Modelling the MAT_024 (Piecewise Linear plasticity) card for Graphite cast iron from given True stress strain test data.

OBJECTIVE:

To extract the data from stress-strain curve given for graphite cast iron.

From the extracted data for various strain rates:

1. Deduce the value of Elastic modulus. (E = 20.9 `times` `10^6` psi)
2. Convert the data into Kg-mm-ms system of units.
3. Process the data and create the material as demonstrated (Curve fitting - Equidistant strain data - Yield by 0.02% offset)
4. Model the material in LSTC LS DYNA.
5. Run uniaxial tensile simulation in solver
6. Verify and validate results.

THEORY AND SOP FOR MATERIAL MODELLING USING RAW TEST DATA:

1. Test data:

Test data acquisition is done using "Get Data Digitizer" app, which could convert the graph image into data points and can export it into excdl format.

Graph 1- The graph for raw data from testing for three different strain rates.

2. Predicted data, Trendline equation and curve fitting.

1. The raw data is collected and cleaned i.e reproducibility and kinks in curves are removed.
2. The graph is plotted in excel using raw data, visual match confirmation from actual graph image (Raw data graph)
3. The predicted data is derived from trendline equation retrieved from the raw data curve.
4. The constants of the equation are manipulated in order to get the best fit for predicted data and raw data curve.
5. The optimum constants for the trendline equations are formulated by curve fitting.
6. Stress are found at equidistant strain points to get the smooth curve for LS DYNA input.
7. Using 0.2% offset method, yield point is calculated.

Graph2 - Visual confirmation of plotted graph in excel with actual graph.

For x-y data, pl refer excel sheet "Data.xlsx. - Sheets(Test data)"

Linearity exist till strain = 0.001 or 0.1% strain.

2.1 Initial trendlines for each strain rate:

An initial trendline equation is formulated for graph plotted in excel using Raw test data for each strain rate saperately.

Strain rate 1

Trendline equation is a cubic polinomial which gives best RSQ value of 99%

Initial Trendline equation:

y = `10^-7``x^3` - 6 `times` `10^-5``x^2`+ 0.0089x - 0.0173

Equation can be written as : y = C1`x^3` + C2`x^2` + C3x + C4

where,

C1 = `10^-7`

C2 = - 6 `times` `10^-5`

C3 = 0.0089

C4 = -0.0173

Strain rate 2

Trendline equation is a cubic polinomial which gives best RSQ value of 99%

Initial Trendline equation:

y = 5`times``10^-7``x^3` - `10^-4``x^2`+ 0.0098x - 0.018

Equation can be written as : y = C1`x^3` + C2`x^2` + C3x + C4

where,

C1 = 5`times``10^-7`

C2 = - `10^-4`

C3 = 0.0098

C4 = - 0.018

Strain rate 3

Trendline equation is a linear relation which gives best RSQ value of 99%

Initial Trendline equation:

y = 0.0067x - 0.013

Equation can be written as : y = C1x+C2

where,

C1 = 0.0067

C2 = -0.013

2.2 Curve Fitting

1. Predicted stress calculation: Initial trendline relations are used to get predicted data for stress from initial strain values using the trendline relations.
2. `R^2` value: Square of difference between Raw test stress and Predicted data for each step.
3. Summation of `R^2` (SSR): All values of `R^2` are summed together.
4. Optimization of SSR: The value of SSR is optimized using Excel 'Solver' Add-in.
5. Best fit curves have least value of SSR.
6. The optimization of SSR is by getting the optimized value of C1, C2, C3 and C4.
7. Thus, We get the optimum curve fit equation for strain rate.

3. Equidistant Strain values

From the equation we get from curve fitting, we get values of stress at strain values at equidistant intervals.

Here, interval is selected as per the curve fitting data.

4. 0.2% Offset

Yield point is very difficult to determine for a continuous curve. hence 0.2% offset strain method is used to get the shifted slope.

The intersection of this slope with origional curve gives the yield stress.

5. Calculation of effective plastic strain and corresponding stress from yield.

Effective plastic strain = Total true strain - (True stress/Youngs Modulus)

Effective plastic strains are calculated corresponding to the true stresses.

True stress values of first step is used for each step.(For metals)

LIST OF TABLES AND GRAPHS:

Graph 1- The graph for raw data from testing for three different strain rates.

