Week - 10 Hyperelastic Material Models

OBJECTIVE:

From given data for Engineering strain and engineering stress.

Calculate the Mooney Rivlin and Ogden material constants from the given data.

Run simulation on a dogbone specimen for uniaxial tensile test for 100% strain.

Compare the given experimental data with:

1. d3hsp file data for curve fit for given order of polynomial.

2. Simulation for given order and associated parameters for Mooney rivian and Ogden.

THEORY:

Hyper elastic Material Models:

Source 1: Material Modelling in FEA by Z yang

Source 2: Finite element Modelling of rubber block exposed to shock loading - A paper presented in Lund University.

Hyperelastic constitutive models describe the elastic property of a material from the strain energy.

It is a mathematical relation between the stress and strain.

For most of the hyperelastic material, stress is not just the function of strain but also depend on certian other factors, like

• strain rate
• magnitude of strain
• temperature
• plasticity
• strain amplitude
• frequency of cyclic loading.

What is strain energy?

Strain energy(W) is the potential energy function, a potential energy functio is independent of path and depends only on the initial and final states (i.e current strain)

The strain energy is defined as per the following expression:

Equation 1

In 1 dimensional stress train curve, strain energy is the area under the curve.

Equation 4.1 is differentiated wrt E to get the expression for second Piola-Kirchoffs stress tensor. To convert the strain energy in the corrseponding yield and these corresponding yields are dependent on strain invariants.

Sij is conjugated to dEij/dt

The strain energy can then be represented as a function dependent on left Cauchy-Green deformation tensor B, which for an isotropic material means that the dependency is reduced to principal stretch.

The strain energy function can be written as:

Strain energy functions:

There are various functions to mimic the stress-stretch curve for different loading cases.

The simplest functions in polynomial form are written in following form:

Equation 2

If incompressibility is considered, then I3 = 1, and above function becomes:

Two of the models which can be deduced from above expressions are:-

1. Monney-Rivlin:

It is a model derived from above expression. It is gained if two terms is included,

It is to model the material behaviour, defined by polynomial of first order.

2. Ogden:

It is also a model which has roots in equation 2, but it uses principal stretches instead of strain invariants.

In addition to integer exponent in the ordinary polynolmial, ogden uses real numbers.

Advantage of Ogden model:

It provides better adjustment possibilities to experimental curves.

Order of polynomial ranges from 0 t0 9

DESIGN OF EXPERIMENT:

1. Plotting the experimental values for engineering stress and strain.

2. Dummy solving for material card with N values.

3. Getting curve fit data for parameters, stretch and True stress from d3hsp file.

4. Fitting the curve for d3hsp data with experimental data after converting it into engineering stress and strain.

5. Modelling the material using the parameters under proper boundary and loading conditions.

6. Fetching the data for true stress and strain. Converting it into Engineering stress and strain.

7. Visual comparison of experimental data with d3hsp and simulated data via plot in Engineering stres and strain.

NOTE: All plots for comparison are in Engineering stress and strain.

NOTE: All tensile tests are performed under quasi-static loading.

UNIAXIAL TENSILE TEST:

Dogbone specimen:

Boundary condition:

One end of the dogbone is constrained in x in translation and rotational.

For providing the stability in direction perpendicular to strain. x constrained nodes are not rigidised in y and z.

Instead Middle node, node along the axis of symmetry is constrained along y and z in translation and rotation.

so for nodes at the left end are constrained as:

	Nodes	x-tran	Y-tran	z-tran	x-rot	y-rot	z-rot
SPC1	All at left end	1	0	0	1	1	1
SPC2	1 along Axis symmetry	0	1	0	0	0	0
SPC2	1 along Axis symmetry	0	0	1	0	0	0

By these boundary conditions, the poissons ratio stays valid.

Compression along y and z is allowed, if strain is in x.

Loading condition:

Displacement loading is defined for 100%strain rate.

Strain rate is 3% (mm/mm).`mm^-1`

i.e total strain = 145/145 mm/mm (standard gauge length)

Strain in 60 seconds = 0.03

Strain in 1 second = 0.03/60 = 0.0005 mm/mm

Strain in 1ms = 5`times``10^-7` mm/mm

time in ms for 100% strain = 1/5`times``10^-7` mm/mm = 2000000 ms = 2000 seconds.

	Time (ms)	Displacement (mm)
1	0.00	0.00
2	2000000	145.00
3	2005000	145.00

Loading is defined beyond termination time

All nodes at the right are displaced as per above displacement loading.

