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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.
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.
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[edit]
Saint Venant–Kirchhoff model[edit]
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
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[edit]
Hyperelastic material models can be classified as:
- phenomenological descriptions of observed behavior
- Fung
- Mooney–Rivlin
- Ogden
- Polynomial
- Saint Venant–Kirchhoff
- Yeoh
- Marlow
- mechanistic models deriving from arguments about underlying structure of the material
- Arruda–Boyce model[5]
- Neo–Hookean model[1]
- Buche–Silberstein model[6]
- 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[edit]
Compressible hyperelastic materials[edit]
First Piola–Kirchhoff stress[edit]
If is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as
Second Piola–Kirchhoff stress[edit]
If is the second Piola–Kirchhoff stress tensor then
Cauchy stress[edit]
Similarly, the Cauchy stress is given by
Incompressible hyperelastic materials[edit]
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:
Expressions for the Cauchy stress[edit]
Compressible isotropic hyperelastic materials[edit]
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
The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by
If, in addition, , we have and hence
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
To express the Cauchy stress in terms of the invariants recall that
To express the Cauchy stress in terms of the stretches recall that
If we express the stress in terms of differences between components,
Incompressible isotropic hyperelastic materials[edit]
For incompressible isotropic hyperelastic materials, the strain energy density function is . The Cauchy stress is then given by
Consistency with linear elasticity[edit]
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[edit]
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:
If the material is incompressible, then the above conditions may be expressed in the following form.
Consistency conditions for incompressible I1 based rubber materials[edit]
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
References[edit]
- ^ a b c d e R.W. Ogden, 1984, Non-Linear Elastic Deformations, ISBN 0-486-69648-0, Dover.
- ^ Muhr, A. H. (2005). "Modeling the stress–strain behavior of rubber". Rubber Chemistry and Technology. 78 (3): 391–425. doi:10.5254/1.3547890.
- ^ 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 Method Biomed Eng. 30 (12): 1597–613. doi:10.1002/cnm.2691. PMC 4278556. PMID 25319496.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.
- ^ Fox & Kapoor, Rates of change of eigenvalues and eigenvectors, AIAA Journal, 6 (12) 2426–2429 (1968)
- ^ Friswell MI. The derivatives of repeated eigenvalues and their associated eigenvectors. Journal of Vibration and Acoustics (ASME) 1996; 118:390–397.