"Knowing the matrix stress profile ahead of an existing crack, we know the distance at which no crack will appear. This length is referred to as shielded length."
...
...
@@ -158,7 +158,7 @@
},
{
"cell_type": "markdown",
"id": "stable-smell",
"id": "maritime-incidence",
"metadata": {},
"source": [
"### **Question:** How long is the shielded length?"
...
...
@@ -166,7 +166,7 @@
},
{
"cell_type": "markdown",
"id": "broke-prevention",
"id": "handmade-manufacturer",
"metadata": {},
"source": [
"Use the crack bridge model to find out the distance $l_\\mathrm{shielded}$ at which the matrix normal force attains the level \n",
...
...
@@ -185,7 +185,7 @@
},
{
"cell_type": "markdown",
"id": "permanent-oasis",
"id": "dominican-uganda",
"metadata": {},
"source": [
"### **Question:** What is the elongation and average strain of a crack segment?"
...
...
@@ -193,7 +193,7 @@
},
{
"cell_type": "markdown",
"id": "informative-powell",
"id": "happy-choir",
"metadata": {},
"source": [
"A simple way how to find out the total elongation of the fiber with a nonlinear strain profile is to integrate the strains, i.e.\n",
...
...
@@ -204,7 +204,7 @@
},
{
"cell_type": "markdown",
"id": "personal-synthesis",
"id": "guided-feedback",
"metadata": {},
"source": [
"so that the average, or **composite** strain is obtained as"
...
...
@@ -212,7 +212,7 @@
},
{
"cell_type": "markdown",
"id": "urban-reform",
"id": "fossil-carpet",
"metadata": {},
"source": [
"\\begin{align}\n",
...
...
@@ -222,7 +222,7 @@
},
{
"cell_type": "markdown",
"id": "mobile-composer",
"id": "spread-chart",
"metadata": {},
"source": [
"**Remark:** This expression is important in view of homogenization of non-uniform strain profiles in a heterogenous material structure. It is one of the fundamental concepts of micromechanics. We will use it later on to derive the effective stress-strain curve of a multiply cracked composite."
"The typical shape of the stress-strain curve of the composite involves three stages:\n",
...
...
@@ -249,7 +249,7 @@
},
{
"cell_type": "markdown",
"id": "surgical-detail",
"id": "equivalent-muscle",
"metadata": {},
"source": [
"How to interpret and characterize these three distinguished phases of composite material behavior?"
...
...
@@ -257,7 +257,7 @@
},
{
"cell_type": "markdown",
"id": "communist-serve",
"id": "literary-auction",
"metadata": {},
"source": [
"The ACK model developed by Aveston, Cooper and Kelly is an analytical model that represents the composite tensile response by a trilinear law as shown in the following figure. This model is based on the following assumptions:\n",
"Probabilistic analysis of the car parking problem delivers the result that the average spacing is 1.337 larger than the car length ([Wikipedia: Random sequential adsorption](https://en.wikipedia.org/wiki/Random_sequential_adsorption))"
...
...
@@ -363,7 +363,7 @@
},
{
"cell_type": "markdown",
"id": "curious-wilson",
"id": "pleased-melbourne",
"metadata": {},
"source": [
"### How long is the car in a concrete tensile specimen?\n",
...
...
@@ -377,7 +377,7 @@
},
{
"cell_type": "markdown",
"id": "twenty-charity",
"id": "nearby-stuff",
"metadata": {},
"source": [
"## Examples"
...
...
@@ -385,7 +385,7 @@
},
{
"cell_type": "markdown",
"id": "generic-budget",
"id": "forbidden-sauce",
"metadata": {},
"source": [
"### **Task 1:** Evaluate the tensile stress-strain curve\n",
...
...
@@ -397,7 +397,7 @@
{
"cell_type": "code",
"execution_count": 1,
"id": "settled-johnston",
"id": "editorial-minority",
"metadata": {},
"outputs": [],
"source": [
...
