Keywords: Hencky's elasticity model, logarithmic strain, finite element procedures, compression test, tension-torsion tests, Poynting effect.

Hencky's elasticity model expresses the stored energy function in terms of the logarithmic strain tensor . Constitutive relations take the form

where is the right stretch tensor and is the conjugate stress. As noted by Anand [1,2], Bruhns et al. [3] and many others this formulation has some remarkable properties. What makes it especially attractive is that the structure of is usually very simple containing only a small number of material constants yet to give an accurate prediction in a wide range of deformation regimes. On the other hand, numerical implementation in the context of the finite element method requires delicate handling of the inherent polar decomposition, consistent tangents and other aspects associated with large strains. Incompressibility also plays an important role since the model is frequently used for rubber-like materials.

The paper starts with a careful proposition of constitutive equations. On the onset a general Lagrangean description, embodied in the above equation, is recalled. A general formula for , first discovered by Hoger in the eighties, is quite complex, nonetheless, the latter equation can be substantially simplified by introducing isotropic symmetry to . Using appropriate symmetry groups, it is proved that the back rotated Kirchhoff stress may be substituted for . Consequently, the Doyle-Ericksen formula enables one to convert this relation to Eulerian setting that involves the left stretch tensor in place of and the Kirchhoff stress in place of . In case of isotropy, we thus have

In the finite element method, the left stretch tensor must be computed at each integration point. Therefore, the computational efficiency of the polar decomposition is crucial. This decomposition is traditionally performed by using analytical formulas dependent on the principal stretches. In this paper, the Jacobi iteration method is employed instead, which yields simultaneously both the principal stretches and eigenvectors. Prescribing reasonable tolerances, the iterative scheme proved to be robust and in some situations even faster than the evaluation of the analytical solution.

It is difficult to derive the consistent tanget operator for the full Newton-Raphson procedure. For this and other reasons the quasi-Newton BFGS method is used to solve the nonlinear equilibrium equations. The method is less effective here than, for instance, in small-strain plasticity problems but it still posssesses very good numerical stability. The residual vector is calculated in the current configuration with the aid of the updated isoparametric transform.

For the solution of test examples, 20-node quadratic elements with eight integration points were used to prevent FE meshes from locking in near incompressible problems. No special precautions were taken against the hourglass instability. This arrangement worked well up to sixty percent strain reached in the simulated compression test.

In conclusion, simple tension, compression, simple shear and tension-torsion tests were analysed. Experimental data acquired by the authors from the Rubena rubber company were added to those discussed in [1,2,3,4]. The results and conclusions of this work can be stated as: i) Anand's judgement [1,2] on the accuracy of Hencky's energy function is supported by additional experiments. ii) Finite element procedures for the model's implementation described in this paper can be recommended.

1

Anand, L., "On H. Hencky's approximate strain-energy function for moderate deformations", ASME J. Appl. Mech., 46, 78-82, 1979.

2

Anand, L., "Moderate deformations in extension-torsion of incompressible isotropic elastic materials", J. Mech. Phys. Solids, 34, 293-304, 1986. doi:10.1016/0022-5096(86)90021-9

3

Bruhns, O.T., Xiao, H., Meyers, A., "Hencky's elasticity model with the logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods", Arch. Mech., 52, 489-509, 2000.

4

Hartmann, S., "Numerical studies on the identification of the material parameters of Rivlin's hyperelasticity using tension-torsion tests", Acta Mech., 148, 129-155, 2001. doi:10.1007/BF01183674

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