Nonlinear Elastic Materials

As opposed to hyperelastic materials, where the stress-strain relationship becomes significantly nonlinear at moderate to large strains, nonlinear elastic materials present nonlinear stress-strain relationships even at infinitesimal strains.

Here, nonlinear effects on the strain tensor are not as relevant as the nonlinearity of the elastic properties. Important materials of this class are Ramberg–Osgood for modeling metal and other ductile materials, and the Duncan–Chang soil model.

The nonlinear elastic materials as such do not include strain-rate nor stress-rate in the constitutive equations. It is however possible to add linear viscoelasticity to these materials.

For a nonlinear material to be “energetically sound” it should be possible to take any path in stress-strain space and return to the undeformed state without producing or dissipating any net energy. A requirement is then that the bulk modulus depends only on the volumetric strain, and the shear modulus depends only on the shear strains.

The splitting into volumetric and deviatoric components of the stress tensor helps ensuring the “path independent” restriction for isotropic nonlinear elastic materials.

For isotropic linear elastic materials, the stress tensor follows Hooke’s law:

For a more detailed discussion, see Equation 3-11.

It is possible to split the stress and elastic strain tensors into the deviatoric and volumetric contributions

and

Assuming only elastic stresses in linear isotropic elastic medium, Hooke’s law simplifies to

where K is the bulk modulus and G is the shear modulus. By using the convention that the pressure is the mean stress defined as positive in compression,

The volumetric strain (positive in tension) is

The linear relation between pressure and volumetric elastic strain is thus

The deviatoric stress and deviatoric elastic strain tensors are related by the shear modulus

By using the contraction of the deviatoric stress and strain tensors, we can alternatively relate the invariants of these tensors

For a body subject to pure torsion on the plane 12, the stress tensor components are zero except the shear stress σ12 = σ21 = τ, and also the elastic strain tensor has zero components beside the shear strains on that plane ε12 = ε21 = γel/2.

We can then write

and

The shear stress on the plane

by the shear modulus

For nonlinear elastic materials, there is a nonlinear relation between shear stress and shear strain and/or a nonlinear relation between pressure and volumetric strain.

For the purpose of this discussion,

and

are used alternatively as variables.

In the most general case:

The tangent shear modulus Gt(γel) and the secant shear modulus Gs(γel) in the most general case depend nonlinearly on the shear strain, and are defined as

Note that the secant modulus is sometimes called the chord modulus between zero and current strain level.

The tangent bulk modulus Kt(εel,vol) and the secant bulk modulus Ks(εel,vol) depend on the elastic volumetric strain, and are defined as

For linear elastic materials, it is clear that Gt = Gs = G and Kt = Ks = K, but this is not the case for nonlinear elastic materials.

At zero strain, the secant and shear moduli are equal to each other Gs(0) = Gt (0) and Ks(0) = Kt (0).

The nonlinear elastic materials described in the next sections are represented by introducing nonlinear secant shear and/or bulk moduli.

The nonlinear elastic material models are primarily intended for small strain analysis. When used in a geometrically nonlinear study step, the strains will be interpreted as Green–Lagrange strains and the stresses will be interpreted as second Piola–Kirchhoff stresses. This is relevant for a situation with large rotations but small strains. If the strains become larger than a few percent, then you must be careful when interpreting input parameters and results since the strain and stress tensors also have a nonlinear dependence on the displacements.

The Ramberg–Osgood material model (Ref. 1) is a nonlinear elastic material commonly used to model plastic deformation in metals, but it also often used in soil engineering. As it is an elastic model, it can only represent plasticity during pure on-loading conditions.

For uniaxial extension, the stress-strain curve is defined by the expression

Here, E means the initial Young’s modulus, and εref is the strain at a reference stress σref. The parameter n is the stress exponent. It is common to use εref = 0.002, so σref is the stress at 0.2% strain, typically denoted by the symbol σ0.2. This parameter has several names depending on the literature: 0.2% offset yield strength, 0.2% proof stress, 0.2% proof strength, or 0.2% yield stress. Typical values for stainless steel are E = 200 GPa, σ0.2 = 600 GPa, and n = 4.8.

