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%% Cell type:markdown id: tags:
# **Brittle-Matrix Composite Structures**
Institute of Structural Concrete; @author: Rostislav Chudoba, Abedulgader Baktheer
%% Cell type:markdown id: tags:
## Expedition investigating the BMCS landscape
%% Cell type:markdown id: tags:
| | | | |
|-|-|-------|------|
| ![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)](pull_out/fragmentation.ipynb) | [![image-10.png](fig/bending.png)](bending/bending_3pt.ipynb) | ![image-11.png](fig/compression.png) |
%% Cell type:markdown id: tags:
## Guided tours provided through BMCS landscape
%% Cell type:markdown id: tags:
| No. | Title |
|- | - |
| **[Tour 1:](#tour1)** | Mixture rule (effective composite stiffness) |
| **[Tour 2:](#tour2)** | Constant bond (friction, pull-out, crack-bridge, multiple cracking) |
| **[Tour 3:](#tour3)** | Nonlinear bond (hardening, softening -> failure modes: anchorage, cracing) |
| **[Tour 4:](#tour4)** | Plastic bond behavior (irreversibility -> energy dissipation) |
| **[Tour 5:](#tour5)** | Damage bond behavior (2D sheet debonding) |
| **[Tour 6:](#tour6)** | Concrete cracking and yielding (bended cross section) |
| **[Tour 7:](#tour7)** | Beam deflection - comparison wth EC2 and Model Code |
%% Cell type:markdown id: tags:
## Tools used in the BMCS
%% Cell type:markdown id: tags:
| Engine and wheels | Further information |
|-- |---|
| Jupyter notebooks | |
| OpenWebApps | |
| Elementary syntax of Python language | |
| Plotting tools | `matplotlib` |
| Computer Algebra System | `sympy` |
%% Cell type:markdown id: tags:
## Knowledge and skills developed during the BMCS expedition
%% Cell type:markdown id: tags:
* 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
%% Cell type:markdown id: tags:
<a id="tour1"></a>
## **Tour 1**: Introduction
### 1.1 A roadmap through the BMCS landscape
### 1.2 Introduction to Jupyter Web Apps and notebooks
Not only static slides and videos but also the possibility to interactively put your fingers on the theory in terms of prepared interactive applications is used throughout the course.
<!-- Basic information how to
use the `jupyter` notebooks are summarized here:</br>
[**Web Apps:** first steps](link)
-->
### 1.3 Interactive computational environment
To demonstrate the theory on examples that can be interactively modified, let us consider an elementary case of mixture rule to determine the effective stiffness of an elastic composite </br>
[**Mixture rule**: example elastic mixture rule](tour1_intro/1_1_elastic_stiffness_of_the_composite.ipynb#top)
%% Cell type:markdown id: tags:
<a id="tour2"></a>
## **Tour 2:** Constant bond - pull-out, crack bridge, fragmentation
%% Cell type:markdown id: tags:
### 2.1 - Pull-out from rigid matrix - test setup and theory
Using the analytical solution of the pullout problem assuming constant bond-slip law, elastic fiber and rigid matrix, we first explore the fundamental relations between the measured pull-out curve of a steel-rebar from the concrete matrix:</br>
[**Pull-out:** analytical constant-bond model](tour2_constant_bond/2_1_1_PO_observation.ipynb#top)
%% Cell type:markdown id: tags:
### 2.2 - Classification of pullout configurations with constant bond stress
Include further configurations of a pull-out to show the differences in their behavior, learning the correspondence between the shape of the pull-out curve and the distribution of slip, shear, fiber and matrix strain and stresses depending on a particular configuration, i.e. elastic matrix, short fiber, short matrix and clamped fiber:</br>
[**Pull-out:** extended analytical constant-bond models - short / long / elastic / clamped](tour2_constant_bond/2_2_1_PO_configuration_explorer.ipynb#top)
%% Cell type:markdown id: tags:
### 2.3 - Multiple cracking - fragmentation
The crack-bridging action of a fiber within a composite loaded in tension is a key to understanding the behavior of brittle-matrix composites. By putting crack bridges in a series, we can directly simulate a tensile test and predict the test response in terms of the stress-strain and crack spacing curves. This helps us to study and understand the relation between reinforcement ratio, bond strength, matrix strenth and the tensile response of the composite:
</br>
[**Multiple cracking:** tensile response of a composite](tour2_constant_bond/fragmentation.ipynb#top)
%% Cell type:markdown id: tags:
<a id="tour3"></a>
## **Tour 3:** Non-linear bond-slip law
### 3.1 - Pull-out with softening and hardening bond behavior
The shape of the bond-slip law is distinguished into hardening and softening leading to a completely different pull-out behavior. A numerical model of the pull-out test is used to monitor and explain the debonding process in two studies for steel and CFRP bond to concrete showing a qualitatively different behavior.</br>
[**Pull-out**: with softening and hardening](tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb#top)
### 3.2 - Effect of bond length: anchorage versus pull-out failure
With the developed understanding, we address questions related to design rules: What is the necessary bond length to avoid or to deliberately induce a fiber pull-out or fiber rupture. At which distance from the loaded end can we expect a matrix crack to appear?</br>
[**Anchorage**: pull-out, fiber rupture, matrix crack](tour3_nonlinear_bond/3_2_anchorage_length.ipynb#top)
### Related optional support material
The numerical vehicles used in the above two trips are provided here in a more detail as a bonus material.</br>
[**Appendix**: Newton iterative scheme](extras/newton_method.ipynb#top)</br>
[**Appendix**: Nonlinear finite-element solver for 1d pullout](extras/pullout1d.ipynb#top)
%% Cell type:markdown id: tags:
<a id="tour4"></a>
## **Tour 4:** Irreversibility due to yielding
### 4.1 - Unloading and reloading
Non-linear behavior can be described by nonlinear functions as we did so far. However,
this description cannot capture the irreversible changes within the material structure.
To demonstrate this, let us revisit the pull-out tests and consider a loading scenario with unloading and reloading.</br>
[Unloading with multi-linear bond-slip law](tour4_plastic_bond/4_1_PO_multilinear_unloading.ipynb#top)
### 4.2 - The basic concept of plasticity
The first option in describing irreversible changes in a material point is the
plasticity. The stick-slip interface represents the simplest possible
type of plastic behavior that can be conveniently used to explain the key concept
behind all material models introducing plastic behavior.</br>
[Ideal, isotropic and kinematic hardening of an interface](tour4_plastic_bond/4_2_BS_EP_SH_I_A.ipynb#top)
%% Cell type:markdown id: tags:
### 4.3 - Cyclic pullout test and plastic material behavior
With the knowledge of the plasticity at a material point level, let us study the effect of unloading and reloading at the level of a structure. In particular, we learn to interpret the meaning of the unloading stiffness at the level of a structure.</br>
[Cyclic pullout of textile fabrics and CFRP sheets](tour4_plastic_bond/4_3_PO_trc_cfrp_cyclic.ipynb#top)
%% Cell type:markdown id: tags:
<a id="tour5"></a>
## **Tour 5:** Irreversibility due to damage
### 5.1 Damage initiation, damage evolution, 2D bond behavior
We are going to define several damage functions which are used in damage models available in finite element codes. Their definition is based on an assumed profile of breakage propagation. The derived damage functions are then used to model a two-dimensional interface with slip and shear stress defined as a vector.</br>
[Damage initiation, damage evolution, 2D bond behavior](tour5_damage_bond/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb)
### 5.2 Pullout behavior governed by damage [in production]
Application of the damage model to pullout test is used to discuss general aspects of damage models applied in finite element calculations.</br>
[Pull out simulation using damage model](tour5_damage_bond/5_2_PO_cfrp_damage.ipynb)
%% Cell type:markdown id: tags:
<a id="tour6"></a>
## **Tour 6:** Energy and fracture
### 6.1 Energy flow - supply, storage, dissipation
Inelastic deformation induces energy dissipation. By distinguishing stored and lost energy within a system we can provide another physical perspective to the changes in the material structure.</br>
[Energy games](tour6_energy/6_1_energy_dissipation.ipynb#top)
### 6.2 Frictional pullout and energy dissipation
The analytical solution of the pullout test provides a suitable model to demonstrate the evaluation of the energy dissipation within an initial boundary value problem with including an inelastic deformation.</br>
[Frictional pullout and energy dissipation](tour6_energy/6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top)
%% Cell type:markdown id: tags:
### 6.3 Localization and fracture energy [in production]
### 6.3 Localization and fracture energy
Application of the local energy evaluation scheme to the pullout simulation with softening behavior provides us a clear interpretation of fracture energy</br>
[Localization and fracture energy](tour6_energy/6_3_localized_energy_dissipation.ipynb#top)
%% Cell type:markdown id: tags:
<div style="background-color:lightgreen;text-align:left"> <img src="icons/rest.png" alt="Step by step" width="40" height="40">
&nbsp; &nbsp; <b>Our current location</b> </div>
%% Cell type:markdown id: tags:
<a id="tour7"></a>
## **Tour 7:** Bending and crack propagation
- 7.1 Notched beam, fracture energy
### 7.1 Notched beam to characterize a tensile crack propagation
- 7.2 Simulation using damage model
- 7.3 Mesh sensitivity due to localization
- 7.4 Regularization techniques to stabilize the simulation
%% Cell type:markdown id: tags:
<a id="tour7"></a>
## **Tour 8:** Reinforced bended cross section
- 8.1 Bended cross section with crack bridging, multiple cracking, concrete yielding
- 8.2 Moment-curvature derived for arbitrary cross-sectional layout
- 8.3 Load deflection curve of reinforced concrete beam
- 8.4 Serviceability limit state beam design
%% Cell type:code id: tags:
``` python
```
......
tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb
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%% Cell type:markdown id: tags:
 