Loading is static and hence, quasi-static loading is defined

Analysis is Implicit.

MOONEY-RIVLIN MATERIAL MODEL:

Step1: Data from the experimental data-

Data is quidistant strain data. Smooth and already cleaned. Data is for Engineering stress and strain.

Step2: Plotting the experimental data:

Step3: Getting the data from dehsp file - parameters for Mooney Rivlin and Curve fitting:

In Material card, *MAT_HYPERELASTIC_RUBBER_(TITLE)(077_H) (0)

Density = 0.001 gm/`mm^3`

Poissons ratio = 0.4995

Order of polynomial = 1 (Mooney Rivlin)

SGL, SW, ST =1 since give loading curve is stress-strain data.

Loading curve:

The above experimental data for which we need to get the curve fit is fed as LCID.

A curve is defined using, *DEFINE_CURVE for the above loading.

Here, LCID = 1

Dummy simulation and data extraction from d3hsp file:

A simulation is run, to get the parameters for Mooney Rivian modelling. This data can be saved for future use.(Cost effective)

This data is fetched from d3hsp file.

Parameters for polynomial-

C10 = 0.17680

C01 = 0.14740

Curve fit data-

Data for True stress and Stretch are obtained from d3hsp file

Calculation:

• Engg strain from stretch, `epsilon` =`lambda` - 1

where,

`lambda` = Stretch

`epsilon` = Engg strain

• Engg stress, `sigma` = `Sigma`/`lambda`

where,

`sigma` = Engg stress

`Sigma` = True stress

`lambda` = Stretch

N1
Stretch	Engg Strain	True Stress	Engg Stress
1.0000	0.0000	0.00000	0.0000000000
1.0250	0.0250	0.04810	0.0469268293
1.0500	0.0500	0.09523	0.0906952381
1.0750	0.0750	0.14150	0.1316279070
1.1000	0.1000	0.18700	0.1700000000
1.1250	0.1250	0.23190	0.2061333333
1.1500	0.1500	0.27630	0.2402608696
1.1750	0.1750	0.32010	0.2724255319
1.2000	0.2000	0.36360	0.3030000000
1.2250	0.2250	0.40660	0.3319183673
1.2500	0.2500	0.44940	0.3595200000
1.2750	0.2750	0.49200	0.3858823529
1.3000	0.3000	0.53440	0.4110769231
1.3250	0.3250	0.57660	0.4351698113
1.3500	0.3500	0.61870	0.4582962963
1.3750	0.3750	0.66080	0.4805818182
1.4000	0.4000	0.70280	0.5020000000
1.4250	0.4250	0.74480	0.5226666667
1.4500	0.4500	0.78680	0.5426206897
1.4750	0.4750	0.82890	0.5619661017
1.5000	0.5000	0.87100	0.5806666667
1.5250	0.5250	0.91330	0.5988852459
1.5500	0.5500	0.95560	0.6165161290
1.5750	0.5750	0.99810	0.6337142857
1.6000	0.6000	1.04100	0.6506250000
1.6250	0.6250	1.08400	0.6670769231
1.6500	0.6500	1.12600	0.6824242424
1.6750	0.6750	1.17000	0.6985074627
1.7000	0.7000	1.21300	0.7135294118
1.7250	0.7250	1.25700	0.7286956522
1.7500	0.7500	1.30000	0.7428571429
1.7750	0.7750	1.34500	0.7577464789
1.8000	0.8000	1.38900	0.7716666667
1.8250	0.8250	1.43300	0.7852054795
1.8500	0.8500	1.47800	0.7989189189
1.8750	0.8750	1.52300	0.8122666667
1.9000	0.9000	1.56900	0.8257894737
1.9250	0.9250	1.61400	0.8384415584
1.9500	0.9500	1.66000	0.8512820513
1.9750	0.9750	1.70700	0.8643037975
2.0000	1.0000	1.75300	0.8765000000

Engg Stress and Strain data is plotted from curve fit data gathered from d3hsp file.

The data from d3hsp file and Experimental data overlaps eachother closely. It is a good fit.

Step4: Uniaxial tensile test

1. MATERIAL MODELLING

Now, based on the parameters for Mooney Rivlin material model fetched from d3hsp file. are entered to create the material model.

C10 and C01 is entered for the *MAT_HYPERELASTIC_RUBBER_(TITLE)(077_H)

Here, N=0

2. LOADING:

Displacemnet loading

	Time (ms)	Displacement (mm)
1	0.00	0.00
2	2000	65.00
3	2005	65.00

This loading is to ensure, 100% strain.

