A finite element for soft tissue deformation based on the absolute nodal coordinate formulation

  • 34 Accesses


This paper introduces an implementation of the absolute nodal coordinate formulation (ANCF) that can be used to model fibrous soft tissue in cases of three-dimensional elasticity. It is validated against results from existing incompressible material models. The numerical results for large deformations based on this new ANCF element are compared to results from analytical and commercial software solutions, and the relevance of the implementation to the modeling of biological tissues is discussed. Also considered is how these results relate to the classical results seen in Treloar’s rubber experiments. All the models investigated are considered from both elastic and static points of view. For isotropic cases, neo-Hookean and Mooney–Rivlin models are examined. For the anisotropic case, the Gasser–Ogden–Holzapfel model, including a fiber dispersion variation, is considered. The results produced by the subject ANCF models agreed with results obtained from the commercial software. For the isotropic cases, in fact, the numerical solutions based on the ANCF element were more accurate than those produced by ANSYS.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    Bauchau, O.A., Han, S., Mikkola, A., Matikainen, M.K.: Comparison of the absolute nodal coordinate and geometrically exact formulations for beams. Multibody Syst. Dyn. 32, 67–85 (2014)

  2. 2.

    Bauchau, O.A., Han, S., Mikkola, A., Matikainen, M.K., Gruber, P.: Experimental validation of flexible multibody dynamics beam formulations. Multibody Syst. Dyn. 34, 373–389 (2015)

  3. 3.

    Bauchau, O.A., Wu, G., Betsch, P., Cardona, A., Gerstmayr, J., Jonker, J.B., Masarati, P., Sonneville, V.: Validation of flexible multibody dynamics beam formulations using benchmark problems. Multibody Syst. Dyn. 37, 29–48 (2016)

  4. 4.

    Betsch, P. (ed.): Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics. In: CISM International Centre for Mechanical Sciences, vol 565, 1st edn. Springer, Berlin (2016)

  5. 5.

    Bozorgmehri, B., Hurskainen, V.V., Matikainen, M.K., Mikkola, A.: Dynamic analysis of rotating shafts using the absolute nodal coordinate formulation. J. Sound Vib. 453, 214–236 (2019)

  6. 6.

    Doll, S., Schweizerhof, K.: On the development of volumetric strain energy functions. J. Appl. Mech. 67, 17–21 (1999)

  7. 7.

    Ebel, H., Matikainen, M.K., Hurskainen, V.V., Mikkola, A.: Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity. Nonlinear Dyn. 88, 1075–1091 (2017)

  8. 8.

    Escalona, J.L., Hussien, H.A., Shabana, A.A.: Application of absolute nodal co-ordinate formulation to multibody system dynamics. J. Sound Vib. 214, 833–851 (1998)

  9. 9.

    Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)

  10. 10.

    Franchi, M., Trirè, A., Quaranta, M., Orsini, E., Ottani, V.: Collagen structure of tendon relates to function. Sci. World J. 7, 404–420 (2007)

  11. 11.

    Gasser, T.C., Ogden, W.R., Holzapfel, G.A.: Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface 3, 15–35 (2006)

  12. 12.

    Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. J. Comput. Nonlinear Dyn. 8, 031016 (2013)

  13. 13.

    Gonçalves, P.B., Pamplona, D., Lopes, S.R.X.: Finite deformations of an initially stressed cylindrical shell under internal pressure. Int. J. Mech. Sci. 5, 92–103 (2008)

  14. 14.

    Grossi, E., Shabana, A.A.: Analysis of high-frequency ANCF modes: Navier-stokes physical damping and implicit numerical integration. Acta Mech. 230, 2581–2605 (2019)

  15. 15.

    Holzapfel, G.: Determination of material models for arterial walls from uniaxial extension tests and histological structure. J. Theor. Biol. 238, 290–302 (2006)

  16. 16.

    Holzapfel, G.A., Gasser, T.C.: A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput. Methods Appl. Mech. Eng. 190, 4379–4403 (2001)

  17. 17.