Graph 2 - Visual confirmation of plotted graph in excel with actual graph.

Graph 3: Visual verification of Test data and simulation data for Effective stress Vs Effective plastic strain

Graph 4: Visual verification of Test data and simulation data for Effective stress Vs True strain

Table 1: Stress-strain data from test and predicted values of stress as per the trendline.

Table 2: The data containing optimized values for curve fitting. (Stress strain and optimized stress values)

Table 3: Constants of trendline equation before and after curve fitting.

Table 4: Effective stress corresponding to Effective plastic strain

Table 5: X-Y(Effective plastic Strain- effective stress) data to LS DYNA

Table 6: X-Y(Effective plastic Strain- effective stress) data from LS DYNA

Table 7: Verification for Youngs Modulus, Yield stress, Ultimate stress and Ultimate plastic strain.

CALCULATIONS AND STEPS IN DETAIL:

For Strain rate 1:

E is calculated using =Slope(Stress values, Strain values) = 21001866.3 psi

E = 21001866.3 psi

1 psi = 6894.76 Pa

1 Pa = 1/`10^9` Kg/mm.`ms^2`

i.e 21001866.3 psi = (21001866.3 `times` 6894.76)/`10^9` Kg/mm.`ms^2`

E = 144.8028277 Kg/mm.`ms^2`

E = 144802.8277 MPa

Initial Trend line-

y = `10^-7``x^3` - 6 `times` `10^-5``x^2`+ 0.0089x - 0.0173

C1 = `10^-7`

C2 = - 6 `times` `10^-5`

C3 = 0.0089

C4 = -0.0173

All values of strains from raw test data is put in initial trend line equation to get predicted values of stress.

Example:

Table 1: Stress-strain data from test and predicted values of stress as per the trendline.

Strain	Stress (Kg/mm.ms2)	Predicted Stress (Kg/mm.ms2)
1.20897E-05	0.000713247	-0.01729989
3.65188E-05	0.005002565	-0.01729967
8.51897E-05	0.011434084	-0.01729924
0.000121833	0.017868062	-0.01729892
0.000170442	0.023583874	-0.01729848
0.00020721	0.031449265	-0.01729816
0.000255819	0.037165077	-0.01729772
0.000292525	0.044314761	-0.0172974
0.000353348	0.052175233	-0.01729686
0.000438413	0.062177904	-0.0172961
0.000487271	0.070756541	-0.01729566
0.000536067	0.078619472	-0.01729523
0.000572898	0.087200569	-0.0172949
0.000657651	0.093624709	-0.01729415
0.000706509	0.102203347	-0.01729371
0.000755305	0.110066278	-0.01729328
0.000804038	0.117213502	-0.01729284

Calculation of `R^2`

For stresses corresponding to Strain = 0.000170442

`R^2` = (`0.025147408^2` - `0.023583874^2`) = 2.44464E-06

Example

Table 2: The data containing optimized values for curve fitting.

Strain	Stress (Kg/mm.ms2)	Predicted Stress (Kg/mm.ms2)	Squared error
1.20897E-05	0.000713247	-0.01729989	0.000324473
3.65188E-05	0.005002565	-0.01729967	0.00049739
8.51897E-05	0.011434084	-0.01729924	0.000825604
0.000121833	0.017868062	-0.01729892	0.001236716
0.000170442	0.023583874	-0.01729848	0.001671367
0.00020721	0.031449265	-0.01729816	0.002376311
0.000255819	0.037165077	-0.01729772	0.002966197
0.000292525	0.044314761	-0.0172974	0.003796058
0.000353348	0.052175233	-0.01729686	0.004826371
0.000438413	0.062177904	-0.0172961	0.006316117
0.000487271	0.070756541	-0.01729566	0.007753191
0.000536067	0.078619472	-0.01729523	0.00919963
0.000572898	0.087200569	-0.0172949	0.010919303
0.000657651	0.093624709	-0.01729415	0.012302993
0.000706509	0.102203347	-0.01729371	0.014279547
0.000755305	0.110066278	-0.01729328	0.016220456
0.000804038	0.117213502	-0.01729284	0.018091957

Summation of Squared error:Similarly, Squared error is calculated for each step.

All `R^2` values are summed together to calculate SSR.