*Prescribed_Motion_Set is defined for displacement.

VAD = 2

LCID = 2

SF = 1

3. STRAIN RATE:

3% strain `min^-1`

i.e 100% strain in T = 2000000 ms = 2000 seconds.

This strain rate ensures Viscoelasticity can be avoided during tensile test.

Hence, the shear strain is minimal.

*Control_Termination = 2`times``10^6` ms

4. IMPLICIT ANALYSIS:

Since displacement loading is quasi-static, hence Implicit analysis cards are defined.

*Control_Implicit_General

IMFLAG =1

DT0 = 2`times``10^4` ms

*Control_Implicit_Auto

IAuto = 1

ITOPT =11

ITWIN =6

*Control_Implicit_Solver and _Solution are defined under default values.

5. GEOMETRIC DEFINITIONS

*Section_Shell

ELFORM =2

NIP =2

*Control_Hourglass

IHQ =2

QH =0.1

*Control_Energy

HGEN =2

6. OUTPUT

*Database_D3plot

Dt = 2000 ms (Binary output file)

*Database_ASCII

ELOUT = 2000ms (To get element data, element defined in *Database_History_shell)

GLSTAT = 2000ms (Global data)

Energy curves.

*Database_History_Shell

ID = 338 (mid shell element for deducing results)

Stress, strain, displacement and velocity curves.

*Database_Extent_Binary

STRFLG = 1 (To get the data for strain tensor in d3plot)

Relevant plots: Lower, upper and mean ipt stress in x,y and z.

Simulation:

Simulation data:

Calculation:

• True strain = Data from Mean Ipt x strain vs time
• True stress = Data from effective stress or X-stress with time.
• Engineering strian = `e^(true strain)` - 1
• Stretch = 1 + Engineering strain
• Engineering stress = True stress/ Stretch

Simulation data
Effective Stress (MPa)	True strain (mm/mm)	Engg Strain (mm/mm)	Engg stress (MPa)
0.00000	0.000000	0.000000	0.000000
0.04812	0.025364	0.025688	0.046913
0.09458	0.050624	0.051927	0.089907
0.13920	0.075755	0.078698	0.129040
0.18284	0.100806	0.106062	0.165310
0.22495	0.125728	0.133973	0.198374
0.26706	0.150650	0.162589	0.229710
0.30670	0.175304	0.191609	0.257381
0.34588	0.199910	0.221293	0.283211
0.38507	0.224516	0.251716	0.307633
0.42426	0.249121	0.282897	0.330701
0.46282	0.273813	0.314969	0.351965
0.50128	0.298521	0.347864	0.371905
0.53973	0.323229	0.381582	0.390662
0.57819	0.347937	0.416143	0.408283
0.61642	0.372699	0.451648	0.424636
0.65462	0.397472	0.488057	0.439913
0.69281	0.422244	0.525380	0.454189
0.73101	0.447016	0.563639	0.467503
0.76922	0.471781	0.602846	0.479908
0.80743	0.496545	0.643034	0.491429
0.84565	0.521308	0.684230	0.502099
0.88387	0.546072	0.726458	0.511953
0.92244	0.570896	0.769852	0.521197
0.96108	0.595732	0.814358	0.529710
0.99973	0.620567	0.859983	0.537492
1.03837	0.645402	0.906754	0.544573
1.07748	0.670247	0.954721	0.551218
1.11667	0.695094	1.003897	0.557251
1.15587	0.719940	1.054310	0.562657
1.19507	0.744787	1.105992	0.567461
1.23485	0.769627	1.158960	0.571965
1.27474	0.794466	1.213258	0.575956
1.31463	0.819305	1.268921	0.579407
1.35452	0.844144	1.325985	0.582342
1.39513	0.868996	1.384515	0.585080
1.43588	0.893850	1.444523	0.587386
1.47663	0.918705	1.506042	0.589226
1.51737	0.943559	1.569110	0.590622
1.56032	0.968839	1.634883	0.592180
1.60368	0.994196	1.702551	0.593395

Plot:

CONCLUSION:

