---
title: "Stress-Strain Curves"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{stress-strain}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 5,
  fig.height = 3
)
```

This vignette demonstrates generation of averaged stress-strain curves from
data from several coupons.
The same procedure can be used to generate bearing load-deformation curves and
other similar curves.

There are two functions provided by the `cmstatrExt` package for doing this.
The function `average_curve_lm()` uses `stats::lm()` behind the scenes to
fit a curve defined by a `formula` to the data.
The function `average_curve_optim()` uses numeric optimization to fit a curve
defined by a user-specified function to the data. Use the former function
for fitting polynomials, use the latter for fitting exponential models or
models with discontinuities. Both will be discussed in more detail in this
vignette.

## Setup
In this example, the following packages need to be loaded:

```{r setup}
knitr::opts_chunk$set(message = FALSE, warning = FALSE)
library(cmstatrExt)
library(tidyverse)
```

## Example Data
The `cmstatrExt` package comes with some example stress-strain data. The first
few rows of this data are:

```{r}
head(pa12_tension)
```

This data is in "tidy" format. This means that each data-point
(stress-strain value) is in a row of the `data.frame`. This example data set
contains stress-strain data from four coupons (Samples 1 through 4).
Each row identifies the coupon that the observation comes from. While the
order of the columns does not matter and you can give each column any
name you wish, the data will need to be in this type of format.

Let's plot the `pa12_tension` example data.

```{r}
pa12_tension %>%
  ggplot(aes(x = Strain, y = Stress, color = Coupon)) +
  geom_point()
```

## Fitting a Polynomial Model
For the first example, we'll fit the following quadratic model:

$$
\sigma = c_1 \epsilon + c_2 \epsilon^2
$$
where $\sigma$ is the stress, $\epsilon$ is the strain and $c_1$ and $c_2$ are
constants that we'll find. We need to write this as an `R` `formula`, which
has slightly different notation. The stress and strain variables in our data
are `Stress` and `Strain`, respectively, so we'll use those variable names in
the formula.

```r
Stress ~ I(Strain) + I(Strain^2) + 0
```

Notice that in this formula, a tilde (`~`) is used instead of an equal sign.
You'll also notice that we've wrapped the terms on the right-hand side inside
the identity function `I()`: the reason for this is that `lm` will treat
`Strain^2` as an interaction, rather than squaring the value of `Strain`,
while `I(Strain^2)` will actually square the value of `Strain`. The formula
doesn't need coefficients (e.g. $c_1$ and $c_2$). Finally, notice that we've
included the term `+0`, which tells `lm` that we want the intercept to be
zero, which will normally be desirable due to the physical notion that
stress ought to be zero at zero strain.

The function `average_curve_lm` takes four arguments. The first is a
`data.frame` with the data. The second is the name of the variable defining the
coupon. The third is the formula that we just discussed. The last argument is
the number of "bins": this has a default of 100 and hence can be omitted.
See the documentation for this function for information about binning the data.

Let's run this function and then execute the `summary` method on the result:

```{r}
curve_quadratic <- average_curve_lm(
  pa12_tension, Coupon,
  Stress ~ I(Strain) + I(Strain^2) + 0
)
summary(curve_quadratic)
```

The `summary` method shows the strain range over which the curve was fit.
This range always starts at zero and ends at the *lowest* maximum
strain of any individual coupon.
The `summary` method also lists the coefficients as well as information about
whether each term is statistically significant, the residuals and R-squared
values.

Next, let's plot the original data and the curve fit. We'll use the
`augment` method to add the curve fit to the original data, then
pass the result to `ggplot`.

```{r}
curve_quadratic %>%
  augment() %>%
  ggplot(aes(x = Strain)) +
  geom_point(aes(y = Stress, color = Coupon)) +
  geom_line(aes(y = .fit))
```

Due to the polynomial model that we chose (quadratic), this curve fit is
poor. We can do better. Let's try a cubic function next.

```{r}
curve_cubic <- average_curve_lm(
  pa12_tension, Coupon,
  Stress ~ I(Strain) + I(Strain^2) + I(Strain^3) + 0
)
summary(curve_cubic)
```


```{r}
curve_cubic %>%
  augment() %>%
  ggplot(aes(x = Strain)) +
  geom_point(aes(y = Stress, color = Coupon)) +
  geom_line(aes(y = .fit))
```

This cubic model is a much better fit. The equation for this curve fit is:

$$
\sigma = 1174 \, \epsilon - 8783 \, \epsilon^2 + 20586 \, \epsilon^3
$$

Strain does not need to be the independent variable and stress does not
need to be the dependent variable. We could fit a model with these reversed.

