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Nonlinear material models

Which stress-strain curve to give the computer model, and what each one is for.

Principle 10 of 10 · 5 min read ·

Nonlinear material models: Hardening curvesThree stress-strain curves from the same start: the flat elastic-perfectly-plastic line, a bilinear line that keeps rising gently, and a smooth true curve that rises and slowly bends over.StrainStressFlat after yield: no hardeningUsed for limit-load collapse checksStrainStressTrue curveBilinearFlat: no hardeningFor elastic-plastic and strain checksσ1σ2von MisesTresca, insideMany stresses, one numberon the boundary, it yieldsStrainStressPast yield, then reversedKinematic: reverse yield soonerIsotropic: yield grows both waysDeflectionLoadLinearPlate under pressure: stiffensCompressed shell: softens
Figure 1. Hardening curves.
Contents

A nonlinear material model tells a finite element analysis how the metal behaves once it yields. The usual choices are elastic-perfectly-plastic, used for limit-load checks because it ignores strain hardening and so gives a cautious collapse load, and hardening curves, bilinear or true stress-strain, used where how far the metal stretches matters, such as elastic-plastic collapse and local strain checks. A yield criterion such as von Mises turns a multiaxial stress into one equivalent value, and a hardening rule says where yield sits when the load reverses.

This article goes through each choice, what it is for and what it can hide, and ends with geometric nonlinearity, the other switch that collapse and buckling checks usually need.

Elastic-perfectly-plastic

The simplest model is a straight elastic line up to yield, then flat: strain can grow with no further rise in stress. It needs only two numbers, the elastic modulus and a yield value, and it is robust in a solver.

Nonlinear material models: Elastic-perfectly-plasticStress against strain: a straight line up to yield, then a flat line running on with no further rise in stress.StrainStressFlat after yield: no hardeningUsed for limit-load collapse checksStrainStressTrue curveBilinearFlat: no hardeningFor elastic-plastic and strain checksσ1σ2von MisesTresca, insideMany stresses, one numberon the boundary, it yieldsStrainStressPast yield, then reversedKinematic: reverse yield soonerIsotropic: yield grows both waysDeflectionLoadLinearPlate under pressure: stiffensCompressed shell: softens
Figure 2. A straight line up to yield, then flat, with no rise in stress as strain grows.

Because it ignores strain hardening, the collapse load it gives is on the cautious side for steels that harden. Limit-load checks use it, with the yield value set by the governing code; that value is often a multiple of the allowable stress rather than the measured yield, so the margin is built into the material curve.

Hardening curves: bilinear and true stress-strain

Real steel gets stronger as it is stretched past yield. A bilinear model draws this as a second, shallower straight line, defined by a tangent modulus. A true stress-strain curve follows the measured shape, or a code's standard formula for it, smoothly bending over as strain grows.

Test data usually come as engineering stress and strain, based on the original cross-section and length. The solver needs true values, based on the current, shrinking section. Up to the start of necking, while the strain is spread evenly along the specimen, they convert as:

σ_true = s · (1 + e) ε_true = ln(1 + e)s, e = engineering stress and strain
Engineering and true stress-strain curvesTwo curves for the same illustrative steel up to the start of necking. The true curve, plotted against true strain, keeps rising. The engineering curve, based on the original cross-section, rises more slowly and reaches its peak where necking starts. A short arrow shows an engineering point at 10 percent strain and about 400 MPa moving up to the true point at about 440 MPa.2004006000.050.10.150.20.25neckingTrue curveEngineering curveStress, MPa (illustrative steel)Strain
Figure 3. The same illustrative steel plotted both ways, up to necking. The true curve keeps rising; the engineering curve peaks where necking starts.

Hardening curves are used for elastic-plastic collapse and local strain checks, where how far the metal stretches matters as much as when it first yields.

One number from many stresses

At any point the metal carries normal stresses in three directions as well as shears. A yield criterion turns them into one equivalent stress to compare with the single yield value from a tensile test. The von Mises criterion, based on the energy that changes the metal's shape, is the usual choice. Tresca, based on the largest shear stress, is slightly more cautious: for the same stresses it never gives a lower equivalent stress, and at most about 15 % higher.

von Mises: √(σ² + 3τ²) Tresca: √(σ² + 4τ²)one normal stress σ with one shear τ
Nonlinear material models: One number from many stressesTwo principal stresses plotted against each other. The von Mises boundary is an ellipse; the Tresca boundary is a hexagon that sits just inside it. Inside the boundary the metal is elastic; on it, it yields.StrainStressFlat after yield: no hardeningUsed for limit-load collapse checksStrainStressTrue curveBilinearFlat: no hardeningFor elastic-plastic and strain checksσ1σ2von MisesTresca, insideMany stresses, one numberon the boundary, it yieldsStrainStressPast yield, then reversedKinematic: reverse yield soonerIsotropic: yield grows both waysDeflectionLoadLinearPlate under pressure: stiffensCompressed shell: softens
Figure 4. Two principal stresses plotted against each other: the von Mises ellipse and the Tresca hexagon just inside it.