Test data		d3hsp data (N=1)				Simulation data
Engg strain (mm/mm)	Engg stress (MPa)	Stretch (mm/mm)	Engg Strain (mm/mm)	True Stress (MPa)	Engg Stress (MPa)	Effective Stress (MPa)	True strain (mm/mm)	Engg Strain (mm/mm)	Engg stress (MPa)
0	0.00000	1.0000	0.0000	0.00000	0.0000000000	0.00000	0.000000	0.000000	0.000000
0.025	0.04460	1.0250	0.0250	0.04810	0.0469268293	0.04812	0.025364	0.025688	0.046913
0.05	0.08670	1.0500	0.0500	0.09523	0.0906952381	0.09458	0.050624	0.051927	0.089907
0.075	0.12700	1.0750	0.0750	0.14150	0.1316279070	0.13920	0.075755	0.078698	0.129040
0.1	0.16400	1.1000	0.1000	0.18700	0.1700000000	0.18284	0.100806	0.106062	0.165310
0.125	0.20000	1.1250	0.1250	0.23190	0.2061333333	0.22495	0.125728	0.133973	0.198374
0.15	0.23400	1.1500	0.1500	0.27630	0.2402608696	0.26706	0.150650	0.162589	0.229710
0.175	0.26700	1.1750	0.1750	0.32010	0.2724255319	0.30670	0.175304	0.191609	0.257381
0.2	0.29800	1.2000	0.2000	0.36360	0.3030000000	0.34588	0.199910	0.221293	0.283211
0.225	0.32800	1.2250	0.2250	0.40660	0.3319183673	0.38507	0.224516	0.251716	0.307633
0.25	0.35600	1.2500	0.2500	0.44940	0.3595200000	0.42426	0.249121	0.282897	0.330701
0.275	0.38300	1.2750	0.2750	0.49200	0.3858823529	0.46282	0.273813	0.314969	0.351965
0.3	0.40900	1.3000	0.3000	0.53440	0.4110769231	0.50128	0.298521	0.347864	0.371905
0.325	0.43400	1.3250	0.3250	0.57660	0.4351698113	0.53973	0.323229	0.381582	0.390662
0.35	0.45800	1.3500	0.3500	0.61870	0.4582962963	0.57819	0.347937	0.416143	0.408283
0.375	0.48100	1.3750	0.3750	0.66080	0.4805818182	0.61642	0.372699	0.451648	0.424636
0.4	0.50300	1.4000	0.4000	0.70280	0.5020000000	0.65462	0.397472	0.488057	0.439913
0.425	0.52400	1.4250	0.4250	0.74480	0.5226666667	0.69281	0.422244	0.525380	0.454189
0.45	0.54500	1.4500	0.4500	0.78680	0.5426206897	0.73101	0.447016	0.563639	0.467503
0.475	0.56500	1.4750	0.4750	0.82890	0.5619661017	0.76922	0.471781	0.602846	0.479908
0.5	0.58400	1.5000	0.5000	0.87100	0.5806666667	0.80743	0.496545	0.643034	0.491429
0.525	0.60300	1.5250	0.5250	0.91330	0.5988852459	0.84565	0.521308	0.684230	0.502099
0.55	0.62000	1.5500	0.5500	0.95560	0.6165161290	0.88387	0.546072	0.726458	0.511953
0.575	0.63800	1.5750	0.5750	0.99810	0.6337142857	0.92244	0.570896	0.769852	0.521197
0.6	0.65500	1.6000	0.6000	1.04100	0.6506250000	0.96108	0.595732	0.814358	0.529710
0.625	0.67100	1.6250	0.6250	1.08400	0.6670769231	0.99973	0.620567	0.859983	0.537492
0.65	0.68700	1.6500	0.6500	1.12600	0.6824242424	1.03837	0.645402	0.906754	0.544573
0.675	0.70200	1.6750	0.6750	1.17000	0.6985074627	1.07748	0.670247	0.954721	0.551218
0.7	0.71700	1.7000	0.7000	1.21300	0.7135294118	1.11667	0.695094	1.003897	0.557251
0.725	0.73200	1.7250	0.7250	1.25700	0.7286956522	1.15587	0.719940	1.054310	0.562657
0.75	0.74600	1.7500	0.7500	1.30000	0.7428571429	1.19507	0.744787	1.105992	0.567461
0.775	0.76000	1.7750	0.7750	1.34500	0.7577464789	1.23485	0.769627	1.158960	0.571965
0.8	0.77300	1.8000	0.8000	1.38900	0.7716666667	1.27474	0.794466	1.213258	0.575956
0.825	0.78600	1.8250	0.8250	1.43300	0.7852054795	1.31463	0.819305	1.268921	0.579407
0.85	0.79900	1.8500	0.8500	1.47800	0.7989189189	1.35452	0.844144	1.325985	0.582342
0.875	0.81100	1.8750	0.8750	1.52300	0.8122666667	1.39513	0.868996	1.384515	0.585080
0.9	0.82300	1.9000	0.9000	1.56900	0.8257894737	1.43588	0.893850	1.444523	0.587386
0.925	0.83500	1.9250	0.9250	1.61400	0.8384415584	1.47663	0.918705	1.506042	0.589226
0.95	0.84700	1.9500	0.9500	1.66000	0.8512820513	1.51737	0.943559	1.569110	0.590622
0.975	0.85800	1.9750	0.9750	1.70700	0.8643037975	1.56032	0.968839	1.634883	0.592180
1	0.86900	2.0000	1.0000	1.75300	0.8765000000	1.60368	0.994196	1.702551	0.593395

