Guide to Nonlinear Material Models

Your linear FEA results say the bracket is fine. The prototype just cracked in testing. What went wrong?

Chances are, the part entered the plastic range. While your linear model flagged the high stress, it couldn’t tell you what happens next.

Does the part redistribute load and survive? Does it deform permanently but still function? Or does it fail catastrophically?

To answer those questions, or to model materials like rubber and polymers that are inherently nonlinear, you need nonlinear material models.

Linear elastic analysis assumes constant stiffness and infinitely small deformations. For a lightly loaded component well below yield, that’s perfectly adequate. But the moment your design sees real-world loads; metals yielding, rubber seals stretching 300%, polymers creeping at temperature, linear analysis quietly stops telling the truth.

This guide cuts through the textbook theory and gives you the practical decision framework for nonlinear material models in FEA: which model to pick, what data it requires, and to build models that will reliably converge. Geometric nonlinearity (large displacements) and contact nonlinearity (changing boundary conditions) are separate topics, though real problems often combine all three.

The Four Nonlinear Material Model Types

There are four broad categories of material model in nonlinear analysis. Understanding the abilities of each one is key to making the right selection for the material you’re dealing with.

1. Elastoplastic Models

The workhorse of nonlinear FEA. These describe materials that deform elastically up to a yield point, then undergo permanent plastic deformation. The fundamental relationship is:

where elastic strain recovers on unloading and plastic strain is permanent. The yield point is typically defined by the von Mises criterion:

SimScale supports several variants of non-linear structural analysis through its Static and Dynamic solvers (using the Code_Aster solver) and Nonlinear Mechanical (Marc) analysis types: elastic-perfectly plastic, bilinear hardening, multilinear hardening, and Johnson-Cook (rate/temperature-dependent).

Best for: Structural steels, aluminum alloys, titanium, copper. See the elasto-plastic materials documentation for setup details and Modeling Inelasticity: Plasticity and Creep for a deeper dive on the fundamentals.

2. Hyperelastic Models

For materials that undergo huge elastic deformations (100–700% strain) and fully recover. Defined by a strain energy density function $W$ rather than a simple modulus. For example, the Neo-Hookean model uses:

where $C_1$ is the shear parameter, $\bar{I}_1$ is the first deviatoric strain invariant, $J$ is the volume ratio, and $D_1$ controls compressibility. Higher-order models like Mooney-Rivlin and Ogden add additional terms to capture the S-shaped stress upturn at large stretches.

SimScale supports Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh/Signorini, and Marlow (no calibration needed — Marc solver only).

Best for: Rubber seals, O-rings, elastomeric bushings, silicone, foams, biological tissue. See How to Choose a Hyperelastic Material Model.

3. Viscoelastic and Viscoplastic Models

Time-dependent behavior: stress response depends on strain rate and duration, not just magnitude. The most common structural application is creep — progressive deformation under constant load at elevated temperature, often modeled with a Norton-Bailey power law:

where $A$ and $n$ are material constants, $Q$ is activation energy, $R$ is the gas constant, and $T$ is absolute temperature. SimScale supports creep modeling for high-temperature components.

Best for: Polymers, high-temperature metals (turbine blades, exhaust manifolds), solder joints, sustained-load applications.

4. Damage and Failure Models

Track progressive stiffness degradation leading to fracture. Johnson-Cook in SimScale can be extended with damage criteria for crash, impact, and composite failure scenarios.

Best for: Crash/impact events, progressive failure assessment, composite damage, fatigue crack growth.

Elastoplastic vs. Hyperelastic: How to Decide

Choosing the right model starts with your material, not the deformation. Here’s a practical decision path:

1. What’s your material group?
   • Metals → linear elastic or elastoplastic (depending on whether you need to predict post-yield behavior).
   • Rubber/elastomers → hyperelastic, unless deformation is very small.
   • Thermoplastics → viscoplastic (or elastoplastic for short-duration loads).
2. For metals: do you need to understand post-yield behavior, or just confirm you’re below yield? If the latter, linear elastic + yield criterion may be enough.
3. For rubber-like materials: are deformations large enough that a linear stiffness assumption breaks down? → hyperelastic.
4. Time or temperature effects? Creep, stress relaxation, rate-dependence → viscoelastic or viscoplastic.
5. Do you have test data? Higher-order models need more calibration data.