```{r}
average_curve_lm(
  pa12_tension, Coupon,
  Strain ~ I(Stress) + I(Stress^2) + I(Stress^3) + I(Stress^4) + 0
) %>%
  augment() %>%
  ggplot(aes(y = Stress)) +
  geom_point(aes(x = Strain, color = Coupon)) +
  geom_line(aes(x = .fit))
```

In this case, the fit is not very good, so it's not of much practical use.
However, note that the curve fit ends at the lowest maximum *stress* value of
any individual coupon this time. The curve fit will always end at the lowest
maximum value of the independent variable (the variable on the right hand
side of the formula).

## Fitting a Bilinear Model
Next, we turn our attention to fitting a model that cannot be represented by
an `R` `formula`. We'll fit the following model:

$$
\sigma = \left\{
\begin{matrix}
c_1 \epsilon & \text{if }\epsilon \le \epsilon_1 \\
c_2 \left(\epsilon - \epsilon_1\right) + c_1\epsilon_1 & \text{otherwise}
\end{matrix}
\right.
$$

This model will thus be a straight line starting from the origin extending to
an unknown value of strain ($\epsilon_1$), then continuing with a different
slope. In order to use this model with `average_curve_optim`, we need to write
this as an `R` function where the first argument is the independent variable
(strain in our case) and the second argument is a vector of parameters. In
this case, there are three parameters, $c_1$, $c_2$ and $\epsilon_1$.

```{r}
bilinear_fn <- function(strain, par) {
  c1 <- par[1]
  c2 <- par[2]
  e1 <- par[3]
  if (strain <= e1) {
    return(c1 * strain)
  } else {
    return(c2 * (strain - e1) + c1 * e1)
  }
}
```

The function `average_curve_optim` takes nine arguments:

- `data` a `data.frame` with the stress-strain data
- `coupon_var` the name of the column representing the coupon
- `x_var` the name of the column representing the independent variable
- `y_var` the name of column representing the dependent variable
- `fn` the function representing the model
- `par` an initial guess at the parameters of the model
- `n_bins` the number of bins to sort the data into. The default is 100 and
  it does not need to be specified to accept the default.
- `method` the method used by `optim()`. Defaults to "L-BFGS-B"
- `...` extra parameters to pass to `optim()`

We'll call this function:

```{r}
curve_bilinear <- average_curve_optim(
  pa12_tension,
  Coupon, Strain, Stress,
  bilinear_fn,
  c(1, 1, 0.04) # the initial guess
)
curve_bilinear
```

The value of the third parameter, $\epsilon_1$ is well outside the range we'd
expect. We'd expect that the "knee" to be somewhere in the range of 0.025-0.100.
We can specify upper and lower bounds on the parameters as follows:

```{r}
curve_bilinear <- average_curve_optim(
  pa12_tension,
  Coupon, Strain, Stress,
  bilinear_fn,
  c(1, 1, 0.04),
  lower = c(0, 0, 0.025),
  upper = c(2000, 2000, 0.100)
)
curve_bilinear
```

We can now plot the curve fit over laid with the original data.

```{r}
curve_bilinear %>%
  augment() %>%
  ggplot(aes(x = Strain)) +
  geom_point(aes(y = Stress, color = Coupon)) +
  geom_line(aes(y = .fit))
```

## Preprocessing Data
The example data in the `pa12_tension` data set is fairly well behaved and does
not need pre-processing. However, most data that you will actually
deal with will require pre-processing.

The `fff_shear` data set that comes with the `cmstatrExt` package is more
typical data that does require pre-processing. Let's start by plotting this
data.

```{r}
fff_shear %>%
  ggplot(aes(x = Strain, y = Stress, color = Specimen)) +
  geom_point()
```

There are three aspects of this data that we'll deal with:

- The post-failure behavior at high strains will be removed. This part of the
  stress-strain curve is not important to most structural analysis and will
  make cure-fitting difficult.
- We'll remove the "toe" at the start of the curves. (The "toe" is the small
  portion of the curves at low strain that have a relatively low slope).
  Depending on the test method, this "toe" can be caused by behavior like
  the test coupon seating itself in the test fixture, or clearance being taken
  up. Depending on the cause of the "toe," it may be desirable to remove it.
  In this case, removing it is desirable.
- The curves, even with the "toe" removed, do no pass through the origin.
  This is due to an offset in the strain measurement that does not correspond
  to the physical behavior of the material, so we'll apply an appropriate
  offset.

None of these adjustments are done using the functionality of `cmstatrExt`,
but this example is included anyways as these types of adjustments are typically
a pre-requisite of curve fitting using `cmstatrExt`.

We'll start by removing the offset from the data. To do this,
we'll fit a straight line to the data
from each coupon over a stress range of 1000 to 3000 `psi`, find the
x-intercept of this line and subtract this x-intercept from the strain value.
This is a bit complicated, so we'll do it in a few steps before combining
everything. We'll start by filtering the data so that the stress values
are in the range 1000 to 3000, grouping by `Specimen` and finding the
x-intercept for each. This code will use the `nest()`, `mutate()` `map()`
pattern.

