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.
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use the `jupyter` notebooks are summarized here:</br>
[**Web Apps:** first steps](link)
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[**Jupyter notebooks and Python**: short introduction, installation instructions](tour1_intro/1_2_jupyter_python.ipynb#top)
### 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_3_elastic_stiffness_of_the_composite.ipynb#top)
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## **Tour 2:** Constant bond - pull-out, crack bridge, fragmentation
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### 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>
### 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)
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### 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)
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## **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)
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### 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>
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)
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## **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)
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### 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)
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## **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)
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## **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>
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)
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### 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)
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## **Tour 7:** Bending and crack propagation
### 7.1 Notched beam with straight crack propagation
With the possibility to evaluate energy dissipation, we study the crack propagation
in a notched bending test using a finite-element model. This study presents the reasoning behind
the identification of fracture energy using the notched bending test.
At the same time, it illuminates how to correctly reflect the crack localization
in the standard finite-element method.</br>
[Propagation of a straight crack](tour7_cracking/7_1_bending3pt_2d.ipynb#top)
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### 7.2 Experimental identification of fracture energy and size effect
The explained correspondence between work supplied to the test and the energy dissipation is exploited to design a simple test setup delivering an estimate of the fracture energy as a fundamental characteristic of concrete behavior. The important consequence of localization behavior inducing a so called size effect is presented using a simple parametric study.</br>
[Fracture energy and size effect](tour7_cracking/7_2_fracture_energy_ident.ipynb#top)
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## **Tour 8:** Reinforced bended cross section
### 8.1 Moment-curvature derived for arbitrary cross-sectional layout
The behavior of a reinforced cross section can be characterized using
a nonlinear moment-curvature relation. This relation describes the effect
of multiple cracking and debonding in the tensile, crack bridging action
including debonding, and nonlinear response of concrete in the compression zone.</br>
