"## Setting up a stage: What is a developing displacement discontinuity?\n",
"## Setting up a stage: What is a displacement discontinuity?\n",
"\n",
"To stimulate our thinking about the behavior of the brittle-matrix composites, we need to learn about evolution of displacement discontinuities in the material structure. Assuming that the material structure can be regarded as continuous in an initial state, we introduce the notion of displacement discontinuity as jumps in the displacement field that have developed during the loading process. In other words, we do not consider initial flaws and cracks. The material components are regarded as homogeneous and flawless. "
"To stimulate our thinking about the behavior of the brittle-matrix composites, we need to learn about **evolution of displacement discontinuities** in the material structure. Assuming that the material structure can be regarded as continuous in an initial state, we introduce the notion of displacement discontinuity as jumps that interrupt an otherwise continuous displacement field. These jumps can develop during the loading process. The material components, in our case concrete, are regarded as **homogeneous and flawless**. In reality, this is not the case at the scale of the material structure. However, from a **macroscopic point of view**, if the size of a structure is much larger than the size of the material heterogeneity, this assumption is justified. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The most obvious example of a displacement discontinuity is a crack evolving in a tensile zone of bended specimen. In this case, the displacement discontinuity is represented by a crack opening \n",
"that is obtained as the difference between the displacement of the crack faces. This displacement can be expressed as a vector that can be expressed ans a normal and tangential displacement jump in each point of the crack path. In fracture mechanics, cracks with dominant normal or tangential displacement jump are distinguished as mode I or mode II cracks, respectively. In simple terms, mode I represents a tensile crack and mode II a shear crack."
"The most obvious example of a **displacement discontinuity is a crack** evolving in a tensile zone of bended specimen. In this case, the displacement discontinuity is represented by a crack opening \n",
"that is obtained as the difference between the displacement of the crack faces. This displacement discontinuity can be expressed in each point of the crack path as a vector consisting of a normal and tangential component. In fracture mechanics, cracks are distinguished as mode I or mode II for cracks with pure normal or tangential displacement jump, respectively. In simple terms, **mode I represents a tensile crack and mode II a shear crack**."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Besides the displacement discontinuities developing in an original homogeneous material that we denoted as cracks, we can also recognize displacement jumps developing along interfaces between two material components. In brittle-matrix composites considered here, this is the case between the reinforcement and matrix. Also here, the displacement jump is a vector with normal and tangential components. "
"Besides the displacement discontinuities developing in an originally homogeneous material that we denoted as cracks, we can also recognize **displacement jumps developing along interfaces** between two material components. In brittle-matrix composites considered here, this is the case between the reinforcement and matrix. Also here, the displacement jump is a vector with normal and tangential components. "
]
},
{
...
...
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<aid="top"></a>
# **2.1 Pull-out of elastic fiber from rigid matrix**
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## Setting up a stage: What is a developing displacement discontinuity?
## Setting up a stage: What is a displacement discontinuity?
To stimulate our thinking about the behavior of the brittle-matrix composites, we need to learn about evolution of displacement discontinuities in the material structure. Assuming that the material structure can be regarded as continuous in an initial state, we introduce the notion of displacement discontinuity as jumps in the displacement field that have developed during the loading process. In other words, we do not consider initial flaws and cracks. The material components are regarded as homogeneous and flawless.
To stimulate our thinking about the behavior of the brittle-matrix composites, we need to learn about **evolution of displacement discontinuities** in the material structure. Assuming that the material structure can be regarded as continuous in an initial state, we introduce the notion of displacement discontinuity as jumps that interrupt an otherwise continuous displacement field. These jumps can develop during the loading process. The material components, in our case concrete, are regarded as **homogeneous and flawless**. In reality, this is not the case at the scale of the material structure. However, from a **macroscopic point of view**, if the size of a structure is much larger than the size of the material heterogeneity, this assumption is justified.
%% Cell type:markdown id: tags:
The most obvious example of a displacement discontinuity is a crack evolving in a tensile zone of bended specimen. In this case, the displacement discontinuity is represented by a crack opening
that is obtained as the difference between the displacement of the crack faces. This displacement can be expressed as a vector that can be expressed ans a normal and tangential displacement jump in each point of the crack path. In fracture mechanics, cracks with dominant normal or tangential displacement jump are distinguished as mode I or mode II cracks, respectively. In simple terms, mode I represents a tensile crack and mode II a shear crack.
The most obvious example of a **displacement discontinuity is a crack** evolving in a tensile zone of bended specimen. In this case, the displacement discontinuity is represented by a crack opening
that is obtained as the difference between the displacement of the crack faces. This displacement discontinuity can be expressed in each point of the crack path as a vector consisting of a normal and tangential component. In fracture mechanics, cracks are distinguished as mode I or mode II for cracks with pure normal or tangential displacement jump, respectively. In simple terms, **mode I represents a tensile crack and mode II a shear crack**.