<a id="top"></a>
# **3.1 Nonlinear bond - softening and hardening**
 
[![title](../fig/bmcs_video.png)](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)&nbsp; part 1
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/start_flag.png" alt="Previous trip" width="40" height="40">
&nbsp; &nbsp; <b>Starting point</b> </div>
 
%% Cell type:markdown id: tags:
 
By saying that we want to capture the _material behavior_ we mean
that we realistically describe the **constitutive relation** between the strain and stress which is **valid for
any material point** of the considered volume. With the focus on a one-dimensional interface between two material
components we can reduce this task to the relation between bond stress and slip.
In Tour 2, we assumed the constitutive bond-slip relation constant. However, as we have learned
in trip [2.1 Pull-out of elastic fiber from rigid matrix](../pull_out/2_1_1_PO_observation.ipynb)
this stick-slip interface behavior cannot realistically describe the experimentally measured
response of steel-concrete pull-out with varied length of the bond length $L_b$.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/destination.png" alt="Previous trip" width="40" height="40">
&nbsp; &nbsp; <b>Where are we heading</b> </div>
 
%% Cell type:markdown id: tags:
 
To improve the quality of the model, in this notebook we introduce and investigate more complex shapes of bond slip laws and their effect on the observed pullout response. This extension will enable a more **realistic
prediction of a wide range of pull-out and crack bridge tests**, including
steel rebars, carbon textile fabrics or carbon fiber reinforced polymer (CFRP) sheets.
Using the models, we will perform automated studies of the pull-out response that can demonstrate the different phenomenology behind hardening and softening constitutive behavior.
These studies indicate how validated models can support the definition of engineering design rules.
 
%% Cell type:markdown id: tags:
 
To proceed in small steps we consider two shapes of constant bond-slip law, referred to as **bond-hardening and bond softening**.
 
![image.png](attachment:9086d2ee-b436-406a-aae8-44f90e71f5b3.png)
 
%% Cell type:markdown id: tags:
 
The increasing/hardening or decreasing/softening trend of the bond-slip law in the second branch introduces the question, what kind of **material structure** within the bond zone can induce such type of behavior. An example of an idealized bond system leading to hardening or softening can be provided using a rough surface with an increasing or decreasing number of asperities. A more detailed classification of the bond systems will be shown in Tour 3 which provides a more physically based description of the debonding process. The question studied in this notebook is **what is the qualitative effect of the second bond-slip slope on the pull-out response.**
 
%% Cell type:markdown id: tags:
 
# **Numerical support necessary**
 
To solve a pullout problem for a generally nonlinear bond-slip law, we have to solve
the initial boundary value problem numerically. In this notebook, we will use a finite-element code
implemented within the BMCS tool to study the behavior for two examples of qualitatively different bond-slip laws.
 
%% Cell type:markdown id: tags:
 
**Solution algorithm:** To study of the effect of the nonlinear bond-slip law on
the pullout response we will use the finite-element method solving the nonlinear response of the pull-out test by stepping through the loading history. Let us therefore briefly touch the topic of the solution algorithm needed to solve such a nonlinear problem boundary value problem of continuum mechanics. Generally, a non-linear finite element solver includes the solution of two separate tasks:
- **Time stepping** algorithm that can identify the material state variables satisfying the constitutive law for a prescribed loadincrement in all points of the domain using an iterative Newton-type algorithm.
- Find the **spatial distribution** of the displacement field satisfying the equilibrium, compatibility and boundary conditions using the finite-element discretization.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/bus.png" alt="Diver" width="40" height="40">
&nbsp; &nbsp; <b>Short sidetrip</b> </div>
 
## Time-stepping - Newton method to solve a set of nonlinear equations
 
The Newton method is the basis of all nonlinear time-stepping algorithms used in finite-element codes.
Let us explain the solution procedure by considering a very short bond length $L_\mathrm{b}$x and denote it as a material point $m$ for which a constant profile of the shear stress $\tau(x) = \tau_m$ and slip $s(x) = s_m$ can be assumed.
 
![image.png](attachment:fcf8bea4-2c06-4931-b39d-47e53c0d6dda.png)
 
The iterative time-stepping algorithm with increasing load levels can now be displayed for single unknown displacement variable $w$ which must satisfy the equilibrium condition $\bar{P}(t) = P(w)$, where $\bar(P)$ is a prescribed history of loading. A simple implementation of the time stepping procedure exemplifying the solution procedure for a nonlinear equation is provided for an interested tourist in an Annex notebook [A.2 Newton method](../extras/newton_method.ipynb).
 
![image.png](../fig/newton_iteration.png)
 
In a real simulation of the pull-out problem, the unknown variable is not a slip but the displacement fields $u_\mathrm{m}, u_\mathrm{f}$ are the primary unknowns. They are transformed to corresponding component strains $\varepsilon_\mathrm{m}=u_{\mathrm{m},x}, \varepsilon_\mathrm{f}=u_{\mathrm{f},x}$, and slip $s = u_\mathrm{m} - u_\mathrm{f}$. In the following examples, the component strains are still assumed linear elastic while the bond/shear stress is assumed generally nonlinear. With the known stress fields, the corresponding forces are obtained using numerical integration which deliver the residuum of the global equilibrium condition. The solution scheme described for a single variable in the notebook [A.2](../extras/newton_method.ipynb#newton_iteration_example) remains the same.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/diver.png" alt="Diver" width="40" height="40">
&nbsp; &nbsp; <b>Deep dive</b> </div>
 
%% Cell type:markdown id: tags:
 
## Spatial solver - boundary value problem solved using the finite element method
 
The identification of the displacements within each equilibrium iteration includes the same conditions that we have applied to derive the analytical solution of the pull-out problem with a constant bond slip law. However, the discrete solution satisfies the equilibrium conditions only approximately in a _week sense_. This means that the local differential equilibrium condition is not satisfied everywhere but only in integration points.
 
%% Cell type:markdown id: tags:
 
To provide an insight into the way how do the finite-element tools solve the problem, an open implementation of the nonlinear solver used in this and later notebooks is described completely with a running example, plots and animation in a notebook [A.3 Finite element solver for a pull-out problem](../extras/pullout1d.ipynb). This notebook is an Annex to the course and is meant for ambitious adventurers who want to see how the most finite-element programs available on the market are implemented. Detailed explanation of the theoretical background is provided in the Master's courses on linear structural analysis focused on the theoretical background of the finite-element method and on the nonlinear structural analysis.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/binoculars.png" alt="Traveler in a hurry" width="40" height="40">
&nbsp; &nbsp; <b>Distant view</b> </div>
 
%% Cell type:markdown id: tags:
 
## Example of the finite-element pull-out simulation
 
To understand the functionality of the finite-element model implemented in the referenced notebook [A.3](../extras/pullout1d.ipynb), its output is provided here in form of the pull-out curve and of the fields along the bond zone. The applied boundary conditions are given as follows, the free length $L_\mathrm{f}=0$, the matrix is supported at the loaded end.
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:image.png)
 
%% Cell type:code id: tags:
 
``` python
from IPython.display import HTML
html_video_file = open('../extras/pull_out_animation.html','r')
HTML(html_video_file.read())
```
 
%% Output
 
<IPython.core.display.HTML object>
 
%% Cell type:markdown id: tags:
 
## What constitutive law can induce such a debonding process?
 
%% Cell type:markdown id: tags:
 
A closer look at the simulated evolution of the shear stress along the bond zone in the bottom right
diagram provides an important phenomenological observation. The level of shear increases
at the right, loaded end in the first stage. After reaching the peak shear stress of $N = 2~\mathrm{N}$ , it
diminishes slowly to a low value of approximately 0.1 N.
 
