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The circumferential shearing of a cylindrical annulus of viscoelastic solids described by implicit constitutive relations

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Abstract

In this paper, we use an implicit generalization of the celebrated Kelvin–Voigt solid constitutive equation to study the problem of circumferential shearing of a cylindrical annulus of viscoelastic material. This generalization of the Kelvin–Voigt model allows one to take into consideration the possibility of the body undergoing shear thinning or shear thickening and allows the material moduli to depend on the mechanical pressure. Most importantly, the constitutive relation is a relation in the true mathematical sense in that neither the stress, nor the strain nor the symmetric part of the velocity gradient can be expressed explicitly in terms of the other quantities, thereby making the balance of linear momentum and the constitutive relation requiring solution simultaneously, unlike the case of the Kelvin–Voigt model where we can substitute the constitutive expression for the stress into the equilibrium equations to obtain an equation for the displacement field.

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Notes

  1. The values for the constants in (3)–(5) can be compared with the values shown in Table 1 in [8] for a similar implicit model.

  2. The above values for \(\zeta \), \(\delta \) and \(\rho _r\) can be compared with the values used in [8].

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Acknowledgements

R. Bustamante would like to express his gratitude for the financial support provided by FONDECYT (Chile) under grant no. 1160030. K. R. Rajagopal thanks the National Science Foundation and the Office of Naval Research for support of this work.

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Authors and Affiliations

  1. Departamento de Ingeniería Mecánica, Universidad de Chile, Beauchef 851, Santiago Centro, Santiago, Chile

    R. Bustamante

  2. Department of Mechanical Engineering, University of Texas A&M, College Station, TX, USA

    K. R. Rajagopal

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  1. R. Bustamante
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  2. K. R. Rajagopal
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Correspondence to R. Bustamante.

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Bustamante, R., Rajagopal, K.R. The circumferential shearing of a cylindrical annulus of viscoelastic solids described by implicit constitutive relations. Acta Mech 232, 2679–2694 (2021). https://doi.org/10.1007/s00707-021-02968-9

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