Hyperelastic Material Models

A hyperelastic material is defined by its elastic strain energy density Ws, which is a function of the elastic strain state. It is often referred to as the energy density. The hyperelastic formulation normally gives a nonlinear relation between stress and strain, as opposed to Hooke’s law in linear elasticity.

Most of the time, the right Cauchy-Green deformation tensor C is used to describe the current state of strain (although one could use the left Cauchy-Green tensor B, the deformation gradient tensor F, and so forth), so the strain energy density is written as Ws(C).

For isotropic hyperelastic materials, any state of strain can be described in terms of three independent variables — common choices are the invariants of the right Cauchy-Green tensor C, the invariants of the Green-Lagrange strain tensor, or the principal stretches.

Once the strain energy density is defined, the second Piola-Kirchhoff stress in the local coordinate system is computed as

In the general case, the expression for the energy Ws is symbolically evaluated down to the components of C using the invariants definitions prior to the calculations of the components of the second Piola-Kirchhoff stress tensor. The differentiation is performed in components on the local coordinate system.

	In Equation View, the definition of the stress components are shown as solid.Sl11 = 2*d(solid.Ws,solid.Cl11), solid.Sl12 = d(solid.Ws,solid.Cl12) etc. The factor 2 in front of the differentiation operator for the shear stresses is omitted, since the symmetry in the Cauchy-Green tensor will cause two equal contributions.

	Modeling Geometric Nonlinearity

If thermal expansion is present, a stress-free volume change occurs. This is a pure volumetric change, so the multiplicative decomposition of the deformation gradient tensor in Equation 3-4 implies

Here, the thermal volume ratio, Jth, depends on the thermal stretch λth, which for linear thermal expansion in isotropic materials can be written in terms of the isotropic coefficient of thermal expansion, αiso, and the absolute change in temperature

Here, the term αiso(T−Tref) is the thermal strain. The isotropic thermal gradient is therefore a diagonal tensor defined as

When the coefficient of thermal expansion α is anisotropic, the thermal strain is computed from

and the anisotropic thermal gradient is defined as

	The internal variables for the thermal stretch and the thermal volume ratio are named solid.stchth and solid.Jth.

Hygroscopic swelling is an internal strain caused by changes in moisture content. This strain depends linearly on the moisture content

where βh is the coefficient of hygroscopic swelling, cmo is the moisture concentration, and cmo,ref is the strain-free reference concentration. The coefficient of hygroscopic swelling can represent isotropic or anisotropic swelling. The anisotropic hygroscopic gradient is defined as

It is possible to combine the hyperelastic material models with plasticity. Since these models are primarily used for large strain applications, only the large strain plasticity formulation is available. The decomposition between elastic and plastic deformation is made using a multiplicative decomposition of the deformation gradient tensor,

Here, the plastic deformation tensor Fpl depends on the plastic flow rule, yield function, and plastic potential.

For some classes of hyperelastic materials it is convenient to split the strain energy density into volumetric (also called dilatational) and isochoric (also called distortional or volume-preserving) contributions. The elastic deformation tensor is then multiplicatively decomposed into the volumetric and isochoric components

with Fel,vol as the volumetric elastic deformation (a diagonal tensor) and

the isochoric elastic deformation gradient. Isochoric deformation means that the volume ratio is kept constant during deformation, so the isochoric elastic deformation is computed by scaling it by the elastic volume ratio. The elastic volume ratio is defined by

and the volumetric deformation as

By using Jel it is possible to define the isochoric-elastic deformation gradient

the isochoric-elastic right Cauchy-Green tensor

and the isochoric-elastic Green-Lagrange strain tensor

This scaling changes the eigenvalues of the tensor, but not its principal directions, so the original and isochoric tensors remain coaxial to each other.

Some authors call

and

the modified tensors. Note that

	The internal variables for the isochoric-elastic Cauchy-Green deformation tensor in local coordinate system are named solid.CIel11, solid.CIel12, and so on.

The other two invariants normally used together with Jel are the first and second invariant of the isochoric-elastic right Cauchy-Green deformation tensor

In these equations:

Since

, the third invariant is never explicitly used.

	The internal variables for the invariants Jel, , and are named solid.Jel, solid.I1CIel, and solid.I2CIel.

The invariants of the isochoric (modified) elastic Green-Lagrange strain tensor are related to the invariants of the isochoric-elastic right Cauchy-Green deformation tensor

	The internal variables for the invariants of the isochoric elastic Green-Lagrange strain tensor are named solid.I1eIel, solid.I2eIel, and solid.I3eIel.

If the is selected in the list in the Hyperelastic Material node, the total strain energy density Ws is split into two parts:

here, Wiso is the isochoric strain energy density and Wvol is the volumetric strain energy density.

