Vol.:(0123456789) 1 3 Journal of Vibration Engineering & Technologies https://doi.org/10.1007/s42417-023-01234-7 ORIGINAL PAPER Characterization of the Preloaded Hyperelastic Materials Subjected to Linear Vibrations Bale Baidi Blaise 1  · Liman Kaoye 2  · Bosco Samon 2  · Gambo Betchewe 1  · Tibi Beda 3 Received: 25 June 2023 / Revised: 17 October 2023 / Accepted: 21 November 2023 © Springer Nature Singapore Pte Ltd. 2023 Abstract Background This paper proposes the study of the visco-hyperelastic behavior of a rubber sample. This rubber, coded as the BX rubber sample, is simultaneously loaded and subjected to linear vibrations. Methods A multiplicative non-separable variables law of the Nashif has been used to model the behavior that depends on both stretch and frequency. This method allows splitting the intricately combined test performed jointly on both stretch and frequency. On the one hand, we use Young’s complex modulus E ∗ () calculated from the experimental data, and on the other hand, the hyperelastic characteristics E() of the same material obtained from the experimental tensile curve. The hyperelastic phenomenological Gent–Thomas model and the hyperelastic molecular Flory–Erman model are used to evaluate the combined complex modulus E ∗ (, ) . Results We obtain results that go in the physical sense, i.e, Young’s modulus increases when the material is stretched, while the damping decreases. Keywords Visco-hyperelastic · Hyperelastic · Vibration · Multiplicative model · Combined modulus Introduction The elastomers belong to the great family of polymers and indicate today generally all rubbers, natural or synthetic. Because of their energy-dissipating nature, these mate- rials are used more in the feld of the vibration mechan- ics [1–6] and large deformation [7–12], more specifcally in the suppression of noise and vibrations and also in the calculations of structures using elastomers. Generally, the mechanical behavior of rubber-like materials depends either on the frequency or the temperature, or even on the stress applied. Thus, the behavior law is sometimes defned in quasi-static mode or the dynamic mode. In quasi-static mode, these materials can undergo large deformations and return to their initial form without permanent deformation [13–16]. In dynamics, elastomeric materials exhibit a fre- quency and time behavior having the characteristics of a spring [17]. The study of elastomer’s behavior in large defor- mations under a dynamics regime remains less known and developed. The most well-known and cited research works in this area are the works of [1, 18, 19]. The expression of the constitutive law of this complex study is defned by two approaches. The frst approach is given by Padovan [18]. It is a theory based on the summation of the efects of the vari- ous deformations, i.e., Small vibrations and hyperelasticity. The second approach is that given by Nashif, Jones, and Henderson [1]. This approach consists of separating the vari- ables, i.e., separating the frequencies of the extensions. Tibi Beda et al. [20] used the second approach to study the linear vibrations of a structure subjected to large static deforma- tion. In their work, they used in the constitutive law the two * Bale Baidi Blaise balebaidi.blaise@yahoo.com Liman Kaoye limankaoylims@gmail.com Bosco Samon jboscosamon@gmail.com Gambo Betchewe gambobetch@yahoo.fr Tibi Beda tbeda@yahoo.com 1 Faculty of science, University of Maroua, Maroua, Cameroon 2 Department of Mechanical Engineering, ENSAI, University of Ngaounderé, Ngaoundéré, Cameroon 3 E.N.S.P.Y. (National Advanced School of Engineering of University of Yaounde I), Yaounde, Cameroon