The linear strain is given by

and the nonlinear strain by

The total strain is the sum of linear and nonlinear strains

In order to avoid a circular dependence of internal variables, the nonlinear strain εnl is defined with an auxiliary degree of freedom, so the stress reads σ = E(ε − εnl).

In soil engineering, it is common to write the Ramberg–Osgood material with the stress-strain expression

so at the reference stress σref, the strain is ε = (1 + α)σref/E. It is common to use α = 3/7, so σref represents the stress level at which the secant Young’s modulus has been reduced to 70% its initial value: E0.7 = E/(1 + α) = 0.7E. At this reference stress the strain is ε = σref/E0.7.

For this type of material, the shear stress is related to the elastic shear strain γ by the strain exponent n and a reference shear strain γref (Ref. 2)

The secant shear modulus is given by the power law relation

The strain exponent controls the nonlinear deformation:

The most commonly mentioned model of “bilinear elastic” material is defined with two different bulk moduli for either tension and compression. Commonly, brittle materials like graphite and ceramics exhibit this behavior. The secant bulk modulus reads:

and

where εel,vol is the volumetric strain, Kc is the bulk modulus for compression, and Kt the bulk modulus for tension.

Many nonlinear stress-strain curves are measured in a tensile test, for which a nonlinear curve of force vs displacement is obtained.

If only the uniaxial behavior is measured, the measurements do not fully define the material behavior. An extra assumption is needed. The Uniaxial data material model allows you to assume either a constant Poisson’s ratio, or a constant bulk modulus. Also, if only uniaxial extension data is available, further assumptions are needed for covering the uniaxial compressive behavior of the material.

For the uniaxial tensile test, the axial stress corresponds to the principal stresses σax = σ1 = σmises, and the other two principal stresses are equal to zero, σ2 = σ3 = 0.

The principal (axial) strain is positive in tension, εax = ε1, and the other two (transverse) strains are negative and related by the Poisson’s ratio ε2 = ε3 = −νε1.

For uniaxial compression, the axial strain is negative, and when the principal strains are sorted as ε1 > ε2 > ε3 it corresponds to the third principal strain, εax = ε3. The other two (transverse) strains are positive and related by the Poisson’s ratio ε1 = ε2 = −νε3. Also, the axial stress is negative in compression, and it corresponds to the third principal stress σax = σ3 = −σmises. The other two transverse stresses are zero σ1 = σ2 = 0.

Other strain measures that can be obtained from the elastic strain tensor or its principal values, are the elastic volumetric strain

and the elastic shear strain,

These are used to define the elastic axial strain variable for multiaxial loading.

The uniaxial test defines the relation between the axial stress and elastic axial strain as

Here, Es is the secant Young’s modulus, and the axial stress σax is considered as a function of the elastic axial strain εax. Thus

At zero strain, the secant Young’s modulus is defined as

Assuming a constant Poisson’s ratio, the secant shear modulus is defined as

and the secant bulk modulus as

Furthermore, if only tensile stress-strain data is available, the elastic axial strain for multiaxial loading is computed from the elastic shear strain γ and Poisson’s ratio ν as

When nonsymmetric stress-strain data is available, the elastic axial strain for multiaxial loading is computed from the elastic volumetric strain and Poisson’s ratio

as this expression captures the change of sign in the elastic axial strain when changing from a tensile to a compressive state.

When using a constant bulk modulus assumption, only the symmetric part from the stress-strain data is considered. The secant shear modulus is instead defined as

and the elastic axial strain is defined from both volumetric and shear elastic strains

It is possible to use any uniaxial data function to define the axial stress as a function of elastic axial strain

provided that

	The elastic axial strain εax can be called in user defined uniaxial stress functions by referencing the variables solid.eax, where solid is the name of the physics interface node. See also the description of the Uniaxial Data material model in the Solid Mechanics interface documentation.