    Holzapfel, G.A., Ogden, R.W.: Constitutive modelling of arteries. Proc. Math. Phys. Eng. Sci. 466, 1551–1597 (2010)

  18. 18.

    Horgan, C.O., Saccomandi, G.: Constitutive modelling of rubber-like and biological materials with limiting chain extensibility. Math. Mech. Solids 7, 353–371 (2002)

  19. 19.

    Khayyeri, H., Longo, G., Gustafsson, A., Isaksson, H.: Comparison of structural anisotropic soft tissue models for simulating achilles tendon tensile behaviour. J. Mech. Behav. Biomed. Mater. 61, 431–443 (2016)

  20. 20.

    Kulkarni, S., Shabana, A.A.: Spatial ANCF/CRBF beam elements. Acta Mech. 230, 929–952 (2019)

  21. 21.

    Li, W.: Biomechanical property and modelling of venous wall. Prog. Biophys. Mol. Biol. 133, 56–75 (2018)

  22. 22.

    Lu, S.H.C., Pister, K.S.: Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. Int. J. Solids Struct. 11, 927–934 (1975)

  23. 23.

    Maqueda, L., Shabana, A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. Multibody Syst. Dyn. 18, 375–396 (2007)

  24. 24.

    Maqueda, L.G., Shabana, A.A.: Nonlinear constitutive models and the finite element absolute nodal coordinate formulation. In: ASME Proc. 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C, vol. 5, pp. 1033–1037 (2007)

  25. 25.

    Maqueda, L.G., Bauchau, O.A., Shabana, A.A.: Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study. Multibody Syst. Dyn. 19, 281–302 (2008)

  26. 26.

    Meister, T.A., Rexhaj, E., Rimoldi, S.F., Scherrer, U., Sartori, C.: Fetal programming and vascular dysfunction. Artery Res. 21, 69–77 (2018)

  27. 27.

    Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9, 283–309 (2003)

  28. 28.

    Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11, 582–592 (1940)

  29. 29.

    Nachbagauer, K.: State of the art of ANCF elements regarding geometric description, interpolation strategies, definition of elastic forces, validation and the locking phenomenon in comparison with proposed beam finite element. Arch. Comput. Methods Eng. 21, 293–319 (2014)

  30. 30.

    Nachbagauer, K., Pechstein, A.S., Irschik, H., Gerstmayr, J.: A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 26(3), 245–263 (2011)

  31. 31.

    Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to static and linearized dynamic examples. J. Comput. Nonlinear Dyn. 8, 021004 (2013)

  32. 32.

    Nah, C., Lee, G.B., Lim, J., Kim, Y., SenGupta, R., Gent, A.: Problems in determining the elastic strain energy function for rubber. Int. J. Non-Linear Mech. 45, 232–235 (2010)

  33. 33.

    Ogden, R.W.: Elastic ddformations of rubberlike solids. In: Hopkins, H.G., Sewell, M.J. (eds.) Mechanics of Solids, Mechanics of Solids: The Rodney Hill 60th Anniversary, pp. 499–537. Elsevier, Amsterdam (1982)

  34. 34.

    Orzechowski, G., Fra̧czeks, J.: Nearly incompressible nonlinear material models in the large deformation analysis of beams using ANCF. Nonlinear Dyn. 82, 451–464 (2015)

  35. 35.

    Patel, M., Shabana, A.A.: Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech. 229, 2923–2946 (2018)

  36. 36.

    Pierre, B., Stéphane, A., Susan, L., Michael, S.: Mechanical identification of hyperelastic anisotropic properties of mouse carotid arteries. In: Proulx, T. (ed.) Mechanics of Biological Systems and Materials, vol. 2, pp. 11–17. Springer, New York (2011)

  37. 37.

    Rachev, A., Greenwald, S.E.: Residual strains in conduit arteries. J. Biomech. 36, 661–670 (2003)

  38. 38.

    Rivlin, R.S.: Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 241, 379–397 (1948)

  39. 39.

    Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997)

  40. 40.