Plot:

The Experimental data is compared with:

1. Engineering stress and strain calculated using d3hsp data (Stretch and Fit True stress)
2. Engineering stress and strain calculated using simulation data (Effective plastic strain and Effective true stress)

d3hsp data fits closely with test data, but simulated data deviates from test data.

OGDEN MATERIAL MODEL:

Boundary conditions and loading conditions are same.

Only difference is Material card.

Material card used here, *Material_OGDEN_RUBBER_(TITLE) (077_O) (1)

Material is defined for N=1, and N=6

For instance, N=1

Parameters retrieved from d3hsp file:

For N=1

mu(1)	0.91076024661337	N1
alpha(1)	1.34272388013520

For N=3

mu(1)	-0.031842578663497	N3
alpha(1)	1.926332140745900
mu(2)	0.261375531884940
alpha(2)	1.550340539119100
mu(3)	0.681991923547980
alpha(3)	1.288782095781300

For N=4

mu(1)	0.0059060374936488	N4
alpha(1)	4.2741110160714000
mu(2)	-0.0474584917059350
alpha(2)	3.2497940783369000
mu(3)	0.3243208777580500
alpha(3)	2.0329872899856000
mu(4)	0.6964783424352800
alpha(4)	0.9944861731474600

d3hsp data:

d3hsp data (N=1)				d3hsp data (N=3)				d3hsp data (N=4)
Stretch (mm)	Engg strain (mm/mm)	True stress (MPa)	Engg stress (MPa)	Stretch (mm)	Engg strain (mm/mm)	True stress (MPa)	Engg stress (MPa)	Stretch (mm)	Engg strain (mm/mm)	True stress (MPa)	Engg stress (MPa)
1.000000	0.000000	0.000000	0	1.000000	0.000000	0.000000	0	1.000000	0.000000	0.000000	0
1.025000	0.025000	0.044563	0.0434756	1.025000	0.025000	0.044560	0.04347303	1.025000	0.025000	0.044565	0.04347763
1.050000	0.050000	0.086679	0.0825512	1.050000	0.050000	0.086674	0.08254663	1.050000	0.050000	0.086681	0.08255305
1.075000	0.075000	0.126550	0.1177212	1.075000	0.075000	0.126544	0.11771524	1.075000	0.075000	0.126551	0.11772167
1.100000	0.100000	0.164358	0.1494161	1.100000	0.100000	0.164350	0.14940922	1.100000	0.100000	0.164356	0.14941456
1.125000	0.125000	0.200263	0.1780117	1.125000	0.125000	0.200255	0.17800434	1.125000	0.125000	0.200259	0.17800804
1.150000	0.150000	0.234413	0.2038371	1.150000	0.150000	0.234404	0.20382962	1.150000	0.150000	0.234406	0.20383147
1.175000	0.175000	0.266938	0.2271814	1.175000	0.175000	0.266929	0.22717395	1.175000	0.175000	0.266929	0.22717403
1.200000	0.200000	0.297959	0.2482988	1.200000	0.200000	0.297950	0.2482917	1.200000	0.200000	0.297948	0.24829022
1.225000	0.225000	0.327582	0.2674139	1.225000	0.225000	0.327574	0.26740735	1.225000	0.225000	0.327571	0.26740462
1.250000	0.250000	0.355907	0.2847254	1.250000	0.250000	0.355899	0.28471951	1.250000	0.250000	0.355895	0.2847159
1.275000	0.275000	0.383022	0.3004094	1.275000	0.275000	0.383016	0.30040432	1.275000	0.275000	0.383010	0.30040019
1.300000	0.300000	0.409009	0.3146225	1.300000	0.300000	0.409004	0.31461832	1.300000	0.300000	0.408998	0.314614
1.325000	0.325000	0.433943	0.3275042	1.325000	0.325000	0.433939	0.32750091	1.325000	0.325000	0.433933	0.32749671
1.350000	0.350000	0.457891	0.3391789	1.350000	0.350000	0.457888	0.33917654	1.350000	0.350000	0.457883	0.3391727