Many nonlinear stress-strain curves are measured in a shear test, for which a nonlinear curve of force vs displacement is obtained.

If only the shear behavior is measured, the measurements do not fully define the material behavior and therefore an extra assumption is needed. The Shear data material model assumes a constant bulk modulus.

Other strain measures that can be obtained from the strain tensor or its principal values, are the elastic volumetric strain

and the elastic shear strain,

These are used to define the nonlinear stress-strain relation for multiaxial loading.

From the shear test one could define the relation between the shear stress τ and the elastic shear strain γ as

Here, the secant shear modulus Gs is constant for Linear Elastic materials, but in general one could use a nonlinear relation. It is possible to use shear data to define the shear stress as a function of elastic shear strain as follows:

provided that

in which case, the secant shear modulus is computed from

With the help of the secant shear modulus Gs computed from shear data, Hooke’s law simplifies to

where K is the bulk modulus.

	The elastic shear strain γ can be called in user defined shear stress functions by referencing the variables solid.esh, where solid is the name of the physics interface node. See also the description of the Shear Data material model in the Solid Mechanics interface documentation.

A hyperbolic relation between shear stress and shear strain is obtained by setting the secant shear modulus

where the strain exponent n and a reference shear strain γref control the shape of the hyperbola.

For hyperbolic material models, the maximum shear modulus occurs at zero shear strain, so practitioners might call G the “maximum shear modulus” and use the notation Gmax. Sometimes it is also called “small strain shear modulus”.

The Hardin–Drnevich model (Ref. 3) is a hyperbolic soil model (with n = 1) defined by two input parameters: the initial shear modulus G and a reference shear strain γref

This nonlinear soil model is commonly used for modeling soil dynamics in earthquake engineering problems.

Since τ = Gsγ, the shear stress is bounded by τmax = Gγref as the shear strain increases.

The hyperbolic Hardin–Drnevich model is normally used for quantifying stiffness reduction curves in soils. Commonly, the reference shear strain γref is replaced by the reference shear strain at which the secant shear modulus has been decreased to 70% its initial value. Calling this shear strain value γ0.7, the reference strain is written as

and the secant shear modulus as

so that when γ = γ0.7 the secant shear modulus is Gs = 0.7G.

The original model was originated by Kondner to fit triaxial test data for undrained soils. Duncan and Chang (Ref. 4) and other coworkers (Ref. 5) developed this hyperbolic model to its current state. The material model is written in terms of the axial and radial stresses σ1 and σ3 and the axial strain ε, and it describes the stress-strain curve by fitting the hyperbola

here a and b are material parameters obtained by curve fitting data from the triaxial test. The parameter a is related to the initial Young's modulus E

and the parameter b defines the asymptote of the hyperbola, which is related to the ultimate value of σ1 − σ3 denoted qult

The ultimate value qult is related to the strength of the soil.

For the triaxial test, the axial strain ε is related to the shear strain γ by the Poisson’s ratio as

and the axial and radial stresses are related to the shear stress as

.

It is possible then to write the relation between shear stress and shear strain as

Since the initial shear modulus is related to the initial Young’s modulus as

, this stress-strain relation can alternatively be written as

which is an hyperbolic law with a secant shear modulus of

The Duncan–Selig model is a combination of the Duncan’s hyperbolic material model (Ref. 4, Ref. 5) and Selig’s model to describe nonlinear bulk modulus behavior. Selig (Ref. 6) further developed the model of Duncan and others in order to include a nonlinear volumetric response in soils.

The model defines the nonlinear volumetric response for the pressure as

where εel,vol is the volumetric strain, and εult is the asymptote of the hyperbola, the maximum value for the volumetric compression. Note that K represents the bulk modulus at zero strain.

The secant (nonlinear) bulk modulus is defined for this material model as

This option allows you to write explicitly how the pressure depends on the elastic volumetric strain. This could be an analytic function or data interpolated from a table.