Reference name: MakeCoTech Hyperelastic Explorer v1.0 (MakeCoTech, 2026). Use this name and version when documenting results generated with the tool.
Stretch λ1.000
Engineering strain0.0 %
Selected nominal stress0.000 MPa
Selected strain energy density0.000 MJ/m³
Educational / engineering exploration use only.
This tool is provided for learning, preliminary material-model evaluation and engineering comparison.
Hyperelastic model fitting can be sensitive to test quality, loading modes, parameter identifiability, extrapolation and solver implementation.
The user is responsible for independently validating all fitted parameters and material behavior before use in design, safety-critical work or production FEA.
MakeCoTech provides no warranty that a fitted model will remain stable, accurate or suitable outside the calibrated range and accepts no responsibility for losses, damage, analysis failure or material-model failure resulting from use of the tool or its outputs.
Stretch range
Per loading mode
Display range controls the graph, slider, generated curves and Excel curve range. Fit range filters experimental points only when “Restrict fit” is checked. “Use Experimental Range” reads the currently selected test data and sets both ranges automatically.
Nominal Stress vs Stretch
Stored Strain Energy Density
Uniaxial Specimen
Incompressible: lateral stretches = λ⁻¹ᐟ².
Material Configuration
In incompressible mode, K0 and the Abaqus volumetric parameter are inactive.
E remains editable as the reference small-strain modulus. For J = 1, matching uses μ = E/3 for Neo-Hookean and for the one-term Ogden convention used here. For compressible behavior, μ is matched using E = 9K0μ/(3K0+μ). The Excel export contains original inputs, converted inputs, all loading-mode curves, fit results and charts.
Ogden convention used in this tool (Abaqus-style normalization)
Uiso = (2μ/α²) Σ(λ̄ᵢα − 1), therefore G₀ = μ for one term and, for J = 1, E₀ = 3μ. The exponent α controls the nonlinear curve shape; it does not multiply the initial shear modulus in this normalization. Positive or negative α values are allowed; the fitter excludes only a narrow neighborhood around α = 0 for numerical conditioning. Alternative Ogden formulations use different prefactors, so their parameter called μ is not numerically equivalent.
Uiso = (2μ/α²) Σ(λ̄ᵢα − 1), therefore G₀ = μ for one term and, for J = 1, E₀ = 3μ. The exponent α controls the nonlinear curve shape; it does not multiply the initial shear modulus in this normalization. Positive or negative α values are allowed; the fitter excludes only a narrow neighborhood around α = 0 for numerical conditioning. Alternative Ogden formulations use different prefactors, so their parameter called μ is not numerically equivalent.
Ogden Hyperfoam / Abaqus HYPERFOAM (N=1)
This is a separate, inherently compressible foam law—not the ordinary compressible Ogden model with D₁. It uses the actual principal stretches and couples distortional and volumetric response: U = (2μ₁/α₁²)[Σ λᵢα₁ − 3 + (1/β₁)(J−α₁β₁ − 1)], with β₁ = ν₁/(1−2ν₁). It has no D or D₁ parameter.
This is a separate, inherently compressible foam law—not the ordinary compressible Ogden model with D₁. It uses the actual principal stretches and couples distortional and volumetric response: U = (2μ₁/α₁²)[Σ λᵢα₁ − 3 + (1/β₁)(J−α₁β₁ − 1)], with β₁ = ν₁/(1−2ν₁). It has no D or D₁ parameter.
Current initial slope3.000 MPa
Kinematicsλ₂ = λ₃ = 1.000
Current modelNeo-Hookean
Formulas & Loading Modes
Incompressible: J = 1. Transverse stretches follow the exact incompressible kinematics.
Compressible hyperelastic models: most models use a separate Abaqus D/D1 volumetric term. Hyperfoam is different: its volumetric and distortional response are coupled directly in the HYPERFOAM potential and no D/D1 is entered.
Uniaxial: λ₁=λ. For compressible behavior, λ₂=λ₃ are solved from zero transverse traction.
Equibiaxial: λ₁=λ₂=λ. The thickness stretch λ₃ is solved from zero transverse traction.
Planar / pure shear: λ₁=λ, λ₂=1. The thickness stretch λ₃ is solved from zero transverse traction.
Simple shear: F₁₂=γ and J=1, so the volumetric D1 term does not contribute.
Mooney–Rivlin identifiability in simple shear: for incompressible simple shear, I₁ = I₂ = 3 + γ². Therefore the full Mooney–Rivlin energy reduces exactly to W = (C10 + C01)γ² and τ = 2(C10 + C01)γ. Simple-shear data alone can identify only the sum C10 + C01, not C10 and C01 independently.
