# Dynamic simulation of flow-induced fiber fracture.

INTRODUCTIONFibers have been widely used in various fields of modem technology, such as fiber-reinforced composites. The principal object of fiber addition is to improve mechanical properties - the strength, modulus, and toughness - of the base plastic, metal, or ceramic. The mechanical properties of the composites depend on the fiber modulus and on the fiber length; longer fibers are very effective in reinforcement. These reinforced materials are usually formed into definite shapes for use as commercial products. For example, fiber-reinforced plastics are formed into automobile parts by injection molding. In these cases, even if long fibers are mixed in a matrix for best reinforcement, the fiber length will be reduced during the forming process. Consequently, an unavoidable reduction in reinforcement efficiency occurs. Although a phenomenon of fiber fracture is a serious problem in fiber-reinforced composites, it has not been understood how fibers fracture and how longer fibers can be maintained.

Forgacs and Mason (1, 2) studied the behavior of a fiber in a viscous fluid subjected to a laminar shear flow and developed the theory of fiber fracture in an infinitely dilute system. They assumed that a fiber was so thin that the disturbance of the surrounding flow field was negligible and neglected inertial effects. They derived an equation to calculate the critical value of the fluid shear stress to buckle a straight fiber based on the aspect ratio and the Young's modulus of a fiber.

Turkovich and Erwin (3) experimentally studied the length reduction of reinforcing fibers in short-fiber reinforced plastics during processing. They noted three causes for fiber fracture during processing - (i) fiber-fiber interaction, (ii) fiber contact with processor surfaces, and (iii) fiber interaction with the polymer - and examined their importance in various conditions. They concluded that fiber fracture could be described by the dilute suspension theory developed by Forgacs and Mason, namely, fiber interaction with fluid is dominant in fiber fracture even in a concentrated system.

A study of fiber fracture during processing contributes to an understanding of how the process can be changed to maintain longer fibers. Although experimental observation of fiber fracture is reliable for this purpose (4), it is often difficult to execute. A computer simulation is one intelligent way to analyze fiber fracture. The dynamic behavior of a rigid fiber in a fluid is usually analyzed with the use of Jeffery's equation (5, 6), which describes the motion of an ellipsoidal particle in Stokes flow of a Newtonian fluid. The fiber orientation in the concentrated suspension with interaction among fibers can be calculated by using Folgar and Tucker's equation with an empirical parameter (7, 8). However, these methods are not applicable to the deformation and fracture of fibers. We proposed the particle simulation method (PSM) to analyze the dynamic behavior of a flexible fiber in a previous paper (9). In this method, a fiber is modeled with bonded spheres, and its flexibility can be altered by changing three parameters of the stretching, bending, and twisting constants, which are formulated using the Young's modulus and shear modulus of the fiber. The motion of the fiber model in a flow field is determined by solving the translational and rotational motion equations for individual spheres under the hydrodynamic force and torque exerted on them. We expand this method to analyze the fracture process of a fiber in a flow field. Stress induced in each bond of the fiber model as a result of stretching, bending, and twisting deformation can be formulated using displacement of the bond distance, bond angle, and torsion angle for each pair of spheres, respectively. Therefore, the occurrence of fiber fracture can be defined as the moment when the stress induced in the fiber model surpasses the strength of the fiber. If the dynamic behavior of the fiber model is simulated in a flow field, we can detect critical conditions of the flow field required to avoid fiber fracture. Then, we can get information about which the size of fibers can be maintained under a certain flow condition.

We applied PSM in describing a deflection curve of a cantilever beam with a concentrated load at the end, and verified the validity of the method and formulation by comparing the simulated deflection and slope of the beam with the theoretical ones. Next, fiber fracture in a two-dimensional simple shear flow was simulated, based on assumptions of an infinitely dilute system, no hydrodynamic interaction, and a low Reynolds number of a particle. The critical conditions of the flow field required to avoid fiber fracture after bending deformation were calculated by simulations. The simulation was also compared with Forgacs and Mason's theory.

