='center')
hb=ht
ax.set_xlim(-1,1)
ax.set_ylim(-1.1*tot/2.0, 1.1*tot/2.0)
ax.get_xaxis().set_visible(False)
ax.plot((-1,-0.8),(0,0),'--',color='black')
ax.plot((0.8,1.0),(0,0),'--',color='black')
plt.show()
#EXAMPLE:
%matplotlib inline
m1={'E1':40000, 'E2':10000, 'v12':0.3, 'G12':3000}
layup1 = [ {'mat':m1 , 'ori': 0 , 'thi':0.5} ,
{'mat':m1 , 'ori': 90 , 'thi':0.5} ,
{'mat':m1 , 'ori': 45 , 'thi':0.5} ,
{'mat':m1 , 'ori':-45 , 'thi':0.5} ,
{'mat':m1 , 'ori':-45 , 'thi':0.5} ,
{'mat':m1 , 'ori': 45 , 'thi':0.5} ,
{'mat':m1 , 'ori': 90 , 'thi':0.5} ,
{'mat':m1 , 'ori': 0 , 'thi':0.5}]
illustrateLayup(layup1,size=(4,3))
In [7]:
def illustrateCurvatures(Kx,Ky,Kxy):
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure(figsize=(6,4))
ax = fig.gca(projection='3d')
# Make data.
X = np.arange(-0.5, 0.6, 0.1)
Y = np.arange(-0.5, 0.6, 0.1)
X, Y = np.meshgrid(X, Y)
Z1 = (- Kx*X**2 - Ky*Y**2 - Kxy*X*Y)/2
surf1 = ax.plot_surface(X, Y, Z1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
#EXAMPLE:
%matplotlib inline
illustrateCurvatures(Kx=-0.5, Ky=0.5, Kxy=0.0)
In [8]:
def plotLayerStresses(results):
h= []
sx, sy, sxy = [], [], []
s1, s2, s12 = [], [], []
for layer in results:
h.append(layer['h']['bot'])
h.append(layer['h']['top'])
sx.append(layer['stress']['xyz']['bot'][0])
sx.append(layer['stress']['xyz']['top'][0])
sy.append(layer['stress']['xyz']['bot'][1])
sy.append(layer['stress']['xyz']['top'][1])
sxy.append(layer['stress']['xyz']['bot'][2])
sxy.append(layer['stress']['xyz']['top'][2])
s1.append(layer['stress']['123']['bot'][0])
s1.append(layer['stress']['123']['top'][0])
s2.append(layer['stress']['123']['bot'][1])
s2.append(layer['stress']['123']['top'][1])
s12.append(layer['stress']['123']['bot'][2])
s12.append(layer['stress']['123']['top'][2])
import matplotlib.pyplot as plt
fig,(ax1,ax2) = plt.subplots(ncols=2,nrows=1,figsize=(10,4))
ax1.grid(True)
ax1.plot(sx,h,'-', color='red',label='$\sigma_x$')
ax1.plot(sy,h,'-',color='blue',label='$\sigma_y$')
ax1.plot(sxy,h,'-',color='green',label=r'$\tau_{xy}$')
ax1.set_xlabel('Stress',fontsize=12)
ax1.set_ylabel('z', fontsize=14)
ax1.legend(loc='best')
ax2.grid(True)
ax2.plot(s1,h,'--', color='red',label='$\sigma_1$')
ax2.plot(s2,h,'--',color='blue',label='$\sigma_2$')
ax2.plot(s12,h,'--',color='green',label=r'$\tau_{12}$')
ax2.set_xlabel('Stress',fontsize=12)
ax2.set_ylabel('z', fontsize=14)
ax2.legend(loc='best')
plt.tight_layout()
plt.show()
#EXAMPLE:
from laminatelib import laminateStiffnessMatrix, solveLaminateLoadCase, layerResults
m1={'E1':140000, 'E2':10000, 'v12':0.3, 'G12':5000, 'XT':1200, 'XC':800, 'YT':50, 'YC':120, 'S12':75, 'S23':50, 'f12':-0.5}
layup1 =[ {'mat':m1 , 'ori': 0 , 'thi':1} ,
{'mat':m1 , 'ori': 90 , 'thi':1} ,
{'mat':m1 , 'ori': 0 , 'thi':1} ]
ABD1=laminateStiffnessMatrix(layup1)
load,defs = solveLaminateLoadCase(ABD=ABD1, Nx=500, Nxy=200)
res = layerResults(layup1,defs)
%matplotlib inline
plotLayerStresses(res)
In [9]:
def plotLayerFailure(results):
h= []
ms, me, tw = [], [], []
for layer in results:
h.append(layer['h']['bot'])
h.append(layer['h']['top'])
ms.append(layer['fail']['MS']['bot'])
