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)
-->
[**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)
%% Cell type:markdown id: tags:
<aid="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>
### 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:
<aid="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)
%% Cell type:markdown id: tags:
### 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)
%% Cell type:markdown id: tags:
<aid="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:
<divstyle="background-color:lightgreen;text-align:left"><imgsrc="icons/rest.png"alt="Step by step"width="40"height="40">
<b>Our current location</b></div>
%% 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:
<aid="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:
<divstyle="background-color:lightgreen;text-align:left"><imgsrc="icons/rest.png"alt="Step by step"width="40"height="40">
<b>Our current location</b></div>
%% Cell type:markdown id: tags:
<aid="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>
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
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:
<aid="tour7"></a>
## **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)
%% Cell type:markdown id: tags:
### 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)
%% Cell type:markdown id: tags:
<aid="tour8"></a>
## **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>
"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. This perspective is particularly important for materials exhibiting softening behavior which which induces an energy dissipation within a small volume of material.\n"
"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. This perspective is particularly important for materials exhibiting softening behavior. As we have already learned, the softening behavior leads to stress concentration within a small volume associated with strain localization leading to a discontinuity, either interface gap or crack. The process of localization is can be characterized by the amount of energy dissipation within a small volume of material, the size of which is characteristic for a particular composition of material structure."
]
},
{
...
...
@@ -59,7 +59,7 @@
"id": "extended-eugene",
"metadata": {},
"source": [
"In the present notebook we will revisit the nonlinear pullout curve and show how to evaluate the stored and released energy. We start with an abstract description explaining the evaluation procedure. To provide an elementary example, we then apply the described concept to an elementary example of an elastic bar with no energy dissipation. This prepares the framework for the next example showing applying the evaluation procedure to the pullout test in the notebook [6.2](6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top) and, most importantly, to the notebook [6.3]() setting up the ground for the description of fracture. Thus, our next tour goes through the following three stations: tensile test, pullout test and bending test. We will show a unified framework for the evaluation of energy dissipation in all three of them."
"In the present notebook we will revisit the nonlinear pullout curve and show how to evaluate the stored and released energy. We will start with an abstract description explaining how to quantify and distinguish work, stored energy and dissipated energy. To provide an elementary example, we will then apply the described concept to an elementary example of an elastic bar with no energy dissipation. This prepares the framework for the next example showing how to apply the quantification procedure to a pullout test. To this end, we will revisit the model with constant bond-slip law [6.2](6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top). After that, we will apply the same procedure to the model presented in the notebook [6.3](6_3_localized_energy_dissipation.ipynb#top) for bond behavior with softening. With the fracture energy quantified for an interface based on a pull-out test, we will demonstrate the general description of fracture mechanics. The gained knowledge will be used for the characterization of tensile crack, propagating through a bending test in the next tour starting in notebook [7.1](../tour7_cracking/7_1_bending3pt_2d.ipynb#top) ."
]
},
{
...
...
@@ -100,8 +100,10 @@
"source": [
"## Supplied energy\n",
"\n",
"Consider a general force displacement curve obtained in an experiment. The curve represents a response of a test setup with an arbitrary geometry of the specimen, material components and their layout, and even the boundary conditions. \n",
"The only information we need from the lab is the time profile of the control displacement $w(t)$ and the profile of force $P(t)$ measured by the hydraulic cell at the point of the imposed displacement. Using these two series of data points, we can plot the force displacement curves. "
"Consider a general force displacement curve obtained in an experiment. The curve represents a measured response of a test setup. It can be any kind of test setup: pull-out, punch-through, bending or wedge splitting test. Besides the geometry, also the material components and their layout, and even the boundary conditions are arbitrary. We consider the structure as a black box that can store and dissipate energy. \n",
"To quantify the amount of these energies, the only information we need from the lab is the time profile of the control displacement $w(t)$ and the corresponding time profile of force $P(t)$, measured by the hydraulic cell at the point of the imposed displacement. Using these two series of data points, we can plot the force displacement curves.\n",
"\n",
"**Question:** Can you think of an answer to the question, why do we control the displacement in the test and not the force? Both are technically possible. What happends, when we use force control and reach the maximum load that can be transfered by a specimen? "
]
},
{
...
