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The energy needed to break the fibers in In order to perform the calculation of the fiber length reduction, the checkbox „Fiber degradation“ must be activated in the calculation window. With the activation, further calculations will be performed, because the calculated values are needed for the fiber length reduction calculation.

The calculation takes place taking into account the temporal regressive decrease of the number average fiber length during shearing. This is described by the following equation, which also uses Kloke to describe the fiber length reduction in the twin screw extrusion process:

Eq.1 $$\frac{dl}{dt_v}=-c \cdot l^2$$

with $$l = \frac{L-L_∞}{L_0-L_∞}, 0 < l ≤ 1$$

The solution of Equation 1 and the insertion of the boundary conditions produces the following descriptive equation:

Eq.2 $$l(t_V) = \frac{1}{\frac{t_V}{t_0}+1}$$

where $t_0$ is the time constant at which a halving of the fiber length occurs. The time constant is determined by an energy-related view of the fiber breakage, taking into account the flow processes during plasticization. Energy is used to break a fiber. This energy causes buckling in line with Euler's second buckling case. Through the buckling process, the fiber breaks in the middle, thus halving the fiber length. The energy needed for the buckling can only be introduced into the process via the screw motion in the form of dissipation. Generally speaking, the dissipation energy generated in the process can be described as follows:

Eq.3 $$E_{diss} = η \cdot \dot γ^2 \cdot V \cdot t_V$$

Where $V$ is the melt volume of the area being studied, $η$ the viscosity of the plastic, $\dot γ$ the shear rate and $t_V$ the dwell time.

The given melt volume can be expressed for Euler's second buckling case as follows, assuming a linear elastic behavior:

Eq.4 $$E_{break} = V \cdot Φ \cdot \frac{E \cdot ε_B}{2}$$

By using the energetic equations the time constant $t_0$ is determined and used in equation 2. This results in the descriptive equation for the fiber length reduction, which is used in REX / PSI (Eq. 5).

Eq.5 $$l(t_V) = \frac{κ \cdot Φ \cdot E \cdot ε_B }{t_V \cdot ζ \cdot 2 \cdot η \cdot \dot γ^2 + κ \cdot Φ \cdot E \cdot ε_B}$$

• Tensile strength
• Elongation at break
• Modulus of elasticity
• Density
• Fibre weight fraction

• Number of weighted fibre lengths:

$$L_n = \frac{∑_in_i \cdot l_i}{∑_i n_i}$$

• Volume-weighted fibre length:

$$L_v = \frac{∑_i n_i \cdot {l_i}^2}{∑_i n_i \cdot l_i}$$

• Frequency distribution of fibre lengths: $l_i$

When calculating the fibre length reduction in the plasticising process, a distinction is made between different zones and the damage mechanisms that occur there.

When processing short fibre-reinforced granulates, fibres are partially exposed from the granulate grain at the interface between the solid bed and the melt film. The melt flows around these fibres, which are anchored on one side, and can fail due to this stress. This failure does not occur with long fibre-reinforced materials. Due to the rod shape of the LGF granules and the high l/d ratio, vertically oriented fibres do not occur in the melt film.

A fracture criterion is defined to check whether the fibres clamped at the interface break. This sets the bending stress exerted by the melt on the individual fibre in relation to the tensile strength of the fibre. If the bending stress of the fibre is higher than the tensile strength of the fibre, the fibre is assumed to have failed.

Breakage condition:

$$\frac{σ_{hydro}}{R_m} > 1$$

In addition to fibre breakage at the interface, damage occurs in the melt film and in the melt vortex to fibres that move freely in the melt. The fibre breakage that occurs here is calculated as follows:

It is fundamentally assumed that the failure occurs due to buckling of the fibres under compressive load. The fracture criterion thus results from the critical buckling force calculated according to the second Euler buckling case and the hydrodynamic force $F_i$ exerted by the melt on the fibre.

$$\frac{F_i}{F_{critical buckling force}} = \frac{8 \cdot ζ \cdot η_m \cdot {l_i}^4}{π^3 \cdot E_f \cdot {d_f}^4} (-D:A) > 1$$

$$F_i = \frac{ζ \cdot η_m \cdot {l_i}^2}{8}(-D:A)$$

$$F_{Buckling} = \frac{π^3 \cdot E_f \cdot {d_f}^4}{64 \cdot {l_i}^2}$$

$D$ = deformation tensor

$A$ = orientation tensor

$ζ$ = drag coefficient

The probability of breakage $P_i$ is defined depending on whether the breakage criterion is fulfilled. The fibre interaction coefficient $C_{FB}$ takes into account the influence of the fibre-fibre interaction or the fibre volume fraction on the fibre breakage. In addition to the fibre volume fraction, the fibre interaction coefficient is also dependent on the flow velocity in the channel and the fibre length.

$$ \mathrm{P_i} = \begin{cases} 0 & \text{für } \frac{F_i}{F_{crit}} \leq 1 \\ C_{FB,i}*\left( 1-exp\left( 1- \frac{F_i}{F_{Buckling}}\right) \right) & \text{für } \frac{F_i}{F_{crit}} > 1 \end{cases} $$

The break position of the fibre along the length $l_i$ is described with the assumption of a normal distribution and the definition of a break transition matrix.

$$R_{ji} = normpdf (l_j, \frac{l_i}{2}, Sl_i)$$

Finally, the transition matrix is normalised to the breakage probability.

$$∑_j R_{ji} = 2 P_i$$

The breakage model used here is based on Phelps' model from 2009 for calculating fibre breakage in the injection mould. However, the model was extended to include the fibre interaction coefficient by modifying the calculation of the probability of breakage.

For the various sections of the screw channel, the fibre breakage that occurs and the resulting fibre lengths in the individual sections are calculated and then added using volumetric weighting.

The following is an overview of the different variants that occur and which breakage calculations are taken into account.

Calculation of fibre breakage in the mixed solid bed + melt film

KGF: Breakage model for calculating fibre breakage at the interface + breakage model for melt flow in the melt film

LFT: Fracture model for melt flow in the melt film

Calculation of fibre breakage in the mixed area of solid bed + melt film + melt vortex

KGF: Fracture model for calculating fibre breakage at the interface + fracture model for melt flow in the melt film + fracture model for melt flow in the melt vortex

LFT: Fracture model for melt flow in the melt film + fracture model for melt flow in the melt vortex

Fibre breakage calculation in the pure melt range

Fracture model for melt flow

• Regular calculation
• Throughput
• Pressure profile
• Melting profile
• Temperature profile
• Power and wall shear stress
• Residence time
• Mixed indicators
• Material degradation
• Fiber length degradation
• Degassing index calculation
• Stress Analysis
• Drag pressure flow
• Solids conveying
• Processing of mixtures
• Wall-slipping Materials
• Grooved bush
• Compression ratio & Pumping ratio