TPMplt is a tool-kit for building and visualizing the dynmaic materials model (DMM), suggested by Prasad and Gegel. It provides an easy approach to calculate constructive functions and other related material constants based on a given strain condiiton. 2D and 3D processing-maps with temperature as its x axis, while logarithm strain rate as its y axis are also available.
Valid data for TPMplt strictly obey the management logic in VBTree package. Additionally, factors for temperature (celsius) and strain rates should be written in the format of pure numeric. All different variables are desired to be connected using “-” symbol. For example, if there’s a strain data collected in the conditions of 900 celsius, \(10^{-3}\) strain rate, “0.001-Strain-900” is one of available column names for this data, rather than “10e-3_Strain_T900C”, “SR0.001-Strain-1173K” or such like. If your data contains some unnecessary patterns, a certain degree of data cleaning for column names is required.
The following codes partially display a typical valid data for TPMplt:
library(TPMplt)
head(TPMdata[,1:3])
#> Strain-900-0.001-60% Stress-900-0.001-60% Strain-900-0.01-60%
#> 1 0.00009 2.81 0.00030
#> 2 0.00026 2.95 0.00052
#> 3 0.00059 3.37 0.00068
#> 4 0.00076 3.37 0.00084
#> 5 0.00093 3.65 0.00112
#> 6 0.00104 3.79 0.00122Users can apply any applicable tools to make the summary table for TPMplt, with the format as above-showed.
Besides, it is common to obtain multiple exported files based on
different experiments. TPMplt affords two functions to automatically
generate a summary table from multiple exported files. For details,
please check the R documents for API4TMZ and
TMZdatainput.
It is necessary to build the conceptions for layers and levels in layer for variables, defined by VBTree package. For example, run the following codes to check all column names in the demo dataset in TPMplt pacakge:
colnames(TPMdata)
#> [1] "Strain-900-0.001-60%" "Stress-900-0.001-60%" "Strain-900-0.01-60%"
#> [4] "Stress-900-0.01-60%" "Strain-900-0.1-60%" "Stress-900-0.1-60%"
#> [7] "Strain-900-1-60%" "Stress-900-1-60%" "Strain-950-0.001-60%"
#> [10] "Stress-950-0.001-60%" "Strain-950-0.01-60%" "Stress-950-0.01-60%"
#> [13] "Strain-950-0.1-60%" "Stress-950-0.1-60%" "Strain-950-1-60%"
#> [16] "Stress-950-1-60%" "Strain-1000-0.001-60%" "Stress-1000-0.001-60%"
#> [19] "Strain-1000-0.01-60%" "Stress-1000-0.01-60%" "Strain-1000-0.1-60%"
#> [22] "Stress-1000-0.1-60%" "Strain-1000-1-60%" "Stress-1000-1-60%"
#> [25] "Strain-1050-0.001-60%" "Stress-1050-0.001-60%" "Strain-1050-0.01-60%"
#> [28] "Stress-1050-0.01-60%" "Strain-1050-0.1-60%" "Stress-1050-0.1-60%"
#> [31] "Strain-1050-1-60%" "Stress-1050-1-60%" "Strain-1100-0.001-60%"
#> [34] "Stress-1100-0.001-60%" "Strain-1100-0.01-60%" "Stress-1100-0.01-60%"
#> [37] "Strain-1100-0.1-60%" "Stress-1100-0.1-60%" "Strain-1100-1-60%"
#> [40] "Stress-1100-1-60%" "Strain-1150-0.001-60%" "Stress-1150-0.001-60%"
#> [43] "Strain-1150-0.01-60%" "Stress-1150-0.01-60%" "Strain-1150-0.1-60%"
#> [46] "Stress-1150-0.1-60%" "Strain-1150-1-60%" "Stress-1150-1-60%"
#> [49] "Strain-1200-0.001-60%" "Stress-1200-0.001-60%" "Strain-1200-0.01-60%"
#> [52] "Stress-1200-0.01-60%" "Strain-1200-0.1-60%" "Stress-1200-0.1-60%"
#> [55] "Strain-1200-1-60%" "Stress-1200-1-60%"As we can see, all column names are arranged by the style of “(Strain&Stress)-(Temperature)-(Strain Rate)-(Other)”. In this case, we attribute the variable the concept ‘layer’, and the order of values in specified layer the ‘level’. The layer for temperature is 2, while the layer for strain rate is 3; the temperature of 1000 is at level 3 in layer 2.
The function epsExtract is capable to export a strain
rate-temperature table by specifying eps as the selected
strain condition. However, lyT and lySR, the
two necessary arguments corresponding to layers for temperature and
strain rate respectively, require correct assignment as well.
