# zbMATH — the first resource for mathematics

Kink-type solitary waves within the quasi-linear viscoelastic model. (English) Zbl 07215428
Summary: The quasi-linear model of viscoelasticity is a constitutive law widely used to investigate the time dependent behaviour of soft tissues and bio-materials. For this model, we study the shearing motion and discuss the existence of kink-type wave solutions. In particular, we derive a nonlinear second-order ordinary differential equation which allows to widen the class of solutions given by A. M. Samsonov [Appl. Anal. 57, No. 1–2, 85–100 (1995; Zbl 0879.35111)]. When the stress relaxation function is a Prony series, kink-wave solutions can exist for strongly elliptic strain energy functions, except for the Mooney-Rivlin model. We provide numerical simulations for the Yeoh model.

##### MSC:
 35 Partial differential equations 74 Mechanics of deformable solids
Full Text:
##### References:
 [1] Fung, Y. C., Biomechanics: Mechanical Properties of Living Tissues (1981), Springer-Verlag: Springer-Verlag New York [4] De Pascalis, R.; Abrahams, I. D.; Parnell, W. J., Simple shear of a compressible quasilinear viscoelastic material, Internat. J. Engrg. Sci., 88, Supplement C, 64-72 (2015), special Issue on Qualitative Methods in Engineering Science. URL http://www.sciencedirect.com/science/article/pii/S0020722514002353 [5] De Pascalis, R.; Parnell, W. J.; Abrahams, I. D.; Shearer, T.; Daly, D. M.; Grundy, D., The inflation of viscoelastic balloons and hollow viscera, Proc. R. Soc. A, 474, 20180102 (2018) [6] Liu, T.-P., Nonlinear waves for viscoelasticity with fading memory, J. Differential Equations, 76, 1, 26-46 (1988) [7] Renardy, M.; Hrusa, W. J.; Nohel, J. A., Mathematical problems in viscoelasticity, (Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 35 (1987)) [8] Catheline, S.; Gennisson, J.-L.; Tanter, M.; Fink, M., Observation of shock transverse waves in elastic media, Phys. Rev. Lett., 91, 164301 (2003), URL https://link.aps.org/doi/101103/PhysRevLett.91164301 [9] Nariboli, G.; Lin, W., A new type of burgers’ equation, ZAMM-J. Appl. Math. Mech. Z. Angew. Math. Mech., 53, 8, 505-510 (1973) [10] Jordan, P.; Puri, A., A note on traveling wave solutions for a class of nonlinear viscoelastic media, Phys. Lett. A, 335, 2-3, 150-156 (2005) [11] Destrade, M.; Jordan, P. M.; Saccomandi, G., Compact travelling waves in viscoelastic solids, Europhys. Lett., 87, 4, 48001 (2009) [12] Pucci, E.; Saccomandi, G., Some remarks about a simple history dependent nonlinear viscoelastic model, Mech. Res. Commun., 68, 70-76 (2015), bruno Boley 90th Anniversary Issue. URL http://www.sciencedirect.com/science/article/pii/S0093641315000749 [13] Samsonov, A., Nonlinear strain waves in elastic waveguides, (Nonlinear Waves in Solids (1994), Springer), 349-382 [14] Dreiden, G. V.; Ostrovskii, I.; Samsonov, A. M.; Semenova, I. V.; Sokurinskaia, E., Formation and propagation of deformation solitons in a nonlinearly elastic solid, Zh. Tekh. Fiz., 58, 2040-2047 (1988) [15] Nariboli, G.; Sedov, A., Burgers’s-korteweg-de vries equation for viscoelastic rods and plates, J. Math. Anal. Appl., 32, 3, 661-677 (1970) [16] Samsonov, A. M., Travelling wave solutions for nonlinear dispersive equations with dissipation, Appl. Anal., 57, 1-2, 85-100 (1995) [17] Pucci, E.; Saccomandi, G., On the nonlinear theory of viscoelasticity of differential type, Math. Mech. Solids, 17, 6, 624-630 (2012) [18] Samsonov, A. M., Travelling wave solutions for nonlinear dispersive equations with dissipation, Appl. Anal., 57, 1-2, 85-100 (1995), arXiv:https://doi.org/101080/00036819508840341 [19] Destrade, M.; Saccomandi, G., Finite amplitude elastic waves propagating in compressible solids, Phys. Rev. E, 72, 1, 016620 (2005) [20] Yeoh, O., Some forms of the strain energy function for rubber, Rubber Chem. Technol., 66, 5, 754-771 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.