ms.append(layer['fail']['MS']['top'])
me.append(layer['fail']['ME']['bot'])
me.append(layer['fail']['ME']['top'])
tw.append(layer['fail']['TW']['bot'])
tw.append(layer['fail']['TW']['top'])
import matplotlib.pyplot as plt
fig,ax = plt.subplots(ncols=1,nrows=1,figsize=(5,4))
ax.grid(True)
ax.plot(ms,h,'-', color='red',label='$f_E (MS)$')
ax.plot(me,h,'-',color='blue',label='$f_E (ME)$')
ax.plot(tw,h,'-',color='green',label='$f_E (TW)$')
ax.set_xlabel('$f_E$',fontsize=12)
ax.set_ylabel('z', fontsize=14)
ax.legend(loc='best')
plt.tight_layout()
plt.show()
#EXAMPLE:
from laminatelib import laminateStiffnessMatrix, solveLaminateLoadCase, layerResults
m1={'E1':140000, 'E2':10000, 'v12':0.3, 'G12':5000, 'XT':1200, 'XC':800, 'YT':50, 'YC':120, 'S12':75, 'S23':50, 'f12':-0.5}
layup1 =[ {'mat':m1 , 'ori': 0 , 'thi':1} ,
{'mat':m1 , 'ori': 90 , 'thi':1} ,
{'mat':m1 , 'ori': 0 , 'thi':1} ,
{'mat':m1 , 'ori': 90 , 'thi':1} ,
{'mat':m1 , 'ori': 0 , 'thi':1} ]
ABD1=laminateStiffnessMatrix(layup1)
load,defs = solveLaminateLoadCase(ABD=ABD1, Ny=500, Mx=200)
res = layerResults(layup1,defs)
%matplotlib inline
plotLayerFailure(res)
In [10]:
def plotHashin(results):
h= []
ff,iff = [], []
for layer in results:
h.append(layer['h']['bot'])
h.append(layer['h']['top'])
ff.append(layer['fail']['HNFF']['bot'])
ff.append(layer['fail']['HNFF']['top'])
iff.append(layer['fail']['HNIFF']['bot'])
iff.append(layer['fail']['HNIFF']['top'])
import matplotlib.pyplot as plt
fig,ax = plt.subplots(ncols=1,nrows=1,figsize=(5,4))
ax.grid(True)
ax.plot(ff,h,'-', color='red',label='$f_E (ff)$')
ax.plot(iff,h,'-',color='blue',label='$f_E (iff)$')
ax.set_xlabel('$f_E$',fontsize=12)
ax.set_ylabel('z', fontsize=14)
ax.legend(loc='best')
plt.tight_layout()
plt.show()
#EXAMPLE:
from laminatelib import laminateStiffnessMatrix, solveLaminateLoadCase, layerResults
m1={'E1':140000, 'E2':10000, 'v12':0.3, 'G12':5000, 'XT':1200, 'XC':800, 'YT':50, 'YC':120, 'S12':75, 'S23':50, 'f12':-0.5}
layup1 =[ {'mat':m1 , 'ori': 0 , 'thi':1} ,
{'mat':m1 , 'ori': 90 , 'thi':1} ,
{'mat':m1 , 'ori': 0 , 'thi':1} ,
{'mat':m1 , 'ori': 90 , 'thi':1} ,
{'mat':m1 , 'ori': 0 , 'thi':1} ]
ABD1=laminateStiffnessMatrix(layup1)
load,defs = solveLaminateLoadCase(ABD=ABD1, Ny=500, Mx=200)
res = layerResults(layup1,defs)
%matplotlib inline
plotHashin(res)
In [11]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.stats import norm
fig,ax = plt.subplots(figsize=(8,4))
meanL = 15
covL = 0.1
stdL = covL*meanL
xL=np.linspace(5,25,1000)
fL=norm.pdf(xL, meanL, stdL)
ax.plot(xL, fL, color='blue', label='Load')
ax.set_xlabel('Load (kg)')
ax.set_ylabel('Frequency')
ax.set_xlim(0,)
ax.set_ylim(0,)
ax.grid(True)
ax.legend(loc='best')
meanR = 18
covR = 0.08
stdR = covR*meanR
xR=np.linspace(5,30,1000)
fR=norm.pdf(xR, meanR, stdR)
ax.plot(xR, fR, color='red', label='Resistance')
ax.set_xlabel('Load, Resistance (kg)')
ax.set_ylabel('Frequency')
ax.set_xlim(0,)
ax.set_ylim(0,0.35)
ax.grid(True)
ax.legend(loc='best')
plt.tight_layout()
plt.savefig('imgs/safety-4.png')
plt.show()
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