Abstract
An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.
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Paetsch C., Trimmer B.A., Dorfmann A.: A constitutive model for active-passive transition of muscle fibers. Int. J. Nonlinear Mech. 47, 377–387 (2012)
Paetsch C., Dorfmann A.: Non-linear modeling of active biohybrid materials. Int. J. Nonlinear Mech. 56, 105–114 (2013)
Paetsch, C., Dorfmann, L.: Stability of active muscle tissue. J. Eng. Math. (2015). doi:10.1007/s10665-014-9750-1
Dorfmann A., Trimmer B.A., Woods W.A.: A constitutive model for muscle properties in a soft-bodied arthropod. J. R. Soc. Interface 4, 257–269 (2007)
Wineman A.S., Rajagopal K.R.: Mechanical response of polymers. Cambridge Press, Cambridge (2000)
Christensen R.M.: Theory of viscoelasticty. Dover, NY (1982)
Proske U., Morgan D.L.: Do cross-bridges contribute to the tension during stretch of passive muscle?. J. Muscle Res. Cell Motil. 20, 433–442 (1999)
Mutungi G., Ranatunga K.W.: Do cross-bridges contribute to the tension during stretch of passive muscle? A response. J. Muscle Res. Cell Motil. 21, 301–302 (2000)
Bagni M.A., Colombini B., Geiger P., Berlinguer Palmini R., Cecchi G.: Non-cross-bridge calcium-dependent stiffness in frog muscle fibers. J. Physiol. Lond. 286, 1353–1357 (2004)
Campbell K.S., Lakie M.: A cross-bridge mechanism can explain the thixotropic short-range elastic component of relaxed frog skeletal muscle. J. Physiol. Lond. 510, 941–962 (1998)
Granzier H.L., Labeit S.: The giant protein titin: a major player in myocardial mechanics, signaling, and disease. Circ. Res. 94, 284–295 (2004)
Granzier H.L., Wang K.: Interplay between passive tension and strong and weak binding cross-bridges in insect indirect flight muscle. A functional dissection by gelsolin-mediated thin filament removal. J. Gen. Physiol. 101, 235–270 (1993)
Granzier H.L., Wang K.: Passive tension and stiffness of vertebrate skeletal and insect flight muscles: The contribution of weak cross-bridges and elastic filaments. Biophys. J. 65, 2141–2159 (1993)
Gosline J., Lillie M., Carrington E., Guerette P., Ortlepp C., Savage K.: Elastic proteins: Biological roles and mechanical properties. Philos. Trans. R. Soc. Lond. B 357, 121–132 (2002)
Lieber R.L., Leonard M.E., Brown-Maupin C.G.: Effects of muscle contraction on the load-strain properties of frog aponeurosis and tendon. Cells Tissues Organs 166, 48–54 (2000)
Powers, K., Schappacher-Tilp, G., JinhaA. Leonard, T., Nishikawa, K., Herzog, W.: Titin force is enhanced in actively stretched skeletal muscle J. Exp. Biol. 217, 3629–3636 (2014)
Gautel M.: The sarcomeric cytoskeleton: who picks up the strain?. Curr. Opin. Cell Biol. 23, 39–46 (2011)
Ortega, J.O., Lindstedt, S.L., Nelson, F.E., Jubrias, S.A., Kushmerick, M.J., Conley, K.E.: Muscle force, work and cost: a novel technique to revisit the Fenn effect. J. Exp. Biol. doi:10.1242/jeb114512 (2015)
Hill A.V.: The heat of shortening and the dynamic constants of muscle. Proc. R. Soc. Lond. B 126, 136–195 (1938)
Martins J.A.C., Pires E.B., Salvado R., Dinis P.B.: A numerical model of passive and active behavior of skeletal muscles. Comput. Methods Appl. Mech. Eng. 151, 419–433 (1998)
Parente M.P.L., Natal Jorge R.M., Mascarenhas T., Fernandes A.A., Martins J.A.C.: The influence of the material properties on the biomechanical behavior of the pelvic floor muscles during vaginal delivery. J. Biomech. 42, 1301–1306 (2009)
Tang C.Y., Zhang G., Tsui C.P.: A 3D skeletal muscle model coupled with active contraction of muscle fibres and hyperelastic behavior. J. Biomech. 126, 865–872 (2009)
Palevski A., Glaich I., Portnoy S., Linder-Ganz E., Gefen A.: Stress relaxation of porcine gluteus muscle subjected to sudden transverse deformation as related to pressure sore modeling. J. Biomech. 128, 782–787 (2006)
Van Loocke M., Lyons C.G., Simms C.K.: Viscoelastic properties of passive skeletal muscle in compression: stress-relaxation behaviour and constitutive modelling. J. Biomech. 41, 1555–1566 (2008)
Meyer G.A., McCulloch A.D., Lieber R.L.: A nonlinear model of passive muscle viscosity. J. Appl. Mech. Trans. ASME 133(091007), 1–9 (2011)
