Hyperelastic Material

The 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 is available in the Solid Mechanics, Layered Shell, Shell, and Membrane interfaces. is available for 3D, 2D, and 2D axisymmetry.

When a hyperelastic material is included in your model, all studies are geometrically nonlinear. The check box in the study settings is selected and cannot be cleared.

By adding the following subnodes to the node you can incorporate many other effects:

	See also Hyperelastic Materials in the Structural Mechanics Theory chapter.

The node is only available with some COMSOL products (see https://www.comsol.com/products/specifications/).

	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.

Select a hyperelastic from the list and then go to the applicable section for more information.

Hyperelastic materials can use a mixed formulation by adding the negative mean pressure as an extra dependent variable or a weak constraint to enforce the incompressibility condition. Depending on the hyperelastic material model, select from the list:

	See also Nearly Incompressible Hyperelastic Materials and Incompressible Hyperelastic Materials in the Structural Mechanics Theory chapter.

All hyperelastic material models have density as an input. The default ρ uses values . For enter another value or expression.

If any material in the model has a temperature dependent mass density, and is selected, the list will appear in the section. As a default, the value of Tref is obtained from a . You can also select to enter a value or expression for the reference temperature locally.

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 list, select or .

	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).

From the list select how the material is specified in terms of the strain energy density — , , , or .

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If the option is selected, specify the — or .

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If the option is selected, specify the — , , , , , or .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select a pair of elastic properties for the isotropic hyperelastic material — , , , , or . 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 or enter a value or expression.

From the list select how the material is specified in terms of the strain energy density — , , or .

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If the option is selected, specify the — , , , , , or .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

From the list select a pair of elastic properties for the isotropic hyperelastic material — , , , , or . For each pair of properties, select from the applicable list to either use the value or enter a 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.

From the list select how the material is specified in terms of the strain energy density — , , or .

The C10 and C01 use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The C10, C01, C20, C02, and C11 all use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The C10, C01, C20, C02, C11, C30, C03, C21, and C12 all use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The c1, c2, and c3 all use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

In the table for the , enter values or expressions in each column: and .

For , in the table for the , enter values or expressions in each column: , , and .

From the list select how the material is specified in terms of the strain energy density — , , or .

The c1, c2, and c3 all use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The default values for the μ0 and the N use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The default values for the μ and the model parameter jm use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The default values for the μ, the λm, the α, and the β use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

For the μ and the β and φ all use values .

For the a and n use values .

For the constants l, m, and n and the λ and μ use values .

From the list select how the material is specified in terms of the strain energy density — , , or .

The a and b use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

The A and c use values .

The A provides the anisotropic material properties in the directions given by the list. The can be specified in either or notation. When is selected, a 6-by-6 symmetric matrix is displayed to enter the coefficients of A.

From the list select how the material is specified in terms of the strain energy density — , , , or .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

From the list select how the material is specified in terms of the strain energy density — , , or .

The Gc, Ge, α, and β use values .

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If the option is selected, specify the — , , , , , or . The K uses values .

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If the option is selected, specify the — , , , , , or . The is selected from the list, and the default value for the κ is 100 times the equivalent shear modulus.

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If the option is selected, an extra variable and weak constraint is added to enforce the incompressibility condition Jel = 1.

If a material is selected from the list, enter an expression for the Ws.

You can also use a mixed formulation by adding the mean pressure as an extra dependent variable. In this case, select either or from the list.

If is selected, enter the Wsiso and the Wvol.

If is selected, enter the Wsiso only. An extra weak constraint is added to enforce the incompressibility condition Jel = 1.

Select the check box to compute the strain energy densities based on components of the elastic deformation gradient Fel. The check box 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.

Select the check box to compute the energy dissipated by .

To display this section, click the button (

) and select in the dialog box.

If the hyperelastic material is nearly incompressible or incompressible, select the discretization for the — , , , or .

Select the check box to reduce the integration points for the weak contribution of the feature. Select a method for — , , or to use in combination with the reduced integration scheme. The default stabilization technique is based on the shape function and shape order of the displacement field.

Control the hourglass stabilization scheme by using the option. Select (default) or .

When is selected, enter a stabilization shear modulus, Gstb. The value should be in the order of magnitude of the equivalent shear modulus.

When is selected, enter a stabilization bulk modulus, Kstb. The value should be in the order of magnitude of the equivalent bulk modulus.

To display this section, click the button (

) and select in the dialog box.

Enter the Eeq and the Geq. The defaults are 1 GPa and Eeq/3, respectively. The equivalent moduli are defined by most hyperelastic material models, but not in the 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.