...
@@ -411,7 +411,7 @@
},
{
"cell_type": "markdown",
"id": "single-parallel",
"id": "injured-eugene",
"metadata": {},
"source": [
"Derived parameters of the cross composite section"
...
...
@@ -420,7 +420,7 @@
{
"cell_type": "code",
"execution_count": 2,
"id": "affecting-crown",
"id": "graduate-moses",
"metadata": {},
"outputs": [],
"source": [
...
...
@@ -435,7 +435,7 @@
{
"cell_type": "code",
"execution_count": 3,
"id": "juvenile-ladder",
"id": "bright-cooperative",
"metadata": {},
"outputs": [
{
...
...
@@ -455,7 +455,7 @@
},
{
"cell_type": "markdown",
"id": "warming-revision",
"id": "right-wilson",
"metadata": {},
"source": [
"Characteristic points of the ACK model deliver the values "
...
...
@@ -464,7 +464,7 @@
{
"cell_type": "code",
"execution_count": 4,
"id": "international-wheel",
"id": "retired-tragedy",
"metadata": {},
"outputs": [],
"source": [
...
...
@@ -478,7 +478,7 @@
},
{
"cell_type": "markdown",
"id": "technical-location",
"id": "electoral-lobby",
"metadata": {},
"source": [
"Plot the composite stress-strain curve"
...
...
@@ -487,7 +487,7 @@
{
"cell_type": "code",
"execution_count": 5,
"id": "accurate-librarian",
"id": "legendary-saying",
"metadata": {},
"outputs": [
{
...
...
@@ -518,7 +518,7 @@
},
{
"cell_type": "markdown",
"id": "funded-daily",
"id": "tested-question",
"metadata": {},
"source": [
"**Note** that the curve does not depend on the bond $\\tau p$."
...
...
@@ -526,7 +526,7 @@
},
{
"cell_type": "markdown",
"id": "other-details",
"id": "immune-somewhere",
"metadata": {},
"source": [
"### **Task 2:** Evaluate the crack spacing\n",
...
...
@@ -537,7 +537,7 @@
{
"cell_type": "code",
"execution_count": 6,
"id": "incomplete-harvard",
"id": "tight-orchestra",
"metadata": {},
"outputs": [
{
...
...
@@ -563,7 +563,7 @@
},
{
"cell_type": "markdown",
"id": "british-emphasis",
"id": "hybrid-halloween",
"metadata": {},
"source": [
"which corresponds to reinforcement ratio $V_f$"
...
...
@@ -572,7 +572,7 @@
{
"cell_type": "code",
"execution_count": 7,
"id": "strategic-wallpaper",
"id": "metallic-projector",
"metadata": {},
"outputs": [
{
...
...
@@ -592,7 +592,7 @@
},
{
"cell_type": "markdown",
"id": "royal-milton",
"id": "civic-colony",
"metadata": {},
"source": [
"# **Model 2:** Random matrix strength\n",
...
...
@@ -604,7 +604,7 @@
{
"cell_type": "code",
"execution_count": 8,
"id": "practical-purchase",
"id": "presidential-grounds",
"metadata": {},
"outputs": [
{
...
...
@@ -631,7 +631,7 @@
},
{
"cell_type": "markdown",
"id": "spatial-physiology",
"id": "differential-belly",
"metadata": {},
"source": [
"The probabilistic multiple cracking model uses the crack bridge model which is inserted at any newly emerging crack position. By assembling the profile along the specimen we obtain the \n",
...
...
@@ -647,7 +647,7 @@
},
{
"cell_type": "markdown",
"id": "heavy-penguin",
"id": "respiratory-terrace",
"metadata": {},
"source": [
"**Further reading:** [Paper describing the general probabilistic multiple cracking model](../papers/pmcm_fragmentation.pdf), Journal of Mathematical Modeling (2021)"
...
...
