Hyperelastic Material
The Hyperelastic Material subnode adds the equations for hyperelasticity at large strains. Hyperelastic materials can be suitable for modeling rubber and other polymers, biological tissue, and also for applications in acoustoelasticity. The Hyperelastic Material is available in the Solid Mechanics, Layered Shell, Shell, and Membrane interfaces. Hyperelastic Material is available for 3D, 2D, 2D axisymmetry, 1D, and 1D axisymmetry.
When a hyperelastic material is included in your model, all studies are geometrically nonlinear. The Include geometric nonlinearity checkbox in the study settings is selected and cannot be cleared.
By adding the following subnodes to the Hyperelastic Material node you can incorporate many other effects:
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Thermal Expansion (for Materials)
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Hygroscopic Swelling
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External Stress
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External Strain
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Inelastic Strain Rate
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Damping
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Viscoelasticity
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Plasticity
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Creep
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Viscoplasticity
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Polymer Viscoplasticity
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Fiber
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Mullins Effect
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Damage
See also Hyperelastic Materials in the Structural Mechanics Theory chapter.
The Hyperelastic Material node is only available with some COMSOL products (see www.comsol.com/products/specifications/).
Shell Properties

 
This section is only present when Hyperelastic Material is used in the Layered Shell interface. See the documentation for the Hyperelastic Material node in the Layered Shell chapter.
Coordinate System Selection
The Global coordinate system is selected by default. The Coordinate system list contains all applicable coordinate systems in the component. The coordinate system is used for interpreting directions of orthotropic and anisotropic material data and when stresses or strains are presented in a local system. The coordinate system must have orthonormal coordinate axes, and be defined in the material frame. Many of the possible subnodes inherit the coordinate system settings.
Hyperelastic Material
Select a hyperelastic Material model from the list and then go to the applicable section for more information.
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Neo-Hookean
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Saint-Venant–Kirchhoff
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Mooney–Rivlin, Two Parameters
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Mooney–Rivlin, Five Parameters
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Mooney–Rivlin, Nine Parameters
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Yeoh
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Ogden
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Storakers
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Varga
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Arruda–Boyce
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Gent
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van der Waals
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Blatz–Ko
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Gao
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Murnaghan
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Delfino
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Fung
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Extended Tube
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User Defined

