Page 42 - MSAM-2-3

Materials Science in Additive Manufacturing Cast and 3D-printed fiber orientations

Appendices Substituting Equation AI into Equation AIV:

Appendix A: Analytical result of one-dimensional ZV cos Pz, d
f
,

boundary constraints Num z, d r 2 2 Pz, d (AV)

For a 1D case with thickness Z, the amount of fiber per f 0
unit length and width can be computed by: Then, the total number of fibers crossing this plane at a
ZV given angle can be integrated through the thickness:
N f (AI)
lr 2 Num 0,Z d 2 0 Z 2 Num ,z , d dz

,

,
ff
This volume can be divided into two regions, that (AVI)
us, Region 1 (z < l/2and Region 2 (z≥ l/2), as shown in Substituting Equation AV into Equation AVI:
f
f
Figure A1, where z is the distance from the boundary. For
different cases, Region 2 may not exist. For Region 1, the 2ZV f cos
,
,

total length of the distribution arc is: Num0,Z d r f 2

Pz,
,
Z 2 Pz d

l z 2 dz (AVII)
f sin 2 arcsin 0 Pz d
,
2 l f 2 0

l z z Let w=z/(l/2) be the dimensionless coordinate,
f
f sinn 2 4 arccos l arcsin l Equations AII and AIII can be combined as:

2 f sin 2
f 2 Pw,

(AII) sin 2 w 1 or w sin
For Region 2, the total length of the distribution arc is: w

Pz, l f sin 2 (AIII) sin 2 4 arccos sin w
1 and w
sin (AVIII)

2
and Equation (AVII) can be rewritten as:
If the distance between the fiber center and one specific 2ZV cos

cross-section plane is more than ±lcosθ/2, the fiber cannot Num0,Z d f
,
,

f
pass through the plane. Thus, the number of fibers crossing r f 2
this plane at a given angle can be computed for any distance Pw
d
,
z from the boundary: 0 Zl f 2 Pw dw (AIX)
Pz,

f
Num z, d N 2 l cos 2 d (AIV) 0 , d
,

2
0 Pz, d For a specific position in region 1, where u <1 :

2 Pw, arcsin w P w, 2 Pw,

0 d 0 d arcsin w d
0 0 arcsinw 2 sin d

w

2
arcsiinw
4arccos sin
sin d
2

(AX)
Let θ =arcsinw and integrate into Equation AX:
w

2 Pw, u
0 d 0 2 sin d

2 w

2 4 arccos
sin d
u sin
2 w

2 4 sinarccos
d (AXI)
Figure A1. The schematic of 1D case with two different regions. u sin
Volume 2 Issue 3 (2023) 18 https://doi.org/10.36922/msam.1603