The isochoric strain energy density can be entered as an expression involving the following:

The volumetric strain energy density, Wvol can only be defined as an expression of the elastic volumetric deformation Jel. The quadratic volumetric strain energy density is defined as:

and Harmann-Neff (Ref. 10) volumetric strain energy as:

where κ is the bulk modulus. From here, the volumetric stress (pressure) is calculated as

When the expression in Equation 3-26 is used, the pressure becomes linearly related to the volume change:

but if the expression in Equation 3-27 is used instead of the quadratic function, the pressure becomes a nonlinear function of the volume change:

Then, an auxiliary variable, pw, is added to map the pressure when computing stresses.

The second Piola-Kirchhoff stress is then given by

and the Cauchy stress tensor by

When the is selected in the list in the Hyperelastic Material node, the volumetric strain energy density Wvol is not defined as for Nearly Incompressible Hyperelastic Materials, but instead, a weak constrain is added to account for the incompressibility condition

The auxiliary variable, pw, acts as Lagrange multiplier to enforce the constrain Jel = 1, this variable is then used as the pressure when computing stresses.

The strain energy density Ws consists only of the isochoric strain energy density Wiso contribution

The second Piola-Kirchhoff stress is then given by

and the Cauchy stress tensor by

A numerical scheme is said to exhibit locking if the accuracy of the approximation deteriorates as a parameter tends to a limiting value (Ref. 12). Finite elements in solid mechanics are said to “lock” when exhibiting an unphysical response to deformation (Ref. 13). Locking can occur for many different reasons. For linear elastic materials, this typically happens as Poisson’s ratio tends to 0.5, or the bulk modulus is much larger than the shear modulus. Numerical errors arise because the shape functions are unable to properly describe the volume preserving deformation.

To avoid the locking problem in computations, the mixed formulation replaces pm in Equation 3-28 with a corresponding interpolated pressure help variable pw, which adds an extra degree of freedom to the ones defined by the displacement vector u.

The general procedure is the same as done in the Linear Elastic Materials node.

Different hyperelastic material models are constructed by specifying different elastic strain energy expressions. This module has several predefined material models and also has the option to enter user defined expressions for the strain energy density.

The strain energy density for the compressible version of the Neo-Hookean material is written in terms of the elastic volume ratio Jel and the first invariant of the elastic right Cauchy-Green deformation tensor I1(Cel) (Ref. 11).

Here, λ and μ are the Lamé parameters.

The nearly incompressible version uses the isochoric invariant

to define the isochoric strain energy density Wiso,

and the elastic volume ratio Jel and the bulk modulus κ to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Neo-Hookean material model in the Solid Mechanics interface documentation.

One of the simplest hyperelastic material models is the St Venant-Kirchhoff material, which is an extension of a linear elastic material into the hyperelastic regime.

The elastic strain energy density is written with two parameters (the two Lamé coefficients) and two invariants of the elastic Green-Lagrange strain tensor, I1(εel) and I2(εel)

Here, λ and μ are Lamé parameters. The bulk modulus κ is calculated from the Lamé parameters κ = λ + 2μ/3.

The nearly incompressible version uses the isochoric invariants

and

to define the isochoric strain energy density

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the St Venant-Kirchhoff material model in the Solid Mechanics interface documentation.

For the nearly incompressible version, the isochoric strain energy density is written in terms of the two isochoric invariants of the elastic right Cauchy-Green deformation tensors

and

The material parameters C10 and C01 are related to the Lamé parameter (shear modulus) μ = 2(C10 + C01). The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Mooney-Rivlin, Two Parameters material model in the Solid Mechanics interface documentation.

Rivlin and Saunders (Ref. 2) proposed a phenomenological model for small deformations in rubber-based materials on a polynomial expansion of the first two invariants of the elastic right Cauchy-Green deformation, so the strain energy density is written as an infinite series

with C00 = 0. This material model is sometimes also called polynomial hyperelastic material.

In the first-order approximation, the material model recovers the Mooney-Rivlin strain energy density

while the second-order approximation incorporates second-order terms

The nearly incompressible version uses the isochoric invariants of the elastic right Cauchy-Green deformation tensors

and

to define the isochoric strain energy density

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Mooney-Rivlin, Five Parameters material model in the Solid Mechanics interface documentation.

The Mooney-Rivlin, nine parameters material model is an extension of the polynomial expression to third order terms and the strain energy density is written as

where

and

are the isochoric invariants of the elastic right Cauchy-Green deformation tensors. The elastic volume ratio Jel and the bulk modulus κ to are used to define the volumetric strain energy density, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Mooney-Rivlin, Nine Parameters material model in the Solid Mechanics interface documentation.

