" <a href=\"../exercises/X0201 - Pull-out with constant bond-slip law - Part 1.pdf\"><b>Exercise X0201:</b></a> <b>Pull-out with constant bond-slip law - Part 1</b> \n",
" <a href=\"../exercises/X0202 - Pull-out with constant bond-slip law - Part 2.pdf\"><b>Exercise X0202:</b></a> <b>Pull-out with constant bond-slip law - Part 2</b> \n",
" <a href=\"../exercises/X0203 - Pull-out of short fiber with constant bond-slip.pdf\"><b>Exercise X0203:</b></a> <b>Pull-out of short fiber with constant bond-slip</b> \n",
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# **2.2: Classification of pull-out configurations**
# **2.2: Classification of pull-out configurations**
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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="../exercises/X0201-X0203.pdf"><b>Exercises X0201-X0203:</b></a><b>Pull-out with constant bond-slip</b>
<ahref="../exercises/X0201 - Pull-out with constant bond-slip law - Part 1.pdf"><b>Exercise X0201:</b></a><b>Pull-out with constant bond-slip law - Part 1</b>
<ahref="../exercises/X0202 - Pull-out with constant bond-slip law - Part 2.pdf"><b>Exercise X0202:</b></a><b>Pull-out with constant bond-slip law - Part 2</b>
<ahref="../exercises/X0203 - Pull-out of short fiber with constant bond-slip.pdf"><b>Exercise X0203:</b></a><b>Pull-out of short fiber with constant bond-slip</b>
<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>
" <a href=\"../exercises/X0204.pdf\"><b>Exercise X0203:</b></a> <b>Tensile behavior of a composite with constant bond-slip law</b> \n",
" <a href=\"../exercises/X0204 - Tensile behavior of a composite with constant bond-slip law.pdf\"><b>Exercise X0204:</b></a> <b>Tensile behavior of a composite with constant bond-slip law</b> \n",
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# **2.3 Tensile behavior of brittle-matrix composite**
# **2.3 Tensile behavior of brittle-matrix composite**
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With a basic understanding of the debonding process in the vicinity of a crack bridge studied for various configurations of a pull-out problem ([2.2 Classification of pull-out configurations](2_2_1_PO_configuration_explorer.ipynb)) we describe and visualize the process of fragmentation (multiple cracking) in a reinforced composite.
With a basic understanding of the debonding process in the vicinity of a crack bridge studied for various configurations of a pull-out problem ([2.2 Classification of pull-out configurations](2_2_1_PO_configuration_explorer.ipynb)) we describe and visualize the process of fragmentation (multiple cracking) in a reinforced composite.
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Consider an example the textile reinforced concrete cross section loaded in tension. The cross-sectional parameters are as follows
Consider an example the textile reinforced concrete cross section loaded in tension. The cross-sectional parameters are as follows
Knowing the matrix stress profile ahead of an existing crack, we know the distance at which no crack will appear. This length is referred to as shielded length.
Knowing the matrix stress profile ahead of an existing crack, we know the distance at which no crack will appear. This length is referred to as shielded length.
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### **Question:** How long is the shielded length?
### **Question:** How long is the shielded length?
**Remark:** This expression is important in view of homogenization of non-uniform strain profiles in a heterogenous material structure. It is one of the fundamental concepts of micromechanics. We will use it later on to derive the effective stress-strain curve of a multiply cracked composite.
**Remark:** This expression is important in view of homogenization of non-uniform strain profiles in a heterogenous material structure. It is one of the fundamental concepts of micromechanics. We will use it later on to derive the effective stress-strain curve of a multiply cracked composite.
The typical shape of the stress-strain curve of the composite involves three stages:
The typical shape of the stress-strain curve of the composite involves three stages:
- elastic stage which is governed by the mixture rule
- elastic stage which is governed by the mixture rule
- stage of matrix fragmentation
- stage of matrix fragmentation
- saturated crack pattern with a linear branch
- saturated crack pattern with a linear branch
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How to interpret and characterize these three distinguished phases of composite material behavior?
How to interpret and characterize these three distinguished phases of composite material behavior?