```{r}
fff_shear %>%
  filter(Stress > 1000 & Stress < 3000) %>%
  group_by(Specimen) %>%
  nest() %>%
  mutate(lm = map(data, ~lm(Strain ~ Stress, data = .))) %>%
  mutate(x_intercept = map(lm, ~predict(.x, data.frame(Stress = 0)))) %>%
  select(-c(lm, data)) %>%
  unnest(x_intercept)
```

Now we'll use `inner_join()` to join the `x_intercept` column to the original
data (matching the appropriate `Specimen`).
We'll use `head()` to just show the first 6 rows for brevity.


```{r}
fff_shear %>%
  filter(Stress > 1000 & Stress < 3000) %>%
  group_by(Specimen) %>%
  nest() %>%
  mutate(lm = map(data, ~lm(Strain ~ Stress, data = .))) %>%
  mutate(x_intercept = map(lm, ~predict(.x, data.frame(Stress = 0)))) %>%
  select(-c(lm, data)) %>%
  unnest(x_intercept) %>%
  inner_join(fff_shear, by = "Specimen") %>%
  head(6)
```

Finally, we'll subtract the x-intercept from the strain for each coupon
to obtain the corrected data (and delete the unneeded `x_intercept` column).

```{r}
fff_shear_offset <- fff_shear %>%
  filter(Stress > 1000 & Stress < 3000) %>%
  group_by(Specimen) %>%
  nest() %>%
  mutate(lm = map(data, ~lm(Strain ~ Stress, data = .))) %>%
  mutate(x_intercept = map(lm, ~predict(.x, data.frame(Stress = 0)))) %>%
  select(-c(lm, data)) %>%
  unnest(x_intercept) %>%
  inner_join(fff_shear, by = "Specimen") %>%
  mutate(Strain = Strain - x_intercept) %>%
  select(-c(x_intercept))
```

We'll plot this now.

```{r}
fff_shear_offset %>%
  ggplot(aes(x = Strain, y = Stress, color = Specimen)) +
  geom_point()
```

There are many approaches to removing the post-failure behavior. The specific
approach that should be used will depend on the data and test method. In some
cases, you might choose to simply manually delete some rows from the data file.
However, if you're processing more data, you may want to truncate the data
using some code. Here, we'll approach the removal of the post-failure behavior
by finding the data that has a local slope that is sufficiently negative. We'll
do this by finding the secant over 5 points and checking if this value is
more negative than a certain threshold. First, let's plot the data and color
it by a logical value indicating whether we'll remove the point. This will
help find an appropriate threshold using iteration for how negative a slope
should cause a point to be removed.


```{r}
fff_shear_offset %>%
  group_by(Specimen) %>%
  mutate(Lead_Stress = lead(Stress, 5),
         Lead_Strain = lead(Strain, 5),
         Slope = (Lead_Stress - Stress) / (Lead_Strain - Strain),
         Remove = Slope < -1e5 | is.na(Slope)) %>%
  ggplot(aes(x = Strain, y = Stress, shape = Specimen, color = Remove)) +
  geom_point()
```

The criteria above seems to cut the data off at about the correct point, but
after the initial cutoff, some subsequent data would be incorrectly retained.
In order to avoid this, we'll add another criteria that for each curve, as
soon as a single data point is removed, all subsequent data points will
also be removed.

```{r}
fff_shear_offset %>%
  group_by(Specimen) %>%
  mutate(Lead_Stress = lead(Stress, 5),
         Lead_Strain = lead(Strain, 5),
         Slope = (Lead_Stress - Stress) / (Lead_Strain - Strain),
         Remove = Slope < -1e5 | is.na(Slope),
         Remove = cumsum(Remove) > 0) %>%
  ggplot(aes(x = Strain, y = Stress, shape = Specimen, color = Remove)) +
  geom_point()
```

We'll save this result to a new variable, remove the groupings, filter out the
points that we intend to remove and drop the unneeded temporary variables.

```{r}
fff_shear_truncated <- fff_shear_offset %>%
  group_by(Specimen) %>%
  mutate(Lead_Stress = lead(Stress, 5),
         Lead_Strain = lead(Strain, 5),
         Slope = (Lead_Stress - Stress) / (Lead_Strain - Strain),
         Remove = Slope < -1e5 | is.na(Slope),
         Remove = cumsum(Remove) > 0) %>%
  ungroup() %>%
  filter(!Remove) %>%
  select(Specimen, Stress, Strain)
```

Next, we'll remove the "toe" of each curve. The "toe" extends to a stress of
somewhat less than 1000 `psi`, so we'll remove the "toe" by simply
removing all the data with a stress less than 1000 `psi`.