Executinge the following codes:
require(VBTree)
#> 载入需要的程辑包：VBTree
dl2vbt(chrvec2dl(colnames(TPMdata)))
#> $tree
#> $tree[[1]]
#> [1] "Strain" "Stress"
#>
#> $tree[[2]]
#> $tree[[2]][[1]]
#> [1] "900" "950" "1000" "1050" "1100" "1150" "1200"
#>
#> $tree[[2]][[2]]
#> $tree[[2]][[2]][[1]]
#> [1] "0.001" "0.01" "0.1" "1"
#>
#> $tree[[2]][[2]][[2]]
#> $tree[[2]][[2]][[2]][[1]]
#> [1] "60%"
#>
#> $tree[[2]][[2]][[2]][[2]]
#> list()
#>
#>
#>
#>
#>
#> $dims
#> [1] 2 7 4 1
#>
#> attr(,"class")
#> [1] "Vector.Binary.Tree"The complete structure for all variables is showed. As the result showed above, there’re 7 temperatures and 4 strain rates in our summary table, therefore the numbers of level for temperature and strain rate are 7 and 4 respectively. Based on all introduced knowledge, we can easily find that all factors will be corresponded with a unique identity with the format as (layer, level). For example, we can define the factor “950” is located in layer 2, level 2.
SSplots is automatic completion for stress-strain curve
plots, using VBTree
package group strategy. It help researchers check the profiles of
their experimental data conveniently. The argument grpby
determines attribute to be grouped for each plot. As an instance,
stress-strain curves grouped by strain rates, separated by temperature
condition in each individual plot can be achieved using the code
SSplots(TPMdata, 3, mfrow=c(3, 3)). The layer for strain
rate is 3, therefore the 2nd argument grpby is 3 (7 figures
totally, therefore mfrow use a 3*3 division to ensure
correct display):
Grouped by temperature is also available, by running
SSplots(TPMdata, 2, mfrow=c(2, 2)), four figures will be
exported as:
However, multiple plots export using graphics is very
sensitive to the Plots pane’s in RStudio. Before plotting, please make
sure the area of this pane is large enough to include all your
output.
Sometimes, curves of flow stress as function of flow strain might be
of high vibration, therefore the fitting process for all data is
necessary. However, in most circumstances, raw stress-strain curves
always present too complicated appreance to be fitted by a single linear
model. The function KFprocess() in this package is designed
to solve this problem. From the aforementioned raw stress-strain curves
generated by SSplots(TPMdata, 2, mfrow=c(2, 2)), it is not
difficult to find for most curves, vibration starts from where the
plastic deformation occurs; Using the dV as 0.3 and
dW as 0.006, the main paths of flow stress vs. flow strain
curves can successfully obtained:
Compared to the raw curves, we can find the fitted curves keep the
most appearance with less vibrations or noises. For desired output, you
may try different combinations of arguments dV and
dW. Help documentation and references might be helpful. The
calculation of adiabatic heating correction will utilize the integral of
those curves, the smoothing by Kalman will make a more robust output
rationally.
Adiabatic heating correction can be implemented after smoothing if it
is necessary. We can use the function TCorrect() to output
the adiabatic heating correction on data by declaring some necessary
material contants. Take the conventional steel as the example, the heat
capacity constant and material density are 502.416 J/(kJ*K) and 7.8
g*cm^(-3). The corrected data of flow strain ranged from 0 to 0.9 with
0.02 increase in each step would be:
TCorrect_data <- TCorrect(TPMdata, 2, 3, 0.9, 7.8, 502.416, seq(0, 0.9, 0.02))
SSplots(TCorrect_data, 2, mfrow=c(2, 2))
The 2nd and 3rd argument in TCorrect() are integers to
specify the layer for temperatures and strain rates.
After temperature correction, it is noted that flow stress increase significantly in low temperature, high strain rate, the two conditions which is experientially corresponding to our common sense that working hardening effect will be even obvious.
In previous content, we understand the column name structure of the
TPMdata dataset. Therefore the two layers for temperature
and strain rate will be 2 and 3 respectively. If we want the strain
rate-temperature table in condition of 0.7 flow strain, by executing the
following codes the result can be returned automatically:
epstable <- epsExtract(TCorrect_data, eps = 0.7, lyT = 2, lySR = 3)
epstable
#> $SRT.table
#> 900 950 1000 1050 1100 1150 1200
#> 0.001 93.20607 72.89237 56.57545 41.81621 28.54170 16.14671 4.628616
#> 0.01 173.00795 138.43819 108.38777 83.91802 57.85085 37.67723 17.191613
#> 0.1 254.18349 205.85830 161.76956 120.09987 95.91450 66.25144 39.840429
#> 1 292.08106 240.83446 201.05563 166.40103 133.61666 103.63520 78.347986
#>
#> $epsilon
#> [1] 0.7
#>
#> attr(,"class")
#> [1] "SR-T.table"Additionally, for correct exporting, the number of levels for numeric
variables must be 2, for ‘Strain’ and ‘Stress’ respectively. In most
cases, the package has a internal function to detect the locations for
‘Strain’ and ‘Stress’. Nevertheless, people might have different
nomenclature for ‘Strain’ and ‘Stress’ in their summary tables. If so,
the argument manual will be helpful to manually assign the
locations for ‘Strain’ and ‘Stress’, to ensure the correct output (see
help documentation for the function epsExtract).
