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# **Brittle-Matrix Composite Structures**
Institute of Structural Concrete; @author: Rostislav Chudoba, Abedulgader Baktheer
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## Expedition investigating the BMCS landscape
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| | | | |
|-|-|-------|------|
| ![image-15.png](attachment:image-15.png) | ![image-4.png](fig/reinforcement.png) | ![image-2.png](fig/bond.png) | ![image.png](fig/matrix.png) |
| ![image-14.png](attachment:image-14.png) | [![image-6.png](fig/pullout.png)](pull_out/pull_out.ipynb) | ![image-7.png](fig/crack_bridge.png)| [![image-8.png](fig/mkappa.png)](mkappa/mkappa.ipynb) |
| ![image-13.png](attachment:image-13.png) | [![image-9.png](fig/tension.png)](tension/fragmentation.ipynb) | [![image-10.png](fig/bending.png)](bending/bending_3pt.ipynb) | ![image-11.png](fig/compression.png) |
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## Guided tours provided through BMCS landscape
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| No. | Title |
|- | - |
| **Tour 1:** | Mixture rule (effective composite stiffness) |
| **Tour 2:** | Constant bond (friction, pull-out, crack-bridge, multiple cracking) |
| **Tour 3:** | Nonlinear bond (hardening, softening -> failure modes: anchorage, cracing) |
| **Tour 4:** | Plastic bond behavior (irreversibility -> energy dissipation) |
| **Tour 5:** | Damage bond behavior (2D sheet debonding) |
| **Tour 6:** | Concrete cracking and yielding (bended cross section) |
| **Tour 7:** | Beam deflection - comparison wth EC2 and Model Code |
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## Tools used in the BMCS
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| Engine and wheels | Further information |
|-- |---|
| Jupyter notebooks | |
| OpenWebApps | |
| Elementary syntax of Python language | |
| Plotting tools | `matplotlib` |
| Computer Algebra System | `sympy` |
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## Knowledge and skills developed during the BMCS expedition
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* Understand nonlinear material behavior, stress-redistribution, objectivity of material laws
* Develop a basic understanding of damage, plasticity, fracture
* Formulate simplified analytical models capturing material and structural behavior
* Compare self-developed analytical models with general finite-element models
* Distinguish model verification, calibration, validation and parametric studies
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## Tour 1
## **Tour 1** - Introduction
- 1.1 A roadmap through the BMCS landscape
- 1.2 Introduction to Jupyter Web Apps and notebooks (mathematics and mechanics support)
- 1.3 [Example Elastic mixture rule](bmcs_course/1_1_elastic_stiffness_of_the_composite.ipynb)
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## Tour 2
## **Tour 2** - Constant bond - pull-out, crack bridge, fragmentation
### Material and cross-section
### 2.1 - Pull-out from rigid matrix - test and theory [65 min]
- 2.1 [Pull-out of long fiber from rigid matrix](bmcs_course/2_1_PO_LF_LM_RG.ipynb)
- 2.2 [Pull-out of long fiber from long elastic matrix](bmcs_course/2_2_PO_LF_LM_EL.ipynb)
- 2.3 [Pull-out of short fiber from rigid matrix](bmcs_course/2_3_PO_SF_M_RG.ipynb)
- 2.4 [Crack-bridge behavior]
Interact - [Pull-out - test the analytical solution $P = \sqrt{\bar{\tau},p,E,A,w}$](2_1_1_PO_observation.ipynb)
### 2.2 - Classification of pullout configurations with constant bond stress [45 min]
Interact - [Pull-out - explorer modified analytical models](2_2_1_PO_configuration_explorer.ipynb)
### Cross section and structure
### 2.3 - Constant bond and multiple cracking
- 2.6 [Anchorage]
- 2.7 [Multiple cracking]
- 2.4 [Crack-bridge behavior]
- 2.5 [Multiple cracking]
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## Tour 3
## **Tour 3**
- 3.1 [Pull-out with softening and hardening](bmcs_course/3_1_PO_LF_LM_EL_FE_CB.ipynb)
- 3.2 [EXTRA - Newton iterative scheme](extras/newton_method.ipynb)
- 3.2 [EXTRA - Nonlinear finite-element solver for 1d pullout](extras/pullout1d.ipynb)
- 3.2 [Anchorage] Failure mechanisms - pull-out, fiber rupture, matrix crack
- 3.3 [EXTRA - Newton iterative scheme](extras/newton_method.ipynb)
- 3.4 [EXTRA - Nonlinear finite-element solver for 1d pullout](extras/pullout1d.ipynb)
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## Tour 4
## **Tour 4**
- 4.1 [Unloading with multi-linear bond-slip law](bmcs_course/4_1_PO_multilinear_unloading.ipynb)
- 4.2 [Basic concept of plasticity, ideal and isotropic hardening](bmcs_course/4_2_BS_EP_SH_I_A.ipynb)
- 4.3 [Basic concept of plasticity, kinematic hardening](bmcs_course/4_3_BS_EP_SH_IK_A.ipynb)
- 4.4 [EXTRA - Generalization of the algorithm using vectors](bmcs_course/4_4_BS_EP_SH_IK_N.ipynb)
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## Tour 5
## **Tour 5**
- 5.1 [Damage initiation, damage evolution, 2D bond behavior](bmcs_course/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb)
- 5.2 [Pull out simulation using damage model](bmcs_course/5_2_PO_DM_FRP_N.ipynb)
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## Tour 6
## **Tour 6**
- 6.1 Crack propagation
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## Tour 7
## **Tour 7**
- 7.1 Beam bending
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``` python
```
......