...
@@ -122,9 +124,9 @@
"id": "olive-ethiopia",
"metadata": {},
"source": [
"This curve can be used to answer the question, how much energy must have been supplied by the testing machine to reach the point $A$ of the measured response. Let us first recall that work is defined as the force needed achieve a unit displacement, i.e.\n",
"The experimentally obtained load-displacement curve can be used to answer the question, how much energy must have been supplied by the testing machine to reach the point $A$ of the measured response. Let us first recall that work is defined as the force needed to achieve a unit displacement, i.e.\n",
"$$\n",
" \\mathcal{W} = P w \\;\\; \\mathrm{[Nm]}\n",
" \\mathcal{W} = P w \\;\\; \\mathrm{[Nm]}.\n",
"$$\n",
"In case of a nonlinear profile obtained from the testing machine we apply this definition to evaluate the work increment needed to achieve an infinitesimal displacement\n",
"$$\n",
...
...
@@ -150,7 +152,10 @@
"id": "verified-diana",
"metadata": {},
"source": [
"We already know, that the nonlinear shape of the force-displacement response is due to irreversible changes in the material structure. These changes are connected with energy dissipation. Thus, in point $A$ some fraction of energy has already been dissipated and some is stored in the specimen. How much energy is stored in the specimen at point $A$? \n",
"**One sentence to thermodynamics:**\n",
"We already know, that the nonlinear shape of the force-displacement response is due to irreversible changes in the material structure. We know from thermodynamics that any irreversible change of a system is connected with energy dissipation. Thus, in point $A$, which is behind the peak of the load-displacement curve, irreverisble changes must have already taken place. As a consequence, some amount of energy must already have been dissipated. At the same time, a complementary amount of the work must have been reversibly stored in the specimen. \n",
"\n",
"**How much energy is stored in the specimen at point $A$?**\n",
"In the experiment, we can answer this question by unloading the specimen to the zero force and evaluating the area remaining below the unloading curve. But since this unloading step has not been done, we have to knowledge we gained during the Tours focused on plasticity and damage type of behavior. We know that assuming damage material behavior, the specimen would unload to the origin while with purely plastic behavior, the unloading material stiffness would be equal to the initial stiffness. Therefore, we can use these two limiting assumptions to obtain two bounds on stored energy."
]
},
...
...
@@ -180,7 +185,7 @@
"id": "protective-resource",
"metadata": {},
"source": [
"A more accurate evaluation of stored energy is possible with the help of a model that can capture the inelastic effects of plasticity and damage. Then, the stored energy can be evaluated as an integral over the product of the stress and elastic strain \n",
"A more accurate evaluation of the stored energy is possible with the help of a model that can capture the inelastic effects of plasticity and damage. Then, the stored energy can be evaluated as an integral over the product of the stress and elastic strain \n",
"Assuming linear elastic material behavior we can employ the equilibrium condition and kinematic relation to link the load directly to the control displacement $w$ as\n",
"Assuming linear elastic material behavior, we can employ the equilibrium condition and kinematic relation to link the load directly to the control displacement $w$ as\n",
"\\begin{align}\n",
"P = A \\sigma = A E \\varepsilon = A E \\frac{1}{L} w,\n",
"\\end{align}\n",
...
...
@@ -257,7 +262,7 @@
"id": "sharp-wales",
"metadata": {},
"source": [
"**The energy supply** is then obtained as\n",
"**The amount of energy supply** is then obtained as\n",
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. This perspective is particularly important for materials exhibiting softening behavior which which induces an energy dissipation within a small volume of material.
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. This perspective is particularly important for materials exhibiting softening behavior. As we have already learned, the softening behavior leads to stress concentration within a small volume associated with strain localization leading to a discontinuity, either interface gap or crack. The process of localization is can be characterized by the amount of energy dissipation within a small volume of material, the size of which is characteristic for a particular composition of material structure.
In the present notebook we will revisit the nonlinear pullout curve and show how to evaluate the stored and released energy. We start with an abstract description explaining the evaluation procedure. To provide an elementary example, we then apply the described concept to an elementary example of an elastic bar with no energy dissipation. This prepares the framework for the next example showing applying the evaluation procedure to the pullout test in the notebook[6.2](6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top) and, most importantly, to the notebook [6.3]() setting up the ground for the description of fracture. Thus, our next tour goes through the following three stations: tensile test, pullout test and bending test. We will show a unified framework for the evaluation of energy dissipation in all three of them.
