Integrated Composite Analyzer (ICAN/JAVA)
Fiber Slicing Computation

Fiber Slicing

Because the fibers are assumed to be evenly spaced, it is sufficient to consider one fiber within a volume that is repeated as required.
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The size of the cube containing the fiber may be computed from the diameter of the fiber, d_f , and the fiber volume ratio, k_f . The fiber volume ratio may be expressed as the ratio of the volume of the fiber to the volume of the cube that may be repeated.
(1.1)   k_f = (/4) (d_f )² / D²
where D is length of the side of the cube over which the fibers are evenly spaced.

Solving for D, one obtains:
(1.2)   D = d_f [(/4)/ k_f ]^(½)

Consequently, one need consider a horizontal slice, possibly containing a piece of the fiber, only within a cube over which the fibers are evenly spaced. One such slice is shown below:
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The fiber slice volume ratio may be expressed as the ratio of the volume of the fiber within the horizontal slice to the volume of the horizontal slice. Further information is available about computing volume ratios for a slice of a cylindrical fiber. The analysis below uses the volumes and volume ratios for the slice.

To analyze further, the horizontal slices may be sliced in the vertical direction; for example, along the indicated lines above. The round edges are approximated by square edges, as shown below:
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Fiber Subslicing

Since the fibers are aligned with the 1 (X) axis, the subslice may be moved anywhere along the 1 (X) axis and the amount of overlap will not be affected. Thus the distance in the 1 (X) direction may be ignored and the problem reduced to the two-dimensional problem of computing the amount of overlap between a circle and a rectangle.
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Let the origin of the cylindrical fiber be denoted by O. The radius, R, is ½ d_f , then ½ d_i , to compute the required intersections. The center point of the subslice, C, and the size of the subslice, (dX,dY,dZ), are specified. The farthest point of the subslice, F, may be computed as F = C + ½ D. The nearest point of the subslice, N, may be computed as N = C - ½ D. Where the size vector, D, is computed as: D = [ 0 , dY sign([C - O]_Y), dZ sign([C - O]_Z) ].

1. If the subslice is entirely inside the cylinder, i.e. the cylinder contains the entire subslice, there is obviously an intersection. The entire subslice is inside the cylinder if the farthest point of the subslice is within a radius from the center of the cylinder, i.e. ||F - O|| < R.

2. If the subslice contains the entire cylinder, i.e. the cylinder is entirely inside the subslice, there is obviously an intersection.
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   For each axis, let T be a unit vector defining the direction to test. The 1 (X) axis is omitted. The cylinder is within the subslice if, for all such directions, T, |(C-O)·T| < ½ |D·T| - R.

3. If the subslice is entirely outside the cylinder, i.e. the cylinder is entirely outside the subslice, there is obviously no intersection.
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   For each axis, let T be unit vector defining the direction to test. The 1 (X) axis is omitted. The cylinder is outside the subslice if, for any such direction, T, |(C-O)·T| > ½ |D·T| + R. This test determines whether the cylinder is outside of the box enlarged by the radius of the cylinder on each side. The "green" areas may be described as the area outside the locus of points made by the center of the cylinder rolling around on the outside of the box and inside a box enlarged by the radius of the cylinder on each side. The "green" areas in the figure, near each corner or edge, are left for the following analysis.

4. If the center of the cylinder is outside the subslice by an amount greater than R, the cylinder does not intersect the subslice. For each axis, let T be unit vector defining the direction to test. The 1 (X) axis is omitted. The distance that the center of the cylinder is outside the subslice in the direction specified by T, is given by:
   d_T = |(C-O)·T| - ½ |D·T|
   If d_T < 0, use zero for this term in the next equation, because one only needs the distance outside the corresponding face of the subslice.
   Let d = SQRT( d_Ty² + d_Tz² ).
   There is no intersection when d > R.

5. Otherwise, there is an intersection.

The above analysis may be used as follows:

• Use (1) for R = ½ d_f , to determine whether the subslice is fiber only. In that case, k_f = 1.
• Use (1) for R = ½ d_i , and (3) and (4) for R = ½ d_f , to determine whether the subslice is interphase only. In that case, k_i = 1.
• Use (2) for R = ½ d_i , to determine whether the subslice contains the entire fiber and interphase to simplify the computation.
• Use (3) and (4) for R = ½ d_i , to determine whether the subslice is matrix only. In that case, k_m = 1.

The volume of the intersection may be computed by integration or approximation. The problem with integration is computing the limits of integration, which occur at either the cylindrical interface or some edge of the subslice, and integrating differently within the different regions (difficult). Approximation allows evaluating the limits along a single line, computing the length of the line within the intersection, and using the weighted sum of many such lengths to represent the integral (easy).

To approximate the integral, sum the vertical lengths over all y (2) values. The z location(s) where the vertical line at y intersects the cylinder may be computed as follows:
z = O_Z ± SQRT[ R² - (y - O_Y)² ]
If the computation gives imaginary results, there is no intersection. If real z value(s) can be computed, there will be two solutions z₁ and z₂, corresponding to the negative and positive signs, respectively. The length inside the cylinder at y is the length of the line from (x,y,z₁) to (x,y,z₂). The amount of this line to use, i.e. the amount inside the subslice, is computed by limiting the (maximum) z₂ value to the maximum for the slice and limiting the (minimum) z₁ value to the minimum for the slice. If the resulting (z₂ - z₁) is positive, sum the corresponding volume: (z₂ - z₁) * dA.

Normal Moduli

Shear Moduli

Poisson's Ratio

Thermal Expansion Coefficients

Thermal Heat Conductivities

Heat Capacity

Moisture Expansion Coefficient

Moisture Diffusivities