Browsing by Author "Şengül, Y."

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    Dispersive transverse waves for a strain-limiting continuum model
    (Sage, 2023-10) Erbay, Hüsnü Ata; Rajagopal, K. R.; Saccomandi, G.; Şengül, Y.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata
    It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.
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    Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity
    (Elsevier, 2020-11-15) Erbay, Hüsnü Ata; Erkip, A.; Şengül, Y.; Natural and Mathematical Sciences; ERBAY, Hüsnü Ata
    In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized strain, the rate of change of the linearized strain and the stress. The model is a generalization of the nonlinear Kelvin-Voigt viscoelastic solid under the assumption that the strain and the strain rate are small. We define an initial-value problem for the stress variable and then, under the assumption that the nonlinear constitutive function is strictly increasing, we convert the problem to a new form for the sum of the strain and the strain rate. Using the theory of variable coefficient heat equation together with a fixed point argument we prove local existence of solutions. Finally, for several constitutive functions widely used in the literature we show that the assumption on which the proof of existence is based is not violated.