Rehorn M.R., Schroer A.K., Blemker S.S.: The passive properties of muscle fibers are velocity dependent. J. Biomech. 47, 687–693 (2014)
Bosboom E.M.H., Hesselink M.K.C., Oomens C.W.J., Bouten C.V.C., Drost M.R., Baaijens F.P.T.: Passive transverse mechanical properties of skeletal muscle under in vivo compression. J. Biomech. 34, 1365–1368 (2001)
Van Loocke M., Lyons C.G., Simms C.K.: A validated model of passive muscle in compression. J. Biomech. 39, 2999–3009 (2006)
Van Loocke M., Lyons C.G., Simms C.K.: Viscoelastic properties of passive skeletal muscle in compression: Cyclic behaviour. J. Biomech. 42, 1038–1048 (2009)
Röhrle O., Pullan A.J.: Three-dimensional finite element modelling of muscle forces during mastication. J. Biomech. 40, 3363–3372 (2007)
Ito D., Tanaka E., Yamamoto S.: A novel constitutive model of skeletal muscle taking into account anisotropic damage. J. Mech. Behav. Biomed. Mater. 3, 85–93 (2010)
Lu Y.T., Zhu H.X., Richmond S., Middleton J.: A visco-hyperelastic model for skeletal muscle tissue under high strain rates. J. Biomech. 43, 2629–2632 (2010)
Calvo B., Sierra M., Grasa J., Mu noz M.J., Pe na E.: Determination of passive viscoelastic response of the abdominal muscle and related constitutive modeling: stress-relaxation behavior. J. Mech. Behav. Biomed. Mat. 36, 47–58 (2014)
Dorfmann A.L., Woods W.A., Trimmer B.A.: Muscle performance in a soft-bodied terrestrial crawler: constitutive modeling of strain-rate dependency. J. R. Soc. Interface 5, 349–362 (2008)
Hunter P.J., McCulloch A.D., ter Keurs H.E.D.J.: Modelling the mechanical properties of cardiac muscle. Prog. Biophys. Mol. Biol. 69, 289–331 (1998)
Hollenstein M., Jabareen M., Rubin M.B.: Modeling a smooth elastic-inelastic transition with a strongly objective numerical integrator needing no iteration. Comput. Mech. 52, 649–667 (2013)
Perzyna P.: The constitutive equations for rate sensitive plastic materials. Q. Appl. Math. 20, 321–332 (1963)
Lubliner L., Taylor R.L., Auricchio F.: A new model of generalized plasticity and its numerical implementation. Int. J. Solids Struct. 30, 3171–3184 (1993)
Panoskaltsis V.P., Polymenakos L.C., Soldatos D.: On large deformation generalized plasticity. J. Mech. Mater. Struct. 3, 441–457 (2008)
Einav I.: The unification of hypoplastic and elasto-plastic theories. Int. J. Solids Struct. 49, 1305–1315 (2012)
Flory P.J.: Thermodynamic relations for highly elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)
Ogden R.W.: Nearly isochoric elastic deformations: application to rubberlike solids. J. Mech. Phys. Solids 26, 37–57 (1978)
Rubin M.B., Attia A.: Calculation of hyperelastic response of finitely deformed elastic-viscoplastic materials. Int. J. Numer. Methods Eng. 39, 309–320 (1996)
Rubin M.B.: A simple and convenient isotropic failure surface. ASCE J. Eng. Mech. 117, 348–369 (1991)
Rubin M.B., Papes O.: Advantages of formulating evolution equations for elastic-viscoplastic materials in terms of the velocity gradient instead of the spin tensor. J. Mech. Mater. Struct. 6, 529–543 (2011)
Simo J.C.: Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput. Methods Appl. Mech. Eng. 99, 61–112 (1992)
Simo J.C., Hughes T.J.R.: Computational inelasticity. Springer, New York (1998)
Papes, O.: Nonlinear continuum mechanics in engineering applications. Ph. D. dissertation DISS ETH NO 19956, ETH Zurich (2012)
Dorfmann A., Ogden R.W.: A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber. Int. J. Solids Struct. 40, 2699–2714 (2003)
Dorfmann A., Ogden R.W.: A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber. Int. J. Solids Struct. 41, 1855–1878 (2004)
Dorfmann A., Pancheri F.Q.: A constitutive model for the Mullins effect with changes in material symmetry. Int. J. Nonlin. Mech. 47, 874–887 (2012)
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Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell’Isola.
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Ahamed, T., Rubin, M.B., Trimmer, B.A. et al. Time-dependent behavior of passive skeletal muscle. Continuum Mech. Thermodyn. 28, 561–577 (2016). https://doi.org/10.1007/s00161-015-0464-z
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DOI: https://doi.org/10.1007/s00161-015-0464-z