1.375000	0.375000	0.480917	0.3497579	1.375000	0.375000	0.480915	0.34975645	1.375000	0.375000	0.480911	0.34975316
1.400000	0.400000	0.503077	0.3593408	1.400000	0.400000	0.503076	0.35934033	1.400000	0.400000	0.503073	0.35933771
1.425000	0.425000	0.524425	0.3680172	1.425000	0.425000	0.524425	0.36801761	1.425000	0.425000	0.524422	0.36801575
1.450000	0.450000	0.545008	0.3758675	1.450000	0.450000	0.545010	0.37586874	1.450000	0.450000	0.545008	0.37586764
1.475000	0.475000	0.564872	0.3829641	1.475000	0.475000	0.564875	0.38296614	1.475000	0.475000	0.564875	0.38296576
1.500000	0.500000	0.584059	0.3893724	1.500000	0.500000	0.584063	0.38937511	1.500000	0.500000	0.584063	0.38937539
1.525000	0.525000	0.602606	0.3951513	1.525000	0.525000	0.602611	0.39515469	1.525000	0.525000	0.602612	0.39515551
1.550000	0.550000	0.620549	0.4003544	1.550000	0.550000	0.620555	0.40035825	1.550000	0.550000	0.620557	0.40035947
1.575000	0.575000	0.637922	0.4050299	1.575000	0.575000	0.637929	0.40503415	1.575000	0.575000	0.637931	0.40503562
1.600000	0.600000	0.654755	0.4092217	1.600000	0.600000	0.654762	0.40922628	1.600000	0.600000	0.654765	0.40922783
1.625000	0.625000	0.671076	0.4129697	1.625000	0.625000	0.671084	0.41297451	1.625000	0.625000	0.671086	0.41297596
1.650000	0.650000	0.686912	0.4163102	1.650000	0.650000	0.686920	0.41631508	1.650000	0.650000	0.686922	0.41631628
1.675000	0.675000	0.702288	0.4192761	1.675000	0.675000	0.702296	0.41928102	1.675000	0.675000	0.702297	0.41928182
1.700000	0.700000	0.717226	0.4218977	1.700000	0.700000	0.717234	0.42190244	1.700000	0.700000	0.717235	0.42190272
1.725000	0.725000	0.731749	0.4242024	1.725000	0.725000	0.731757	0.42420682	1.725000	0.725000	0.731756	0.42420649
1.750000	0.750000	0.745877	0.4262152	1.750000	0.750000	0.745884	0.42621927	1.750000	0.750000	0.745882	0.42621828
1.775000	0.775000	0.759628	0.4279593	1.775000	0.775000	0.759634	0.42796277	1.775000	0.775000	0.759631	0.42796112
1.800000	0.800000	0.773020	0.4294556	1.800000	0.800000	0.773025	0.42945836	1.800000	0.800000	0.773021	0.4294561
1.825000	0.825000	0.786070	0.4307233	1.825000	0.825000	0.786074	0.43072533	1.825000	0.825000	0.786069	0.43072256
1.850000	0.850000	0.798794	0.4317803	1.850000	0.850000	0.798796	0.43178137	1.850000	0.850000	0.798790	0.43177829
1.875000	0.875000	0.811205	0.4326427	1.875000	0.875000	0.811205	0.43264274	1.875000	0.875000	0.811199	0.43263957
1.900000	0.900000	0.823319	0.4333256	1.900000	0.900000	0.823316	0.43332437	1.900000	0.900000	0.823311	0.43332145
1.925000	0.925000	0.835147	0.4338426	1.925000	0.925000	0.835142	0.43384002	1.925000	0.925000	0.835138	0.43383774
1.950000	0.950000	0.846702	0.4342063	1.950000	0.950000	0.846695	0.43420233	1.950000	0.950000	0.846692	0.43420119
1.975000	0.975000	0.857996	0.4344286	1.975000	0.975000	0.857985	0.43442298	1.975000	0.975000	0.857986	0.43442354
2.000000	1.000000	0.869040	0.43452	2.000000	1.000000	0.869025	0.4345127	2.000000	1.000000	0.869031	0.43451564

Plot:

Plot depicting that d3hsp data doesnot fit closely with experiment data.

