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en:grundlagenhandbuch:faserbruchberechnung [2026/05/28 09:43] – deppe2en:grundlagenhandbuch:faserbruchberechnung [2026/09/04 13:17] (aktuell) – [Modeling] paal
Zeile 27: Zeile 27:
 These tests showed that the profiles of the average glass fiber length along the mixing zone show an exponential decrease. This profile is qualitatively shown in Figure 2. These tests showed that the profiles of the average glass fiber length along the mixing zone show an exponential decrease. This profile is qualitatively shown in Figure 2.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_002.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_002.svg?nolink&500 |}}
  
 **Figure 2:** Qualitative profile of the glass fiber length **Figure 2:** Qualitative profile of the glass fiber length
Zeile 33: Zeile 33:
 For a description of this behavior the following differential equation is used: For a description of this behavior the following differential equation is used:
  
-$$\frac{dl}{dt} = c \cdot l^{\infty} ; \text{with} l = \frac{L - L_{\infty}}{L_0 - L_{\infty}} \tag{1}$$+$$\frac{dl}{dt} = c \cdot l^{\infty} ; \text{with } l = \frac{L - L_{\infty}}{L_0 - L_{\infty}} \tag{1}$$
  
 The origin of this equation refers to the initial glass fiber length $L_0$ with a dimensionless fiber length $l$, which describes the relation of the current glass fiber length $L$ to the initial fiber length $L_0$. The equation profile asymptotically approaches a process-limited fiber length $L_{\infty}$. The further parameters describe the speed $c$ and the sensitivity $α$ in terms of a fiber breakage. The origin of this equation refers to the initial glass fiber length $L_0$ with a dimensionless fiber length $l$, which describes the relation of the current glass fiber length $L$ to the initial fiber length $L_0$. The equation profile asymptotically approaches a process-limited fiber length $L_{\infty}$. The further parameters describe the speed $c$ and the sensitivity $α$ in terms of a fiber breakage.
Zeile 41: Zeile 41:
 For a solution of this differential equation and in particular for determining the process-limited fiber length $L_{\infty}$ the preliminary considerations about flow processes in twin-screw extruders are used. For a solution of this differential equation and in particular for determining the process-limited fiber length $L_{\infty}$ the preliminary considerations about flow processes in twin-screw extruders are used.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_003.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_003.svg?nolink&500 |}}
  
 **Figure 3:** Shear flow stress on a fiber **Figure 3:** Shear flow stress on a fiber
Zeile 55: Zeile 55:
 Here $E$ is the fiber-Young's-modulus, $I$ the minimal moment of inertia of the fiber and $L$ the fiber length [[en:grundlagenhandbuch:faserbruchberechnung#references |[BG97]]]. Here $E$ is the fiber-Young's-modulus, $I$ the minimal moment of inertia of the fiber and $L$ the fiber length [[en:grundlagenhandbuch:faserbruchberechnung#references |[BG97]]].
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_004.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_004.svg?nolink&300 |}}
  
 **Figure 4:** Euler buckling **Figure 4:** Euler buckling
Zeile 61: Zeile 61:
 If the force from the flow dominates the critical compressive force, the fiber will break in the centre. In the next step half of the fiber length is regarded. If the ratio of these forces is still bigger than one, the fiber will again break in the centre. This procedure will go on until the ratio is smaller than one, so that the critical compressive force dominates the force from the flow and the process-limited fiber length is found. This approach shows that the final fiber length basically depends on the shear stress in the system and the fiber geometry. This procedure is illustrated in the following figure. If the force from the flow dominates the critical compressive force, the fiber will break in the centre. In the next step half of the fiber length is regarded. If the ratio of these forces is still bigger than one, the fiber will again break in the centre. This procedure will go on until the ratio is smaller than one, so that the critical compressive force dominates the force from the flow and the process-limited fiber length is found. This approach shows that the final fiber length basically depends on the shear stress in the system and the fiber geometry. This procedure is illustrated in the following figure.
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_005.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_005.svg?nolink&500 |}}
  
 **Figure 5:** Calculation process for the final fiber length **Figure 5:** Calculation process for the final fiber length
Zeile 71: Zeile 71:
 $$w\left(\frac{x}{L}\right) = \frac{F \cdot L^2}{48 \cdot E \cdot I} \left(3 \cdot \frac{x}{L} - 4 \cdot \left(\frac{x}{L}\right)^2\right) \tag{4}$$ $$w\left(\frac{x}{L}\right) = \frac{F \cdot L^2}{48 \cdot E \cdot I} \left(3 \cdot \frac{x}{L} - 4 \cdot \left(\frac{x}{L}\right)^2\right) \tag{4}$$
  
-{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_006.png?nolink |}}+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_faserbruchberechnung_006.svg?nolink&500 |}}
  
 **Figure 6:** Resulting bending line at stressed fiber **Figure 6:** Resulting bending line at stressed fiber