Mooney–Rivlin identifiability in planar / pure shear: with λ₁=λ, λ₂=1, λ₃=λ⁻¹, I₁ = I₂ = λ² + 1 + λ⁻². Planar data alone therefore also constrains only C10 + C01. Combining uniaxial and/or equibiaxial data is needed to separate the two parameters.
Volumetric: J is prescribed. For the quadratic Abaqus volumetric term, compression pressure p = K0(1 − J). The sign convention used here is pressure positive in compression: J<1 → p>0, J=1 → p=0, J>1 → p<0.
Isochoric Neo-Hookean: Uiso = ½μ(Ī₁ − 3) = C10(Ī₁ − 3), with μ = 2C10.
Isochoric Mooney–Rivlin: Uiso = C10(Ī₁ − 3) + C01(Ī₂ − 3).
Isochoric one-term Ogden: Uiso = (2μ/α²)(λ̄₁^α + λ̄₂^α + λ̄₃^α − 3).
Arruda–Boyce: Abaqus eight-chain form using μ and locking stretch λm, with a fifth-order series in Ī₁. The Abaqus parameter μ is not exactly the initial shear modulus; μ₀ = μ[1 + 3/(5λm²) + 99/(175λm⁴) + 513/(875λm⁶) + 42039/(67375λm⁸)].
Yeoh: Uiso = C10(Ī₁−3) + C20(Ī₁−3)² + C30(Ī₁−3)³. Initial shear modulus μ₀ = 2C10.
Reduced Polynomial N=2: Uiso = C10(Ī₁−3) + C20(Ī₁−3)². Initial shear modulus μ₀ = 2C10.
Van der Waals / Kilian: Abaqus form based on Ĩ = (1−β)Ī₁ + βĪ₂, locking stretch λm, initial shear modulus μ, global interaction parameter a and mixture parameter β.
Ogden Hyperfoam / Abaqus HYPERFOAM, N=1: U = (2μ₁/α₁²)[λ₁^α₁ + λ₂^α₁ + λ₃^α₁ − 3 + (1/β₁)(J^(−α₁β₁) − 1)]. The law is inherently compressible and uses the total mechanical principal stretches, not isochoric stretches. β₁ = ν₁/(1−2ν₁). For N=1, G₀ = μ₁ and K₀ = 2μ₁(β₁ + 1/3). No D/D₁ parameter is used.
Abaqus-style calibration convention: non-volumetric hyperelastic coefficients are fitted under full incompressibility; volumetric pressure–volume data identify D1 or D separately. The fitting objective can reproduce Abaqus' relative stress-error measure or use ordinary absolute least squares.
Abaqus-family scope in this version: coefficient-based Neo-Hookean, Mooney–Rivlin, Ogden (1-term), Arruda–Boyce, Yeoh, Reduced Polynomial N=2, Van der Waals and Abaqus Ogden Hyperfoam N=1 are implemented. Marlow and Valanis–Landel are test-data-defined forms and are intentionally left for a later dedicated implementation; Hencky is also not included in the fitter.
Ogden parameter convention used here: this is the Abaqus-style normalization in which the initial shear modulus is G₀ = Σμᵢ. For one term, G₀ = μ, so for an incompressible material E₀ = 3μ. The exponent α changes the nonlinear shape but does not change the initial shear modulus in this convention. Do not mix this μ with alternative Ogden normalizations that use a different prefactor.
Compressibility does not require FEA here.
These tests are homogeneous, so the tool evaluates the constitutive energy directly. For uniaxial, biaxial and planar loading it numerically solves the free transverse stretch before calculating the nominal stress.
Linear elastic curve: this is a small-strain reference only. It is deliberately extrapolated on the same axes for comparison and should not be interpreted as a finite-strain constitutive law.
In compressible mode, axial Cauchy-stress input is intentionally not offered because converting Cauchy stress to nominal stress requires J. Without measured lateral strain/volume change, J is not known from the axial data alone.
Fit-quality warning: a low RMSE or high R² does not guarantee constitutive stability outside the calibration range. Fitted parameters should be checked for stability and sensible extrapolation before use in large-strain FEA.
W is strain-energy density. With stress parameters in MPa, W is numerically in MPa = MJ/m³.
These tests are homogeneous, so the tool evaluates the constitutive energy directly. For uniaxial, biaxial and planar loading it numerically solves the free transverse stretch before calculating the nominal stress.
Linear elastic curve: this is a small-strain reference only. It is deliberately extrapolated on the same axes for comparison and should not be interpreted as a finite-strain constitutive law.