```{r}
fff_shear_truncated_no_toe <- fff_shear_truncated %>%
  filter(Stress > 1000)
```

Now, let's plot this pre-processed data.

```{r}
fff_shear_truncated_no_toe %>%
  ggplot(aes(x = Strain, y = Stress, color = Specimen)) +
  geom_point() +
  xlim(c(0, NA)) +
  ylim(c(0, NA))
```

Now, let's try fitting an averaged curve to this data.

```{r}
curve_fff_shear <- fff_shear_truncated_no_toe %>%
  average_curve_lm(
    Specimen,
    Stress ~ I(Strain) + I(Strain^2) + I(Strain^3) + 0
  )
curve_fff_shear
```

And we'll plot this curve fit overlaid on the data with only the strain
offset corrected (leaving the "toe" and the post-failure behavior intact).

```{r}
curve_fff_shear %>%
  augment(fff_shear) %>%
  ggplot(aes(x = Strain)) +
  geom_point(aes(y = Stress, color = Specimen)) +
  geom_line(aes(y = .fit))
```


## Plots for Publication

The last part of this vignette will focus on creating plots for publication
in reports. We'll focus on the following items as examples. It will be up to
you how you want to customize your plots for your particular publication.

- Changing the theme
- Overlaying curves and data from multiple environmental conditions
- Adding a secondary y-axis to show both US Customary and SI units

The `cmstatrExt` package does not come with any data sets with multiple
environmental conditions. We'll "fake it" for illustration purposes
by scaling the `pa12_tension` stress by 50% and strain by 125%
to generate data for the "Fake ETA"
environmental condition. We'll create a new data frame by stacking the original
`pa12_tension` data frame and a version of the `pa12_tension` data frame with
the stress scaled. Before stacking these data frames, we'll add a new column
for condition.

```{r}
pa12_tension_conditions <-
  bind_rows(
    pa12_tension %>%
      mutate(Condition = "RTA"),
    pa12_tension %>%
      mutate(Condition = "Fake ETA",
             Stress = 0.50 * Stress,
             Strain = 1.25 * Strain)
  )
```

We already have a cubic model for the original `pa12_tension` data, but that
model was missing the `Condition` column, so we'll fit it again.

```{r}
curve_cubic_rta <- pa12_tension_conditions %>%
  filter(Condition == "RTA") %>%
  average_curve_lm(
    Coupon,
    Stress ~ I(Strain) + I(Strain^2) + I(Strain^3) + 0
  )
curve_cubic_rta
```

We'll do the same for the "Fake ETA" data.

```{r}
curve_cubic_fake_eta <- pa12_tension_conditions %>%
  filter(Condition == "Fake ETA") %>%
  average_curve_lm(
    Coupon,
    Stress ~ I(Strain) + I(Strain^2) + I(Strain^3) + 0
  )
curve_cubic_fake_eta
```

Now, we can plot the two curves.

```{r}
bind_rows(
  augment(curve_cubic_rta),
  augment(curve_cubic_fake_eta)
) %>%
  ggplot(aes(x = Strain, y = .fit, color = Condition)) +
  geom_line()
```

In some cases, you'd also want to show the raw data, which can be done as
follows. Note that we needed to set the `group` aestetic in the call to
`geom_line()`.

```{r}
bind_rows(
  augment(curve_cubic_rta),
  augment(curve_cubic_fake_eta)
) %>%
  group_by(Condition) %>%
  ggplot(aes(x = Strain)) +
  geom_point(aes(y = Stress, color = Condition)) +
  geom_line(aes(y = .fit, group = Condition))
```

Next, we'll add a secondary y-axis. Since the primary y-axis is in units of
`MPa`, the secondary y-axis will be in units of `ksi`. To do this, we'll use
a call to `scale_y_continuous()`

```{r}
bind_rows(
  augment(curve_cubic_rta),
  augment(curve_cubic_fake_eta)
) %>%
  ggplot(aes(x = Strain, y = .fit, color = Condition)) +
  geom_line() +
  scale_y_continuous(
    "Stress [MPa]",
    sec.axis = sec_axis(~ . * 0.1450377377, name = "Stress [ksi]")
  )
```

And it's likely that you'd want to change the theme of the plot using,
for example, `theme_bw()` or your own custom theme.

```{r}
bind_rows(
  augment(curve_cubic_rta),
  augment(curve_cubic_fake_eta)
) %>%
  ggplot(aes(x = Strain, y = .fit, color = Condition)) +
  geom_line() +
  scale_y_continuous(
    "Stress [MPa]",
    sec.axis = sec_axis(~ . * 0.1450377377, name = "Stress [ksi]")
  ) +
  theme_bw()
```