A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials


Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress \(\mathsf{P}_{\text{act}}\) in the active stress case and a multiplicative strain \(\mathsf{F}_{a}\) in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears.

Considering an incompressible and transversely isotropic material, we design constitutive relations for \(\mathsf{P}_{\text{act}}\) and \(\mathsf{F}_{a}\) so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data.

Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials.

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  1. 1.

    Ambrosi, D., Pezzuto, S.: Active stress vs. active strain in mechanobiology: constitutive issues. J. Elast. 107, 199–212 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Ambrosi, D., Arioli, G., Nobile, F., Quarteroni, A.: Electromechanical coupling in cardiac dynamics: the active strain approach. SIAM J. Appl. Math. 71(2), 605–621 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Blemker, S.S., Pinsky, P.M., Delp, S.L.: A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii. J. Biomech. 38(4), 657–665 (2005)

    Article  Google Scholar 

  4. 4.

    Ehret, A.E., Böl, M., Itskov, M.: A continuum constitutive model for the active behaviour of skeletal muscle. J. Mech. Phys. Solids 59(3), 625–636 (2011)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Giantesio, G., Musesti, A.: A continuum model of skeletal muscle tissue with loss of activation. In: Gerisch, A., Penta, R., Lang, J. (eds.) Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications. Lecture Notes in Computational Science and Engineering, vol. 122, pp. 139–159. Springer, Berlin (2017)

    Chapter  Google Scholar 

  6. 6.

    Giantesio, G., Musesti, A.: Strain-dependent internal parameters in hyperelastic biological materials. Int. J. Non-Linear Mech. 95, 162–167 (2017)

    ADS  Article  Google Scholar 

  7. 7.

    Giantesio, G., Marzocchi, A., Musesti, A.: Loss of mass and performance in skeletal muscle tissue: a continuum model. Commun. Appl. Ind. Math. 9(1), 1–19 (2018)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Hawkins, D., Bey, M.: A comprehensive approach for studying muscle-tendon mechanics. ASME J. Biomech. Eng. 116, 51–55 (1994)

    Article  Google Scholar 

  9. 9.

    Heidlauf, T., Röhrle, O.: Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library OpenCMISS. Comput. Math. Methods Med. 2013, 1–14 (2013)

    MATH  Article  Google Scholar 

  10. 10.

    Heidlauf, T., Röhrle, O.: On the treatment of active behaviour in continuum muscle mechanics. PAMM 13(1), 71–72 (2013)

    MATH  Article  Google Scholar 

  11. 11.

    Heidlauf, T., Röhrle, O.: A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements. Front. Physiol. 5, 498 (2014)

    Article  Google Scholar 

  12. 12.

    Hernández-Gascón, B., Grasa, J., Calvo, B., Rodríguez, J.: A 3D electro-mechanical continuum model for simulating skeletal muscle contraction. J. Theor. Biol. 335, 108–118 (2013)

    MATH  Article  Google Scholar 

  13. 13.

    Horgan, C.O., Murphy, J.G.: Simple shearing of soft biological tissues. Proc. R. Soc., Math. Phys. Eng. Sci. 467, 760–777 (2011)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Kondaurov, V.I., Nikitin, L.V.: Finite strains of viscoelastic muscle tissue. J. Appl. Math. Mech. 51(3), 346–353 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Martins, J., Pires, E., Salvado, R., Dinis, P.: A numerical model of passive and active behavior of skeletal muscles. Comput. Methods Appl. Mech. Eng. 151(3–4), 419–433 (1998)

    ADS  MATH  Article  Google Scholar 

  16. 16.

    Morrow, D.A., Donahue, T.L.H., Odegard, G.M., Kaufman, K.R.: Transversely isotropic tensile material properties of skeletal muscle tissue. J. Mech. Behav. Biomed. Mater. 3(1), 124–129 (2010)

    Article  Google Scholar 

  17. 17.

    Nardinocchi, P., Teresi, L.: On the active response of soft living tissues. J. Elast. 88(1), 27–39 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Neff, P.: Some results concerning the mathematical treatment of finite plasticity. In: Deformation and Failure in Metallic Materials, pp. 251–274. Springer, Berlin (2003)

    Chapter  Google Scholar 

  19. 19.

    Odegard, G.M., Haut Donahue, T.L., Morrow, D.A., Kaufman, K.R.: Constitutive modeling of skeletal muscle tissue with an explicit strain-energy function. J. Biomech. Eng. 130, 061017 (2008)

    Article  Google Scholar 

  20. 20.

    Paetsch, C., Dorfmann, L.: Stability of active muscle tissue. J. Eng. Math. 95(1), 193–216 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Pathmanathan, P., Chapman, S.J., Gavaghan, D.J., Whiteley, J.P.: Cardiac electromechanics: the effect of contraction model on the mathematical problem and accuracy of the numerical scheme. Q. J. Mech. Appl. Math. 63(3), 375 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Pezzuto, S., Ambrosi, D., Quarteroni, A.: An orthotropic active-strain model for the myocardium mechanics and its numerical approximation. Eur. J. Mech. A, Solids 48, 83–96 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Rossi, S., Ruiz-Baier, R., Pavarino, L.F., Quarteroni, A.: Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int. J. Numer. Methods Biomed. Eng. 28(6–7), 761–788 (2012)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Taber, L.A., Perucchio, R.: Modeling heart development. J. Elast. 61(1), 165–197 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Weickenmeier, J., Itskov, M., Mazza, E., Jabareen, M.: A physically motivated constitutive model for 3D numerical simulation of skeletal muscles. Int. J. Numer. Methods Biomed. Eng. 30(5), 545–562 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Wilkie, D.R.: The mechanical properties of muscle. Br. Med. Bull. 12(3), 177–182 (1956)

    Article  Google Scholar 

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The authors thank the anonymous reviewers for their comments and suggestions.

This work has been partially supported by National Group of Mathematical Physics (GNFM-INdAM).

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Correspondence to Alessandro Musesti.

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Giantesio, G., Musesti, A. & Riccobelli, D. A Comparison Between Active Strain and Active Stress in Transversely Isotropic Hyperelastic Materials. J Elast 137, 63–82 (2019). https://doi.org/10.1007/s10659-018-9708-z

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  • Activation
  • Hyperelasticity
  • Simple shear

Mathematics Subject Classification

  • 74B20
  • 74L15
  • 74E10