In compressible mode, axial Cauchy-stress input is intentionally not offered because converting Cauchy stress to nominal stress requires J. Without measured lateral strain/volume change, J is not known from the axial data alone.
Fit-quality warning: a low RMSE or high R² does not guarantee constitutive stability outside the calibration range. Fitted parameters should be checked for stability and sensible extrapolation before use in large-strain FEA.
W is strain-energy density. With stress parameters in MPa, W is numerically in MPa = MJ/m³.
Experimental Curve Fitting
Each loading mode stores its own 30-point table. Paste two columns directly from Excel. Use the checkboxes to decide which datasets participate in a combined fit.
Which test data should be used for fitting?
The checkboxes below control which experimental datasets are included together when you press Fit Selected Model or Fit All Models. The dataset selector above only chooses which 30-point table you are currently viewing/editing. Unticking a test excludes it from the fit without deleting its entered data. In compressible mode, volumetric data can identify K0/D1 independently.
The checkboxes below control which experimental datasets are included together when you press Fit Selected Model or Fit All Models. The dataset selector above only chooses which 30-point table you are currently viewing/editing. Unticking a test excludes it from the fit without deleting its entered data. In compressible mode, volumetric data can identify K0/D1 independently.
Choose an example for the material family:
Hyperelastic benchmark: Abaqus SI Hypertest rubber data: 23 uniaxial, 16 biaxial, 13 planar and 4 volumetric points, displayed in MPa.
Hyperfoam demo: synthetic data generated by the one-term Ogden Hyperfoam model with μ₁ = 0.5 MPa, α₁ = 2 and ν₁ = 0.20. Loads all five test modes and selects Hyperfoam with compressible behavior. Use these generated data to practice fitting; supplier or measured material data are required for material identification.
Hyperelastic benchmark: Abaqus SI Hypertest rubber data: 23 uniaxial, 16 biaxial, 13 planar and 4 volumetric points, displayed in MPa.
Hyperfoam demo: synthetic data generated by the one-term Ogden Hyperfoam model with μ₁ = 0.5 MPa, α₁ = 2 and ν₁ = 0.20. Loads all five test modes and selects Hyperfoam with compressible behavior. Use these generated data to practice fitting; supplier or measured material data are required for material identification.
| # | Stretch λ | Stress (MPa) |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| 8 | ||
| 9 | ||
| 10 | ||
| 11 | ||
| 12 | ||
| 13 | ||
| 14 | ||
| 15 | ||
| 16 | ||
| 17 | ||
| 18 | ||
| 19 | ||
| 20 | ||
| 21 | ||
| 22 | ||
| 23 | ||
| 24 | ||
| 25 | ||
| 26 | ||
| 27 | ||
| 28 | ||
| 29 | ||
| 30 |
Paste from Excel: copy deformation + stress columns, click the first light-blue cell, then Ctrl+V. Blank rows are ignored. Decimal commas are accepted.
Enter or paste experimental data. The active loading-mode data will appear as purple markers on the stress plot.
| Model | Fitted parameters | RMSE | R² |
|---|---|---|---|
| Linear elastic — small-strain reference | — | — | — |
| Neo-Hookean | — | — | — |
| Mooney–Rivlin | — | — | — |
| Ogden (1-term, Abaqus norm.) | — | — | — |
| Arruda–Boyce | — | — | — |
| Yeoh | — | — | — |
| Reduced Polynomial N=2 | — | — | — |
| Van der Waals | — | — | — |
| Ogden Hyperfoam N=1 | — | — | — |
Combined fitting uses only the checked datasets. Abaqus relative stress error minimizes Σ(1 − Tmodel/Ttest)², matching the Abaqus calibration objective; exact zero-stress points are omitted from that relative objective because the denominator is undefined, but they remain in RMSE/R² reporting. Standard least squares minimizes absolute stress residuals. For the standard hyperelastic coefficient models, this tool follows the Abaqus convention of assuming full incompressibility for the non-volumetric tests and fitting volumetric data separately to K0/D or D1. Ogden Hyperfoam is treated separately as an inherently compressible coupled law; if volumetric data are checked, those data participate directly in the Hyperfoam fit. For incompressible stretch modes, Cauchy stress is converted to nominal stress using P = σ/λ. Simple shear and planar / pure shear each identify only C10 + C01 for Mooney–Rivlin; uniaxial and/or equibiaxial data are required to separate C10 and C01. Fit All keeps K0 fixed so the deviatoric models can be compared consistently. Abaqus performs a full Drucker stability check; this browser tool does not reproduce the full tangent positive-definiteness test, so fitted parameters should still be checked in Abaqus before production FEA.