MODEL

The details of the fiber model are described in the previous paper (9). Here, we explain its outline and modifications for estimating the stress induced in the fiber model. Considering a cylindrical fiber of length 2 aN, diameter 2 a, and an aspect ratio of N, it is assumed to be made up of N spheres of radius a, as shown in Fig. 1. Each pair of adjacent spheres is bonded, and can stretch, bend, and twist by changing the bond distance, bond angle, and torsion angle, respectively. When the fiber model is deformed, a force or torque is exerted on each sphere to recover the equilibrium position.

For the stretching deformation, the following force [F.sup.s], in proportion to displacement r of the bond distance between adjacent spheres is exerted on each sphere to recover the equilibrium distance,

[F.sup.s] = -[k.sub.s]r (1)

where [k.sub.s] is the stretching force constant. This force corresponds to the rigidity of stretching deformation, and is represented by using the Young's modulus E,

[k.sub.s] = [Pi]a/2 E (2)

The stretching stress [[Sigma].sub.s] induced in the fiber model is represented by

[[Sigma].sub.s] = E/2 a r (3)

If the stretching strength of the fiber is [Mathematical Expression Omitted], the occurrence of fiber fracture can be defined as the moment when displacement of the bond distance between adjacent spheres equals the following value:

[Mathematical Expression Omitted]

Next, for the bending deformation, the following torque [T.sup.b] in proportion to displacement [[Theta].sub.b] of the bond angle between adjacent spheres exerted on each sphere to restore the equilibrium bond angle,

[T.sup.b] = -[k.sub.b][[Theta].sub.b] (5)

where [k.sub.b] is the bending torque constant. The bending torque constant corresponds to the rigidity of bending deformation. It is also represented by using the Young's modulus, E,

[k.sub.b] = [Pi][a.sup.3]/8 E (6)

The bending stress [[Sigma].sub.b] induced in the fiber model is represented by

[[Sigma].sub.b] = E/2 [[Theta].sub.b] (7)

The occurrence of fiber fracture for the bending strength [Mathematical Expression Omitted] can be defined as the moment when displacement of the bond angle between adjacent spheres equals the following value;

[Mathematical Expression Omitted]

Finally, for the twisting deformation, the following torque [T.sup.t] in proportion to displacement [[Theta].sub.t] of the torsion angle between adjacent spheres exerted on each sphere to recover the equilibrium torsion angle

[T.sup.t] = -[k.sub.t][[Theta].sub.t] (9)

where [k.sub.t] is the twisting constant, corresponding to the rigidity of twisting deformation. It is represented by using the shear modulus G,

[k.sub.t] = [Pi][a.sup.3]/4 G (10)

The twisting stress [[Sigma].sub.t] induced in the fiber model is represented by

[[Sigma].sub.t] = G/2 [[Theta].sub.t] (11)

Likewise, if the twisting strength of the fiber is [Mathematical Expression Omitted], the occurrence of fiber fracture can be defined as the moment when displacement of the torsion angle between adjacent spheres equals the following value

[Mathematical Expression Omitted]

The fiber model made up of bonded spheres can stretch, bend, and twist like a flexible fiber. The flexibility of the fiber model can be changed from rigid to flexible by altering three parameters, [k.sub.s], [k.sub.b], and [k.sub.t]. The fracture process of a fiber can be analyzed by considering the strength of a fiber and the stress induced in it for each deformation, as described above. The force and torque are formulated using linear functions of displacement of the bond distance, bond angle, and torsion angle, in Eqs 1, 5, and 9, respectively. However, other factors may apply. In practice, most fibers show a nonlinear behavior of deformation and fracture. Therefore, other nonlinear functions of displacement, which are not described in this paper, can be also applicable.

The fiber is, in a sense, discretized into an array of spheres. For a discretized model like FEM, convergence normally depends on the level of discretization. In the fiber model of PSM, however, the number of spheres is uniformly defined for a fiber of given aspect ratio, not length. Therefore, convergence is always constant, and is guaranteed at a level of the comparison with a cantilever result described in the following section.