SSR = Sum(All `R^2` values) = 7.013600395

Curve is:

The above value of SSR is to be optimized to get the best fit.

It is done using SOlver add-in

Go to - Data tab >> Solver >> Solver Panel opens>> Enter values >> Solve.

Choose cell with SSR value for minimum value.

Choose constants (C1. C2, C3 and C4) values.

Choose a solving method - GRG Non linear.

Optimized SSR = 0.000370408

Fit curve:

Optimized values of Constants:

C1	1902063.588
C2	30468.62156
C3	174.0439469
C4	0.003641266

Optimized relation between, Stress and strain:

y = 1902063.588 `x^3` + 30468.62156 `x^2` + 174.0439469x + 0.003641266

Table 3: Constants of trendline equation before and after curve fitting:

BEFORE	AFTER
C1 = `10^-7`	C1 = 1902063.588
C2 = -5`times``10^-5`	C2 = 30468.62156
C3 = 0.0089	C3 = 174.0439469
C4 = -0.0176	C4 = 0.003641266
SSR = 7.013600395	SSR = 0.000370408
RSQ = 0.923691	RSQ = 0.99964045
Equation: y = `10^-7`-5`times``x^3` -5`times` `10^-5``x^2`+ 0.0089x -0.0176	Equation: y = 1902063.588`x^3` + 30468.621`x^2`+ 174.043x + 0.99964045

RSQ is the percentage of curve fit.

Using the relation: y = 1902063.588`x^3` + 30468.621`x^2`+ 174.043x + 0.99964045

For strain intervals- 0.0001 stress values are calculated.

Equidistant strain	Calculated stress (kg/mm.ms2)
0	0
0.0001	0.017101611
0.0002	0.033605261
0.0003	0.049522364
0.0004	0.064864331
0.0005	0.079642576
0.0006	0.09386851
0.0007	0.107553546
0.0008	0.120709096
0.0009	0.133346573
0.001	0.145477389
0.0011	0.157112956
0.0012	0.168264687
0.0013	0.178943994
0.0014	0.18916229
0.0015	0.198930986

Curve:

Calculation of Yield stress:

0.2% strain is offseted and Stress is calculated by multiplying 'E' with strain.

Stress = 144.80 Kg/mm.`ms^2` `times` 0.0004 = 0.064864331 Kg/mm.`ms^2`

An offset value is used to get the intersection.

Yield stress = 0.3456 Kg/mm.`ms^2`

All calculate data:

	Youngs Modulus		Yield Stress		Ultimate stress
Strain rate 1	144.8028277 GPa		210 MPa		377.752 MPa

EFFECTIVE PLASTIC STRAIN CALCULATIONS.

Step1: True Stress and True strain at proportionality limit are tabulated.

Step2: All the values of true stress and strain are are tabulated after proportionality limit.

Step3: Effective plastic strain is calculated.

Effective plastic strain = Total True strain - (True Stress/ Youngs Modulus)

Step4: Corresponding values of effective plastic strain and effective stress are calculated.

Note: True stress is taken for first step only for every step because in metals, E is very large.

