2. Introduction

9.5.1 Direct Simulation Methods

Long fibers are not perfectly rigid, and therefore may exhibit deformation (bending and/or twisting) during processing. Many authors use direct simulations to model the flexible fiber motion at the particle level, where the individual flexible fiber model consists of multiple rigid parts connected with each other. The bending and twisting deformation is reflected by allow-ing movement of the rigid parts relative to each other.

Several physical models (Yamamota and Matsuoka [405–407]; Ross and Klingenberg [316];

Joung et al. [183, 184]; Tang and Advani [355]; Schmid et al. [323]; Wang et al. [391]; Kittipoom-wong and Jabbarzadeh [204]) have been employed to represent flexible fibers (Figure 9.2), from which mathematical models can be constructed based on force balances. We use the model developed by Joung et al. [183, 184] as an example, where the flexible fiber suspension is mod-eled by a chain of spherical beads in viscous flow. The beads are linked with each other by inextensible connectors. Only the beads interact with the fluid. The connectors do not inter-act with the fluid, but serve to transmit internal forces and maintain the configuration of the fiber. Within each bead is a joint linking two connectors. Each joint allows limited bending and torsion, and produces a restoring moment acting to straighten the fibers when they are deflected from equilibrium. Given the Young’s and shear moduli, one can calculate the inter-nal bending moment, the torsion moment, and the bend and twist angles for each joint. Joint deflections are caused by fiber interaction forces.

There are two types of forces acting on each sphere: external forces and internal forces. Ex-ternal forces arise from viscous drag (including long-range velocity disturbances from other

beads), and short-range lubrication forces between beads when they are in close proximity.

The internal force is the connector tension. Since the connectors are rigid and inextensible, the “tension” is simply the reactive force that exactly opposes external forces acting in the di-rection of the connector. Once the external forces are determined, one can also calculate the moments acting at each joint. The internal moment at a joint is equal to the moment bal-ance generated by forces on beads to either side of the joint. The external drag forces can be calculated by

F^µ_drag= −ζ[˙r^µ− (u^µ+ ˜u^µ)]. (9.16)

This equation describes the drag force acting on beadµ located at position r^µ. In Equation 9.16,ζ is the Stokes drag coefficient given by ζ = 6πηsa, whereηsis the solvent viscosity and a is the radius of the spherical bead. According to the expression of Equation 9.16, the force is proportional to the difference between the bead velocity ˙r^µand the solvent velocity (u^µ+ ˜u^µ) at beadµ. The velocity u^µis determined assuming a homogeneous velocity gradient L tensor:

u^µ= L · r^µ. The velocity ˜u^µis the cumulative perturbation velocity at beadµ from all other spheresν in the suspension, described by

˜ u^µ=X

ν̸=µΩ^µν· F_drag^ν , (9.17)

where F_drag^ν is the drag force acting on all other suspension spheresν, and Ω^µνis called the Oseen tensor and is given by

Ω^µν= 1

Short-range lubrication forces are calculated using the following equation:

F_lub^µν= −3πηs

Since all interactions are between beads only, and interactions between beads are governed by the same laws regardless of whether they belong to the same fiber or different fibers, one does not need to distinguish beads of different fibers while determining external forces.

For a rigid fiber, the orientation of the fiber is described by a single unit direction vector along the major axis of the fiber. Similarly, the orientation of a flexible fiber can be characterized by a normalized end-to-end vector p of the bead-chain model. In simulations, once the fiber rotation and translation rates are known, their new position and orientation can be calculated for the current time step. The vector p is overlaid by the configuration of chain-of-beads. The interaction forces are calculated using Equations 9.16 and 9.19. Then an additional calculation is performed to determine the deflection of the bead-chain along the fiber length. Fiber defor-mation may lead to a further small change in the end-to-end orientation vector p. The orien-tation results can be used to predict rheological properties of the suspension. Joung et al. [184]

have shown that there is a distinct increase in bulk viscosity as fiber curvature is increased.

138 9 Long Fiber-Filled Materials

Figure 9.2 Schematic representations of flexible fiber models

Figure 9.3 Bead-rod model of Strautins and Latz [346]

9.5.2 Continuum Modeling

Direct simulations are useful to explore the particle-particle interactions, such as long and short-range hydrodynamic effects. However, they are currently not efficient for modeling the entire molding process. A continuum model that accounts for the orientation evolution of semi-flexible fibers is that proposed by Strautins and Latz [346], which is derived from the so-called bead-rod model. In this model, a semi-flexible fiber is modeled as two connected rods of orientation p and q, each of length l_rodthat may flex about a central pivot point, as shown in Figure 9.3.