METHOD

As described in the previous paper, governing equations are derived to determine the motion of the fiber model in a flow field on assumptions of an infinitely dilute system, no hydrodynamic interaction, and a low Reynolds number of a particle. Consider the fiber model immersed in a Newtonian fluid of viscosity [[Eta].sub.0] and subjected to a macroscopic flow v(r),

v(r) = [Kappa] [center dot] r, (13)

where [Kappa] is the velocity gradient tensor, and r is the global coordinate. The hydrodynamic force and torque operate on each sphere of the fiber model, and tangential friction force operates at the contact point between paired spheres. The hydrodynamic force and torque are assumed to be proportional to the relative velocity and the angular velocity of the sphere with respect to the macroscopic flow, respectively (10). Let [r.sub.i] and [[Theta].sub.i] be the position and the angle of the sphere i. For example, Cartesian coordinates (x, y, z) are used for [r.sub.i] and the Euler angles ([Theta], [Phi], [Psi]) are used for [[Theta].sub.i]. Also, let [v.sub.i] and [[Omega].sub.i] be the translational velocity and the angular velocity of the sphere i. Then the translational friction force and the angular friction torque are represented by

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

where 6[Pi][[Eta].sub.0] a is the Stokes' friction constant, 8[Pi][[Eta].sub.0] [a.sup.3] is the rotational friction constant, and [Omega]([r.sub.i]) is the macroscopic rotational velocity of the fluid,

[Omega]([r.sub.i]) = 1/2 rotv([r.sub.i]). (16)

This hydrodynamic drag is, of course, a simple estimation. A similar approximation using point force and torque was utilized in the theory of dynamics of rodlike polymers (10), in which the sphere array was called a "shish-kebab model." In that discussion, the rotational friction constant of cylinder was estimated for the sphere array model, and the results agreed with the precise hydrodynamic calculation for the cylinder by a correction factor. We recognize that the coefficients of Eqs 14 and 15 are not strict for a cylindrical particle. With the help of these simple approximations, however, PSM can conveniently treat deformation and fraction of fibers in a flow field. We shall resolve the discrepancy between the fiber model and a strict cylinder, and improve the accuracy of PSM in future work.

Let [f.sub.ij] be the tangential friction force exerted on sphere i by sphere j at their contact point. The friction force [f.sub.ji] exerted on sphere j by sphere i is equal to [-f.sub.ij]. Then, the translational and rotational motion equations are written as

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted],

where m is the mass of the sphere of density [Rho], or 4[Pi] [a.sup.3][Rho]/3, and [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted] are the stretching force, bending torque, and twisting torque, respectively. The summation terms in Eqs 17 and 18 consider both sides of the neighbor of the sphere i.

To determine [f.sub.ij], we impose the nonslip condition, that is, the translational velocities of each sphere are the same at their contact point. Then this condition is written as

[v.sub.i] + a[[Omega].sub.i] x [n.sub.ij] = [v.sub.j] + a[[Omega].sub.j] x [n.sub.ji] (19)

where [n.sub.ij] is the unti vector defined as

[n.sub.ij] = ([r.sub.j] - [r.sub.i])/[absolute of [r.sub.j] - [r.sub.i]] (20)

Although two adjacent spheres are not in contact when the fiber model is stretched, we still employ the nonslip conditions because of continuity of a real fiber. In practice, the tangential components of the translational velocity are considered to be always the same at the point of contact.