 
In the Solid Mechanics, Explicit Dynamics interface:
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The material model User defined is not available.
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The options in the Compressibility list is, for all material models, limited to Compressible, coupled and Compressible, uncoupled.
Density
All hyperelastic material models have density as an input. The default Density ρ uses values From material. For User defined enter another value or expression.
If any material in the model has a temperature dependent mass density, and From material is selected, the Volume reference temperature list will appear in the Model Input section. As a default, the value of Tref is obtained from a Common model input. You can also select User defined to enter a value or an expression for the reference temperature locally.
Mixed Formulation
For a material with a very low compressibility, using only displacements as degrees of freedom may lead to a numerically ill-posed problem. You can then use a mixed formulation, which adds an extra dependent variable for either the pressure or for the Lagrange multiplier to enforce incompressibility, see the Nearly Incompressible Hyperelastic Materials and Incompressible Hyperelastic Materials sections in the Structural Mechanics Theory chapter.
From the Use mixed formulation list, select None or Pressure formulation. It is also possible to select an Implicit formulation when an assumption of plane stress is used.
The density is needed for dynamic analysis or when the elastic data is given in terms of wave speed. It is also used when computing mass forces for gravitational or rotating frame loads, and when computing mass properties (Computing Mass Properties).
See also
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Mass Density and Volume Reference Temperature.
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Using Common Model Input
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Default Model Inputs and Model Input in the COMSOL Multiphysics Reference Manual.
Neo-Hookean
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, coupled; Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, coupled option is selected, specify the Volumetric strain energy density — Simo–Pister or Miehe.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
From the Specify list select a pair of elastic properties for the isotropic hyperelastic material — Young’s modulus and Poisson’s ratio, Young’s modulus and shear modulus, Bulk modulus and shear modulus, Lamé parameters, or Pressure-wave and shear-wave speeds. Each of these pairs define the Lamé parameters at infinitesimal deformation as it is possible to convert from one set of properties to another, see Specification of Elastic Properties for Isotropic Materials. For each property, select from the applicable list to either use the value From material or enter a User defined value or expression.
Saint-Venant–Kirchhoff
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, coupled; Compressible, uncoupled; or Nearly incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
From the Specify list select a pair of elastic properties for the isotropic hyperelastic material — Young’s modulus and Poisson’s ratio, Young’s modulus and shear modulus, Bulk modulus and shear modulus, Lamé parameters, or Pressure-wave and shear-wave speeds. For each pair of properties, select from the applicable list to either use the value From material or enter a User defined value or expression. Each of these pairs define the Lamé parameters at infinitesimal deformation as it is possible to convert from one set of properties to another, see Specification of Elastic Properties for Isotropic Materials.
Mooney–Rivlin, Two Parameters
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
The Model parameters C10 and C01 use values From material.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
Mooney–Rivlin, Five Parameters
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters C10, C01, C20, C02, and C11 all use values From material.
Mooney–Rivlin, Nine Parameters
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters C10, C01, C20, C02, C11, C30, C03, C21, and C12 all use values From material.
Yeoh
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters c1, c2, and c3 all use values From material.
Ogden
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
In the table for the Ogden parameters, enter values or expressions in each column: Shear modulus (Pa) and Alpha parameter.
Storakers
For Storakers, in the table for the Storakers parameters, enter values or expressions in each column: Shear modulus (Pa), Alpha parameter, and Beta parameter.
Varga
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters c1, c2, and c3 all use values From material.
Arruda–Boyce
From the Isochoric strain energy list select how the strain energy density of the material is specified in terms of the invariants — Five terms expansion or Inverse Langevin function.
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The default values for the Macroscopic shear modulus μ0 and the Number of segments N use values From material.
Gent
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The default values for the Macroscopic shear modulus μ and the model parameter jm use values From material.
van der Waals
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The default values for the Shear modulus μ, the Maximum chain stretch λm, the Chain network interaction α, and the Weight β use values From material.
Blatz–Ko
For Blatz–Ko the Shear modulus μ and the Model parameters β and ϕ all use values From material.
Gao
For Gao the Model parameters a and n use values From material.
Murnaghan
For Murnaghan the Murnaghan third-order elastic moduli constants l, m, and n and the Lamé parameters λ and μ use values From material.
Delfino
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters a and b use values From material.
Fung
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, coupled; Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Coefficient matrix A and Fung parameter c use values From material.
The Coefficient matrix A provides the anisotropic material properties in the directions given by the Coordinate system list. The Material data ordering can be specified in either Standard or Voigt notation. When User defined is selected, a 6-by-6 symmetric matrix is displayed to enter the coefficients of A.
Extended Tube
From the Compressibility list select how the material is specified in terms of the strain energy density — Compressible, uncoupled; Nearly incompressible; or Incompressible.
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If the Compressible, uncoupled option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The Bulk modulus K, and for the Polynomial option the parameters K2 and K3, use values From material.
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If the Nearly incompressible option is selected, specify the Volumetric strain energy density — Quadratic, Polynomial, Logarithmic, Hartmann–Neff, Miehe, Simo–Taylor, or User defined. The pressure formulation is selected from the Use mixed formulation list, and the default value for the Bulk modulus κ is 100 times the equivalent shear modulus.
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If the Incompressible option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.
The Model parameters Gc, Ge, α, and β use values From material.
User Defined
If a Compressible material is selected from the Compressibility list, enter an expression for the Elastic strain energy density Ws.
You can also use a mixed formulation by adding the mean pressure as an extra dependent variable. In this case, select either Nearly incompressible or Incompressible from the Compressibility list.
If Nearly incompressible is selected, enter the Isochoric strain energy density Wsiso and the Volumetric strain energy density Wvol.
If Incompressible is selected, enter the Isochoric strain energy density Wsiso only. An extra weak constraint is added to enforce the incompressibility condition Jel = 1.
Select the Use elastic deformation gradient checkbox to compute the strain energy densities based on components of the elastic deformation gradient Fel. The checkbox is not selected by default, so it is assumed that Ws, Wsiso, and Wvol are expressions of the components or invariants of the elastic right Cauchy–Green tensor Cel.
For examples of:
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Mooney–Rivlin, two-parameters and Ogden, see Inflation of a Spherical Rubber Balloon. Application Library path Nonlinear_Structural_Materials_Module/Hyperelasticity/balloon_inflation.
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Murnaghan, see Elasto-Acoustic Effect in Rail Steel. Application Library path Nonlinear_Structural_Materials_Module/Hyperelasticity/rail_steel.
Energy Dissipation
Select how to compute the energy dissipated by Creep, Plasticity, Viscoplasticity, or other dissipative processes.
Select how to Store dissipation — From physics interface, Individual contributions, Total, Domain ODEs (legacy), or Off.