The constitutive law valid at each material point has thus a first ascending and second descending branch. Such kind of behavior is called **softening**. Constitutive behavior exhibiting softening has a severe impact on the structural behavior by introducing the phenomena of strain localization to discrete shear and tensile cracks, accompanied with stress redistribution during the debonding or crack propagation process. The pull-out problem can be conveniently used to visualize the correspondence between the **softening** material law and the structural response with a debonding propagation, as opposed to **hardening** material law.
 
%% Cell type:markdown id: tags:
 
# **Setting up the model components - new material model**
 
%% Cell type:markdown id: tags:
 
For the purpose of this comparison, let us introduce a simple piece-wise linear bond-slip law, that can be inserted into the non-linear finite-element code to investigate the effect of the type of nonlinearity on the pull-out response.
 
%% Cell type:markdown id: tags:
 
<a id="trilinear_material_model"></a>
<div style="background-color:lightgray;text-align:left"> <img src="../icons/work.png" alt="Coding intermezzo" width="40" height="40">
&nbsp; &nbsp; <b>Coding intermezzo</b> </div>
 
%% Cell type:markdown id: tags:
 
## Construct a material model with tri-linear bond-slip law
To indicate how the below examples are implemented let us define a a piece-wise linear function with three branches constituting the bond-slip behavior. It can be used to exemplify how to implement material models in standard non-linear finite-element codes for structural analysis. In codes like `ANSYS, Abaqus, ATENA, Diana`, the spatial integration of the stresses and stiffnesses is based on the so called **predictor**, **corrector** scheme.
 
%% Cell type:markdown id: tags:
 
This simply means that the material model must provide two functions
1. the stress evaluation for a given strain increment
2. the derivative of stress with respect to the strain increment, i.e. the material stiffness.
In our case of a bond-slip law, we need to provide two functions
\begin{align}
\tau(s) \\
\frac{\mathrm{d} \tau}{ \mathrm{d} s}
\end{align}.
 
%% Cell type:markdown id: tags:
 
**Let's import the packages:**
 
%% Cell type:code id: tags:
 
``` python
%matplotlib widget
import sympy as sp # symbolic algebra package
import numpy as np # numerical package
import matplotlib.pyplot as plt # plotting package
sp.init_printing() # enable nice formating of the derived expressions
```
 
%% Cell type:markdown id: tags:
 
The tri-linear function can be readily constructed using the already known `Piecewise` function provied in `sympy`
 
%% Cell type:code id: tags:
 
``` python
s = sp.symbols('s')
tau_1, s_1, tau_2, s_2 = sp.symbols(r'tau_1, s_1, tau_2, s_2')
tau_s = sp.Piecewise(
(tau_1 / s_1 * s, s <= s_1), # value, condition
(tau_1 + (tau_2-tau_1) / (s_2-s_1) * (s - s_1), s <= s_2), # value, condition
(tau_2, True) # value, otherwise
)
tau_s
```
 
%% Output
 
$\displaystyle \begin{cases} \frac{s \tau_{1}}{s_{1}} & \text{for}\: s \leq s_{1} \\\tau_{1} + \frac{\left(s - s_{1}\right) \left(- \tau_{1} + \tau_{2}\right)}{- s_{1} + s_{2}} & \text{for}\: s \leq s_{2} \\\tau_{2} & \text{otherwise} \end{cases}$
⎧ s⋅τ₁
⎪ ──── for s ≤ s₁
⎪ s₁
⎪
⎨ (s - s₁)⋅(-τ₁ + τ₂)
⎪τ₁ + ─────────────────── for s ≤ s₂
⎪ -s₁ + s₂
⎪
⎩ τ₂ otherwise
 
%% Cell type:markdown id: tags:
 
The derivative is obtained as
 
%% Cell type:code id: tags:
 
``` python
d_tau_s = sp.diff(tau_s, s)
d_tau_s
```
 
%% Output
 
$\displaystyle \begin{cases} \frac{\tau_{1}}{s_{1}} & \text{for}\: s \leq s_{1} \\\frac{- \tau_{1} + \tau_{2}}{- s_{1} + s_{2}} & \text{for}\: s \leq s_{2} \\0 & \text{otherwise} \end{cases}$
⎧ τ₁
⎪ ── for s ≤ s₁
⎪ s₁
⎪
⎨-τ₁ + τ₂
⎪──────── for s ≤ s₂
⎪-s₁ + s₂
⎪
⎩ 0 otherwise
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/evaluate.png" alt="Evaluate" width="40" height="40">
&nbsp; &nbsp; <b>How to get numbers?</b> </div>
 
%% Cell type:markdown id: tags:
 
**The above results are symbols! How to transform them to numbers and graphs?**
 
`sympy` offers the possibility to generate executable code from symbolic expression (`C`, `Fortran`, or `Python`).
To get `Python` functions that accept the characteristic points `tau_1`, `tau_2`, `s_1`, `s_2`
and evaluating the above defined expressions `tau_s` and `d_tau_s`, we need the following two lines:
 
%% Cell type:code id: tags:
 
``` python
get_tau_s = sp.lambdify((s, tau_1, tau_2, s_1, s_2), tau_s, 'numpy')
get_d_tau_s = sp.lambdify((s, tau_1, tau_2, s_1, s_2), d_tau_s, 'numpy')
```
 
%% Cell type:markdown id: tags:
 
The parameter `numpy` enables us to evaluate both functions for arrays of values, not only for a single number. As a result, an array of slip values can be directly sent to the function `get_tau_s` to obtain an array of corresponding stresses
 
%% Cell type:code id: tags:
 
``` python
get_tau_s(np.array([0, 0.5, 1, 1.5, 2]), 1, 0.1, 1, 2)
```
 
%% Output
 
array([0. , 0.5 , 1. , 0.55, 0.1 ])
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/view.png" alt="Evaluate" width="40" height="40">
&nbsp; &nbsp; <b>How to to plot it?</b> </div>
 
%% Cell type:markdown id: tags:
 
Let us now show that the implemented bond-slip function provides a sufficient range of qualitative shapes to demonstrate and discuss the effect of softening and hardening behavior of the interface material. Let us setup a figure `fig` with two axes `ax1` and `ax2` to verify if the defined function is implemented correctly
 
%% Cell type:code id: tags:
 
``` python
fig, (ax1, ax2) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)
fig.canvas.header_visible = False
s_range = np.linspace(0, 3, 1050)
for tau_2 in [0, 0.5, 1, 1.5, 2]:
ax1.plot(s_range, get_tau_s(s_range, 1, tau_2, 0.1, 2));
ax2.plot(s_range, get_d_tau_s(s_range, 1, tau_2, 0.1, 2));
ax1.set_xlabel(r'$s$ [mm]'); ax1.set_ylabel(r'$\tau$ [MPa]');
ax2.set_xlabel(r'$s$ [mm]'); ax2.set_ylabel(r'$\mathrm{d}\tau/\mathrm{d}s$ [MPa/mm]');
```
 
%% Output
 
 
%% Cell type:markdown id: tags:
 
## Preconfigured pullout model provided in BMCS Tool Suite
The presented function is the simplest model provided in a general-purpose nonlinear finite-element simulator `BMCS-Tool-Suite`.
The package `bmcs_cross_section` provides several preconfigured models that can be used to analyze and visualize the behavior of a composite cross-section. The analysis of the pullout problem discussed here can be done using the class `PullOutModel1D` that can be imported as follows
 
%% Cell type:code id: tags:
 
``` python
from bmcs_cross_section.pullout import PullOutModel1D
```
 
%% Cell type:markdown id: tags:
 
An instance of the pullout model can be constructed using the following line
 
%% Cell type:code id: tags:
 
``` python
po = PullOutModel1D()
```
 
%% Cell type:markdown id: tags:
 
For convenience, let us summarize the model parameters before showing how to assign them to the model instance
 
%% Cell type:markdown id: tags:
 
**Geometrical variables:**
 
| Python | Parameter | Description |
| :- | :-: | :- |
| `A_f` | $A_\mathrm{f}$ | Cross section area modulus of the reinforcement |
| `A_m` | $A_\mathrm{m}$ | Cross section area modulus of the matrix |
| `P_b` | $p_\mathrm{b}$ | Perimeter of the reinforcement |
| `L_b` | $L_\mathrm{b}$ | Length of the bond zone of the pulled-out bar |
 
**Material parameters of a tri-linear bond law:**
 
| Python | Parameter | Description |
| :- | :-: | :- |
| `E_f` | $E_\mathrm{f}$ | Young's modulus of the reinforcement |
| `E_m` | $E_\mathrm{m}$ | Young's modulus of the matrix |
| `tau_1` | $\tau_1$ | bond strength |
| `tau_2` | $\tau_2$ | bond stress at plateu |
| `s_1` | $s_1$ | slip at bond strengh |
| `s_2` | $s_1$ | slip at plateau stress |
 
%% Cell type:markdown id: tags:
 
**Fixed support positions:**
 
| Python |
| :- |
| `non-loaded end (matrix)` |
| `loaded end (matrix)` |
| `non-loaded end (reinf)` |
| `clamped left` |
 
%% Cell type:markdown id: tags:
 
Even more conveniently, let us render the interaction window generated by the model to directly see the structure and the naming of the parameters
 
%% Cell type:code id: tags:
 
``` python
po.interact()
```
 
%% Output
 
 
%% Cell type:markdown id: tags:
 
The tree structure at the top-left frame shows the individual model components. Parameters of each component are shown in the bottom-left frame. By nagivating through tree, the parameter frame and the plotting frame are updated to see the corresponding part of the model. The control bar at the bottom can be used to start, stop and reset the simulation.
 
