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.
Thermal Expansion
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
and
Here, the term αiso(TTref) 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
Hygroscopic Swelling
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
Hyperelasticity with Plasticity
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.
Isochoric Elastic Deformation
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 other two invariants normally used together with Jel are the first and second invariant of the isochoric-elastic right Cauchy-Green deformation tensor
and
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
s
Nearly Incompressible Hyperelastic Materials
If the Nearly incompressible material is selected in the Compressibility 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:
(3-26)
and Harmann-Neff (Ref. 10) volumetric strain energy as:
(3-27)
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
(3-28)
and the Cauchy stress tensor by
Incompressible Hyperelastic Materials
When the Incompressible material is selected in the Compressibility 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
(3-29)
and the Cauchy stress tensor by
(3-30)
The Locking Problem
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.
Theory for the Predefined Hyperelastic Material Models
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.
Neo-Hookean
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.
St Venant-Kirchhoff
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, I1el) and I2el)
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.
Mooney-Rivlin, Two Parameters
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.
Mooney-Rivlin, Five Parameters
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.
Mooney-Rivlin, Nine Parameters
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
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.
Ogden
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.
See also the description of the Ogden material model in the Solid Mechanics interface documentation.
Storakers
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
and
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.
Varga
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.
Arruda-Boyce
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:
c1
c2
c3
c4
c5
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.
Gent
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.
Blatz-Ko
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
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:
and
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.
Murnaghan
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, I1el), I2el), and I3el):
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.
User Defined
It is possible to define the strain energy density for compressible, nearly incompressible or incompressible hyperelastic materials when selecting the user defined option.
For Compressible hyperelastic materials, enter an expression for the elastic strain energy Ws, which can include any expressions involving the following:
Components of Cel, the elastic right Cauchy-Green deformation tensor in the local material coordinate system.
The internal variables for these invariants are named solid.I1Cel, solid.I2Cel, and solid.I3Cel.
Principal elastic stretches λel1, λel2, and λel3, which are the square-root of the eigenvalues of the elastic right Cauchy-Green deformation tensor Cel.
The internal variables for the principal elastic stretches are named solid.stchelp1, solid.stchelp2, and solid.stchelp3.
the invariants of εel are written in terms of the invariants of Cel:
The internal variables for these invariants are named solid.I1eel, solid.I2eel, and solid.I3eel.
When the Nearly incompressible material option is selected for the Hyperelastic Material node, the elastic strain energy is decoupled into the volumetric and isochoric components.:
The volumetric strain energy Wvol, which can be an expression involving the elastic volume ratio
The isochoric strain energy, Wiso, as an expression involving the invariants of the isochoric elastic right Cauchy-Green tensor and ; the invariants of the isochoric elastic Green-Lagrange strain , , and ; or the principal isochoric elastic stretches defined as
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 Incompressible material option is selected, enter an expression for the isochoric elastic strain energy Wiso, as done for the Nearly incompressible material 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.
Mullins Effect
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.