ALEX 2018 Workshop: Abstracts

One-dimensional nonlinear viscoelasticity with limited strain

H. A. Erbay ${}^{(1)}$ , A. Erkip ${}^{(2)}$ , and Yasemin Şengül ${}^{(2)}$

(1) Özyeğin University, Department of Natural and Mathematical Sciences, Istanbul (Turkey)

(2) Sabancı University, Faculty of Engineering and Natural Sciences, Istanbul (Turkey)

We are interested in finding solutions of nonlinear differential equations describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal [1, 2, 3]. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We first look at traveling wave solutions that correspond to the heteroclinic connections between the two constant states, and establish conditions for the existence of such solutions, and find them explicitly, implicitly or numerically, for various forms of the non-linear constitutive relation [4]. Then we consider corresponding Cauchy and boundary-value problems from both modelling and analysis points of view.

Acknowledgments: This work is partially supported by TÜBİTAK-1001 Grant 116F093.

References

1 K. R. Rajagopal, On implicit constitutive theories, Appl. Math. 48 (2003), 279–319.
2 – , On a new class of models in elasticity, J. Math. Comput. Appl. 15 (2010), 506–528.
3 – , On the nonlinear elastic response of bodies in the small strain range, Acta. Mech. 225 (2014), 1545–1553.
4 H. A. Erbay, Y. Şengül, Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity, Int. J. Non-Linear Mech. 77 (2015), 61-68.