	Elastoplastic	Hyperelastic
Deformation	Permanent above yield	Fully recoverable
Strain range	0.1–20%	10–700%+
Stiffness	Young’s modulus + hardening slope	Strain energy density function
Unloading	Returns along elastic slope, leaves residual strain	Returns to original shape
Typical materials	Steel, aluminum, titanium	Rubber, silicone, elastomers, foams

Model Selection Quick Reference

Application	Materials	Recommended Model
Structural overload	Steel, aluminum	Bilinear / multilinear elastoplastic
Metal forming	Sheet metal	Multilinear or Johnson-Cook
Rubber seal / bushing	Elastomers, silicone	Mooney-Rivlin, Ogden, or Marlow
Snap-fit assembly	ABS, nylon	Multilinear elastoplastic
High-temp piping	Stainless, Inconel	Elastoplastic + creep
Crash / impact	Metals, foams	Johnson-Cook + damage

Choosing Your Model Parameters

Once you’ve identified the material type, three further decisions shape your simulation setup.

1. Which Yield Criterion?

The yield criterion determines when the material transitions from elastic to plastic. For metals, this is almost always von Mises. You only need alternatives for pressure-dependent materials:

Criterion	Best For	When to Use
von Mises	Ductile metals (steel, aluminum, copper)	Default for engineering metals. Yielding depends on distortional energy.
Tresca	Ductile metals (conservative)	~15% more conservative than von Mises. Required by some pressure vessel codes.
Mohr-Coulomb	Soils, concrete, rock	Shear strength depends on normal stress. Standard for geotechnical work.
Drucker-Prager	Soils, concrete, polymers, foams	Smooth 3D version of Mohr-Coulomb. Easier FEA implementation.

2. Bilinear or Multilinear Hardening?

Bilinear uses a single post-yield slope defined by three parameters $(E), (\sigma_y), (E_T)$. Fast to set up, good for initial sizing. Multilinear traces the actual stress-strain curve with multiple data points — always more accurate when test data is available.

Aspect	Bilinear	Multilinear
Parameters	EE E, σy\sigma_y σy​, ETE_T ET​ (3 values)	EE E + *n* stress-strain data points
Accuracy	Good for small plastic strains	Follows actual material curve
Setup time	Minutes	Moderate (need test data)
Best when…	Quick check, limited data	Accuracy matters, data available

Quick estimate when you only have yield and UTS:

For mild steel, $E_T$ is typically 1–2 GPa compared to $E \approx 200$ GPa.

3. Which Hardening Rule?

This governs how the yield surface evolves with plastic strain — critical for cyclic loading accuracy:

Rule	Yield Surface Behavior	Use When
Isotropic	Expands uniformly	Monotonic loading (single overload)
Kinematic	Translates (shifts position)	Cyclic loading, seismic, fatigue
Combined	Expands + translates	Complex histories, accurate cyclic modeling

Rule of thumb: load applied once → isotropic. Load reversals (bolt tightening/loosening, pressure cycling, seismic) → kinematic or combined. Getting this wrong gives non-conservative cyclic results.

Getting the Material Data Right

A nonlinear model is only as good as its input data.

For metals: the uniaxial tensile test (ASTM E8 / ISO 6892) gives you everything – $(E), (\sigma_y), UTS$​, and the full stress-strain curve. The critical step is converting to true stress and true strain:

These formulas are valid up to the onset of necking. SimScale’s documentation provides detailed conversion guidance.

For rubber/elastomers: you need multiple test modes (uniaxial, equibiaxial, shear) because hyperelastic behavior varies with loading mode. Rule of thumb: at least as many independent tests as model parameters. A comprehensive comparison of 85 hyperelastic constitutive models by Dal et al. (2021) provides useful guidance on which strain energy functions best fit different rubber compounds [1].

No test data? Use material databases (MatWeb, MMPDS, CES EduPack) or SimScale’s built-in library. Just verify alloy designation, temper condition, and temperature range — database values are typical properties, not guaranteed minimums.