Differentiating Eq 19 with respect to time t,

d[v.sub.i]/dt + a d[[Omega].sub.i]/dt x [n.sub.ij] + a[[Omega].sub.i] x ([[Omega].sub.i] x [n.sub.ij]) = d[v.sub.j]/dt + a d[[Omega].sub.j]/dt x [n.sub.ji] + a[[Omega].sub.j] x ([[Omega].sub.j] x [n.sub.ji]) (21)

and substituting Eqs 17 and 18 into it using the translational velocity [v.sub.i](t) and the angular velocity [[Omega].sub.i](t) at time t, the 3(N - 1) dimensional simultaneous equations for [f.sub.ij] are obtained. After they are solved, Eqs 17 and 18 determine [v.sub.i](t + [Delta]t) and [[Omega].sub.i](t + [Delta]t) at time t + [Delta]t by using a finite differential technique. Also, the position [r.sub.i](t + [Delta]t) and the angle [[Theta].sub.i](t + [Delta]t) at time t + [Delta]t are obtained by

[r.sub.i](t + [Delta]t) = [r.sub.i](t) + [Delta]t[v.sub.i](t) + 1/2 [([Delta]t).sup.2] d[v.sub.i]/dt (22)

and

[[Theta].sub.i](t + [Delta]t) = [[Theta].sub.i](t) + [Delta]t[[Omega].sub.i](t) + 1/2 [([Delta]t).sup.2] d[[Omega].sub.i]/dt (23)

Thus, the time evolution of the fiber model is determined by the following procedure.

(i) An initial configuration of the fiber model and a flow field are set at the initial time, t = 0.

(ii) For a given conformation of each sphere of the fiber model, the stretching force, bending torque, and twisting torque are calculated by Eqs 1, 5, and 9.

(iii) The hydrodynamic force and torque are calculated by Eqs 14 and 15, using the translational and angular velocity of spheres at that time.

(iv) Then, simultaneous equations are constructed for [f.sub.ij] by Eqs 17, 18, and 21, and are solved.

(v) New translational and angular velocities are calculated at time t + [Delta]t by Eqs 17 and 18, and the position and angle of spheres are also calculated at time t + [Delta]t by Eqs 22 and 23.

(vi) The angle of spheres is adjusted to satisfy the nonslip condition of Eq 19.

(vii) The stress induced in each bond of the fiber model is estimated, and the occurrence of fiber fracture is judged.

(viii) By repeating the procedure from (ii) to (vii), we can follow the motion of all spheres, or the motion of the whole of the fiber model, and fracture process of a fiber.

Procedure (vi) is introduced to remove numerical errors, as described in the previous paper. The non-slip conditions may be broken by numerical errors because of a finite differential technique. To remove this problem, the angles of spheres are slightly adjusted at each time step so that the nonslip conditions are satisfied, in a fashion analogous to that adopted by Doi and Chen (11) for the simulation of the aggregating system. They adjusted the positions of spheres in order to maintain contact. The spheres are slightly rotated from [[Theta].sub.i] to [[Theta].sub.i] + [Delta][[Theta].sub.i] in such a way that the value of the following function becomes a minimum:

[Epsilon] = [Sigma] [([Delta][[Theta].sub.i]).sup.2] (24)

APPLICATION

Deflection Curve

The validity of the fiber model and formulation described above was confirmed by using a theoretical solution. The deflection curve of a cantilever beam with a load concentrated at its end is well known for its theoretical solution (12). Therefore, we calculate the deflection curve of a cantilever beam using the fiber model by the particle simulation method.

Consider a cylindrical beam, of radius a, length L, and Young's modulus E, supported at one end, as shown in Fig. 2. When the external force F is exerted at the other end, the beam deflects slightly. Choosing the coordinates as shown in Fig. 2, the deflection y and the slope [Theta] of the central axis at position x are represented by the following equations:

y = 2F/3[Pi][a.sub.4]E ([x.sup.3] - 3 L[x.sup.2]) (25)

and

[Theta] = -2F/[Pi][a.sup.4]E ([x.sup.2] - 2 Lx) (26)

Theory assumes that the beam is uniform, or the flexural rigidity of the member is constant throughout the beam. Of course, Eqs 25 and 26 are valid for small curvatures.

On the other hand, this problem can be treated by the particle simulation method, as shown in Fig. 3. Here, we consider the beam of length L = 10a by using the fiber model constructed from six spheres. One end sphere is constrained at the wall by freezing translation and rotational motion. With force exerted at the other end sphere, the fiber model bends like a cantilever beam. The deflection y and the slope [Theta] at each sphere are calculated by simulation.