Table 4: Effective stress corresponding to Effective plastic strain

Equidistant strain	True stress (MPa)	Effective plastic strain	Effective stress
0	0	0	0
0.001	141.8361233	0.00000	141.8361233
0.0011	153.4716905	0.00020	153.4716905
0.0012	164.6234215	0.00030	164.6234215
0.0013	175.3027286	0.00040	175.3027286
0.0014	185.5210243	0.00050	185.5210243
0.0015	195.2897208	0.00060	195.2897208
0.0016	204.6202307	0.00070	204.6202307
0.0017	213.5239662	0.00080	213.5239662
0.0018	222.0123398	0.00090	222.0123398
0.0019	230.0967638	0.00100	230.0967638
0.002	237.7886506	0.00110	237.7886506
0.0021	245.0994126	0.00120	245.0994126
0.0022	252.0404622	0.00130	252.0404622
0.0023	258.6232118	0.00140	258.6232118
0.0024	264.8590737	0.00150	264.8590737
0.0025	270.7594604	0.00160	270.7594604
0.0026	276.3357841	0.00170	276.3357841
0.0027	281.5994574	0.00180	281.5994574
0.0028	286.5618925	0.00190	286.5618925
0.0029	291.2345018	0.00200	291.2345018
0.003	295.6286978	0.00210	295.6286978
0.0031	299.7558928	0.00220	299.7558928
0.0032	303.6274992	0.00230	303.6274992
0.0033	307.2549294	0.00240	307.2549294
0.0034	310.6495958	0.00250	310.6495958
0.0035	313.8229106	0.00260	313.8229106
0.0036	316.7862864	0.00270	316.7862864
0.0037	319.5511355	0.00280	319.5511355
0.0038	322.1288703	0.00290	322.1288703
0.0039	324.5309032	0.00300	324.5309032
0.004	326.7686465	0.00310	326.7686465
0.0041	328.8535126	0.00320	328.8535126
0.0042	330.796914	0.00330	330.796914
0.0043	332.6102629	0.00340	332.6102629
0.0044	334.3049719	0.00350	334.3049719
0.0045	335.8924531	0.00360	335.8924531
0.0046	337.3841191	0.00370	337.3841191
0.0047	338.7913823	0.00380	338.7913823
0.0048	340.1256549	0.00390	340.1256549
0.0049	341.3983494	0.00400	341.3983494
0.005	342.6208782	0.00410	342.6208782
0.0051	343.8046536	0.00420	343.8046536
0.0052	344.9610881	0.00430	344.9610881
0.0053	346.1015939	0.00440	346.1015939
0.0054	347.2375836	0.00450	347.2375836
0.0055	348.3804694	0.00460	348.3804694
0.0056	349.5416637	0.00470	349.5416637
0.0057	350.732579	0.00480	350.732579
0.0058	351.9646277	0.00490	351.9646277
0.0059	353.249222	0.00500	353.249222
0.006	354.5977744	0.00510	354.5977744
0.0061	356.0216972	0.00520	356.0216972
0.0062	357.5324029	0.00530	357.5324029
0.0063	359.1413039	0.00540	359.1413039
0.0064	360.8598124	0.00550	360.8598124
0.0065	362.6993409	0.00560	362.6993409
0.0066	364.6713018	0.00570	364.6713018
0.0067	366.7871074	0.00580	366.7871074
0.0068	369.0581702	0.00590	369.0581702
0.0069	371.4959025	0.00600	371.4959025
0.007	374.1117166	0.00610	374.1117166

Yield is calculated at 0.02% offset for strain.

Effective plastic strain at Yield	Yield Stress	Plastic strain
Strain Rate 1	210.304 MPa	0.0015

Effective plastic strain at Failure	Ultimate Stress	Plastic strain
Strain Rate 1	374.111 MPa	0.00610

DATA IMPORT TO LS DYNA

For strain rate 1:

A curve is imported in LS DYNA using Define>>Curve>>X-Y data import

This curve defines the non linear region in LS DYNA as plastic region.

Linear part of curve is defined using the Young's Modulus.

E = 144000 MPa

v = 0.29

`rho` = 7.1 `times` `10^-6` Kg/`mm^3`

SIGY = 210.304 MPa

FAIL = 0.00601

Table 5: X-Y(Effective plastic Strain- effective stress) data to LS DYNA

Effective plastic strain	Effective stress (MPa)
0.00010	141.8361233
0.00020	153.4716905
0.00030	164.6234215
0.00040	175.3027286
0.00050	185.5210243
0.00060	195.2897208
0.00070	204.6202307
0.00080	213.5239662
0.00090	222.0123398
0.00100	230.0967638
0.00110	237.7886506
0.00120	245.0994126
0.00130	252.0404622
0.00140	258.6232118
0.00150	264.8590737
0.00160	270.7594604
0.00170	276.3357841
0.00180	281.5994574
0.00190	286.5618925
0.00200	291.2345018
0.00210	295.6286978
0.00220	299.7558928
0.00230	303.6274992
0.00240	307.2549294
0.00250	310.6495958
0.00260	313.8229106
0.00270	316.7862864
0.00280	319.5511355
0.00290	322.1288703
0.00300	324.5309032
0.00310	326.7686465
0.00320	328.8535126
0.00330	330.796914
0.00340	332.6102629
0.00350	334.3049719
0.00360	335.8924531
0.00370	337.3841191
0.00380	338.7913823
0.00390	340.1256549
0.00400	341.3983494
0.00410	342.6208782
0.00420	343.8046536
0.00430	344.9610881
0.00440	346.1015939
0.00450	347.2375836
0.00460	348.3804694
0.00470	349.5416637
0.00480	350.732579
0.00490	351.9646277
0.00500	353.249222
0.00510	354.5977744
0.00520	356.0216972
0.00530	357.5324029
0.00540	359.1413039
0.00550	360.8598124
0.00560	362.6993409
0.00570	364.6713018
0.00580	366.7871074
0.00590	369.0581702
0.00600	371.4959025
0.00610	374.1117166