Yeoh proposed (Ref. 1) a phenomenological model in order to fit experimental data of filled rubbers, where Mooney-Rivlin and Neo-Hookean models were to simple to describe the stiffening effect in the large strain regime. The strain energy was fitted to experimental data by means of three parameters, and the first invariant of the elastic right Cauchy-Green deformation tensors I1(Cel)

The shear modulus depends on the deformation, and it is calculated as

This imposes a restriction on the coefficients

, since

.

The nearly incompressible version uses the isochoric invariant of the elastic right Cauchy-Green deformation tensor

to define the isochoric strain energy density

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Yeoh material model in the Solid Mechanics interface documentation.

The Neo-Hookean material model usually fits well to experimental data at moderate strains but fails to model hyperelastic deformations at high strains. In order to model rubber-like materials at high strains, Ogden adapted (Ref. 1) the energy of a Neo-Hookean material to

Here αp and μp are material parameters, and λel1, λel2, and λel3 are the principal elastic stretches such as Jel = λel1λel2λel3.

The Ogden model is empirical, in the sense that it does not relate the material parameters αp and μp to physical phenomena. The parameters αp and μp are obtained by curve-fitting measured data, which can be difficult for N > 2. The most common implementation of Ogden material is with N = 2, so four parameters are needed.

The nearly incompressible version uses the isochoric elastic stretches

so the isochoric strain energy density is defined as

The isochoric elastic stretches define a volume preserving deformation, since

The shear modulus is then defined from

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	Sometimes a slightly different definition of the strain energy function for the Ogden material is used. If you have material data for an Ogden material given, be careful that the definitions used are the same. If not, you will need to rescale the μk coefficients.

	See also the description of the Ogden material model in the Solid Mechanics interface documentation.

The Storakers material (Ref. 13 and Ref. 16) is used to model highly compressible foams. The strain energy density is written in a similar fashion as in Ogden material:

The initial shear and bulk moduli are computed from the parameters μk and βk as

for constant parameters βk = β, the initial bulk modulus becomes κ = 2μ(β + 1/3), so a stable material requires μ > 0 and β > −1/3. In this case, the Poisson's ratio is given by ν = β/(2β+2/3), which means that for a Poisson’s ratio larger than −1, β > −2/9 is needed.

	See also the description of the Storakers material model in the Solid Mechanics interface documentation.

The Varga material model (Ref. 1) describes the strain energy in terms of the elastic stretches as

The nearly incompressible version uses the isochoric elastic stretches defined as

so the isochoric strain energy density is defined as

The simplest Varga model is obtained by setting c1 = μ and c2 = 0:

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Varga material model in the Solid Mechanics interface documentation.

The other hyperelastic materials described are phenomenological models in the sense that they do not relate the different material parameters (normally obtained by curve-fitting experimental data) to physical phenomena.

Arruda and Boyce (Ref. 3) derived a material model based on Langevin statistics of polymer chains. The strain energy density is defined by

Here, μ0 is the initial macroscopic shear modulus, I1(Cel) is the first invariant of the elastic right Cauchy-Green deformation tensor, and the coefficients cp are obtained by series expansion of the inverse Langevin function.

Arruda and Boyce truncated the series and used only the first five terms of the series. The coefficients cp are listed in Table 3-2:

Table 3-2: First Five Coefficients of Arruda-Boyce Material Model
c1	c2	c3	c4	c5
1/2	1/20 N	11/1050 N2	19/7000 N3	519/673750 N4

Other authors (Ref. 1) use only the first three coefficients of the series. The number of segments in the polymeric chain is specified by the parameter N so the material model is described by only two parameters, μ0 and N. This material model is sometimes also called the eight-chain model.

The nearly incompressible version uses the isochoric invariant

to define the isochoric strain energy density

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	See also the description of the Arruda-Boyce material model in the Solid Mechanics interface documentation.

Many hyperelastic material models are difficult to fit to experimental data. Gent material (Ref. 14 and Ref. 15) is a simple phenomenological constitutive model based on only two parameters, μ and jm, which defines the strain energy density as:

Here, μ is the shear modulus and jm is a limiting value for I1− 3, which takes care of the limiting polymeric chain extensibility of the material.

Since the strain energy density does not depend on the second invariant I2, Gent model is often classified as a generalized Neo-Hookean material. The strain energy density tends to be the one of incompressible Neo-Hookean material as

.

The nearly incompressible formulation uses the isochoric invariants

to define the isochoric strain energy density

The elastic volume ratio Jel and the bulk modulus κ are used to define the volumetric strain energy density Wvol, see Nearly Incompressible Hyperelastic Materials. The incompressible option uses the same isochoric strain energy as when selecting the nearly incompressible option, but an extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials. Gent material is the simplest model of the limiting chain extensibility family.

	See also the description of the Gent material model in the Solid Mechanics interface documentation.