%% Cell type:markdown id: tags:
Besides the displacement discontinuities developing in an original homogeneous material that we denoted as cracks, we can also recognize displacement jumps developing along interfaces between two material components. In brittle-matrix composites considered here, this is the case between the reinforcement and matrix. Also here, the displacement jump is a vector with normal and tangential components.
Besides the displacement discontinuities developing in an originally homogeneous material that we denoted as cracks, we can also recognize **displacement jumps developing along interfaces** between two material components. In brittle-matrix composites considered here, this is the case between the reinforcement and matrix. Also here, the displacement jump is a vector with normal and tangential components.
%% Cell type:markdown id: tags:
Our primary focus in this explanation is on the **evolution and propagation** disconstinuities. To show this propagation in a most simple setting, we will consider the pull-out problem, in which only the tangential displacement appears and the normal displacement jump can be neglected.
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**Assumptions made to describe the pull-out behavior:**
- For the explanatory purpose, it is sufficient to consider only the tangential displacement jump along an interface, or simply speaking the *slip* between the matrix and reinforcement evolving.
- The stress distribution over the cross section of the reinforcement will be considered uniform. Therefore, a uni-axial idealization of the pulled-out bar is justified.
- For simplicity, we will also first consider the matrix as rigid, i.e. infinitely stiff first.
- The shear stress between the reinforcement and matrix will be considered constant
An analytical solution of the pull-out problem is obtained by combining the (1) equilibrium condition, (2) constitutive assumption and (3) kinematic relation. These conditions must be fullfilled **at any point** along the interface, or more generally, in the whole material domain under consideration. Besides these three **local** conditions, **global** conditions (4) must be introduced to reflect the supports and the loads of the so called **initial boundary value problem**.
Let us consider a fiber with the cross-sectional area $A_\mathrm{f}$ and perimeter $p_\mathrm{f}$. The stress in the fiber at each point $x$ along the bond zone is denoted as $\sigma_\mathrm{f}(x)$ and the shear stress between the fiber as $\tau(x)$. As specified in the assumption (4) above, for simplicity, we will first explicitly set $\tau(x) = \bar{\tau}$ constant in the debonded zone $x \in (0, a)$ with $a$ represents the debonded length. The parameter $\bar{\tau}$ represents the frictional (or yield) stress. Then, a mathematical model of a pullout can be exemplified using the four introduced model ingredients as follows:
1.**Equilibrium condition:** Differential equilibrium equation relates the shear flow $p_\mathrm{f} \tau$ to the change of the normal force
$A_\mathrm{f} \mathrm{d}\sigma_\mathrm{f}$ in the fiber on an infinitesimal element $\mathrm{d}x$
3.**Kinematic relation:** Local strain in the fiber $\varepsilon_\mathrm{f}(x)$ is given as a derivative of the displacement field $u_\mathrm{f}(x)$ of the pulled fiber, i.e.
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 |
|:- |:- |:- |
| $E_\mathrm{f}$ | MPa | Young's modulus of reinforcement |
| $\bar{\tau}$ | MPa | Bond stress |
| $A_\mathrm{f}$ | mm$^2$ | Cross-sectional area of reinforcement |
| $p$ | mm | Perimeter of contact between concrete and reinforcement |
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### **Observation**
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- The measured displament at the loaded and unloaded end are different
- 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
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### **Question**
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- Can the above derived model describe the debonding process correctly?
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# Look inside the specimen using the model
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The parameters of the above experiment are specified as follows
%% Cell type:code id: tags:
``` python
ds=16
A_f=(ds/2)**2*3.14# mm^2 - reinforcement area
L_b=5*ds# mm - bond length
E_f=210000# MPa - reinforcement stiffness
p_b=3.14*ds# mm - bond perimeter
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
**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:
``` python
po.interact()
```
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## Let's learn from the model
Exercise the relation between $P$ and $\tau(x)$ and between $w$ and $\varepsilon(x)$.
1. What is the meaning of the green area?
2. What is the meaning of the red area?
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. What is the role of debonded length $a$ in view of general non-linear simulation?
5. When does the pull-out fail?
5. What happends with $a$ upon unloading?
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# Further material showing the sympy derivation
(not necessary for the BMCS exam)
- 2.1.2 [EXTRA - Pull-out of elastic long fiber from rigid long matrix](2_1_2_PO_ELF_RLM_CAS.ipynb)</br>
How to use the symbolic computer algebra system (CAS) to derive a symbolic model and make it executable.