Linearity terminates at 0.1% strain rate i.e E is to be calculated at strain = 0.001

Effective stress vs Effective plastic strain from LS Dyna data

Table 6: X-Y(Effective plastic Strain- effective stress) data from LS DYNA

Simulation
Effective Plastic strain	Effective stress (MPa)
0.00000	0.00
0.00000	20.58
0.00000	41.14
0.00000	61.71
0.00000	82.27
0.00000	102.81
0.00000	122.99
0.00006	137.08
0.00014	146.84
0.00022	156.03
0.00030	165.00
0.00039	173.78
0.00047	182.39
0.00055	190.82
0.00064	199.07
0.00073	207.13
0.00082	215.00
0.00091	222.65
0.00100	230.10
0.00109	237.34
0.00119	244.36
0.00129	251.17
0.00139	257.76
0.00149	264.12
0.00159	270.24
0.00170	276.13
0.00180	281.79
0.00191	287.20
0.00203	292.37
0.00214	297.28
0.00226	301.94
0.00238	306.36
0.00250	310.53
0.00262	314.45
0.00275	318.13
0.00288	321.56
0.00301	324.77
0.00315	327.75
0.00329	330.50
0.00343	333.06
0.00357	335.43
0.00372	337.63
0.00387	339.69
0.00402	341.61
0.00417	343.45
0.00432	345.21
0.00447	346.95
0.00462	348.67
0.00477	350.40
0.00492	352.18
0.00506	354.02
0.00519	355.93
0.00529	170.61
0.00533	2.20
0.00533	2.13
0.00533	2.12
0.00533	2.11
0.00533	2.09
0.00533	2.07
0.00533	2.06
0.00533	2.08
0.00533	2.08
0.00533	2.08
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.08
0.00533	2.08
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.08
0.00533	2.08
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.08
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.07
0.00533	2.08
0.00533	2.07
0.00533	2.07

Graph 3: Visual verification of Test data and simulation data for Effective stress Vs Effective plastic strain

Calculations:

Youngs Modulus, E = 144000 MPa

Effective stress at linearity = 144000 `times` 0.001 = 144 MPa

Value closest to above stress from LS DYNA = 146.840 at strain = 0.00014

True strain = 0.001 + 0.00014 = 0.00114

So Youngs Modulus = 146.840/0.00114

E = 128807.89 MPa

Ultimate stress, U = 355.93 MPa

Ultimate plastic strain = 0.00519

Youngs Modulus	Ultimate stress	Ultimate plastic strain
128807.89 MPa	355.93 MPa	0.00519

COMPARISON OF TEST DATA AND SIMULATED DATA.

Graph 4: Visual verification of Test data and simulation data for Effective stress Vs True strain

	Test Data	Simulated data
Youngs Modulus (MPa)	144802	128807.89
Yield Stress (MPa)	210.304 (0.02% strain)	225 (0.02% strain)
Ultimate Stress (MPa)	374.111	355.93
Ultimate Plastic strain	0.00610	0.00519

Table 7: Verification for Youngs Modulus, Yield stress, Ultimate stress and Ultimate plastic strain.

LS DYNA Keywords:

*Control_Termination

T = 100ms

*Control_Hourglass

IHQ = 2

QH = 0.1

*Control_Energy

HGEN = 2

*Section_Shell

ELFORM = 2

Thickness = 1.5mm

*Mat_Piecewise_linear_plasticity_MAT_024

E = 144802 MPa

SIGY = 210.304 MPa

FAIL = 0.00610

v = 0.29

`rho` = 7.1 `times` `10^-6` Kg/`mm^3`

LCSS = 1

Curve is defined above in Table5.

Displacement loading is defined as per below.

Time (ms)	Displacement (mm)
0	0
100	1
101	1