    Shen, Z., Li, P., Liu, C., Hu, G.: A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation. Nonlinear Dyn. 77(3), 1019–1033 (2014)

  41. 41.

    Shmurak, M.I., Kuchumov, A.G., Voronova, N.O.: Hyperelastic models analysis for description of soft human tissues behavior. Master’s J. 1, 230–243 (2017)

  42. 42.

    Simo, J.C., Taylor, R.L., Pister, K.S.: Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 51, 177–208 (1985)

  43. 43.

    Skarel, P., Bursa, J.: Comparison of constitutive models of arterial layers with distributed collagen fibre orientations. Acta Bioeng. Biomech. 16, 47–58 (2014)

  44. 44.

    Sokhanvar, S., Dargahi, J., Packirisamy, M.: Hyperelastic modelling and parametric study of soft tissue embedded lump for mis applications. Int. J. Med. Robot. Comput. Assist. Surg. 4, 232–241 (2008)

  45. 45.

    Steinmann, P., Hossain, M., Possart, G.: Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Arch. Appl. Mech. 89, 1183–1217 (2012)

  46. 46.

    Treloar, L.R.G.: Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70 (1944)

  47. 47.

    Weiss, J.A., Gardiner, J.C.: Computational modeling of ligament mechanics. Crit. Rev. Biomed. Eng. 29, 303–371 (2001)

  48. 48.

    Weiss, J.A., Maker, B.N., Govindjee, S.: Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Methods Appl. Mech. Eng. 135, 107–128 (1996)

Download references


We would like to thank the Research Foundation of the Lappeenranta University of Technology and the Academy of Finland (Application No. 299033 for funding 519 of Academy Research Fellow) for the generous grants that made this work possible.

Author information

Correspondence to Ajay B. Harish.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Analytical solution

Appendix A: Analytical solution

The analytical solution which we use in this paper has been outlined here. Firstly, we assume an object without any holes or different types of imperfections. Also, we assume that the material is isotropic in nature. Let us now consider the total volumes of them in the initial configuration, which can be expressed for cylindrical or rectangular bars in the forms

$$\begin{aligned} V_\mathrm{cyl}=\pi LR^2, V_\mathrm{rec}=HWL. \end{aligned}$$

Upon application of load, the largest dimensions of them, i.e., let us assume it to be L, changes by \(\lambda \) times. If we consider the material to be incompressible, then the volume of the object does not change; from here for the cylinder, we have received \(r=\frac{R}{\sqrt{\lambda }}\), and in the rectangular cross-sectional case, we have \(w=\frac{W}{\sqrt{\lambda }}\), \(h=\frac{H}{\sqrt{\lambda }}\), where rhw are dimensions of circular and rectangular cross sections in actual configurations.

The Cauchy stress tensor for the incompressible solids can be given as

$$\begin{aligned} \sigma =-p\mathbf{I }+2\mathbf{F }\left( \frac{\partial {\varPsi }}{\partial \mathbf{C }}\right) \mathbf{F }^T \end{aligned}$$

where \({\varPsi }\) is potential density function, p is a function of hydrostatic stress (which is not determined by the deformation). \(\mathbf{C }\) is the right Cauchy–Green tensor. However, p is not established from deformation; it is possible to receive it from boundary conditions. Our deformation occurs along one of the axes, let’s name this axis z, the components of stress tensors in others, i.e., x and y are equal to zero. From this condition, the form of p is possible to derive and then substitute into \(\sigma _{zz}\). The final expressions for the applied loads from which we can obtain the values of \(\lambda \) and as a result define the total displacements are

$$\begin{aligned} N_\mathrm{cyl}=2\pi \int _{0}^{r} \sigma _{zz} r\mathrm{d}r, N_\mathrm{rec}=\int _{0}^{h} \int _{0}^{w} \sigma _{zz} \mathrm{d}x\mathrm{d}y. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Obrezkov, L.P., Matikainen, M.K. & Harish, A.B. A finite element for soft tissue deformation based on the absolute nodal coordinate formulation. Acta Mech (2020) doi:10.1007/s00707-019-02607-4

Download citation