%% Cell type:markdown id: tags:
 
**Example interaction:** Develop some confidence into the correctness of the model. Change the stiffness of the components such that they have the same area and stiffness modulus. Run the simulation and watch the profile of the shear flow along the bond length. Increase the bond length, reset the calculation and run it anew. Change the position support and verify the profile of the displacements.
 
%% Cell type:markdown id: tags:
 
# **Studies 1: Hardening bond-slip law**
 
%% Cell type:markdown id: tags:
 
[![title](../fig/bmcs_video.png)](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)&nbsp; part 2
 
%% Cell type:markdown id: tags:
 
## RILEM Pull-Out Test revisited
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:image.png)
 
%% Cell type:code id: tags:
 
``` python
po_rilem = PullOutModel1D(n_e_x=300, w_max=0.12) # n_e_x - number of finite elements along the bond zone
po_rilem.n_e_x=400
```
 
%% Cell type:markdown id: tags:
 
To configure the model such that it reflects the RILEM test we can either use the interactive editor above, or assign the
attributes directly. As apparent from the editor frame above, attributes `fixed_boundary` and `material model` are dropdown boxes offering several options. To assign these parameters we can use the following scheme
- assign one of the options available in the dropdown box to the attribute `attribute` as a string
- the option object is then available as an attribute with the name `attribute_` with the trailing underscore.
Thus, to define a trilinear bond-slip law we can proceed as follows
 
%% Cell type:code id: tags:
 
``` python
po_rilem.material_model = 'trilinear' # polymorphis attribute - there are several options to be chosen from
# set the parameters of the above defined tri-linear bond-slip law - add also the matrix and fiber stiffness
po_rilem.material_model_.E_m=28000 # [MPa]
po_rilem.material_model_.E_f=210000 # [MPa]
po_rilem.material_model_.tau_1=4
po_rilem.material_model_.s_1=1e-3
po_rilem.material_model_.tau_2=8
po_rilem.material_model_.s_2=0.12
```
 
%% Cell type:markdown id: tags:
 
To set several parameters of the model component at once, the `trait_set` method can be used as an alternative to one-by-one assignement
 
%% Cell type:code id: tags:
 
``` python
d = 16.0 # [mm]
po_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)
po_rilem.geometry.L_x=5*d
po_rilem.fixed_boundary='loaded end (matrix)'
```
 
%% Cell type:markdown id: tags:
 
The configured model can be rendered anytime as a web-app to check the input parameters and to adjust them.
 
%% Cell type:code id: tags:
 
``` python
po_rilem.run()
po_rilem.interact()
```
 
%% Output
 
 
%% Cell type:markdown id: tags:
 
## Bond-slip law calibration/validation
 
**Can we find just one material law that predicts all three tests?**
- The preconfigured bond-slip law with an ascending branch after reaching the strength of $\tau_1 = 4$ MPa with the parameters $\tau_1 = 4$ MPa, $\tau_2 = 8$ MPa, $s_1 = 0.001$ mm, $s_1 = 0.12$ mm
can reproduce the test with $d = 16$ mm and $L_b = 5d = 80$ mm.
- To see the prediction for the test with $L_b = 10d = 160$ mm, modify the parameter `geometry.L_x = 160`. The result shows a good match with the experimentally observed response.
 
%% Cell type:markdown id: tags:
 
**Can we compare the differences in one plot?**
 
- The interactive user interface is illustrative and provides a quick orientation in the scope and functionality of the model. Once we have learned its structure, we can use the programming interface to run simulations in a loop and plot them in a single graph to see the similar picture as in the output of the RILEM test above.
 
- Try to compare the third test with $d = 28$ mm and $L_b = 5d$ mm.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/step_by_step.png" alt="Step by step" width="40" height="40">
&nbsp; &nbsp; <b>Plot step by step</b> </div>
 
%% Cell type:code id: tags:
 
``` python
fig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)
fig.canvas.header_visible = False
 
print('calculate d=16 mm, L=5d')
d = 16.0 # [mm]
po_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)
po_rilem.w_max = 0.12
po_rilem.geometry.L_x=5*d
po_rilem.reset() # it is like pressing the reset button in the above window
po_rilem.run() # like pressing the run button
po_rilem.history.plot_Pw(ax, color='blue')
 
print('calculate d=16 mm, L=10d')
d = 16.0 # [mm]
po_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)
po_rilem.w_max = 0.12
po_rilem.geometry.L_x=10*d
po_rilem.reset()
po_rilem.run()
po_rilem.hist.plot_Pw(ax, color='red')
 
print('calculate d=28 mm, L=3d')
d = 16.0 # [mm]
po_rilem.cross_section.trait_set(A_m=100*100, A_f=3.14*(d/2)**2, P_b=3.14*d)
po_rilem.geometry.L_x=3*d
po_rilem.w_max = 0.05
po_rilem.reset()
po_rilem.run()
po_rilem.hist.plot_Pw(ax, color='green')
po_rilem.material_model_.plot(ax_bond_slip)
 
# The code sequence can be certainly shortened by using the loop.
# It is deliberately omitted here as the focus is not on programming.
```
 
%% Output
 
 
calculate d=16 mm, L=5d
calculate d=16 mm, L=10d
calculate d=28 mm, L=3d
 
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-27-efa9a6e3b430> in <module>
28 po_rilem.run()
29 po_rilem.hist.plot_Pw(ax, color='green')
---> 30 po_rilem.material_model_.plot(ax_bond_slip)
31
32 # The code sequence can be certainly shortened by using the loop.
~/PycharmProjects/bmcs_utils/bmcs_utils/model.py in plot(self, axes)
37 def plot(self, axes):
38 """Alias to update plot - to be overloaded by subclasses"""
---> 39 self.update_plot(axes)
40
41 def update_plot(self, axes):
~/PycharmProjects/bmcs_ibvpy/ibvpy/tmodel/mats1D5/vmats1D5_bondslip1D.py in update_plot(self, axes)
30
31 def update_plot(self, axes):
---> 32 ax_tau, ax_d_tau = axes
33 s_max = self.s_max
34 n_s = 100
TypeError: cannot unpack non-iterable AxesSubplot object
%% Cell type:markdown id: tags:
 
## **Comments** on the study
- Note that the bond-slip law that can fit all three pull-out tests exhibits hardening.
- The maximum control displacement `w_max` is set equal to the one applied in the test as no information beyond this value is provided by the tests.
- The trilinear bond-slip law does not give us the flexibility to reproduce the pull-out failure
as it ends with a plateu.
 
## **Need for a more flexible bond-slip law**
- A more flexibility is provided by a `multilinear` material model for which a list of `s_data` and `tau_data`
can be specified.
- The `multilinear` material model is used in the following code to show how to achieve a pull-out failure by introducing a descending branch in the bond-slip law.
- Note that for bond-slip laws with descending branch, convergence problems can occur when approaching the pullout failure. The convergence behavior can be improved by refining the spatial discretization given by the number of finite elements along the bond zone `n_e_x` and by the size of the time step
`time_line.step`.
 