In this stimulation, a, a[[Rho].sup.1/2]/[E.sup.1/2], and [a.sup.2]E are defined as the units of length, time, and force, respectively. The simulation was done by using nondimensional formulas of Eqs 17, 18, and 19. The external force F is chosen to be 0.001 [a.sup.2]E. The deflection curve doesn't depend on the fluid viscosity [[Eta].sub.0] because it is a stationary configuration, so the hydrodynamic force and torque have no physical meaning in this simulation. However, if the external force F is loaded at the initial time t = 0 without the hydrodynamic force and torque, the fiber model would continue to swing endlessly. So, the hydrodynamic force and torque of Eqs 14 and 15, with [[Eta].sub.0] chosen to be a[[Rho].sup.1/2] [E.sup.1/2] and no macroscopic flow, are used for damping the swing, and the stationary configuration of the fiber model is calculated. The time step [Delta]t used in the simulation is 0.01 a[[Rho].sup.1/2]/[E.sup.1/2].

Fiber Fracture

The fracture process of a fiber is simulated in two-dimensional shear flow. Most fibers fracture as a result of the bending stress imparted by the fluid after bending deformation. The fiber model is initially set in the direction of the y-axis, and the center of the fiber model is at the origin of coordinates as shown in Fig. 4. For time t [greater than] 0, a two-dimensional shear flow is applied; the Cartesian components of the flow are given by

[Mathematical Expression Omitted]

where [v.sub.x](r) and [v.sub.y](r) are the velocities in the x direction and the y direction, respectively, and [Mathematical Expression Omitted] is the shear rate. Then the fiber model begins to rotate clockwise, with bending deformation such as shown in Fig. 4. The calculations are carried out for various aspect ratios from 3 to 30 and several bending strengths of a fiber. For each aspect ratio, simulation was carried out for various conditions of the fluid shear stress relative to the Young's modulus of the fiber [Mathematical Expression Omitted], and the critical condition of fiber fracture was detected. Since the presented simulation is carried out in two dimensions, the twisting motion can be neglected.

In this simulation, a, [Mathematical Expression Omitted], and [Mathematical Expression Omitted] are defined as the units of length, time, and force, respectively. Time step [Delta]t used in the simulation is [Mathematical Expression Omitted]. Nondimensional formulas of Eqs 17, 18, and 19 are as follows:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

where Re is the Reynolds number of a sphere, defined as [Mathematical Expression Omitted], and the asterisk means the nondimensional value. The Reynolds number used in the simulation is 0.1 because of neglecting the inertia of a sphere.

Forgacs and Mason derived the theoretical relation of the fluid shear stress required to buckle a fiber to the aspect ratio and the Young's modulus given by

[Mathematical Expression Omitted]

where [r.sub.c] is the aspect ratio of a fiber. This relation is derived on the assumption that a fiber is sufficiently long and thin, with a large [r.sub.c]. This relation means that the threshold shear stress decreases as the length of fiber becomes longer. Equation 31 indicates that the fracture condition depends only on [r.sub.c] and not on the absolute dimension of the fiber. Simulated values of fracture condition of the fluid shear stress related to the Young's modulus of a fiber were compared with this theoretical one.

RESULTS AND DISCUSSION

Deflection Curve

The deflection and the slope of the cantilever beam with concentrated load at the end are shown in Figs. 5 and 6, respectively. The simulated data are discrete because they can be calculated only at the center of each sphere in the particle simulation method. In both Figures, the number in parentheses represents the relative error of the simulated value to the theoretical one. Good agreement was found between simulation and theory in both the deflection and slope of the beam. Relative error is mainly caused by the method of approximation of the fiber model by the cantilever beam. That is to say, the deflection curve of the cantilever beam between adjacent spheres is approximated by an arc in the particle simulation method, while the theoretical one is not an arc. Considering that the relative error is small, the validity of the fiber model and formulation is verified by this simulation.