In the present notebook we will revisit the nonlinear pullout curve and show how to evaluate the stored and released energy. We will start with an abstract description explaining how to quantify and distinguish work, stored energy and dissipated energy. To provide an elementary example, we will then apply the described concept to an elementary example of an elastic bar with no energy dissipation. This prepares the framework for the next example showing how to apply the quantification procedure to a pullout test. To this end, we will revisit the model with constant bond-slip law [6.2](6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top). After that, we will apply the same procedure to the model presented in the notebook [6.3](6_3_localized_energy_dissipation.ipynb#top) for bond behavior with softening. With the fracture energy quantified for an interface based on a pull-out test, we will demonstrate the general description of fracture mechanics. The gained knowledge will be used for the characterization of tensile crack, propagating through a bending test in the next tour starting in notebook [7.1](../tour7_cracking/7_1_bending3pt_2d.ipynb#top).
Consider a general force displacement curve obtained in an experiment. The curve represents a response of a test setup with an arbitrary geometry of the specimen, material components and their layout, and even the boundary conditions.
The only information we need from the lab is the time profile of the control displacement $w(t)$ and the profile of force $P(t)$ measured by the hydraulic cell at the point of the imposed displacement. Using these two series of data points, we can plot the force displacement curves.
Consider a general force displacement curve obtained in an experiment. The curve represents a measured response of a test setup. It can be any kind of test setup: pull-out, punch-through, bending or wedge splitting test. Besides the geometry, also the material components and their layout, and even the boundary conditions are arbitrary. We consider the structure as a black box that can store and dissipate energy.
To quantify the amount of these energies, the only information we need from the lab is the time profile of the control displacement $w(t)$ and the corresponding time profile of force $P(t)$, measured by the hydraulic cell at the point of the imposed displacement. Using these two series of data points, we can plot the force displacement curves.
**Question:** Can you think of an answer to the question, why do we control the displacement in the test and not the force? Both are technically possible. What happends, when we use force control and reach the maximum load that can be transfered by a specimen?
This curve can be used to answer the question, how much energy must have been supplied by the testing machine to reach the point $A$ of the measured response. Let us first recall that work is defined as the force needed achieve a unit displacement, i.e.
The experimentally obtained load-displacement curve can be used to answer the question, how much energy must have been supplied by the testing machine to reach the point $A$ of the measured response. Let us first recall that work is defined as the force needed to achieve a unit displacement, i.e.
$$
\mathcal{W} = P w \;\;\mathrm{[Nm]}
\mathcal{W} = P w \;\;\mathrm{[Nm]}.
$$
In case of a nonlinear profile obtained from the testing machine we apply this definition to evaluate the work increment needed to achieve an infinitesimal displacement
$$
\mathrm{d}\mathcal{W} = P \;\mathrm{d}w
$$
so that the total work needed to achieve the point $A$ with the displacement $w$ is obtained as an integral
$$
\mathcal{W} = \int_0^w P(w) \;\mathrm{d}w
$$
representing the area below the force displacement curve.
%% Cell type:markdown id:amazing-monitor tags:
## Stored energy
%% Cell type:markdown id:verified-diana tags:
We already know, that the nonlinear shape of the force-displacement response is due to irreversible changes in the material structure. These changes are connected with energy dissipation. Thus, in point $A$ some fraction of energy has already been dissipated and some is stored in the specimen. How much energy is stored in the specimen at point $A$?
**One sentence to thermodynamics:**
We already know, that the nonlinear shape of the force-displacement response is due to irreversible changes in the material structure. We know from thermodynamics that any irreversible change of a system is connected with energy dissipation. Thus, in point $A$, which is behind the peak of the load-displacement curve, irreverisble changes must have already taken place. As a consequence, some amount of energy must already have been dissipated. At the same time, a complementary amount of the work must have been reversibly stored in the specimen.