However, d3hsp data fits closely for N=1 and N=3.

Simulation:

Animation:

Simulated data:

• True strain data = Mean Ipt stress data with time from LS DYNA
• Engineering strain = `e^(True strain)` - 1
• True stress = Effective stress (v-m) with time.
• Engineering stress = True stress/ (1+Engineering strain)

N=1				N=3				N=4
True stress (MPa)	True strain (mm/mm)	Engg strain (mm/mm)	Engg stress (MPa)	True stress (MPa)	True strain (mm/mm)	Engg strain (mm/mm)	Engg stress (MPa)	True stress (MPa)	True strain (mm/mm)	Engg strain (mm/mm)	Engg stress (MPa)
9.33E-10	0.000000	0	9.33293E-10	9.33E-10	0.000000	0	9.33293E-10	9.33E-10	0	0	9.33361E-10
4.57E-02	0.025360	0.025684372	0.044548722	4.57E-02	0.025360	0.025684372	0.044548722	0.045695	0.025360182	0.025684487	0.044550704
9.04E-02	0.050610	0.051912922	0.085906009	9.04E-02	0.050610	0.051912922	0.085906009	0.090368	0.050610457	0.051913048	0.08590789
1.34E-01	0.075724	0.078665061	0.124094955	1.34E-01	0.075724	0.078665061	0.124094955	0.133857	0.075724341	0.078665189	0.124095245
1.77E-01	0.100755	0.106005191	0.159751716	1.77E-01	0.100755	0.106005191	0.159751716	0.176684	0.100754716	0.106005323	0.159750121
2.18E-01	0.125651	0.133886696	0.192666902	2.18E-01	0.125651	0.133886696	0.192666902	0.218458	0.125651404	0.133886831	0.192662989
2.60E-01	0.150548	0.162471063	0.223866775	2.60E-01	0.150548	0.162471063	0.223866775	0.260232	0.150548086	0.162471202	0.223860646
3.00E-01	0.175174	0.191452994	0.251928032	3.00E-01	0.175174	0.191452994	0.251928032	0.300151	0.175173655	0.191453101	0.251920355
3.40E-01	0.199749	0.22109678	0.278227	3.40E-01	0.199749	0.22109678	0.278227	0.339731	0.199749514	0.221096853	0.278217929
3.79E-01	0.224325	0.251478114	0.303100611	3.79E-01	0.224325	0.251478114	0.303100611	0.379311	0.224325374	0.251478151	0.303090219
4.19E-01	0.248901	0.282615347	0.326602545	4.19E-01	0.248901	0.282615347	0.326602545	0.418891	0.248901248	0.282615366	0.326590946
4.58E-01	0.273565	0.314642762	0.348562165	4.58E-01	0.273565	0.314642762	0.348562165	0.45822	0.273564816	0.314642566	0.348550678
4.98E-01	0.298245	0.347491619	0.369217662	4.98E-01	0.298245	0.347491619	0.369217662	0.497503	0.298244476	0.347491177	0.36920657
5.37E-01	0.322925	0.381161309	0.388658934	5.37E-01	0.322925	0.381161309	0.388658934	0.536786	0.322924137	0.381160568	0.388648225
5.76E-01	0.347605	0.415672259	0.40693293	5.76E-01	0.347605	0.415672259	0.40693293	0.576069	0.347603798	0.415671247	0.406922611
6.15E-01	0.372334	0.451117362	0.423949157	6.15E-01	0.372334	0.451117362	0.423949157	0.615189	0.37233448	0.451118271	0.423941211
6.54E-01	0.397072	0.487463448	0.439866801	6.54E-01	0.397072	0.487463448	0.439866801	0.654279	0.397074491	0.487466728	0.439861383
6.93E-01	0.421811	0.524719893	0.454753386	6.93E-01	0.421811	0.524719893	0.454753386	0.69337	0.421814501	0.524725664	0.454750336
7.32E-01	0.446549	0.562909499	0.468649874	7.32E-01	0.446549	0.562909499	0.468649874	0.73246	0.446554482	0.562917836	0.468648938
7.71E-01	0.471333	0.602128928	0.481534518	7.71E-01	0.471333	0.602128928	0.481534518	0.771496	0.471345514	0.602148457	0.481538321
8.10E-01	0.496126	0.642346327	0.493496351	8.10E-01	0.496126	0.642346327	0.493496351	0.810522	0.496145874	0.642379121	0.493504861
8.50E-01	0.520918	0.683573282	0.504583559	8.50E-01	0.520918	0.683573282	0.504583559	0.849548	0.520946264	0.683620046	0.504596166
8.89E-01	0.545711	0.725835188	0.514831858	8.89E-01	0.545711	0.725835188	0.514831858	0.888574	0.545746565	0.725896395	0.514848032