%% Cell type:code id: tags:
 
``` python
fig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)
fig.canvas.header_visible = False
d = 32.0 # [mm]
po_rilem.w_max = 0.12
po_rilem.time_line.step = 0.05
po_rilem.material_model='multilinear'
po_rilem.material_model_.trait_set(E_m=28000, E_f=210000, tau_data='0, 4, 6, 0, 0', s_data='0, 1e-3, 0.08, 0.12, 0.2')
po_rilem.geometry.L_x= 1*d
po_rilem.reset()
po_rilem.run()
po_rilem.hist.plot_Pw(ax, color='magenta')
po_rilem.material_model_.plot(ax_bond_slip)
```
 
%% Output
%% Cell type:markdown id: tags:
 
## **Questions:** Effect of bond length on the pullout response - **bond hardening**
 
%% Cell type:markdown id: tags:
 
- The iterative trial and error fitting is tedious. **How to design a test from which we can directly obtain the bond-slip law?**
Comparing the test with $L_b = 5d$ and $L_b = 10d$, we recognize that the shorter bond length resembles more the shape of the bond-slip law. To verify this, set the bond length in the above example to $L_\mathrm{b} = 1d$.
- On the other hand, if we increase the length, the maximum pull-out will increase. **How can we determine the bond length at which the steel bar will yield?**. A simple and quick answer to this question can be provided by reusing the analytical pull-out model with a constant bond-slip law as a first approximation. The maximum achievable pull-out force of a test with an embedded length $L_\mathrm{b}$ is given as
\begin{align}
\label{EQ:MaxEmbeddedLength}
P_{L} = \bar{\tau} p_\mathrm{b} L_\mathrm{b}
\end{align}
where $p_\mathrm{b}$ denotes the perimeter, equal in all experiments. The force at which the reinforcement attains the strength $\sigma_{\mathrm{f},\mathrm{mu}}$ and breaks is
\begin{align}
P_{\mathrm{f},\mathrm{mu}} = \sigma_{\mathrm{f},\mathrm{mu}} A_\mathrm{f}
\end{align}
so that the bond length at which the reinforcement will fail is obtained by requiring $P_L = P_{\mathrm{f},\mathrm{mu}}$ which renders
\begin{align}
\label{EQ:ConstantBondAnchorageLength}
L_{\mathrm{b}} = \frac{\sigma_{\mathrm{f},\mathrm{mu}} A_\mathrm{f} }
{\bar{\tau} p}.
\end{align}
For a generally nonlinear bond-slip law, we need to evaluate the maximum load numerically. Two examples quantifying the effect of the bond-length for bond-hardening and bond-softening systematically are provided in the notebook [3.2 Anchorage length](3_2_anchorage_length.ipynb)
 
%% Cell type:markdown id: tags:
 
<a id="cfrp_sheet_test"></a>
# **Studies 2: Softening bond-slip law**
 
%% Cell type:markdown id: tags:
 
[![title](../fig/bmcs_video.png)](https://moodle.rwth-aachen.de/mod/page/view.php?id=551816)&nbsp; part 3
 
%% Cell type:markdown id: tags:
 
The presence of the descending branch in a constitutive law is the key for understanding the propagating debonding. Let us use the established framework to study the different phenomenology that occurs for constitutive laws exhibiting softening.
Consider an interface between a fiber reinforced polymer (FRP) sheet used for retrofitting of RC structure. The study is based on a paper by [Dai et al. (2005)](../papers/dai_frp_pullout_2005.pdf). The goal of the paper is to derive constitutive laws capturing the bond-slip behavior of the adhesive between the FRP sheet and concrete surface. We will use selected experimental pullout curves from the paper to reproduce them using the numerical pullout model introduced above by verifying the bond-slip law derived in the paper.
 
%% Cell type:markdown id: tags:
 
## Test setup
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:974c60ea-69f6-4175-ab89-cea30a6bf1d6.png)
 
%% Cell type:markdown id: tags:
 
The width of the sheet was $p_b = 100$ mm, the attached bond length is also $L_b = 100$ mm.
The properties of the tested sheets are summarized in the table. The dimensions of the concrete block were $200 \times 400 \times 200$ mm.
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:fb43c5d7-b9f6-4d4d-8406-9d7610039506.png)
 
%% Cell type:markdown id: tags:
 
The pull-out curves measured for different adhesives used to realize the bond to the matrix were evaluated as the strain in the FRP sheet at the loaded end versus slip displacement.
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:0e4f48fa-0996-4c4a-be4c-7a2f893ba91d.png)
 
%% Cell type:markdown id: tags:
 
To compare with our studies, we can transfer the results to pullout force $P$ by evaluating
\begin{align}
P = p_\mathrm{b} t_\mathrm{f} E_\mathrm{f} \varepsilon_\mathrm{f}
\end{align}
yielding for the strain 0.010
 
%% Cell type:code id: tags:
 
``` python
p_b = 100 # [mm]
t_f = 0.11 # [mm]
E_f = 230000 # [MPa]
eps_f_max = 0.01 # [-]
P_max = p_b * t_f * E_f * eps_f_max / 1000 # [kN]
P_max # [kN]
```
 
%% Output
$\displaystyle 25.3$
25.3
%% Cell type:markdown id: tags:
 
The bond-slip law reported by the authors of the paper has the following shape
<a id="cfrp_bond_slip"></a>
 
%% Cell type:markdown id: tags:
 
![image.png](attachment:766b18c8-7f58-49a7-9406-ec2022c29188.png)
 
%% Cell type:markdown id: tags:
 
## Model for CFRP pullout test
 
%% Cell type:markdown id: tags:
 
Let us construct another pullout model named `po_cfrp` with a bond slip law exhibiting strong softening. Such kind of behavior is observed in tests between FRP sheets and concrete. An example of such an experimental study
<a id="cfrp_trilinear_bond"></a>
 
%% Cell type:code id: tags:
 
``` python
po_cfrp = PullOutModel1D(n_e_x=300, w_max=1.5) # mm
po_cfrp.geometry.L_x=100 # mm
po_cfrp.time_line.step = 0.02
po_cfrp.cross_section.trait_set(A_m=400*200, A_f=100*0.11, P_b=100)
po_cfrp.material_model='trilinear'
po_cfrp.material_model_.trait_set(E_m=28000, E_f=230000, tau_1=5.5, tau_2=0, s_1=0.08, s_2=0.4)
po_cfrp.interact()
```
 
%% Output
%% Cell type:markdown id: tags:
 
## **Conclusions:** to interactive study of CFRP sheet debonding
- The bond-slip law reported in the paper can reproduce well the pullout response measured in the test.
- The study of the debonding process shows that the adhesive is only active within an effective length of approximately 40-50 mm.
- As a consequence, in contrast to steel rebar studied above, the maximum pullout load cannot be increased by an increasing bond length.
- In the studied case there will FRP rupture is not possible because its strength is larger than $P_\max$ of 25 kN. To verify this, we use the strength of 3550 MPa given in the above table and multiply with the cross-sectional area of the sheet, i.e. $f_t t_f p_b$ to obtain
 
%% Cell type:code id: tags:
 
``` python
f_t = 3550 # CFRP sheet strength in [MPa] - see the table above
f_t * t_f * p_b / 1000 # breaking force of the sheet 100 x 100 mm in [kN]
```
 
%% Output
$\displaystyle 39.05$
39.05
%% Cell type:markdown id: tags:
 
## **Question:** Effect of bond length on the pullout response - **bond softening**
- Similarly to the the example with bond hardening above, we ask the question what happens with the pullout curve if we reduce the bond length to a minimum. The answer is the same - we will recover a bond-slip law multiplied by the bond area.
- However, if we increase the bond length, the trend will be different as already mentioned above. Once the length exceeds the effective bond length, there will be no increase in the pullout force and the pullout curve will exhibit a plateau. Let us show this trend by running a simple parametric study. Instead of doing it step by step we now run a loop over the list of length and colors, change the parameter `geometry.L_x` within the loop, `reset`, `run`, and `plot` the pullout curve in a respective color.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/run.png" alt="Run" width="40" height="40">
&nbsp; &nbsp; <b>Run in a loop to see the effect of bond length</b> </div>
 
%% Cell type:markdown id: tags:
 
Note that a list in python is defined by the brackets
```[1,2,3,4]```. Two lists can be "zipped" together so that we can run
a loop over the lengths and colors as shown in the third line of the cell
<a id="crfp_study"></a>
 
%% Cell type:code id: tags:
 
