Defining Flow Stress from Interpolated Data - Using Lambda Functions

In this notebook it is shown, how to define a flow stress function from interpolated data using the interpolation functions from scipy. The concept can be transferred to any other material data as well.

This notebook is essentially the same as Interpolating_Flow_Stress.ipynb, with one main difference: we will use lambda functions as hook values instead of real hook implementations. This is appropriate for usage in input files and for quick testing purposes, plugins shall and can not use this approach.

First, we need to import some packages.

import pyroll.basic as pr # PyRolL itself

import pandas as pd # pandas for data import
import numpy as np # numpy for numeric calculations
import scipy.interpolate as interpolate # scipy.interpolate for interpolations

We import the data to interpolate from a CSV file using pandas. The result is a dataframe containing all data from the file. Note, that the column names are derived from the first row of the file.

flow_stress_data = pd.read_csv("Interpolating_Flow_Stress_Data.csv")
flow_stress_data

	phi	kf
0	0.000000	100.000000
1	0.020202	106.203666
2	0.040404	107.637609
3	0.060606	108.625511
4	0.080808	109.403000
...	...	...
95	1.919192	124.320109
96	1.939394	124.396628
97	1.959596	124.472591
98	1.979798	124.548008
99	2.000000	124.622888

100 rows × 2 columns

We can interpolate the data using scipy to get an interpolation function, which we can evaluate as any other function.

flow_stress_function = interpolate.interp1d(flow_stress_data.phi, flow_stress_data.kf)

Here comes the difference, we do not define hook implementations, but give the interpolation function directly to the in profile. The flow_stress hook is the main hook for calculation flow stress, it returns the flow stress depending on the current state of the profile. Another hook regarding flow stress is flow_stress_function. It returns a function of the classic flow stress parameters strain, strain rate and temperature to enable plugins to calculate their own variations of flow stress. Here it needs a nested lambda function to be defined.

in_profile = pr.Profile.square(
    side=45e-3,
    corner_radius=3e-3,
    temperature=1100 + 273.15,
    strain=0,
    density=7.5e3,
    specific_heat_capacity=690,
    flow_stress=lambda self: flow_stress_function(self.strain),
    flow_stress_function=lambda self: lambda strain, strain_rate, temperature: flow_stress_function(strain)
)
in_profile

SquareProfile classifiers 'square' 'diamond' corner_radius 0.003 cross_section Polygon area 0.0020172289364149133 height 0.061154328932550704 perimeter 0.17484198694172845 width 0.061154328932550704 density 7500.0 diagonal np.float64(0.06363961030678927) flow_stress <function <lambda> at 0x702d976b9da0> flow_stress_function <function <lambda> at 0x702d976b9d00> side 0.045 specific_heat_capacity 690 strain 0 t 0 temperature 1373.15

And the groove, the working roll and the roll pass…

groove1 = pr.DiamondGroove(
    usable_width=76.55e-3,
    tip_depth=22.1e-3,
    r1=12e-3,
    r2=8e-3,
)
groove1

DiamondGroove alpha1 np.float64(0.5236363668157296) alpha2 np.float64(0.5236363668157296) classifiers 'generic_elongation' 'diamond' contour_line LineString height 0.020862195195594648 length 0.12428942750252188 width 0.11360126410252692 depth np.float64(0.020862195195594648) flank_angle np.float64(0.5236363668157296) ground_width np.float64(0.004287509401684611) r1 0.012 r2 0.008 tip_angle None tip_depth 0.0221 usable_width 0.07655

roll1 = pr.Roll(
    groove=groove1,
    nominal_radius=328e-3,
)
roll1

Roll

groove

DiamondGroove alpha1 np.float64(0.5236363668157296) alpha2 np.float64(0.5236363668157296) classifiers 'generic_elongation' 'diamond' contour_line LineString height 0.020862195195594648 length 0.12428942750252188 width 0.11360126410252692 depth np.float64(0.020862195195594648) flank_angle np.float64(0.5236363668157296) ground_width np.float64(0.004287509401684611) r1 0.012 r2 0.008 tip_angle None tip_depth 0.0221 usable_width 0.07655