 
The option Domain ODEs (legacy) is not available in the interfaces intended for time-explicit dynamic analysis.
Use From physics interface to treat the dissipative processes as specified in the settings of the physics interface, see for instance Energy Dissipation in the Solid Mechanics interface.
Use Individual contributions to treat each dissipative process independently. Selecting this option gives a more flexible implementation for problems where dissipation occurs at different time scales, and you want to distinguish each phenomenon separately.
Use Total to accumulate all the dissipative processes into one common variable.
Use Domain ODEs to accumulate the dissipative processes into ODE variables instead of internal state variables.
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Dissipated Energy
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Energy Variables
Discretization
If the hyperelastic material is nearly incompressible or incompressible, select the discretization for the Auxiliary pressure — Automatic, Discontinuous Lagrange, Continuous, Linear, or Constant.
The Discretization section is available when you use mixed formulation. To display the section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog.
Quadrature Settings
Select the Reduced integration checkbox to reduce the number of integration points used for the weak contribution in the material. Reduced integration give faster computations at the element level and help mitigate locking issues. However, reduced integration may also introduce numerical instabilities, which then require an additional stabilization term.
By default, the Reduced integration checkbox is cleared, except in interfaces designed for explicit dynamic analysis, where it is always selected.
It is possible to define different hourglass stabilization methods per mesh element type, for instance, when combining hexahedral and prisms elements within the same domain.
Select a method for Hourglass stabilization — Automatic, Energy sampling, Hessian, Flanagan–Belytschko, Manual, or None to be used in combination with the reduced integration scheme.

 
The Energy sampling and Hessian methods are available in the Solid Mechanics and Solid Mechanics, Explicit Dynamics interfaces.
The Flanagan–Belytschko method is only available in the Solid Mechanics, Explicit Dynamics interface and should only be used with the Explicit Dynamics study step.
The Automatic option selects the stabilization method and its properties based on the physics interface, space dimension, shape function type and order of the displacement field, and the study type.
For the Energy sampling, Hessian, and Flanagan–Belytschko methods, the hourglass stiffness can be scaled by specifying a stiffness multiplier fstb. This multiplier can be an expression of any parameter or variable in the model. This multiplier can be expressed as any parameter or variable in the model. However, to maintain computational efficiency, avoid expensive definitions such as nonlocal coupling operators.
For the hourglass stabilization methods, the stiffness is automatically adjusted during the solution process using an internal multiplier. This accounts for the effect of inelastic deformations, such as damage and plasticity. This correction can be disabled by setting Inelastic deformations to Ignore. Ignoring inelastic deformations may be necessary for scenarios like cyclic loading, where the hourglass stiffness could become too small during unloading. Alternatively, adjust the value for the Minimum stiffness multiplier., which acts as a lower bound for the hourglass stiffness, which is applied in addition to any expression entered for fstb.
For the Energy sampling method, the default stabilization uses an isotropic linear elastic potential. Set the Energy Sampling Potential to Hyperelastic for large deformations. When selected, a neo-Hookean potential is used for nonlinear strains, while linear strains use the linear elastic potential.
The Hessian method evaluates a weak contribution using the reduced integration order. For hexahedral mesh elements, this approach may not suppress all hourglass modes. In most cases, non-stabilized modes are suppressed by constraints adjacent to the domain, but if they are not, select Use full integration to increase the integration order of the stabilization equations to suppress all modes. The higher integration order will increase the computational cost of the stabilization method.
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When using the Manual option, select Shear stabilization or Volumetric stabilization. When Shear stabilization is selected, enter a stabilization shear modulus, Gstb. The value should be in the order of magnitude of the equivalent shear modulus.
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When Volumetric stabilization is selected, enter a stabilization bulk modulus, Kstb. The value should be in the order of magnitude of the equivalent bulk modulus.
See also Reduced Integration and Hourglass Stabilization in the Structural Mechanics Theory chapter and Using Reduced Integration in the Structural Mechanics Modeling chapter.
Estimate of Maximum Wave Speed

 
This section is only present in the Solid Mechanics, Explicit Dynamics interface.
Select how the maximum wave speed in the material is determined — Automatic or User defined. For User defined, enter the maximum wave speed, cmax.
Advanced
This section is only displayed when User defined has been selected as the material model.
Enter the Equivalent Young’s modulus Eeq and the Equivalent shear modulus Geq. The defaults are 1 GPa and Eeq/3, respectively. The equivalent moduli are defined by most hyperelastic material models, but not in the User defined option. The characteristic stiffness is needed in expressions for the default penalty factors in contact methods, and it should be representative for the stiffness of the destination domain material in a direction normal to the boundary.
Location in User Interface
Context Menus
Solid Mechanics > Material Models > Hyperelastic Material
Solid Mechanics, Explicit Dynamics > Material Models > Hyperelastic Material
Layered Shell > Material Models > Hyperelastic Material
Shell > Material Models > Hyperelastic Material, Layered
Membrane > Material Models > Hyperelastic Material
Ribbon
Physics tab with Solid Mechanics or Solid Mechanics, Explicit Dynamics selected:
Domains > Material Models > Hyperelastic Material
Physics tab with Shell, Layered Shell or Membrane selected:
Boundaries > Material Models > Hyperelastic Material