Publisher Summary

Short-fiber composites are reinforced by particles that are slender, but whose length is small compared to the overall dimensions of the part. They include fiber-reinforced thermoplastics for injection molding and extrusion, sheet molding compounds and short-fiber GMTs (glass-mat reinforced thermoplastics). Short-fiber composites can be processed by fast, highly-automated methods such as injection or compression molding. In these processes, the fibers and the resin are transported as a suspension into the mold cavity. The properties of a short-fiber composite part are highly dependent on the way the part is manufactured. As the resin or molding compound deforms to achieve the desired shape, the orientation of the fibers is changing. Fiber orientation changes stop when the matrix solidifies and the orientation pattern becomes part of the microstructure of the finished article. To design short-fiber composite parts effectively, the way that processing-induced fiber orientation influences the properties of the finished part must be anticipated. Hence, the flow of fiber suspensions needs to be understood in order to predict the orientation distribution of the fibers.

2.1 Introduction

Short-fiber composites are reinforced by particles that are slender, but whose length is small compared to the overall dimensions of the part. They include fiber-reinforced thermoplastics for injection molding and extrusion, sheet molding compounds and short-fiber GMTs (glass-mat reinforced thermoplastics). Short-fiber composites can be processed by fast, highly-automated methods such as injection or compression molding. In these processes, the fibers and the resin are transported as a suspension into the mold cavity. The properties of a short-fiber composite part are highly dependent on the way the part is manufactured. As the resin or molding compound deforms to achieve the desired shape, the orientation of the fibers is changing. Fiber orientation changes stop when the matrix solidifies and the orientation pattern becomes part of the microstructure of the finished article. To design short-fiber composite parts effectively, the way that processing-induced fiber orientation influences the properties of the finished part must be anticipated. Hence, the flow of fiber suspensions needs to be understood in order to predict the orientation distribution of the fibers.

The best way to learn about fiber motion is first to analyse how a single fiber behaves in a flowing suspension. Hence, in this chapter, a brief overview is provided first of the background to the motion of single fibers in an infinite sea of liquid and basic orientation characterization methods as a foundation for the later topics. A review of the early work on fiber–fiber interactions, most notably those of S. G. Mason and co-workers is presented. This is followed by a description of more recent work on fiber–fiber interactions. The current state of the art as far as fiber–wall interactions are concerned is examined in the following section. Finally, a summary and future outlook on the subject of fiber–fiber and fiber–wall interactions is discussed. The reader is referred to Chapters 3 and 6 in Advani1 for additional discussion on short-fiber suspensions.

During the flow of suspensions, the suspended fibers, which are typically non-spherical, tend to orient themselves in certain preferred directions and translate with the fluid. The particles are normally assumed to translate affinely with the fluid, that is, the velocity of the center of each particle is equal to the value of the unperturbed velocity of the fluid at that spatial location. However, the rotation of the particles may lead to a distribution of orientations at different locations, due to the flow field and hence is referred to as flow-induced orientation.

A fiber, which is immersed in a flowing fluid, is subjected to the local velocity gradients in the fluid. Due to these gradients a fluid element will experience deformation which includes rotation and stretching. A fiber will also rotate as the fluid deforms but will not change in length. The disturbance in the flow field caused by an immersed fiber decays as the distance from it increases. If there is another fiber or a wall present within the area where the disturbance is felt, the motion of the fiber is affected. The hydrodynamic force and torque felt by the fiber depend on its shape and the local rheological properties of the suspending fluid. This force and torque determine the rotational and translational velocities of the fiber.

The problem of motion of particles suspended in a flowing liquid has been studied in detail.2–4 However, most of the existing theoretical models pertain only to the motion of a single particle. Analytical results are not available to predict the motion of particles in flows in which they are allowed to interact with other particles. Several researchers have observed that the qualitative behavior of rigid rod-like fibers varies as the concentration of the fibers in the suspension increases.5–7 This is mainly because the fibers interact with each other. It is necessary to understand the mechanics of fiber–fiber interactions to be able to quantify its effect and judge its impact on the structure of a flowing suspension. The orientation state of a fiber suspension plays an important role in deciding the material properties in both the processing phase when it is in a fluid form and in the post-manufacturing phase when it is in a solid form. It is well known that physical properties such as conductivity and mechanical properties such as elastic modulus are greater in the direction parallel to a fiber than in a direction perpendicular to the fiber. Similarly, the viscosity of a suspension is also a strong function of the orientation state. Therefore the orientation state of a short fiber suspension/composite is a topic of immense interest.

Usually, fiber suspensions are classified into three concentration regimes according to the fiber volume fraction, φv, and fiber aspect ratio, rf defined as the length l/diameter d (Fig. 2.1):

v<1rf2Dilute [1a]

rf2<ϕv<1rfsemi-concentrated [1b]

rf<ϕvconcentrated [1c]

The semi-concentrated and the concentrated regimes constitute what was previously referred to in this chapter as the non-dilute regime. For the case of cylindrical fibers

l3=4πrf2ϕv [2]

where n is the number of fibers per unit volume.

The effect of the fiber concentration on the fiber–fiber interactions may be summarized as follows. In the dilute region, the spacing between the fibers is greater than their length and the fibers are free to move and rotate and hydrodynamic interactions between the fibers are rare. Hence, one may use Jeffery’s theory to model the fiber rotations in dilute suspensions. In semiconcentrated suspensions, the spacing between the fibers is less than l but greater than d and hydrodynamic interactions are frequent. In the concentrated region, the spacing between fibers is of the order of d and non-hydrodynamic effects, such as ‘physical’ collisions between fibers may become important. The physics of the angular motion of particles in dilute and non-dilute suspensions is vastly different.

2.2 Single fiber motion

In the absence of Brownian motion, the translation and rotation of a neutrally buoyant ellipsoid suspended in a Newtonian fluid undergoing a homogeneous flow was solved by Jeffery.8 Consider an ellipsoid of aspect ratio re whose centroid is located at the origin and whose major axis is in the direction p shown in Fig. 2.2.

Jeffery’s solution for the angular velocity of this ellipsoid immersed in a simple shear flow x=γ˙y is given by

˙=(re2−1re2+1)γ˙4sin 2θ sin 2ϕ [3a]

˙=−γ˙(re2+1)(re2sin2ϕcos2ϕ). [3b]

From the above, it can be seen that the angular velocities depend linearly on the shear rate. This is as expected because the problem being solved is linear (Stokes flow). Therefore, the angular change in the ellipsoids over a period of time is dependent only on the total strain γ˙t) and not on the time t and strain rate ˙ individually. It can also be seen that ˙ is maximum at ϕ = π/2 when the ellipsoid is perpendicular to the flow and minimum at...