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The ACK model developed by Aveston, Cooper and Kelly is an analytical model that represents the composite tensile response by a trilinear law as shown in the following figure. This model is based on the following assumptions:
The ACK model developed by Aveston, Cooper and Kelly is an analytical model that represents the composite tensile response by a trilinear law as shown in the following figure. This model is based on the following assumptions:
- The bond behavior is governed by a constant frictional bond in the debonded interface
- The bond behavior is governed by a constant frictional bond in the debonded interface
- The constitutive law for both reinforcement and matrix is assumed to be linear-elastic with brittle failure upon reaching their strengths
- The constitutive law for both reinforcement and matrix is assumed to be linear-elastic with brittle failure upon reaching their strengths
- Multiple cracking occurs at a constant level of applied stress, inducing a horizontal branch in the stress-strain behavior
- Multiple cracking occurs at a constant level of applied stress, inducing a horizontal branch in the stress-strain behavior
The trilinear curve of ACK model represents the composite tensile response by identifying the following characteristic points:
The trilinear curve of ACK model represents the composite tensile response by identifying the following characteristic points:
- [$\sigma_{1}, \varepsilon_{1}$]: The inital values of stress and strain are set to zero.
- [$\sigma_{1}, \varepsilon_{1}$]: The inital values of stress and strain are set to zero.
- [$\sigma_{2}, \varepsilon_{2}$]: In the first stage, the matrix is uncracked and perfect bond between matrix and fabric is assumed up to the first cracking stress $\sigma_{2}$ , which is defined as
- [$\sigma_{2}, \varepsilon_{2}$]: In the first stage, the matrix is uncracked and perfect bond between matrix and fabric is assumed up to the first cracking stress $\sigma_{2}$ , which is defined as
\begin{align}
\begin{align}
\sigma_{2} = E_\mathrm{c} \varepsilon_{2}
\sigma_{2} = E_\mathrm{c} \varepsilon_{2}
\end{align}
\end{align}
where $\varepsilon_{2}$ is the composite strain value at which the matrix cracks and $E_\mathrm{c}$ is the composite stiffness.
where $\varepsilon_{2}$ is the composite strain value at which the matrix cracks and $E_\mathrm{c}$ is the composite stiffness.
- [$\sigma_{4}, \varepsilon_{4}$]: Finally, in the third stage, when the crack pattern is stabilized the load increases linearly up to the ultimate tensile stress $\sigma_4$ with a slope equal to $E_\mathrm{r}$
- [$\sigma_{4}, \varepsilon_{4}$]: Finally, in the third stage, when the crack pattern is stabilized the load increases linearly up to the ultimate tensile stress $\sigma_4$ with a slope equal to $E_\mathrm{r}$
The ultimate tensile stress is given as
The ultimate tensile stress is given as
\begin{align}
\begin{align}
\sigma_4 = \sigma_\mathrm{fu} \; V_\mathrm{f}
\sigma_4 = \sigma_\mathrm{fu} \; V_\mathrm{f}
\end{align}
\end{align}
where $\sigma_\mathrm{fu}$ is the tensile strength of the fiber.
where $\sigma_\mathrm{fu}$ is the tensile strength of the fiber.
The slope $E_\mathrm{r}$ represents the effective stiffness of the reinforcement with respect to the whole cross section and is given as
The slope $E_\mathrm{r}$ represents the effective stiffness of the reinforcement with respect to the whole cross section and is given as
\begin{align}
\begin{align}
E_\mathrm{r} = E_\mathrm{f} \; V_\mathrm{f}
E_\mathrm{r} = E_\mathrm{f} \; V_\mathrm{f}
\end{align}
\end{align}
The composilte strain at failure $\varepsilon_{4}$ is given as
The composilte strain at failure $\varepsilon_{4}$ is given as
Probabilistic analysis of the car parking problem delivers the result that the average spacing is 1.337 larger than the car length ([Wikipedia: Random sequential adsorption](https://en.wikipedia.org/wiki/Random_sequential_adsorption))
Probabilistic analysis of the car parking problem delivers the result that the average spacing is 1.337 larger than the car length ([Wikipedia: Random sequential adsorption](https://en.wikipedia.org/wiki/Random_sequential_adsorption))
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### How long is the car in a concrete tensile specimen?
### How long is the car in a concrete tensile specimen?
The final average crack spacing $l_\mathrm{cs}$ is given as
The final average crack spacing $l_\mathrm{cs}$ is given as
## **Question:** How to evaluate the corresponding crack opening?
## **Question:** How to evaluate the corresponding crack opening?
In the design of steel reinforced concrete or carbon concrete, it is necessary to limit the maximum crack opening to a prescribed value, i.e. $w_\max < 0.1$ mm. As an exercise propose a formula for $w_\max$.