When loads cycle: hardening rules

Once the metal has yielded, the model must know where yield sits when the load reverses. Isotropic hardening grows the yield range evenly in both directions, so after stretching to a higher stress the metal must be pushed equally far the other way before it yields again. Kinematic hardening shifts the yield range instead, so stretching past yield lowers the yield in reverse, the Bauschinger effect seen in real steels. Combined models mix the two.

Nonlinear material models: When loads cycle: hardening rulesStress against strain, loaded past yield and then reversed. With kinematic hardening the reverse yield comes early; with isotropic hardening it comes only at the same stress as the peak, but negative.StrainStressFlat after yield: no hardeningUsed for limit-load collapse checksStrainStressTrue curveBilinearFlat: no hardeningFor elastic-plastic and strain checksσ1σ2von MisesTresca, insideMany stresses, one numberon the boundary, it yieldsStrainStressPast yield, then reversedKinematic: reverse yield soonerIsotropic: yield grows both waysDeflectionLoadLinearPlate under pressure: stiffensCompressed shell: softens
Figure 5. Loaded past yield and reversed: with kinematic hardening the reverse yield comes early; with isotropic hardening only at the same stress as the peak, but negative.

For cycling loads and ratcheting checks, kinematic hardening is closer to how steel behaves. Isotropic hardening can make later cycles look elastic and hide ratcheting, so it should be used for cyclic checks only where the governing code allows it. Codes may prescribe the model for these checks, in some cases elastic-perfectly-plastic.

Geometry is another nonlinearity

Material nonlinearity is about the metal; geometric nonlinearity is about the shape. When deflections grow large, the loads act on a changed shape. A flat plate under pressure stiffens as it bulges and starts to carry the load as a membrane, while a compressed shell softens and can buckle. Pressure that follows the deforming surface is part of the same effect.

Nonlinear material models: Geometry is another nonlinearityLoad against deflection. The small-deflection answer is a straight line; a plate under pressure curves above it as it stiffens, and a compressed shell curves below it and levels off.StrainStressFlat after yield: no hardeningUsed for limit-load collapse checksStrainStressTrue curveBilinearFlat: no hardeningFor elastic-plastic and strain checksσ1σ2von MisesTresca, insideMany stresses, one numberon the boundary, it yieldsStrainStressPast yield, then reversedKinematic: reverse yield soonerIsotropic: yield grows both waysDeflectionLoadLinearPlate under pressure: stiffensCompressed shell: softens
Figure 6. The small-deflection answer is a straight line; a plate under pressure curves above it, a compressed shell curves below and levels off.

The two are separate switches in the model. Collapse and buckling checks usually need both on: large displacements, so that the shape change is captured, and a yielding material, so that plasticity is.

Matching the model to the check

CheckMaterial modelGeometry
Limit loadElastic-perfectly-plastic, code yield valueSmall displacement
Elastic-plastic collapseTrue stress-strain with hardeningLarge displacement
Local strainTrue stress-strain with hardeningLarge displacement
RatchetingPerfectly plastic or kinematic hardeningAs the code directs
Buckling with imperfectionsYielding material, often with hardeningLarge displacement
Typical pairings. The governing code may prescribe the material curve, the yield value and the geometry setting for each check.

Key points

  • Elastic-perfectly-plastic is simple and cautious, and suits limit-load checks.
  • Hardening curves, bilinear or true stress-strain, suit collapse and local strain checks; convert test data to true values.
  • von Mises is the usual yield criterion; Tresca is slightly more cautious.
  • Kinematic hardening is closer to steel under cycling; isotropic hardening can hide ratcheting.
  • Geometric nonlinearity is a separate switch; collapse and buckling checks usually need it on.

Where this matters in practice

  • Limit-load checks of nozzles, openings and junctions in heads and shells.
  • Elastic-plastic collapse and local strain checks at heavily loaded lugs, supports and attachments.
  • Buckling checks of thin shells, cones and heads with an imposed imperfection.
  • Ratcheting checks at attachments and junctions under cycling loads.

Corix includes a limit-load analysis on a shell model, using the elastic-perfectly-plastic model with small displacements. In early access, by invitation.

See the specialty module

Related principles

Next principle: Local loads on pressure vessel shells

Educational article. It explains principles and is not design guidance; follow your governing code and edition.