nominal_radius

0.328

pass1 = pr.RollPass(
    roll=roll1,
    gap=3e-3,
    velocity=1,
)
pass1

TwoRollPass classifiers 'generic_elongation' 'symmetric' 'diamond' contour_lines MultiLineString height 0.0447243903911893 length 0.24857885500504376 width 0.11360126410252692 convergence_history disk_elements engine TwoRollPass.Engine gap 0.003 global_iterator 1 in_profile None label '' out_profile None roll TwoRollPass.Roll contour_points array([[-5.68006321e-02, 0.00000000e+00], [-4.14906321e-02, 0.00000000e+00], [-4.11906125e-02, 3.75107461e-06], [-4.08905930e-02, 1.50113425e-05], [-4.05905735e-02, 3.38020023e-05], [-4.02905539e-02, 6.01586074e-05], [-3.99905344e-02, 9.41314051e-05], [-3.96905149e-02, 1.35785821e-04], [-3.93904953e-02, 1.85203103e-04], [-3.90904758e-02, 2.42481133e-04], [-3.87904563e-02, 3.07735434e-04], [-3.84904367e-02, 3.81100396e-04], [-3.81904172e-02, 4.62730744e-04], [-3.78903977e-02, 5.52803299e-04], [-3.75903781e-02, 6.51519075e-04], [-3.72903586e-02, 7.59105780e-04], [-3.69903391e-02, 8.75820795e-04], [-3.66903195e-02, 1.00195473e-03], [-3.63903000e-02, 1.13783572e-03], [-3.60902805e-02, 1.28383451e-03], [-3.57902609e-02, 1.44037079e-03], [-3.54902414e-02, 1.60792071e-03], [-4.00026044e-03, 1.97902481e-02], [-3.80024741e-03, 1.99019480e-02], [-3.60023439e-03, 2.00063055e-02], [-3.40022137e-03, 2.01036381e-02], [-3.20020835e-03, 2.01942254e-02], [-3.00019533e-03, 2.02783147e-02], [-2.80018231e-03, 2.03561247e-02], [-2.60016928e-03, 2.04278491e-02], [-2.40015626e-03, 2.04936597e-02], [-2.20014324e-03, 2.05537080e-02], [-2.00013022e-03, 2.06081283e-02], [-1.80011720e-03, 2.06570382e-02], [-1.60010417e-03, 2.07005411e-02], [-1.40009115e-03, 2.07387265e-02], [-1.20007813e-03, 2.07716713e-02], [-1.00006511e-03, 2.07994409e-02], [-8.00052087e-04, 2.08220895e-02], [-6.00039066e-04, 2.08396605e-02], [-4.00026044e-04, 2.08521876e-02], [-2.00013022e-04, 2.08596945e-02], [ 0.00000000e+00, 2.08621952e-02], [ 2.00013022e-04, 2.08596945e-02], [ 4.00026044e-04, 2.08521876e-02], [ 6.00039066e-04, 2.08396605e-02], [ 8.00052087e-04, 2.08220895e-02], [ 1.00006511e-03, 2.07994409e-02], [ 1.20007813e-03, 2.07716713e-02], [ 1.40009115e-03, 2.07387265e-02], [ 1.60010417e-03, 2.07005411e-02], [ 1.80011720e-03, 2.06570382e-02], [ 2.00013022e-03, 2.06081283e-02], [ 2.20014324e-03, 2.05537080e-02], [ 2.40015626e-03, 2.04936597e-02], [ 2.60016928e-03, 2.04278491e-02], [ 2.80018231e-03, 2.03561247e-02], [ 3.00019533e-03, 2.02783147e-02], [ 3.20020835e-03, 2.01942254e-02], [ 3.40022137e-03, 2.01036381e-02], [ 3.60023439e-03, 2.00063055e-02], [ 3.80024741e-03, 1.99019480e-02], [ 4.00026044e-03, 1.97902481e-02], [ 3.54902414e-02, 1.60792071e-03], [ 3.57902609e-02, 1.44037079e-03], [ 3.60902805e-02, 1.28383451e-03], [ 3.63903000e-02, 1.13783572e-03], [ 3.66903195e-02, 1.00195473e-03], [ 3.69903391e-02, 8.75820795e-04], [ 3.72903586e-02, 7.59105780e-04], [ 3.75903781e-02, 6.51519075e-04], [ 3.78903977e-02, 5.52803299e-04], [ 3.81904172e-02, 4.62730744e-04], [ 3.84904367e-02, 3.81100396e-04], [ 3.87904563e-02, 3.07735434e-04], [ 3.90904758e-02, 2.42481133e-04], [ 3.93904953e-02, 1.85203103e-04], [ 3.96905149e-02, 1.35785821e-04], [ 3.99905344e-02, 9.41314051e-05], [ 4.02905539e-02, 6.01586074e-05], [ 4.05905735e-02, 3.38020023e-05], [ 4.08905930e-02, 1.50113425e-05], [ 4.11906125e-02, 3.75107461e-06], [ 4.14906321e-02, 0.00000000e+00], [ 5.68006321e-02, 0.00000000e+00]]) groove DiamondGroove alpha1 np.float64(0.5236363668157296) alpha2 np.float64(0.5236363668157296) classifiers 'generic_elongation' 'diamond' contour_line LineString height 0.020862195195594648 length 0.12428942750252188 width 0.11360126410252692 depth np.float64(0.020862195195594648) flank_angle np.float64(0.5236363668157296) ground_width np.float64(0.004287509401684611) r1 0.012 r2 0.008 tip_angle None tip_depth 0.0221 usable_width 0.07655 nominal_radius 0.328 velocity 1

Lets run the solution procedure of the roll pass. We see no errors claiming that a flow stress value is missing.

pass1.solve(in_profile)

WARNING:root:No coulomb_friction_coefficient hook found. Continuing with 0.

WARNING:root:No coulomb_friction_coefficient hook found. Continuing with 0.

WARNING:root:No coulomb_friction_coefficient hook found. Continuing with 0.

WARNING:root:No coulomb_friction_coefficient hook found. Continuing with 0.

WARNING:root:No coulomb_friction_coefficient hook found. Continuing with 0.

Profile classifiers 'generic_elongation' 'symmetric' 'diamond' cross_section Polygon area 0.0017909735754438032 height 0.0447243903911893 perimeter 0.16262870637092414 width 0.06560167360406854 cross_section_error -0.06535133698736628 cross_section_filling_ratio 0.9346486630126337 density 7500.0 filling_error np.float64(-0.14302189935900012) filling_ratio np.float64(0.8569781006409999) flow_stress <function <lambda> at 0x702d976b9da0> flow_stress_function <function <lambda> at 0x702d976b9d00> length 0.0 specific_heat_capacity 690 strain np.float64(0.20420123818243657) t np.float64(0.07056039637464354) technologically_orientated_cross_section Polygon area 0.0017909735754438032 height 0.0447243903911893 perimeter 0.16262870637092414 width 0.06560167360406854 temperature np.float64(1369.1203272133532) velocity 1

Lets check, if our new hooks are really used. We check that the flow stress value stored in the out profile of the pass is numerically equal to a manually calculated value using the respective strain. If everything went fine, we shall se no output of this statement, otherwise an error message.

assert np.isclose(pass1.out_profile.flow_stress, flow_stress_function(pass1.out_profile.strain))