3. Substitute the result into the kinematic relation stating that the pull-out displacement is equal to the integral of fiber strain along the debonded length
3. Substitute the result into the kinematic relation stating that the pull-out displacement is equal to the integral of fiber strain along the debonded length
4. Identify the integration constants by applying boundary conditions: equilibrium at loaded end and compatibility and smoothness at the end of the debonded zone $x = a$
4. Identify the integration constants by applying boundary conditions: equilibrium at loaded end and compatibility and smoothness at the end of the debonded zone $x = a$
These step deliver the pull-out curve as a square root function
These step deliver the pull-out curve as a square root function
\begin{align}
\begin{align}
P = \sqrt{p \bar{\tau} E_\mathrm{f} A_\mathrm{f} w}
P = \sqrt{p \bar{\tau} E_\mathrm{f} A_\mathrm{f} w}
Let us utilize the the derived model to simulate the test results of the RILEM pull-out test
Let us utilize the the derived model to simulate the test results of the RILEM pull-out test
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| Symbol | Unit |Description |
| Symbol | Unit |Description |
|:- |:- |:- |
|:- |:- |:- |
| $E_\mathrm{f}$ | MPa | Young's modulus of reinforcement |
| $E_\mathrm{f}$ | MPa | Young's modulus of reinforcement |
| $\bar{\tau}$ | MPa | Bond stress |
| $\bar{\tau}$ | MPa | Bond stress |
| $A_\mathrm{f}$ | mm$^2$ | Cross-sectional area of reinforcement |
| $A_\mathrm{f}$ | mm$^2$ | Cross-sectional area of reinforcement |
| $p$ | mm | Perimeter of contact between concrete and reinforcement |
| $p$ | mm | Perimeter of contact between concrete and reinforcement |
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### **Observation**
### **Observation**
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- The measured displament at the loaded and unloaded end are different
- The measured displament at the loaded and unloaded end are different
- Their difference increases with increasing bond length $L_\mathrm{b}$
- Their difference increases with increasing bond length $L_\mathrm{b}$
- The shape of the pull-out curve has a shape of a square root function
- The shape of the pull-out curve has a shape of a square root function
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### **Question**
### **Question**
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- Can the above derived model describe the debonding process correctly?
- Can the above derived model describe the debonding process correctly?
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# Look inside the specimen using the model
# Look inside the specimen using the model
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The parameters of the above experiment are specified as follows
The parameters of the above experiment are specified as follows
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
ds=16
ds=16
A_f=(ds/2)**2*3.14# mm^2 - reinforcement area
A_f=(ds/2)**2*3.14# mm^2 - reinforcement area
L_b=5*ds# mm - bond length
L_b=5*ds# mm - bond length
E_f=210000# MPa - reinforcement stiffness
E_f=210000# MPa - reinforcement stiffness
p_b=3.14*ds# mm - bond perimeter
p_b=3.14*ds# mm - bond perimeter
w_max=0.12# mm - maximum displacement
w_max=0.12# mm - maximum displacement
```
```
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**Construct the model:** To study the model behavior import the class `PO_ELF_RLM`, construct it with the defined parameters and run the `interact` method
**Construct the model:** To study the model behavior import the class `PO_ELF_RLM`, construct it with the defined parameters and run the `interact` method
----> 3 po = PO_ELF_RLM(E_f=E_f, L_b=L_b, p=p_b, A_f=A_f, w_max=w_max)
NameError: name 'E_f' is not defined
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**Remark:** that the length $L_b$ is not the end of the bond zone. It only measures the slip at the position $x = L_\mathrm{b}$ from the loadedend. However, the debonding process can continue beyond this length.
**Remark:** that the length $L_b$ is not the end of the bond zone. It only measures the slip at the position $x = L_\mathrm{b}$ from the loadedend. However, the debonding process can continue beyond this length.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
po.interact()
po.interact()
```
```
%% Output
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## Let's learn from the model
## Let's learn from the model
Exercise the relation between $P$ and $\tau(x)$ and between $w$ and $\varepsilon(x)$.
Exercise the relation between $P$ and $\tau(x)$ and between $w$ and $\varepsilon(x)$.
1. What is the meaning of the green area?
1. What is the meaning of the green area?
2. What is the meaning of the red area?
2. What is the meaning of the red area?
3. What is the meaning of the slope of the green curve?
3. What is the meaning of the slope of the green curve?
4. Is it possible to reproduce the shown RILEM test response using this "frictional" model?
4. Is it possible to reproduce the shown RILEM test response using this "frictional" model?
4. What is the role of debonded length $a$ in view of general non-linear simulation?
4. What is the role of debonded length $a$ in view of general non-linear simulation?
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The analytical solution of the pull-out from rigid matrix for constant bond-slip law
The analytical solution of the pull-out from rigid matrix for constant bond-slip law
explained in the notebook [2.1 Pull-out of elastic fiber from rigid matrix](2_1_1_PO_observation.ipynb) can be adapted to
explained in the notebook [2.1 Pull-out of elastic fiber from rigid matrix](2_1_1_PO_observation.ipynb) can be adapted to
several practically relevant configurations that occur in brittle-matrix compostes. The notebook
several practically relevant configurations that occur in brittle-matrix compostes. The notebook
addresses the following four configuration of pull-out
addresses the following four configuration of pull-out
- Rigid matrix
- Rigid matrix
- Elastic matrix
- Elastic matrix
- Short fiber
- Short fiber
- Clamped fiber
- Clamped fiber
This notebook summarizes these four configurations using interactive web-apps to show their qualitatively different behavior. Using the prepared models,the correspondence between the pull-out curve $P(w)$ and the debonding process is visualized in terms of the stress and strain profiles along the bond length. In all models, the following material parameters are be used.
This notebook summarizes these four configurations using interactive web-apps to show their qualitatively different behavior. Using the prepared models,the correspondence between the pull-out curve $P(w)$ and the debonding process is visualized in terms of the stress and strain profiles along the bond length. In all models, the following material parameters are be used.
- The four configurations of the pull-out show that the pull-out curves have qualitatively different shape and explain the externally observered by visualizing the stress field development during the loading history.
- The four configurations of the pull-out show that the pull-out curves have qualitatively different shape and explain the externally observered by visualizing the stress field development during the loading history.
- The pull-out curves calculated using the models 1 Rigid Matrix (PO-ELF-RLM) and 2 Elastic Matrix (PO-ELF-ELM) are not affected by the bond length $L_\mathrm{b}$.
- The pull-out curves calculated using the models 1 Rigid Matrix (PO-ELF-RLM) and 2 Elastic Matrix (PO-ELF-ELM) are not affected by the bond length $L_\mathrm{b}$.
- On the other hand, bond length strongly affects the maximum force and descending branch in in model 3 Short Fiber (PO-ESF-RLM). Bond length also affects the value of the final stiffness in the model 4 Clamped Fiber (PO-ECF-ECM)
- On the other hand, bond length strongly affects the maximum force and descending branch in in model 3 Short Fiber (PO-ESF-RLM). Bond length also affects the value of the final stiffness in the model 4 Clamped Fiber (PO-ECF-ECM)
<ahref="fragmentation.ipynb#top">2.3 Tensile behavior of a composite</a> <imgsrc="../icons/next.png"alt="Previous trip"width="50"height="50"></div>
<ahref="fragmentation.ipynb#top">2.3 Tensile behavior of a composite</a> <imgsrc="../icons/next.png"alt="Previous trip"width="50"height="50"></div>