``` python
fig, (ax, ax_bond_slip) = plt.subplots(1,2, figsize=(10,4), tight_layout=True)
fig.canvas.header_visible = False
for L, color in zip([5, 10, 50, 100, 200], ['red','green','blue','black','orange']):
print('evaluating pullout curve for L', L)
po_cfrp.geometry.L_x=L
po_cfrp.reset()
po_cfrp.run()
po_cfrp.history.plot_Pw(ax, color=color)
po_cfrp.material_model_.plot(ax_bond_slip)
```
 
%% Output
evaluating pullout curve for L 5
evaluating pullout curve for L 10
evaluating pullout curve for L 50
evaluating pullout curve for L 100
evaluating pullout curve for L 200
%% Cell type:markdown id: tags:
 
# **Remark to structural ductility:** how to make the plateau useful?
 
The softening bond cannot exploit the full strength of the CFRP sheet which might seem uneconomic at first sight. On the other hand it can be viewed as a mechanism that increases the deformation capacity of the structure with a constant level of load. This property can be effectively used to enhance the ductility of the structure, i.e. induce large deformation before the structural collapse required in engineering designs. This documents the importance of knowledge of the stress redistribution mechanisms available in the material. In steel reinforced structure, the ductility is provided inherently by the steel yielding property. In In case of brittle reinforcement, e.g. carbon fabrics, CFRP sheets, glass fabrics, other sources of ductility must be provided to ensure the sufficient deformation capacity between the serviceability and ultimate limit states.
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/exercise.png" alt="Run" width="40" height="40">
&nbsp; &nbsp; <a href="../exercises/X0301 - Pull-out curve versus shear stress profiles.pdf"><b>Exercise X0301:</b></a> <b>Pull-out curve versus shear stress profiles - part 1</b>
<a href="https://moodle.rwth-aachen.de/mod/page/view.php?id=551821"><img src="../icons/bmcs_video.png" alt="Run"></a>
</div>
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left"> <img src="../icons/exercise.png" alt="Run" width="40" height="40">
&nbsp; &nbsp; <a href="../exercises/X0302 - Pull-out curve versus shear stress profiles.pdf"><b>Exercise X0302:</b></a> <b>Pull-out curve versus shear stress profiles - part 2</b>
<a href="https://moodle.rwth-aachen.de/mod/page/view.php?id=551823"><img src="../icons/bmcs_video.png" alt="Run"></a>
</div>
 
%% Cell type:markdown id: tags:
 
<div style="background-color:lightgray;text-align:left;width:45%;display:inline-table;"> <img src="../icons/previous.png" alt="Previous trip" width="50" height="50">
&nbsp; <a href="../tour2_constant_bond/fragmentation.ipynb#top">2.3 Tensile behavior of composite</a>
</div><div style="background-color:lightgray;text-align:center;width:10%;display:inline-table;"> <a href="#top"><img src="../icons/compass.png" alt="Compass" width="50" height="50"></a></div><div style="background-color:lightgray;text-align:right;width:45%;display:inline-table;">
<a href="3_2_anchorage_length.ipynb#top">3.2 Pullout curve versus bond length</a>&nbsp; <img src="../icons/next.png" alt="Previous trip" width="50" height="50"> </div>
......
tour6_energy/6_3_localized_energy_dissipation.ipynb
+ 4
− 4
View file @ c3fc8367
......@@ -246,7 +246,7 @@
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "8bc97b6764834c199a1e50d65207907e",
"model_id": "8e4d3116f22b45d0a599ac0d0862c181",
"version_major": 2,
"version_minor": 0
},
......@@ -271,7 +271,7 @@
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "31f39ca22bf941dfb37a2c1fef6a5208",
"model_id": "0d14959e36ad4d15b69f11cef352f7d3",
"version_major": 2,
"version_minor": 0
},
......@@ -434,14 +434,14 @@
},
{
"cell_type": "code",
"execution_count": 8,
"execution_count": 7,
"id": "a8acac48-f466-473c-8de9-fef4187d3b81",
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "42f6fec2662046e4ad3f31f50b997e23",
"model_id": "9c762b72a49342a0b187626bc2978a31",
"version_major": 2,
"version_minor": 0
},
......
tour7_cracking/7_1_bending3pt_2d.ipynb
+ 247
− 152
View file @ c3fc8367
%% Cell type:markdown id:1cdb7314-aaae-4b45-8ab9-20c2e6d6e1cc tags:
# **7.1 Propagation of a straight crack**
%% Cell type:markdown id:1c5177ac-cfcd-4fba-965c-fea4ddb87881 tags:
[still under construction]
%% Cell type:markdown id:0465afaf-c8d1-4604-a9bd-3bd91bb67ca4 tags:
<div style="background-color:lightgray;text-align:left"> <img src="../icons/start_flag.png" alt="Previous trip" width="50" height="50">
&nbsp; &nbsp; <b>Starting point</b> </div>
%% Cell type:markdown id:564277ba-99b5-43ec-ab94-b448eed57ee0 tags:
In the notebook [6.3](../tour6_energy/6_3_localized_energy_dissipation.ipynb#top) we have analyzed the correspondence between the damage function inducing softening and the propagation of discontinuity macroscopically observable as debonding in a pullout test. We mentioned the correspondence between the locally and globally energy evaluated energy dissipation indicating the energy dissipated at a material point can be used to estimate the total amount of dissipated energy by relating it to a localized crack of a known known surface area. In case of an interface area between two material components, like CFRP sheet and concrete, the area is well defined.
Let us now apply the same concept to a crack propagation through a concrete specimen. Even though additional aspects, like two-dimensional stress and strains states, need to be considered, the general concept of energy dissipation introduced previously remains the similar.
%% Cell type:markdown id:b9526df2-893a-4125-b029-d843b224bae6 tags:
<div style="background-color:lightgray;text-align:left"> <img src="../icons/destination.png" alt="Previous trip" width="50" height="50">
&nbsp; &nbsp; <b>Where are we heading</b> </div>
%% Cell type:markdown id:cc776d2c-f89a-4c69-8374-93447c23eda8 tags:
In the present notebook we apply the damage model inducing material softening to a two-dimensional case of stable crack propagation. In particular, we are going to use the three-point bending test as a means of rendering an elementary crack-propagation scenario, namely a straight crack propagating through a cross section. After discussing the test setup, we provide the test results in terms of load-deflection curve. Then, we will construct a numerical model that can reproduce such a behavior. Similarly to the previous lecture, we will consider the link between fracture and damage concepts that are both included in any finite element computation involving strain-softening material behavior
%% Cell type:markdown id:d098050e-94a4-4ea3-b817-20e14a132fef tags:
# **Why notched three-point bending test?**
%% Cell type:markdown id:b1e064b7-2688-40a4-83ce-491b6dcabe6d tags:
An isolated tensile crack propagation can be initiated using a notched specimen. The most common configurations used to study the cracking behavior for a tensile crack denoted as mode I are the wedge splitting test and notched, three-point bending test. Both these tests aim to characterize the material behavior in terms of the softening law describing the relation between the tensile stress transmitted across the localization zone and the corresponding crack opening.
Due to its simplicity, three-point-bending test on a notched concrete has become a standard (RILEM) to determine the fracture energy $G_\mathrm{f}$ characterizing the cracking behavior of concrete. The test induces a single crack in the notched section propagating straight upwards from the notch in a stable manner. The energy is dissipated in a local region in the crack vicinity so that it can be directly ascribed to the area of emerging crack surface.