On the basis of dynamic materials modeling (DMM), constructive
function and all material constants can be calculated from a certain
strain rate-temperature table. The function DMMprocess
includes all calculation steps for DMM processing. Call it on our
previous result (strain rate-temperature table through specified value
of strain), the complete output can be observed as following:
DMM <- DMMprocess(epstable)
DMM
#> $MaterialCoefficients
#> $MaterialCoefficients$base
#> [1] 10
#>
#> $MaterialCoefficients$epsilon.strain
#> [1] 0.7
#>
#>
#> $tablelist
#> $tablelist$SRTtable
#> 900 950 1000 1050 1100 1150 1200
#> 0.001 93.20607 72.89237 56.57545 41.81621 28.54170 16.14671 4.628616
#> 0.01 173.00795 138.43819 108.38777 83.91802 57.85085 37.67723 17.191613
#> 0.1 254.18349 205.85830 161.76956 120.09987 95.91450 66.25144 39.840429
#> 1 292.08106 240.83446 201.05563 166.40103 133.61666 103.63520 78.347986
#>
#> $tablelist$etatable
#> 900 950 1000 1050 1100 1150 1200
#> 0.001 0.4821623 0.4989540 0.5143547 0.5917198 0.5232698 0.6239722 0.8350506
#> 0.01 0.3589281 0.3674713 0.3650931 0.3429635 0.4142833 0.4550536 0.6160868
#> 0.1 0.2056066 0.2140968 0.2290617 0.2246296 0.3048442 0.3439621 0.4698911
#> 1 0.0104753 0.0329846 0.1208608 0.3033972 0.1983005 0.3234620 0.4645742
#>
#> $tablelist$xitable
#> 900 950 1000 1050 1100 1150
#> 0.001 0.224795781 0.22189737 0.19878697 0.03604194 0.24504314 0.1575992
#> 0.01 0.156394203 0.15762142 0.15510319 0.12251926 0.19505528 0.1866591
#> 0.1 0.083198325 0.08768193 0.10260601 0.12190650 0.14289945 0.1783580
#> 1 0.003901329 0.01276194 0.05634805 0.23643104 0.09207504 0.2042067
#> 1200
#> 0.001 -0.2421264
#> 0.01 0.1417022
#> 0.1 0.2411929
#> 1 0.3590626
#>
#>
#> attr(,"class")
#> [1] "DMMresult"If the constructive function is also the thing we concerned, set the
argument ConsFunc in DMMprocess as
TRUE:
DMM <- DMMprocess(epstable, ConsFunc = TRUE)
#> Constitutive equation in0.7strain:
#> Epsilon=A{[sinh(Alpha*Sigma)]^n}*exp[-Q/(RT)]
#> Where A =1458354157082817s^-1
#> Alpha =0.0125766595139204MPa^-1 = 1.257666e-02Pa^-1
#> Q =427084.271412688J/mol = 427.084271412688kJ/mol
#> R = 8.314J/(mol*K)
#> Epsilon is strain rate
#> Sigma is flow stress
#> T is Temperature (K)Additionally, if the argument InteractMode is TRUE, all
computation process and corresponding fitting figures can be output step
by step. There shows some results in following figures:
All workflow control will be prompted in the console, something like:
As we can see, the “etatable” and “xitable” in tablelist are two constitutive parts for building processing-map. The former is for power dissipation efficiency factor eta, and the later is for rheological stability coefficient xi. However, since the values in both are discrete, regression process for these tables is necessary.
Support vector regression (SVR)
is suggested in this package. The function SVRModel will
return a regression result which can be utilized by following functions
for plots.
The following demonstration save the returned result into the vairable “PLTbd”, for the meaning of “Plots builder”.
Function TPM2dplt can generate a thermal processing map
from regression result easily. Following codes demonstrates the
processing map from “PLTbd” with default settings:
In this processing map plot, customarily the power dissipation efficiency factor eta is denoted by contour with gradient blue, while the rheological stability coefficient xi is represented by a gradient colours from blue (low) to red (high). Annotation in top-left show the strain condition for current processing map.
Locations for annotation, gradient colors for rheological stability
coefficient xi is customizable in TPM2dplt function. Refer
the documentation for details.
Function TPM3dplt can build a thermal processing results
as similar as TPM2dplt did. The difference is, eta and xi
will be separately plotted.
By running the following code, we can easily obtain the results as shown in the next figure.
These two 3d surfaces are generated through the rgl package, therefore it will be very convenient to utilize 3d rotation on our plots for further analysis in details.
Especially, the xi plots is coupled with a specific plane, z=0, in order to make divisions for the stable and unstable regions (doesn’t included in demo figure). If the prediction for xi contained certain negative values, this plan will be visible in 3d surface plot.