TMM4175 Polymer Composites

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CASE STUDY: Fiber diameter distribution¶

While carbon fibers and Kevlar fibers usually have consistent diameter with just a minor variation, the diameter within single batches of glass fibers varies significantly [1].

This case study presents ideas on how to numerically process and visualize fiber diameter distribution and randomly placed fibers in a unidirectional configuration. The diameters of a representative sample of 600 glass fibers are listed in the text file fiberDiameters.txt.

The simple numpy method for reading the data into a numpy array is:

import numpy as np
fibd=np.loadtxt('data/fiberDiameters.txt')

Then, visualize the fiber diameter distribution:

%matplotlib inline

def plotDistribution(d):
    import matplotlib.pyplot as plt
    fig,ax = plt.subplots(figsize=(8,6))
    n,bins,patches=plt.hist(d, 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()
    
plotDistribution(fibd)

Relevant statistical data:

print('Min:',np.amin(fibd))
print('Max:',np.amax(fibd))
print('Range:',np.ptp(fibd))
print('Mean:',np.mean(fibd))
print('Stdev:',np.std(fibd))

Min: 7.54
Max: 17.0
Range: 9.46
Mean: 12.025266666666667
Stdev: 1.5298298147905938

The basic idea of the following code is to simulate a cross section of UD composite with randomly placed fibers with varying diameters according to the previous distribution:

def plotRandomDistributionOfFibers(dx,dy,d,n):
    import numpy as np
    rx=dx*np.random.rand(n)
    ry=dy*np.random.rand(n)
    x, y = [], []  #lists of coordinates verified to fit
    dm=[] 
    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])
            dm.append(d[k])
            k=k+1
    print('Number of fibers:',k)
    import matplotlib.pyplot as plt
    fig,ax=plt.subplots(figsize=(12,12))
    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]), dm[i]/2, color='black',fc='silver')
        ax.add_artist(circle1)

plotRandomDistributionOfFibers(dx=200,dy=200,d=fibd,n=20000)

Number of fibers: 191

Although the sample of fibers is the same, another random sample of positions will produce a different result, and likely a different number of fibers in the cross section:

plotRandomDistributionOfFibers(dx=200,dy=200,d=fibd,n=20000)

Number of fibers: 188

The same code for a sample of fibers having identical diameters:

d12=[12.025 for i in range(0,300)]
plotRandomDistributionOfFibers(dx=200,dy=200,d=d12,n=20000)

Number of fibers: 197

The function scipy.stats.norm() creates a normal distribution based on the mean and the standard deviation which can be added to the plot:

def plotDistributionB(d):
    import matplotlib.pyplot as plt
    from scipy.stats import norm
    x=np.linspace(min(d),max(d))
    y=norm.pdf(x,np.mean(d),np.std(d))
    fig,ax = plt.subplots(figsize=(8,6))
    n,bins,patches=plt.hist(d, 20, density=True,facecolor='teal', alpha=0.6 )
    ax.plot(x,y,color='black')
    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()
    
plotDistributionB(fibd)

Finally, creating a fictive distribution is simple. For example, natural fibers show typically diameters within a large range as in the following example:

def plotFakeDistribution(mean,std,n):
    import matplotlib.pyplot as plt
    from scipy.stats import norm
    dis = np.random.normal(mean, std, n)
    fig,ax = plt.subplots(figsize=(8,6))
    n,bins,patches=plt.hist(dis, 15, density=True,facecolor='orange', alpha=0.6 )
    x=np.linspace(min(dis),max(dis))
    y=norm.pdf(x,mean,std)
    ax.plot(x,y,color='black')
    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()

plotFakeDistribution(40,10,500)

References and further readings¶

Thomason, J. L. "Structure–property Relationships in Glass Reinforced Polyamide, Part 2: The Effects of Average Fiber Diameter and Diameter Distribution." Polymer Composites 28, no. 3 (2007): 331-43.

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