Fiber Fracture

Representations of the fiber model during simulation are shown in Fig. 7 for the fiber of bending strength 0.1E and an aspect ratio of 20. Simulating conditions of the fluid shear stress to the Young's modulus of a fiber, or [Mathematical Expression Omitted], are (7a) 6.4 x [10.sup.-5] and (7b) 8.4 x [10.sup.-5]. The fiber rotates clockwise with an S-shaped bending deformation and doesn't fracture in the case of (7a). On the other hand, the fiber fracture occurs at the time [t.sup.*] = 45 in the case of (7b). The fiber model decomposes into two six-sphere models and an eight-sphere one. Simulation was carried out for various conditions of [Mathematical Expression Omitted], and the critical value of fiber fracture was detected.

Figure 8 shows the behavior of the fiber model for the bending strength 0.2E and an aspect ratio of 20. Fiber fracture doesn't occur in the case of (8a) [Mathematical Expression Omitted], but occurs at the time [t.sup.*] = 45 in the case of (8b) 1.1 x [10.sup.-4]. The fiber model fractures after larger deformation in comparison with the case of the bending strength 0.1E in Fig. 7b. This tendency is notable for the case of the bending strength 0.3E. Figure 9 illustrates the fiber model for the bending strength 0.3E and an aspect ratio of 20. Simulating conditions of [Mathematical Expression Omitted] are (9a) 1.3 x [10.sup.-4] and (9b) 1.4 x [10.sup.-4]. In the case of (9b), the fiber model fractures at the time [t.sup.*] = 46 after large deformation in a snake regime, and decomposes into two seven-sphere models and a six-sphere one.

Critical values of [Mathematical Expression Omitted] calculated by simulation and theoretical ones are plotted as a function of aspect ratios of the fiber in Fig. 10. The fiber model fractures at two nodes and decomposes into three parts in our simulation after bending in an S-shaped manner because of the symmetry of the flow field. However, the fiber buckles at the center of the fiber in Frogacs and Mason's theory. Therefore, the plotted data of Forgacs and Mason's relation were used for twice the aspect of a fiber. The theoretical value of [Mathematical Expression Omitted] breaks down at small aspect ratios because of the assumption of large aspect ratio [r.sub.c]. It is natural that the value of [Mathematical Expression Omitted] at each aspect ratio becomes larger as the strength of the fiber increases. Simulated values of [Mathematical Expression Omitted] required to avoid fiber fracture for the bending strength 0.1E agree with theoretical ones over an aspect ratio of 15. It is understood that the fracture condition after slight deformation for the bending strength 0.1E may be almost the same as the buckling condition.

The critical radius of curvature R for fiber fracture is expressed using thin rod theory as

R/a = E/T (32)

where T is the ultimate tensile strength of the fiber. According to Salinas and Pittman (4), reinforcing materials such as glass fiber and carbon fiber have E/T values between about 30 and 300. On the other hand, the critical radius of curvature for fiber fracture in PSM can be expressed using Eq 8 as

[Mathematical Expression Omitted]

Note the displacement of the bond angle [[Theta].sub.b] is identical to the central angle of curvature at each pair of spheres, R[[Theta].sub.b] = 2a. The fracture condition for the bending strength 0.01E, which is also shown in Fig. 10, is in the range of E/T of reinforcing materials (E/T = 100). The values [Mathematical Expression Omitted] for the fracture condition of 0.01E are slightly shifted to lower values from those of 0.1E because of the brittleness of the fiber. It is concluded that the fracture condition for reinforcing materials lie in the range around 0.1E and 0.01E.

