Keywords: hyperelasticity, splines, nonlinear elasticity, incompressible transversely isotropic materials, living tissues.

Hyperelastic formulations based on a stored energy function are essential in order to preserve energy during large but purely elastic deformations and to recover the original strains once stresses are released (or vice-versa). On the other hand, any constitutive model should be able to represent the experimental behavior as closely as possible. Ideally, the user of the model would simply prescribe some stress-strain data and, when possible, the three dimensional model should be able to replicate them when analysing the particular cases given by the experiments. Frequently, hyperelastic models based on explicit, given, stored energy functions do not have the desired flexibility to retain arbitrary experimental behaviour and, hence, the user sacrifices constitutive behaviour accuracy for mathematical and physical correctness.

In this sense, the Sussman-Bathe model [1] based on splines bridges this gap and presents a relevant advance in the WYPIWYG (what you prescribe is what you get) philosophy. The user prescribes some stress-strain experimental data and a hyperelastic model based on splines is created which fits that data, regardless of the complexity of the behaviour being modelled. An equally spaced domain discretisation gives the procedure an efficiency equivalent to that of models in which the shape of the stored energy function is given a priori.

However, the original Sussman-Bathe model is only valid for isotropic materials. In this work we extend their procedure for transversely isotropic incompressible materials, as for example one fibre composite materials or many living tissues. The procedure is developed using logarithmic stress and strain measures. Of course, as one should expect, the isotropic hyperelastic material model of Sussman and Bathe is completely recovered by our model when the test data corresponds to those of an isotropic material. The model has been used to obtain predictions for the experiments of Diani et al. [2] for transversely isotropic materials.

The conclusions given by Sussman and Bathe for their model are also applicable to our model, as the ability to capture some behaviour effects for which other models show less proficiency. Furthermore, the same limitations also apply to our model. Both tension and compression test data should be available and there is also a possibility of reproducing experimental errors arising from the close interpolation of the test data during the first steps of the procedure.

1

T. Sussman, K.J. Bathe, "A Model of Incompressible Isotropic Hyperelastic Material Behavior using Spline Interpolations of Tension-Compression Test Data", Communications in Numerical Methods in Engineering, 25, 53-63, 2009. doi:10.1002/cnm.1105

2

J. Diani, M. Brieu, J.M. Vacherand, A. Rezgui, "Directional model for isotropic and anisotropic hyperelastic rubber-like materials", Mechanics of Materials, 36, 313-321, 2004. doi:10.1016/S0167-6636(03)00025-5

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