In the design of steel reinforced concrete or carbon concrete, it is necessary to limit the maximum crack opening to a prescribed value, i.e. $w_\max < 0.1$ mm. As an exercise propose a formula for $w_\max$.
In reality, the matrix strength $\sigma_\mathrm{mu}$ is random. Its profile along the tensile specimen can be described by the probability distribution function.
In reality, the matrix strength $\sigma_\mathrm{mu}$ is random. Its profile along the tensile specimen can be described by the probability distribution function.
Weibull probability distribution is used to describe the strength of materials.
Weibull probability distribution is used to describe the strength of materials.
%% Cell type:code id:martial-sense tags:
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``` python
``` python
%matplotlib widget
%matplotlib widget
from pmcm import PMCM
from pmcm import PMCM
pm = PMCM()
pm = PMCM()
pm.interact()
pm.interact()
```
```
%% Output
%% Output
%% Cell type:markdown id:ahead-lafayette tags:
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The probabilistic multiple cracking model uses the crack bridge model which is inserted at any newly emerging crack position. By assembling the profile along the specimen we obtain the
The probabilistic multiple cracking model uses the crack bridge model which is inserted at any newly emerging crack position. By assembling the profile along the specimen we obtain the
- matrix stress field $\sigma_\mathrm{m}(x, \sigma_\mathrm{c})$, and
- matrix stress field $\sigma_\mathrm{m}(x, \sigma_\mathrm{c})$, and
- reinforcement strain field $\varepsilon_\mathrm{f}(x, \sigma_\mathrm{c})$.
- reinforcement strain field $\varepsilon_\mathrm{f}(x, \sigma_\mathrm{c})$.
for any state of loading $\sigma_\mathrm{c}$. No finite element calculation is needed.
for any state of loading $\sigma_\mathrm{c}$. No finite element calculation is needed.
The algorithm used in the above web-app can identify the individual cracks exactly. For a given load level $\sigma_\mathrm{c}$ and crack distribution along the specimen, the composite strain of a specimen with a length $L$ is obtained using the averaging formula introduced above
The algorithm used in the above web-app can identify the individual cracks exactly. For a given load level $\sigma_\mathrm{c}$ and crack distribution along the specimen, the composite strain of a specimen with a length $L$ is obtained using the averaging formula introduced above
**Further reading:** [Paper describing the general probabilistic multiple cracking model](../papers/pmcm_fragmentation.pdf), Journal of Mathematical Modeling (2021)
**Further reading:** [Paper describing the general probabilistic multiple cracking model](../papers/pmcm_fragmentation.pdf), Journal of Mathematical Modeling (2021)
Apply the model to a composite cross section using carbon textile fabrics as specified in the notebook [**Mixture rule**](1_1_elastic_stiffness_of_the_composite.ipynb)
Apply the model to a composite cross section using carbon textile fabrics as specified in the notebook [**Mixture rule**](1_1_elastic_stiffness_of_the_composite.ipynb)
### Task 3: Use the model to validate its prediction
### Task 3: Use the model to validate its prediction
Given the cross sectional areas $A_\mathrm{m}, A_\mathrm{f}$, concrete and reinforcement stiffness $E_\mathrm{m}, E_\mathrm{f}$ and strength $\sigma_\mathrm{mu}, \sigma_\mathrm{fu}$, reinforcement ratio $V_\mathrm{f}$, bond stress $\tau$, and perimeter $p$ calculate the average crack width at failure of a tensile specimen.
Given the cross sectional areas $A_\mathrm{m}, A_\mathrm{f}$, concrete and reinforcement stiffness $E_\mathrm{m}, E_\mathrm{f}$ and strength $\sigma_\mathrm{mu}, \sigma_\mathrm{fu}$, reinforcement ratio $V_\mathrm{f}$, bond stress $\tau$, and perimeter $p$ calculate the average crack width at failure of a tensile specimen.
### Task 4: Evaluate the average crack width
### Task 4: Evaluate the average crack width
Compare the crack width obtained using the ACK model and the PMCM model for a given reinforced cross section design
Compare the crack width obtained using the ACK model and the PMCM model for a given reinforced cross section design
<a href="../exercises/X0204.pdf"><b>Exercise X0203:</b></a> <b>Tensile behavior of a composite with constant bond-slip law</b>
<a href="../exercises/X0204 - Tensile behavior of a composite with constant bond-slip law.pdf"><b>Exercise X0204:</b></a> <b>Tensile behavior of a composite with constant bond-slip law</b>