The numerical simulation of this model can be readily performed using the material model with the damage function derived in previous notebooks. An example of the geometry and boundary conditions of the three-point bending test is provided by the Petersen test series using the following setup.
%% Cell type:markdown id:0a4edb61-8c07-492a-8f08-82af92bb684d tags:
![image.png](attachment:aa0b3375-f959-416c-ad75-641d96347767.png)
%% Cell type:markdown id:969cea85-7c46-4444-8847-d793f565bc43 tags:
# **Boundary value problem and its FE discretization**
%% Cell type:markdown id:780cfe67-6a6f-435d-afce-dbb2bc491e76 tags:
# **Material model**
%% Cell type:code id:83e0cf34-4db1-41bf-ac41-7840d0ba44f1 tags:
``` python
%matplotlib widget
from bmcs_bending.bending3pt_2d import BendingTestModel
from ibvpy.tmodel.mats2D import MATS2DScalarDamage
```
%% Cell type:markdown id:d93e03f8-4d44-4316-b8db-c160517b8430 tags:
\begin{align}
\sigma_{ab} = (1 - \omega(\kappa(\varepsilon_{cd}))) D_{abcd}^\mathrm{el} \varepsilon_{cd}
\end{align}
%% Cell type:markdown id:d6ab78b9-00d0-4a8d-803a-111b10e920b4 tags:
\begin{align}
D_{abcd} &= \dfrac{\mathrm{d} \sigma_{ab}}{\mathrm{d}\varepsilon_{cd}} \\
& =
\left(1 - \omega(\kappa(\varepsilon_{cd}))\right) D_{abcd}^\mathrm{el} -
\dfrac{\mathrm{d} \omega}{\mathrm{d}\kappa}
\dfrac{\partial \kappa}{\partial\varepsilon_{cd}} D_{abcd}^\mathrm{el} \, \varepsilon_{cd} \, \theta_\mathrm{alg,stiff}
\end{align}
%% Cell type:code id:cda0ba20-eb96-48de-a904-1376d9376a73 tags:
``` python
bt = BendingTestModel(material_model='scalar damage',
n_e_x=8, n_e_y=16, w_max=-2, k_max=500)
n_e_x=6, n_e_y=16, w_max=-2, k_max=500)
E = 30000
f_ct = 3.3
kappa_0 = f_ct / E
bt.time_line.step=0.01
bt.time_line.step=0.03
bt.history.warp_factor=100
bt.cross_section.trait_set(b=50)
bt.geometry.trait_set(L=2000, H=200, a=100, L_c=1)
bt.material_model_.trait_set(E = E, nu = 0.0)
bt.material_model_.trait_set(E = E, nu = 0.0) # note nu = 0.0 to avoid compressive failure
bt.material_model_.omega_fn = 'exp-slope'
bt.material_model_.omega_fn_.trait_set(kappa_0=kappa_0, kappa_f=0.0336)
bt.material_model_.trait_set(D_alg=1, eps_max=0.1);
bt.material_model_.trait_set(D_alg=1, eps_max=1);
```
%% Cell type:markdown id:7541585c-f32d-4954-acfd-d81432fdfd29 tags:
## Strain norm
%% Cell type:markdown id:2a6b3a48-27f9-4fdf-9752-bd42bb9e861a tags:
The scalar damage material model can combine different measures of equivalent strain with a different type of damage function.
The task of the energy norm is to transform the two-dimensional strain tensor
$$
\varepsilon_{ab}
=
\left[
\begin{array}{cc}
\sigma_{xx} & \sigma_{xy} \\
\sigma_{yx} & \sigma_{yy}
\end{array}
\right]
$$
to a scalar value $\kappa$. The energy norms already mentioned in the notebook [5.1](../tour5_damage_bond/5_1_Introspect_Damage_Evolution_Damage_initiation.ipynb#strain_norm) play the role of an elastic limit. Their visual inspection
can be performed using the model component of the material model `MATSScalarDamage`.
%% Cell type:code id:0b3773c1-edc6-4952-a414-a0e5f1e41f91 tags:
``` python
bt.material_model_.strain_norm_.name
bt.material_model_.strain_norm = 'Rankine'
bt.material_model_.strain_norm_.interact()
```
%% Output
'Rankine strain norm'
%% Cell type:markdown id:db2febf3-5dbb-4199-bf91-495614a79d4d tags:
Further strain norms can be chosen from the options
- Rankine strain norm
- Masars strain norm
- Energy norm
These norms can have to be combined with an elastic threshold to distinguish the elastic and inelastic domains.
Inspect the visual representation of the equivalent strain measure by changing the selector in the material model app. (It is necessary to select the respective tree node to render the visualization of the currently selected option.)
%% Cell type:code id:8dd4a96a-e8b6-4cbc-9f71-bdf0f680b35d tags:
``` python
bt.material_model_.interact()
```
%% Output
%% Cell type:markdown id:d5e41956-14c6-4ad1-ac42-c4b8b66b15fc tags:
## The importance of algorithmic stiffness
%% Cell type:markdown id:382aac37-21a3-4c94-b245-48e353f576d3 tags:
The rate of convergence strongly depends on the quality of prediction of the stress state for the next step. The parameter `D_alg` can be used ton control the calculation by choosing between the secant and algorithmic, tangential stiffness. Derivation of the material stiffness might appear tedious at the first sight but in fact, it only requires the evaluation of the chain derivatives involved in the constitutive law.
%% Cell type:markdown id:d06b9560-012b-40ce-918a-7f93c60dbc97 tags:
Let us exemplify the derivation of the algorithmic material stiffness on the example of the isotropic damage model. In contrast to the version presented in notebook [5.1]() for the bond-slip law, we are applying the damage model to a 2D continuum described by the strain and stress tensor tensors $\boldsymbol{\varepsilon} = \varepsilon_{ab}$
and $\boldsymbol{\sigma} = \sigma_{ab}$, where letter indexes $a,b,c,d = [1,2]$ enumerate the orthogonal directions of the cartesian coordinate system. For example, $\varepsilon_{11} = \varepsilon_{xx}$. Then the stress-strain relation prescribed by the damage model is given as
%% Cell type:markdown id:d93e03f8-4d44-4316-b8db-c160517b8430 tags:
\begin{align}
\sigma_{ab} = (1 - \omega(\kappa(\varepsilon_{cd}))) D_{abcd}^\mathrm{el} \varepsilon_{cd}
\end{align}
%% Cell type:markdown id:0db20967-ba8d-426e-a1f8-dfbf6780fc45 tags:
where $\kappa(\varepsilon_{cd})$ represents the equivalent strain described above, and $\omega(\kappa)$ represents the particular damage function. The algorithmic stiffness is derived as the derivative of the above stress tensor with respect to a strain tensor. This means that every stress component must be differentiated with respect to every strain component. The result of this expression is a rank 4 tensor $D_{abcd}$ describing the instantaneous stiffness of a material point undergoing damage. The resulting expression delivers the material stiffness in the following form
%% Cell type:markdown id:d6ab78b9-00d0-4a8d-803a-111b10e920b4 tags:
\begin{align}
D_{abcd} &= \dfrac{\mathrm{d} \sigma_{ab}}{\mathrm{d}\varepsilon_{cd}} \\
& =
\left(1 - \omega(\kappa(\varepsilon_{cd}))\right) D_{abcd}^\mathrm{el} -
\dfrac{\mathrm{d} \omega}{\mathrm{d}\kappa}
\dfrac{\partial \kappa}{\partial\varepsilon_{cd}} D_{abcd}^\mathrm{el} \, \varepsilon_{cd} \, \theta_\mathrm{alg,stiff}
\end{align}
%% Cell type:markdown id:4795a44a-a0cd-41cf-a55f-32be19e84454 tags:
The parameter `D_alg` is denoted here as $\theta_{\mathrm{alg,stiff}}$ and is provided as an input parameter in the material model. By setting this parameter to zero, the second term in the above expression is canceled. Then, only the secant stiffness is available. The importance of the algorithmic stiffness can be studied in the simulation of the bending test. The default value $\theta_\mathrm{alg,stiff} = 1$ is activating the algorithmic stiffness. By setting $\theta_\mathrm{alg,stiff} = 0$ one can see the strong reduction of the performance.
%% Cell type:markdown id:a33174d1-fd84-4ca9-924c-a500ee83f47f tags:
However, in some situation, for example when simulating simultaneous crack propagation in reinforced specimens the tangential stiffness might induce divergence. The reason for such behavior is the fact that the bond stress can rapidly change the sign. In these situations, advanced control strategies are required.
%% Cell type:code id:2a956afb-0425-4318-9335-628675151680 tags:
``` python
#bt.run()
```
%% Cell type:code id:bf30823c-3ed8-4e8b-a84c-23cc25c8e094 tags:
``` python
bt.interact()