When Do You Actually Need Nonlinear Materials?

Now that you know the model types, here’s a quick decision filter for whether you need them at all:

Indicator	Linear Is Fine	Go Nonlinear
Max stress vs. yield	Well below yield (FoS > 2)	Near, at, or above yield
Material class	Metals under light loads	Rubber, elastomers, polymers, foams
Strain magnitude	< 1–2%	> 2–5% or material-dependent
Design function	Stiffness-driven	Energy absorption, forming, sealing
Code compliance	Allowable stress only	Plastic analysis (ASME, Eurocode, AISC)
Loading	Static, monotonic	Cyclic, impact, path-dependent

The trade-off is compute time: nonlinear runs take 5–20× longer than linear. Cloud platforms like SimScale (up to 192 cores per run) shrink that gap significantly.

Getting It to Converge

Nonlinear problems require iterative solvers (typically Newton-Raphson), which means convergence isn’t guaranteed. Here’s how to fix the most common failures:

Problem	Likely Cause	Fix
Divergence at yielding	Step too large across elastic-plastic transition	Reduce step size; use automatic stepping
Oscillation without converging	Perfect plasticity $(E_T = 0)$	Add small tangent modulus; increase max iterations
Element distortion errors	Excessive strain in coarse mesh	Refine mesh in plastic zones
Slow convergence (rubber)	Volumetric locking from incompressibility	Reduced integration elements; check D1​ parameter (Code_aster) or bulk modulus (Marc)
Non-convergence with contact	Combined nonlinearities	Smaller load steps; verify contact; use Marc auto-detect

Design Code Limits

A critical question in any nonlinear analysis: how much plastic strain is acceptable? Design codes provide the answer:

Code	Application	Allowable Plastic Strain	Notes
ASME BPVC Sec. VIII Div. 2	Pressure vessels	5% local membrane + bending	Triaxiality and forming strain considered
EN 1993-1-6	Steel shells	Up to 5% equivalent	Buckling interaction must also be checked
DNV-RP-C208	Offshore structures	2–5% depending on criticality	Fracture mechanics screening above 2%
AISC 360	Steel buildings	Implicit via shape factor	Full plastification of compact sections allowed

These are design margins, not material failure limits. They account for fabrication imperfections, residual stresses, and fracture toughness. Exceeding them doesn’t mean immediate failure — it means the design no longer meets code safety requirements.

Getting Started with SimScale

Setting up a nonlinear material analysis follows five steps:

1. Upload your CAD and choose Nonlinear Static (Code_Aster) or Nonlinear Mechanical (Marc).
2. Define your material — select from the library or enter custom parameters / stress-strain data.
3. Set up contacts, BCs, and loads — Marc offers automatic contact detection for assemblies.
4. Configure mesh and numerics — refine where plastic strains are expected; adjust load stepping as needed.
5. Run and post-process — von Mises stress, plastic strain, displacements, and reaction forces available in the browser.

Start from a template: try the snap-fit assembly tutorial or the FSAE impact attenuator crash test.

Related Research Papers

1. Dal, H., Açıkgöz, K., & Badienia, Y. (2021). “On the performance of isotropic hyperelastic constitutive models for rubber-like materials: A state of the art review.” Applied Mechanics Reviews, 73(2).
2. Simo, J. C., & Taylor, R. L. (1985). “Consistent tangent operators for rate-independent elastoplasticity.” Computer Methods in Applied Mechanics and Engineering, 48(1), 101–118.
3. Simo, J. C., & Armero, F. (1992). “Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes.” International Journal for Numerical Methods in Engineering, 33(7), 1413–1449.
4. Chaboche, J. L. (1986). “Time-independent constitutive theories for cyclic plasticity.” International Journal of Plasticity, 2(2), 149–188.
5. Ogden, R. W. (1972). “Large deformation isotropic elasticity — on the correlation of theory and experiment for incompressible rubberlike solids.” Proceedings of the Royal Society of London A, 326(1567), 565–584.
6. Bathe, K. J. (1996). Finite Element Procedures. Prentice Hall.

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