Hyperelastic material

Summary

A hyperelastic or Green elastic material[1] is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.

Stress–strain curves for various hyperelastic material models.

For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials.[2] The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues[3][4] are also often modeled via the hyperelastic idealization. In addition to being used to model physical materials, hyperelastic materials are also used as fictitious media, e.g. in the third medium contact method.

Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda–Boyce model.

Hyperelastic material models

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Saint Venant–Kirchhoff model

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The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. This model has the general form and the isotropic form respectively   where   is tensor contraction,   is the second Piola–Kirchhoff stress,   is a fourth order stiffness tensor and   is the Lagrangian Green strain given by     and   are the Lamé constants, and   is the second order unit tensor.

The strain-energy density function for the Saint Venant–Kirchhoff model is  

and the second Piola–Kirchhoff stress can be derived from the relation  

Classification of hyperelastic material models

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Hyperelastic material models can be classified as:

  1. phenomenological descriptions of observed behavior
  2. mechanistic models deriving from arguments about underlying structure of the material
  3. hybrids of phenomenological and mechanistic models

Generally, a hyperelastic model should satisfy the Drucker stability criterion. Some hyperelastic models satisfy the Valanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of the principal stretches  :  

Stress–strain relations

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Compressible hyperelastic materials

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First Piola–Kirchhoff stress

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If   is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as   where   is the deformation gradient. In terms of the Lagrangian Green strain ( )   In terms of the right Cauchy–Green deformation tensor ( )  

Second Piola–Kirchhoff stress

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If   is the second Piola–Kirchhoff stress tensor then   In terms of the Lagrangian Green strain   In terms of the right Cauchy–Green deformation tensor   The above relation is also known as the Doyle-Ericksen formula in the material configuration.

Cauchy stress

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Similarly, the Cauchy stress is given by   In terms of the Lagrangian Green strain   In terms of the right Cauchy–Green deformation tensor   The above expressions are valid even for anisotropic media (in which case, the potential function is understood to depend implicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of the left Cauchy-Green deformation tensor as follows:[7]  

Incompressible hyperelastic materials

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For an incompressible material  . The incompressibility constraint is therefore  . To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:   where the hydrostatic pressure   functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes   This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy stress tensor which is given by  

Expressions for the Cauchy stress

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Compressible isotropic hyperelastic materials

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For isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left Cauchy–Green deformation tensor (or right Cauchy–Green deformation tensor). If the strain energy density function is   then   (See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

Proof 1

The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by   where   is the right Cauchy–Green deformation tensor and   is the deformation gradient. The Cauchy stress is given by   where  . Let   be the three principal invariants of  . Then   The derivatives of the invariants of the symmetric tensor   are   Therefore, we can write   Plugging into the expression for the Cauchy stress gives   Using the left Cauchy–Green deformation tensor   and noting that  , we can write   For an incompressible material   and hence  .Then   Therefore, the Cauchy stress is given by   where   is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.

If, in addition,  , we have   and hence   In that case the Cauchy stress can be expressed as  

Proof 2

The isochoric deformation gradient is defined as  , resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor  . The invariants of   are   The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add   into the fray to describe the volumetric behaviour.

To express the Cauchy stress in terms of the invariants   recall that   The chain rule of differentiation gives us   Recall that the Cauchy stress is given by   In terms of the invariants   we have   Plugging in the expressions for the derivatives of   in terms of  , we have   or,   In terms of the deviatoric part of  , we can write   For an incompressible material   and hence  .Then the Cauchy stress is given by   where   is an undetermined pressure-like Lagrange multiplier term. In addition, if  , we have   and hence the Cauchy stress can be expressed as  

Proof 3

To express the Cauchy stress in terms of the stretches   recall that   The chain rule gives   The Cauchy stress is given by   Plugging in the expression for the derivative of   leads to   Using the spectral decomposition of   we have   Also note that   Therefore, the expression for the Cauchy stress can be written as   For an incompressible material   and hence  . Following Ogden[1] p. 485, we may write   Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.[8][9] A rigorous tensor derivative can only be found by solving another eigenvalue problem.

If we express the stress in terms of differences between components,   If in addition to incompressibility we have   then a possible solution to the problem requires   and we can write the stress differences as  

Incompressible isotropic hyperelastic materials

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For incompressible isotropic hyperelastic materials, the strain energy density function is  . The Cauchy stress is then given by   where   is an undetermined pressure. In terms of stress differences   If in addition  , then   If  , then  

Consistency with linear elasticity

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Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparing Hooke's law with linearized hyperelasticity at small strains.

Consistency conditions for isotropic hyperelastic models

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For isotropic hyperelastic materials to be consistent with isotropic linear elasticity, the stress–strain relation should have the following form in the infinitesimal strain limit:   where   are the Lamé constants. The strain energy density function that corresponds to the above relation is[1]   For an incompressible material   and we have   For any strain energy density function   to reduce to the above forms for small strains the following conditions have to be met[1]  

If the material is incompressible, then the above conditions may be expressed in the following form.   These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.

Consistency conditions for incompressible I1 based rubber materials

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Many elastomers are modeled adequately by a strain energy density function that depends only on  . For such materials we have  . The consistency conditions for incompressible materials for   may then be expressed as   The second consistency condition above can be derived by noting that   These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.

References

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  1. ^ a b c d e R.W. Ogden, 1984, Non-Linear Elastic Deformations, ISBN 0-486-69648-0, Dover.
  2. ^ Muhr, A. H. (2005). "Modeling the stress–strain behavior of rubber". Rubber Chemistry and Technology. 78 (3): 391–425. doi:10.5254/1.3547890.
  3. ^ Gao, H; Ma, X; Qi, N; Berry, C; Griffith, BE; Luo, X (2014). "A finite strain nonlinear human mitral valve model with fluid-structure interaction". Int J Numer Methods Biomed Eng. 30 (12): 1597–613. doi:10.1002/cnm.2691. PMC 4278556. PMID 25319496.
  4. ^ Jia, F; Ben Amar, M; Billoud, B; Charrier, B (2017). "Morphoelasticity in the development of brown alga Ectocarpus siliculosus: from cell rounding to branching". J R Soc Interface. 14 (127): 20160596. doi:10.1098/rsif.2016.0596. PMC 5332559. PMID 28228537.
  5. ^ Arruda, E.M.; Boyce, M.C. (1993). "A three-dimensional model for the large stretch behavior of rubber elastic materials" (PDF). J. Mech. Phys. Solids. 41: 389–412. doi:10.1016/0022-5096(93)90013-6. S2CID 136924401.
  6. ^ Buche, M.R.; Silberstein, M.N. (2020). "Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble". Phys. Rev. E. 102 (1): 012501. arXiv:2004.07874. Bibcode:2020PhRvE.102a2501B. doi:10.1103/PhysRevE.102.012501. PMID 32794915. S2CID 215814600.
  7. ^ Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.
  8. ^ Fox & Kapoor, Rates of change of eigenvalues and eigenvectors, AIAA Journal, 6 (12) 2426–2429 (1968)
  9. ^ Friswell MI. The derivatives of repeated eigenvalues and their associated eigenvectors. Journal of Vibration and Acoustics (ASME) 1996; 118:390–397.

See also

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