```
%% Output
%% Cell type:markdown id:9c3028e4-f16b-4d7c-8020-0fa13fade796 tags:
If a standard finite element discretization is combined with a softening material model,
1. it will always induce localization into a band of finite elements at the propagating stress concentration.
2. the area below the stress strain curve of a material model represents the energy dissipated by a unit volume of a material.
As a consequence, the total amount of the dissipated energy can be calculated as a product of the specific dissipation energy $G_\mathrm{f}$ $\mathrm{[J/mm^3]}$ and the volume of elements constituting the crack band $V_\mathrm{diss}$ $\mathrm{[mm^2]}$. In general,
$$
G_\mathrm{total} = G_\mathrm{f} V_\mathrm{diss}
$$
In a notched bending test, the crack band is running from the notch straight through the cross section. Therefore, it is possible to directly calculate the dissipative volume $V_\mathrm{diss}$ as
$$
V_\mathrm{diss} = (H - a) B L_\mathrm{c}
$$
%% Cell type:code id:752906fd-2a8a-44ec-b77c-9e3a496f9c0f tags:
``` python
V_diss = (bt.geometry.H - bt.geometry.a)*bt.cross_section.b * bt.geometry.L_c
V_diss
```
%% Output
5000.0
%% Cell type:markdown id:9dc0bd1a-7a3e-497b-bb63-0e0c239c1baa tags:
The specific energy $G_\mathrm{f}$ is available as an attribute of the material model. It has the value
%% Cell type:code id:ec96f0ce-0f91-4d3a-8d59-d3cedb3a9f9c tags:
``` python
bt.material_model_.G_f # J/mm^2
```
%% Output
0.10508670091190059
0.1092317778327407
%% Cell type:markdown id:21b49bb5-cc8d-43d7-82f9-74a80a44f06b tags:
Thus, the total dissipated energy should read
%% Cell type:code id:8265609d-ed22-4240-a1ac-22bd025137a8 tags:
``` python
G_total = V_diss * bt.material_model_.G_f
G_total
```
%% Output
525.4335045595029
546.1588891637035
%% Cell type:markdown id:9a257535-82e4-451e-8b8f-82ee2a4cc686 tags:
The total dissipated energy is available in the `hist` model component as a last element of the recorded array, so that we can pick it using the index `-1`
%% Cell type:code id:e5e6854c-6ec7-43a7-b4bc-d6a725af6b79 tags:
``` python
bt.hist['energy'].G_t[-1]
#bt.hist['energy'].G_t[-1]
```
%% Output
377.64457017857245
%% Cell type:markdown id:2e354956-a60f-445f-8e8f-a7b21ce4c1ce tags:
# **Parametric study**
%% Cell type:markdown id:3479d206-6b40-4b8b-aef7-95143d0bb3fa tags:
<div style="background-color:lightgray;text-align:left"> <img src="../icons/run.png" alt="Run" width="40" height="40">
&nbsp; &nbsp; <b>Run in a loop along the changing width of the crack band</b> </div>
%% Cell type:code id:fd5696fa-725e-4c7b-bac8-f263bdf532d3 tags:
``` python
Fw_dict = {}
G_dict = {}
```
%% Cell type:code id:c83a2693-4fbe-4ce8-a3eb-0a281430f1b7 tags:
``` python
L_c_list = [0.5, 1, 2, 4]
L_c_list = [1, 2, 4]
for L_c in L_c_list:
if Fw_dict.get(L_c):
continue
print('L_c', L_c)
print('calculating F-w and G for crack band L_c = %g [mm]' % L_c)
bt.geometry.L_c = L_c
bt.reset()
try:
bt.run()
except StopIteration:
print('not converging for', L_c)
print('simulation interupted due to slow convergence', L_c)
Fw_dict[L_c] = bt.hist['Fw'].Fw
G_dict[L_c] = bt.hist['energy'].G_t[-1]
```
%% Output
L_c 0.5
not converging for 0.5
L_c 1
L_c 2
L_c 4
calculating F-w and G for crack band L_c = 1 [mm]
calculating F-w and G for crack band L_c = 2 [mm]
calculating F-w and G for crack band L_c = 4 [mm]
%% Cell type:markdown id:6bdc3565-8107-4459-81ca-653b7c179da0 tags:
<div style="background-color:lightgray;text-align:left"> <img src="../icons/view.png" alt="Run" width="40" height="40">
&nbsp; &nbsp; <b>... let us put the results into a diagram</b> </div>
%% Cell type:code id:cf30a1f8-7eae-4aa4-8101-fe8337c6cba8 tags:
``` python
import matplotlib.pylab as plt
fig, ax = plt.subplots(1,1)
fig, (ax, ax_G) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)
fig.canvas.header_visible=False
for L_c, (F, w) in Fw_dict.items():
ax.plot(-w,-F,label='L_c = %g' % L_c)
ax.legend()
ax.set_ylabel(r'$F$')
ax.set_xlabel(r'$w$ [mm]');
ax.set_ylabel(r'$F$ [N]');
G_list = [G_dict[L_c] for L_c in L_c_list]
ax_G.plot(L_c_list, G_list, marker='H')
ax_G.set_xlabel(r'$L_\mathrm{c}$ [mm]')
ax_G.set_ylabel(r'$G_\mathrm{total}$ [kJ]');
```
%% Output
<matplotlib.legend.Legend at 0x7ff52059da60>
%% Cell type:markdown id:68f2a220-525a-4fc1-9209-eee1a98dd276 tags:
%% Cell type:code id:a34f8596-23be-4a6a-bb88-ae0c851d4104 tags:
# **How to make the results mesh independent?**
``` python
import sympy as sp
```
%% Cell type:markdown id:83a6019b-7909-499c-8050-b2748563d382 tags:
%% Cell type:code id:cd940d2c-99aa-4c97-b973-aabab76bc777 tags:
The finite element codes solve this problem by adjusting the slope of the softening branch to keep the energy dissipation within a crack band invariant for a changed size of an element. This is the most common technique used in commercial finite element codes.
``` python
eps, D = sp.symbols(r'\varepsilon, D')
```
%% Cell type:markdown id:f4a93e45-ad34-4485-bf75-b7e7a46150ee tags:
Let us demonstrate the concept of mesh-adjusted softening response on the running example.
%% Cell type:code id:32656538-f9d7-498f-993a-36ad88c2998f tags:
%% Cell type:code id:0e49628e-0d75-4a92-a1e9-74386de668fc tags:
``` python
eps2 = sp.sqrt(D * eps**2)
eps2
Fw_reg_dict = {}
G_reg_dict = {}
L_c_list = [1, 2, 4]
kappa_f = 0.0336
for L_c in L_c_list:
if Fw_reg_dict.get(L_c):
continue
print('calculating F-w and G for crack band L_c = %g [mm]' % L_c)
bt.reset()
bt.geometry.L_c = L_c
bt.material_model_.omega_fn_.kappa_f = kappa_f / L_c #### REGULARIZATION ####
print('G_f', bt.material_model_.G_f)
try:
bt.run()
except StopIteration:
print('simulation interupted due to slow convergence', L_c)
Fw_reg_dict[L_c] = bt.hist['Fw'].Fw
G_reg_dict[L_c] = bt.hist['energy'].G_t[-1]
```
%% Cell type:markdown id:e9f08de9-0aaa-4ae3-8c0a-3d35a7d98d30 tags:
$$
(D \varepsilon^2)^{-\frac{1}{2}} \cdot 2 D \varepsilon
=
D^{-\frac{1}{2}} \varepsilon^{-1}\cdot 2 D \varepsilon
$$
%% Cell type:markdown id:cf16c894-7de4-4409-b5e2-3161aafcefe4 tags:
$$
\dfrac{1}{2 \sqrt{D \varepsilon^2}} \cdot 2 D \varepsilon = \dfrac{D\varepsilon}{\sqrt{D \varepsilon^2}}
$$
%% Cell type:code id:8f01a1d4-a63d-48bb-84bc-f0f14e5a206f tags:
%% Output
``` python
a = sp.symbols('a')
(sp.sqrt(a).diff(a) * (D*eps**2).diff(eps)).subs(a, D*eps**2)
```
calculating F-w and G for crack band L_c = 1 [mm]
G_f 0.1092317778327407
calculating F-w and G for crack band L_c = 2 [mm]
G_f 0.053795255499048136
calculating F-w and G for crack band L_c = 4 [mm]
G_f 0.02608239216637152
%% Cell type:code id:0c8b38f7-407a-4910-9328-d07f444c1d5a tags:
%% Cell type:code id:bc7eb155-af93-45db-b1c8-7237afbffb1f tags:
``` python
eps2.diff(eps)
import matplotlib.pylab as plt
fig, (ax, ax_G) = plt.subplots(1,2, figsize=(8,3), tight_layout=True)
fig.canvas.header_visible=False
for L_c, (F, w) in Fw_reg_dict.items():
ax.plot(-w,-F,label='L_c = %g' % L_c)
ax.legend()
ax.set_xlabel(r'$w$ [mm]');
ax.set_ylabel(r'$F$ [N]');
G_list = [G_reg_dict[L_c] for L_c in L_c_list]
ax_G.plot(L_c_list, G_list, marker='H')
ax_G.set_xlabel(r'$L_\mathrm{c}$ [mm]')
ax_G.set_ylabel(r'$G_\mathrm{total}$ [kJ]');
```
%% Cell type:markdown id:90a5f095-b384-4a86-97bd-2752518c2081 tags:
$$
\kappa = \sqrt{ \varepsilon_{ab} D_{abcd} \varepsilon_{cd}}
$$
%% Cell type:markdown id:939af17a-2f68-4e3b-8150-0e239bfad745 tags:
%% Output
$$
\dfrac{\partial \kappa}{\partial \varepsilon_{cd}}
=
\dfrac{1}{\kappa} D_{abcd} \varepsilon_{cd}
$$
%% Cell type:code id:0dff0fb6-5d1b-4caa-874f-556ad1d1e0c1 tags:
%% Cell type:markdown id:eea93b93-7e10-49f8-8537-d90b5c042463 tags:
``` python
```
......
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