@@ -655,7 +655,7 @@
},
{
"cell_type": "markdown",
"id": "governing-listening",
"id": "selected-conflict",
"metadata": {},
"source": [
"## PMCM application"
...
...
@@ -663,7 +663,7 @@
},
{
"cell_type": "markdown",
"id": "hybrid-support",
"id": "flexible-wayne",
"metadata": {},
"source": [
"Apply the model to a composite cross section using carbon textile fabrics as specified in the notebook [**Mixture rule**](1_1_elastic_stiffness_of_the_composite.ipynb)"
With a basic understanding of the debonding process in the vicinity of a crack bridge studied for various configurations of a pull-out problem ([2.2 Classification of pull-out configurations](2_2_1_PO_configuration_explorer.ipynb)) we describe and visualize the process of fragmentation (multiple cracking) in a reinforced composite.
%% Cell type:markdown id:extended-japan tags:
%% Cell type:markdown id:approved-clarity tags:
Consider an example the textile reinforced concrete cross section loaded in tension. The cross-sectional parameters are as follows
Knowing the matrix stress profile ahead of an existing crack, we know the distance at which no crack will appear. This length is referred to as shielded length.
%% Cell type:markdown id:stable-smell tags:
%% Cell type:markdown id:maritime-incidence tags:
### **Question:** How long is the shielded length?
**Remark:** This expression is important in view of homogenization of non-uniform strain profiles in a heterogenous material structure. It is one of the fundamental concepts of micromechanics. We will use it later on to derive the effective stress-strain curve of a multiply cracked composite.
The typical shape of the stress-strain curve of the composite involves three stages:
- elastic stage which is governed by the mixture rule
- stage of matrix fragmentation
- saturated crack pattern with a linear branch
%% Cell type:markdown id:surgical-detail tags:
%% Cell type:markdown id:equivalent-muscle tags:
How to interpret and characterize these three distinguished phases of composite material behavior?
%% Cell type:markdown id:communist-serve tags:
%% Cell type:markdown id:literary-auction tags:
The ACK model developed by Aveston, Cooper and Kelly is an analytical model that represents the composite tensile response by a trilinear law as shown in the following figure. This model is based on the following assumptions:
- The bond behavior is governed by a constant frictional bond in the debonded interface
- The constitutive law for both reinforcement and matrix is assumed to be linear-elastic with brittle failure upon reaching their strengths
- Multiple cracking occurs at a constant level of applied stress, inducing a horizontal branch in the stress-strain behavior
The trilinear curve of ACK model represents the composite tensile response by identifying the following characteristic points:
- [$\sigma_{1}, \varepsilon_{1}$]: The inital values of stress and strain are set to zero.
- [$\sigma_{2}, \varepsilon_{2}$]: In the first stage, the matrix is uncracked and perfect bond between matrix and fabric is assumed up to the first cracking stress $\sigma_{2}$ , which is defined as
\begin{align}
\sigma_{2} = E_\mathrm{c} \varepsilon_{2}
\end{align}
where $\varepsilon_{2}$ is the composite strain value at which the matrix cracks and $E_\mathrm{c}$ is the composite stiffness.
- [$\sigma_{4}, \varepsilon_{4}$]: Finally, in the third stage, when the crack pattern is stabilized the load increases linearly up to the ultimate tensile stress $\sigma_4$ with a slope equal to $E_\mathrm{r}$
The ultimate tensile stress is given as
\begin{align}
\sigma_4 = \sigma_\mathrm{fu} \; V_\mathrm{f}
\end{align}
where $\sigma_\mathrm{fu}$ is the tensile strength of the fiber.
The slope $E_\mathrm{r}$ represents the effective stiffness of the reinforcement with respect to the whole cross section and is given as
\begin{align}
E_\mathrm{r} = E_\mathrm{f} \; V_\mathrm{f}
\end{align}
The composilte strain at failure $\varepsilon_{4}$ is given as
Consider a process where particles (cars) are randomly introduced in a system (along the street).