Finally, some simulation results for long fibers of higher aspect ratio are presented. Figure 11a shows the fiber fracture for the bending strength of 0.01E and an aspect ratio of 100 in the case of [Mathematical Expression Omitted]. In this simulation, the fiber was initially set in the x-axis in order to save computing time. The fiber decomposes into three parts - two 20-sphere models and a 60-sphere one. The shortened fiber will continue to decompose if the fluid shear stress is large enough to include the fracture. Figure 11b shows motion of highly flexible fiber of aspect ratio 100. The fiber coiled and entangled as in experimental observations (1). In this simulation overlap was permitted to nonbonded spheres of the fiber model. It is assumed that the overlapped spheres are slightly out of place in the normal direction. The PSM is applicable to long fibers like these; though computing time increases with aspect ratio.

CONCLUSIONS

In this paper, we have developed a method for dynamic simulation of the fracture process of a fiber in a flow field. In this method, a fiber is modeled with bonded spheres and can stretch, bend, and twist, by changing the bond distance, bond angle, and torsion angle between adjacent spheres, respectively. The stress induced in each bond of the fiber model as a result of stretching, bending, and twisting deformation can be formulated using displacement of the bond distance, bond angle, and torsion angle for each pair of spheres, and the Young's modulus or the shear modulus of the fiber. Therefore, the occurrence of fiber fracture can be defined as the moment when the stress induced in the fiber model surpasses the strength of the fiber for each deformation. The motion of the fiber model in a flow field is determined by solving the translational and rotational motion equations for individual spheres under the hydrodynamic force and the torque exerted on them. A critical condition of the flow field required to avoid fiber fracture can be detected by simulation in a flow field. We can obtain information about which size of fibers can be maintained under a certain flow condition.

We verified the validity of this method and formulation by calculating the deflection curve of a cantilever beam with concentrated load at the end, and by comparing the simulated deflection and slope of the beam with the theoretical ones. Good agreement was found in both the deflection and slope of the beam. The fracture process of a fiber in a two-dimensional simple shear flow as simulated on assumptions of an infinitely dilute system, no hydrodynamic interaction, and a low Reynolds number of a particle. The calculated critical conditions of the flow field required to avoid fiber fracture were compared to those of Forgacs and Mason's theory. Simulated values of fracture condition of the fluid shear stress to the Young's modulus of a fiber for the bending stress 0.1E agree with the theoretical ones over an aspect ratio of 15. From these results, it was confirmed that the developed method was able to simulate the fracture process of a fiber in a flow field. The fracture process of fibers in a concentrated system will be elucidated by consideration of interaction among fibers in future paper.

ACKNOWLEDGMENTS

We would like to thank Dr. O. Kamigaito, Dr. T. Kurauchi, and Dr. H. Takahashi of Toyota Central Research & Development Laboratories, Inc., for helpful suggestions throughout the work.

REFERENCES

1. O. L. Forgacs and S. G. Mason, J. Colloid Sci., 14, 457 (1959).

2. O. L. Forgacs and S. G. Mason, J. Colloid Sci., 14, 473 (1959).

3. R. von Turkovich and L. Erwin, Polym. Eng. Sci., 23, 743 (1983).

4. A. Salinas and J. Pittman, Polym. Eng. Sci., 21, 23 (1981).

5. G. B. Jeffery, Proc. R. Soc., A102, 161 (1922).

6. C. A. Stover, D. Koch, and C. Cohen, J. Fluid Mech., 238, 277 (1992).

7. F. Folgar and C. L. Tucker, J. Reinf. Plastics Compos., 17, 330 (1983).

8. F. Folgar and C. L. Tucker, J. Reinf. Plastics Compos., 3, 98 (1984).

9. S. Yamamoto and T. Matsuoka, J. Chem. Phys., 98, 644 (1993).

10. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University, Oxford, England (1986).

11. M. Doi and D. Chen, J. Chem. Phys., 90, 5271 (1989).

12. E. J. Hearn, Mechanics of Materials, 2nd Ed., Pergamon Press, Oxford, England (1985).

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Author: | Yamamoto, Satoru; Matsuoka, Takaaki |
---|---|

Publication: | Polymer Engineering and Science |

Date: | Jun 1, 1995 |

Words: | 5118 |

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