The moments of the p and q vectors, with respect to the orientation distribution function ψ(p,q,t), are defined as

a^(m,n)= Z Z

p^mqⁿψ(p,q,t)dpdq, (9.20)

where p^mqⁿis the (m + n)-th rank tensor. Only the following 4 cases are important:

a_{i j}^(2,0)= Z Z

pipjψ(p,q,t)dpdq, (9.21)

a_{i j}^(1,1)= Z Z

piqjψ(p,q,t)dpdq, (9.22)

a_i^(1,0)= Z Z

p_iψ(p,q,t)dpdq, (9.23)

a_{i j kl}^(4,0)= Z Z

p_ip_jp_kp_lψ(p,q,t)dpdq. (9.24)

One can then obtain the equation for orientation tensor of the long fibers as follows (see Ort-man et al. [274] for details):

D a^(2,0)_{i j}

D t − Li ka^(2,0)_{k j} − Lj ka^(2,0)_ki + 2 Lkla_{i j kl}^(4,0)

=l_rod 2

³

a^(1,0)_i m_j+ mia^(1,0)_j − 2mka^(1,0)_k a^(2,0)_{i j} ´

− 2κ³

a^(1,1)_{i j} − a^(1,1)_kk a^(2,0)_{i j} ´

+ 2CIγ¡δ˙ i j− 3a^(2,0)_{i j} ¢ , (9.25)

140 9 Long Fiber-Filled Materials

D a_{i j}^(1,1)

D t − Li ka_{k j}^(1,1)− Lj ka_ki^(1,1)+ 2 Lkla^(2,0)_{l k} a^(1,1)_{i j}

=l_rod 2

³

a_i^(1,0)mj+ mia^(1,0)_j − 2mka^(1,0)_k a_{i j}^(1,1)´

− 2κ

³

a^(2,0)_{i j} − a^(1,1)_kk a_{i j}^(1,1)´

− 4CIγa˙ ^(1,1)_{i j} , (9.26)

D a_i^(1,0)

D t − Li ka_k^(1,0)+ a_kl^(2,0)L_{l k}a_i^(1,0)=l_rod 2

³

mi− a^(1,0)_i mja^(1,0)_j ´

− κa_i^(1,0)

³

1 − a_kk^(1,1)´

− 2CIγa˙ ^(1,0)_i , (9.27) with

m_i= ∂²u_i

∂xj∂xk

a^(2,0)_{j k} , (9.28)

where uiis the velocity vector. Within these equations, CI is the interaction coefficient as de-fined in the Folgar-Tucker model, andκ is the resistive bending potential coefficient. When the value ofκ decreases, the fiber becomes more flexible. Conversely, as the value of κ in-creases, the model behaves more like a rigid fiber, and whenκ → ∞, the solution of the model converges towards the Folgar-Tucker model for high aspect ratio rigid fibers.

Ortman et al. [274] proposed a stress law for the bead-rod model as follows:

σ^bead−rod= −pI + 2ηm¡D + f1φD + f2a^(4,0)₄ : D¢

+ 3

4l_rod² ηmκφartr(r)¡a^(2,0)− R¢, (9.29)

where p is the isotropic pressure,ηmis the matrix viscosity, D is the deformation rate tensor, f1and f2are empirical parameters to be determined by fitting rheological data of long-fiber suspensions,φ is the volume fraction of fibers, r is a second order, dimensional end-to-end orientation tensor, defined as the second moment of the end-to-end vector, l_rod(p − q), with respect toψ:

r ≡ Z Z

l_rod² (p − q)(p − q)ψ(p,q, t)dpdq = 2l_rod² (a^(2,0)− a^(1,1)), (9.30) and R is defined as a normalized and dimensionless end-to-end orientation tensor:

R ≡ r

tr(r)=a^(2,0)− a^(1,1)

1 − tr(a^(1,1)). (9.31)

The last term in Equation 9.29 is the bending stress term. For a perfectly straight fiber, p = −q, we have a^(2,0)= R, so that the bending stress term vanishes and the equation reduces to the constitutive model for rigid fiber suspensions (Lipscomb et al. [227]).

10 Crystallization

An essential requirement for improving injection molding simulation is the incorporation of crystallization effects. In Section 5.3.5, we remarked that the use of a single no-flow or transi-tion temperature is unjustified for semi-crystalline materials. It can lead to errors in fill pattern prediction and calculation of shrinkage and warpage. Moreover, if we are to be able to predict properties of semi-crystalline materials after molding, and when subjected to variable envi-ronments, we need to incorporate crystallization calculations explicitly.

A major research trend in the simulation of injection molding is to predict properties of the fi-nal product through the prediction of the processing-induced material morphology. For amor-phous polymers, the term morphology refers to molecular orientation and deformation. For semi-crystalline polymers, morphology also includes the degree of crystallinity, the shape, the sizes, and the orientation of the crystalline structures. This chapter focuses on prediction of morphology for semi-crystalline polymers in injection molding.

10.1 Crystallization Kinetics in