They must not overlap any previously parked car. What is the average distance between two neighbouring cars?
%% Cell type:markdown id:signal-trustee tags:
%% Cell type:markdown id:british-wilson tags:
Probabilistic analysis of the car parking problem delivers the result that the average spacing is 1.337 larger than the car length ([Wikipedia: Random sequential adsorption](https://en.wikipedia.org/wiki/Random_sequential_adsorption))
%% Cell type:markdown id:curious-wilson tags:
%% Cell type:markdown id:pleased-melbourne tags:
### How long is the car in a concrete tensile specimen?
The final average crack spacing $l_\mathrm{cs}$ is given as
**Note** that the curve does not depend on the bond $\tau p$.
%% Cell type:markdown id:other-details tags:
%% Cell type:markdown id:immune-somewhere tags:
### **Task 2:** Evaluate the crack spacing
What does ACK model predict for a specimen with the dimensions $100 \times 100$ mm reinforced with 1% ratio.
%% Cell type:code id:incomplete-harvard tags:
%% Cell type:code id:tight-orchestra tags:
``` python
A_c = 100 * 100
d = 16
p = 3.14 * d
A_f = 3.14 * (d/2)**2
A_m = A_c - A_f
tau = 8
1.337 * A_m * sig_mu / (p * tau)
```
%% Output
97.79047929936306
%% Cell type:markdown id:british-emphasis tags:
%% Cell type:markdown id:hybrid-halloween tags:
which corresponds to reinforcement ratio $V_f$
%% Cell type:code id:strategic-wallpaper tags:
%% Cell type:code id:metallic-projector tags:
``` python
A_f / A_c
```
%% Output
0.020096
%% Cell type:markdown id:royal-milton tags:
%% Cell type:markdown id:civic-colony tags:
# **Model 2:** Random matrix strength
In reality, the matrix strength $\sigma_\mathrm{mu}$ is random. Its profile along the tensile specimen can be described by the probability distribution function.
Weibull probability distribution is used to describe the strength of materials.
%% Cell type:code id:practical-purchase tags:
%% Cell type:code id:presidential-grounds tags:
``` python
%matplotlib widget
from pmcm import PMCM
pm = PMCM()
pm.interact()
```
%% Output
%% Cell type:markdown id:spatial-physiology tags:
%% Cell type:markdown id:differential-belly tags:
The probabilistic multiple cracking model uses the crack bridge model which is inserted at any newly emerging crack position. By assembling the profile along the specimen we obtain the
- matrix stress field $\sigma_\mathrm{m}(x, \sigma_\mathrm{c})$, and
- reinforcement strain field $\varepsilon_\mathrm{f}(x, \sigma_\mathrm{c})$.
for any state of loading $\sigma_\mathrm{c}$. No finite element calculation is needed.
The algorithm used in the above web-app can identify the individual cracks exactly. For a given load level $\sigma_\mathrm{c}$ and crack distribution along the specimen, the composite strain of a specimen with a length $L$ is obtained using the averaging formula introduced above
**Further reading:** [Paper describing the general probabilistic multiple cracking model](../papers/pmcm_fragmentation.pdf), Journal of Mathematical Modeling (2021)
Apply the model to a composite cross section using carbon textile fabrics as specified in the notebook [**Mixture rule**](1_1_elastic_stiffness_of_the_composite.ipynb)
### Task 3: Use the model to validate its prediction
Given the cross sectional areas $A_\mathrm{m}, A_\mathrm{f}$, concrete and reinforcement stiffness $E_\mathrm{m}, E_\mathrm{f}$ and strength $\sigma_\mathrm{mu}, \sigma_\mathrm{fu}$, reinforcement ratio $V_\mathrm{f}$, bond stress $\tau$, and perimeter $p$ calculate the average crack width at failure of a tensile specimen.
### Task 4: Evaluate the average crack width
Compare the crack width obtained using the ACK model and the PMCM model for a given reinforced cross section design