9.28E-01	0.570580	0.769292848	0.524273532	9.28E-01	0.570580	0.769292848	0.524273532	0.927573	0.570560038	0.769257626	0.524272178
9.67E-01	0.595463	0.813870208	0.532940096	9.67E-01	0.595463	0.813870208	0.532940096	0.966566	0.595375896	0.813712583	0.532921047
1.01E+00	0.620346	0.859570695	0.540864006	1.01E+00	0.620346	0.859570695	0.540864006	1.005559	0.620191753	0.859284531	0.540831009
1.04E+00	0.645229	0.906422607	0.548076594	1.04E+00	0.645229	0.906422607	0.548076594	1.044552	0.64500761	0.906001534	0.548032995
1.08E+00	0.670147	0.954524152	0.55462539	1.08E+00	0.670147	0.954524152	0.55462539	1.08378	0.669969618	0.954177949	0.554596336
1.12E+00	0.695071	1.003852262	0.560522892	1.12E+00	0.695071	1.003852262	0.560522892	1.123051	0.694958389	1.003625699	0.560509464
1.16E+00	0.719996	1.054425432	0.565793916	1.16E+00	0.719996	1.054425432	0.565793916	1.162322	0.719947159	1.054324656	0.565792947
1.20E+00	0.744921	1.106274842	0.570465797	1.20E+00	0.744921	1.106274842	0.570465797	1.201594	0.74493593	1.106306479	0.570474264
1.24E+00	0.769874	1.159494348	0.574601154	1.24E+00	0.769874	1.159494348	0.574601154	1.240919	0.76991421	1.159580975	0.574611005
1.28E+00	0.794832	1.214070038	0.578192692	1.28E+00	0.794832	1.214070038	0.578192692	1.280254	0.794890583	1.214198712	0.578202055
1.32E+00	0.819791	1.27002499	0.581257992	1.32E+00	0.819791	1.27002499	0.581257992	1.319589	0.819866955	1.27019778	0.581266314
1.36E+00	0.844749	1.32739406	0.583820919	1.36E+00	0.844749	1.32739406	0.583820919	1.358925	0.844843328	1.327613114	0.583827535
1.40E+00	0.869805	1.386444973	0.585955616	1.40E+00	0.869805	1.386444973	0.585955616	1.398473	0.869887352	1.386641988	0.585958344
1.44E+00	0.894878	1.447037889	0.587636041	1.44E+00	0.894878	1.447037889	0.587636041	1.43806	0.894943833	1.447198335	0.587635266
1.48E+00	0.919952	1.509169137	0.588874053	1.48E+00	0.919952	1.509169137	0.588874053	1.477647	0.920000255	1.50929103	0.588870437
1.52E+00	0.945025	1.572877922	0.589690405	1.52E+00	0.945025	1.572877922	0.589690405	1.517235	0.945056736	1.572959355	0.589684622
1.56E+00	0.970446	1.639119962	0.590485341	1.56E+00	0.970446	1.639119962	0.590485341	1.558592	0.970602989	1.639535591	0.59047948
1.60E+00	0.995929	1.707239496	0.590935896	1.60E+00	0.995929	1.707239496	0.590935896	1.600273	0.996239066	1.708077751	0.590925845

Plot:

The simulated data is plotted.

The data doesnot fit closely with Test data. Niether for d3hsp nor simulated data.

CONCLUSION:

1. The stress for hyperelastic material doesnot only depends on strain. There are various other parameters which define the stress i.e strain rate, plasticity, temperature, amplitude and magnitude of strain, frequency of cyclic load etc.

2. The material behaviour of hyperelastic material is represented using strain energy density and stretch.

3. The Mooney-Rivlin material model is used to define the hyperelastic material represented by a polynomial of order 1. This model is represented as function of strain invariants. This model has two parameters for incompressible model.

4. The Ogden material model is used to define the hyperelastic material represented by multi degree polynomial. This model is represented as a function of principal streches. This model has parameters - mu and alpha. mu and alpha are real numbers for which stretches are raised.

5. Ogden model has advantage when it comes to fitting the test curves as parameters are rasied to power which are real number.

6. Comparison of Mooney Rivlin and Ogden Material model.