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en:grundlagenhandbuch:faserbruchberechnung [2026/06/13 14:48] – [Process-limited Fiber Length] 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 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 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