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# Example 2.1: PO-ELF-RLM
Pull-out of a elastic long fiber (ELF) from rigid long matrix (RLG)
[Video - pullout with constant bond](add link)
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# Observation
Let us try to utilize the depicted idealization for the derivation of a model that can help us simulate the test results of the RILEM pull-out test
![image.png](attachment:image.png)
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![image.png](attachment:image.png)
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### Boundary value problem
![image.png](attachment:image.png)
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# Look inside the specimen using the model
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``` python
import numpy as np
```
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``` python
ds = 16
A_f = (ds/2)**2 * np.pi # mm^2
L_b = 5 * ds # mm
E_f = 210000 # MPa
p_b = np.pi * ds
w_max = 0.12 # mm
```
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``` python
%matplotlib widget
from pull_out import PullOutAModel, PO_ELF_RLM_Symb
po = PullOutAModel(symb_class=PO_ELF_RLM_Symb)
po.trait_set(E_f=E_f, L_b=L_b, p=p_b, A_f=A_f, w_max=w_max)
from pull_out import PO_ELF_RLM
po = PO_ELF_RLM(E_f=E_f, L_b=L_b, p=p_b, A_f=A_f, w_max=w_max)
po.interact()
```
%% Output
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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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``` python
```
......
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# 2.2: PO classification of configurations
Pull-out of a elastic long fiber (ELF) from rigid long matrix (RLG)
[Video - pullout with constant bond](add link)
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![image.png](attachment:image.png)
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### Boundary value problem
![image.png](attachment:image.png)
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# Look inside the specimen using the model
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``` python
%matplotlib widget
```
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![image.png](attachment:9993286b-4057-4b22-b695-3e395e3ffeef.png)
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``` python
from pull_out import PO_ELF_RLM
po_explorer = PO_ELF_RLM()
po_explorer = PO_ELF_RLM(E_f=1, E_m=1, tau=1, p=1, A_m=1, A_f=1, w_max=0.5, L_b=1, t=0.5)
po_explorer.interact()
```
%% Output
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![image.png](attachment:19c4f327-1799-4e88-b100-96d66ebf2dff.png)
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``` python
from pull_out import PO_ELF_ELM
po_explorer = PO_ELF_ELM()
po_explorer = PO_ELF_ELM(E_f=1, E_m=1, tau=1, p=1, A_m=1, A_f=1, w_max=0.5, L_b=1, t=0.5)
po_explorer.interact()
```
%% Output
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![image.png](attachment:95a04ae4-2cb4-46a8-8581-9de442b93ee4.png)
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``` python
from pull_out import PO_ESF_RLM
po_explorer = PO_ESF_RLM()
po_explorer.interact()
```
%% Output
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![image.png](attachment:9de172b7-49d2-43f9-8aa3-90031c3b9197.png)
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``` python
from pull_out import CB_ELF_ELM
po_explorer = CB_ELF_ELM()
po_explorer.interact()
```
%% Output
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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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``` python
```
......
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Contents: The lecture introduces the extensions of the analytical model of pull-out based on the assumption of a constant bond-slip law.
# Pull-out of elastic fiber from rigid matrix
# 2.1 Pull-out of elastic fiber from rigid matrix
- 2.1.1 [Pull-out observation, virtual experiment](2_1_1_PO_observation.ipynb)
- 2.1.2 [EXTRA - Pull-out of elastic long fiber from rigid long matrix](2_1_2_PO_ELF_RLM.ipynb)
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# Classification of pull-out configurations
# 2.2 Classification of pull-out configurations xxx
- 2.2.1 [Pull-out configuration explorer](2_2_1_PO_configuration_explorer.ipynb)
Preserve these changes
- 2.2.2 [EXTRA - Pull-out of elastic long fiber from elastic long matrix](2_2_2_PO_ELF_ELM.ipynb)
- 2.2.3 [EXTRA - Pull-out of elastic short fiber from rigid long matrix](2_2_3_PO_ESF_RM.ipynb)
- 2.2.3 [EXTRA - Pull-out of elastic short fiber from rigid long matrix](2_2_4_PO_ESF_RM.ipynb)
- 2.2.4 [EXTRA - Pull-out of elastic long fiber from rigid short matrix](2_2_5_PO_ELF_RSM.ipynb)
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``` python
```
......
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