**How much energy is stored in the specimen at point $A$?**
In the experiment, we can answer this question by unloading the specimen to the zero force and evaluating the area remaining below the unloading curve. But since this unloading step has not been done, we have to knowledge we gained during the Tours focused on plasticity and damage type of behavior. We know that assuming damage material behavior, the specimen would unload to the origin while with purely plastic behavior, the unloading material stiffness would be equal to the initial stiffness. Therefore, we can use these two limiting assumptions to obtain two bounds on stored energy.
The two bounds on the stored energy are depicted in the figure as the stored energy corresponding to the plastic lower bound $\mathcal{U}_\pi$ below the dashed curve, which is parallel to the initial stiffness $K_0$, and damage upper bound below the unloading curve, which connects the point $A$ with the origin.
A more accurate evaluation of stored energy is possible with the help of a model that can capture the inelastic effects of plasticity and damage. Then, the stored energy can be evaluated as an integral over the product of the stress and elastic strain
A more accurate evaluation of the stored energy is possible with the help of a model that can capture the inelastic effects of plasticity and damage. Then, the stored energy can be evaluated as an integral over the product of the stress and elastic strain
The difference between supplied and stored energy delivers the dissipated energy
The difference between the supplied and stored energy delivers the amount dissipated energy
$$
G = \mathcal{W} - \mathcal{U}
$$
The described procedure has been implemented in the numerical simulator used in the model applications presented during the course so that it is possible to monitor the supplied, stored and released energy for all inelastic material models shown so far. We will use this feature to develop an understanding of the energy development using the pullout test and the bending test later.
To start in a transparent way, let us first address two elementary examples that can be solved analytically: elastic bar test and a pullout test with constant bond-slip law which has been analytical solved in Tour 2.
The first exercise demonstrating the above equations presents the elastic bar for which we can readily derive the force-displacement curve $P(w)$ and the stored energy function $\mathcal{W}$.
Assuming linear elastic material behavior we can employ the equilibrium condition and kinematic relation to link the load directly to the control displacement $w$ as
Assuming linear elastic material behavior, we can employ the equilibrium condition and kinematic relation to link the load directly to the control displacement $w$ as
\begin{align}
P = A \sigma = A E \varepsilon = A E \frac{1}{L} w,
\end{align}
where $E$ denotes the Young’s modulus, $\sigma$ the stresses, $A$ the cross-sectional area and $\varepsilon$ the strain.
%% Cell type:markdown id:sharp-wales tags:
**The energy supply** is then obtained as
**The amount of energy supply** is then obtained as
\begin{align}
\label{eq:elastic_energy_supply}
\mathcal{W} = \int^w_0 P(w) \;\mathrm{d}w =
A E \frac{1}{L} \int^w_0 w \;\mathrm{d}w
=
\frac{1}{2} A E \frac{1}{L} w^2
=
\frac{1}{2} P w
\end{align}
%% Cell type:markdown id:fossil-fetish tags:
**Stored elastic energy** is
represented by the area below the unloading branch integrated over the whole volume of the structure.
In case of the tensile specimen with uniform profile of stress $\sigma$ and strain $\varepsilon$, the integral reduces to
\begin{align}
\mathcal{U}
=
\frac{1}{2}
\int_\Omega
\sigma
\cdot
\varepsilon
\;
\mathrm{d}x
=
\frac{1}{2}
A L \;\sigma \cdot \varepsilon
\end{align}
%% Cell type:markdown id:identical-shaft tags:
Realizing that $P = A \sigma$ and $w = L \varepsilon$ we obtain the expression for stored energy
%% Cell type:markdown id:planned-protocol tags:
\begin{align}
\mathcal{U}
=
\frac{1}{2}
P w
\end{align}
%% Cell type:markdown id:included-melbourne tags:
which means that the supplied and stored energy is equivalent, i.e.
\begin{align}
\mathcal{W} = \mathcal{U} \implies G = 0
\end{align}
recovering the obvious fact that the supplied energy is stored in the elastic specimen without any loss.
<ahref="../tour6_energy/6_2_Energy_released_in_pullout_constant_bond_and_rigid_matrix.ipynb#top">6.2 Frictional pullout and energy dissipation</a> <imgsrc="../icons/next.png"alt="Previous trip"width="50"height="50"></div>