The Blatz-Ko material model was developed for foamed elastomers and polyurethane rubbers, and it is valid for compressible isotropic hyperelastic materials (Ref. 1).

The elastic strain energy density is written with three parameters and the three invariants of the elastic right Cauchy-Green deformation tensor, I1(Cel), I2(Cel), and I3(Cel)

Here,

is an interpolation parameter bounded to

, the parameter μ is the shear modulus, and β is an expression of Poisson’s ratio.

When the parameter

, the strain energy simplifies to a similar form of the Mooney-Rivlin material model

In the special case of

, the strain energy reduces to a similar form of the Neo-Hookean model

	See also the description of the Blatz-Ko material model in the Solid Mechanics interface documentation.

Gao proposed (Ref. 17) a simple hyperelastic material where the strain energy density is defined by two parameters, a and n, and two invariants of the elastic right Cauchy-Green deformation tensors Cel:

Here, the invariant I-1(Cel) is calculated as:

Gao proposed that the material is unconditionally stable when the parameters are bounded to 1 < n < 3 and 0 < a, and related these parameters under small strain to the Young’s modulus and Poisson’s ratio by:

Since n = (1+ν)/(1−2ν) and it is bounded to 1 < n < 3, this material model is stable for materials with an initial Poisson’s ratio in the range of 0 < ν < 2/7.

	See also the description of the Gao material model in the Solid Mechanics interface documentation.

The Murnaghan potential is used in nonlinear acoustoelasticity. Most conveniently, it is expressed in terms of the three invariants of the elastic Green-Lagrange strain tensor, I1(εel), I2(εel), and I3(εel):

Here, l, m, and n are the Murnaghan third-order elastic moduli, which can be found experimentally for many commonly encountered materials such as steel and aluminum, and λ and μ are the Lamé parameters.

	See also the description of the Murnaghan material model in the Solid Mechanics interface documentation.

It is possible to define the strain energy density for compressible, nearly incompressible or incompressible hyperelastic materials when selecting the user defined option.

For hyperelastic materials, enter an expression for the elastic strain energy Ws, which can include any expressions involving the following:

	The internal variables for these invariants are named solid.I1Cel, solid.I2Cel, and solid.I3Cel.

	The internal variables for the principal elastic stretches are named solid.stchelp1, solid.stchelp2, and solid.stchelp3.

	The internal variables for these invariants are named solid.I1eel, solid.I2eel, and solid.I3eel.

When the option is selected for the Hyperelastic Material node, the elastic strain energy is decoupled into the volumetric and isochoric components.:

	The internal variables for Jel, , and are named solid.Jel, solid.I1CIel, and solid.I2CIel. The internal variables for , , and are named solid.I1eIel, solid.I2eIel, and solid.I3eIel.

When the option is selected, enter an expression for the isochoric elastic strain energy Wiso, as done for the option. An extra variable is added to enforce the incompressibility condition Jel = 1, see Incompressible Hyperelastic Materials.

	The strain energy density must not contain any other expressions involving displacement or their derivatives. Examples of such expressions are components of the displacement gradient ∇u and deformation gradient F = ∇u + I tensors, their transpose, inversions, as well as the global material system components of C and ε. If they occur, such variables are treated as constants during symbolic differentiations.

	See also the description of the User defined material model in the Solid Mechanics interface documentation.

Some nonlinear effects observed in rubbers, such as hysteresis in stress-stretch curves, residual strains, and stress softening effects, are not accounted in the formulation of common hyperelastic materials. The Mullins effect (Ref. 22-24) describes the stress-softening phenomenon observed under cyclic loading in elastomers and biological materials.

Ogden and Roxburgh (Ref. 25) used an additional state variable to model the Mullins effect. The state variable η is introduced to memorize the microstructural damage on reinforced rubber after repeated loading-unloading cycles. The modified isochoric strain energy density reads

here, Wiso is the isochoric strain energy of the undamaged material, and

is referred as the damage function. The choice of the damage function

is completely arbitrary as long as the some constrains are fulfilled. The authors proposed a state function based on the error function, which defined how the state variable η varies as a function of the isochoric strain energy

here, erf(.) is the error function, r and m are positive parameters, and Wmax is the maximum attained value of the isochoric strain energy density on the loading path.

The associated microstructural damage is computed from

and the isochoric strain energy modified by the Mullins effect reads

Over the years others authors (Ref. 26-27) have proposed different flavors of Ogden-Roxburgh model, the version implemented in COMSOL Multiphysics uses the hyperbolic tangent function instead of the error function, and a parameter to define the maximum allowed damage d∞=1/r. The microstructural damage is then computed from

Here, d∞, α, and Wsat are positive parameters.

Miehe (Ref. 28) proposed an exponential expression for the damage variable in order to model the Mullins effect

where Wsat and d∞ are positive parameters.