The description of viscoelastic media with memory requires a thermodynamic formalism capable of capturing irreversible processes. In this work, a physicalmathematical model for anisotropic viscoelastic materials is developed based on the V. Ciancio—G.A. Kluitenberg theory. The generalized evolution equation is derived, showing its connection with modified Euler-Lagrange equations and the Rayleigh dissipation function. A direct and explicit reduction to the classical Maxwell model for viscoelasticity is demonstrated in the appropriate limit, confirming the consistency of the proposed framework. This work aims to offer a unified theoretical framework, providing a solid foundation for future developments.
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On the Evolution of Internal Variables for Anisotropic Viscoelastic Media
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Received: 02 May 2026 | Revised: 03 Jun 2026 | Accepted: 25 Jun 2026 | Published: 14 Jul 2026
Abstract
Keywords
thermodynamics of continuum | rheology | materials of stress-rate and internal-variable type | anisotropic | heat flow
References
- 1.
Meixner, J. Zur Thermodynamik der irreversiblen Prozesse. Ann. Phys. 1943, 435, 244–270.
- 2.
Kluitenberg, G.A. Thermodynamical theory of elasticity and plasticity. Physica 1962, 28, 217–232.
- 3.
Kluitenberg, G.A. On the thermodynamics of viscosity and plasticity. Physica 1963, 29, 633–652.
- 4.
Kluitenberg, G.A.; Ciancio, V. On linear dynamical equations of state for isotropic media -I- General formalism. Physica 1978, 93, 273–286.
- 5.
Ciancio, V.; Kluitenberg, G.A. On linear dynamical equations of state for isotropic media -II- Some cases of special interest. Physica 1979, 99, 592–600.
- 6.
Ciancio, V. Propagation and attenuation of singular wave surfaces in linear inelastic media. Physica 1979, 97, 127–138.
- 7.
Kluitenberg, G.A. On vectorial internal variables and dielectric and magnetic relaxation phenomena. Physica 1981, 109, 91–122.
- 8.
Ciancio, V. On the general Debye equation for media with dielectric relaxation phenomena described by vectorial internal variables. J. Non-Equilib. Thermodyn. 1989, 14, 239–250.
- 9.
Ciancio, V.; Kluitenberg, G.A. On electromagnetic waves in isotropic media with dielectric relaxation. Acta Phys. Hung. 1989, 66, 251–276.
- 10.
Ciancio, V.; Ciancio, A.; Farsaci, F. On general properties of phenomenological and state coefficients for isotropic viscoanelastic media. Phys. B Condens. Matter 2008, 403, 3221–3227.
- 11.
Ciancio, V.; Palumbo, A. A thermodynamical theory with internal variables describing thermal effects in viscous fluids. J. Non-Equilib. Thermodyn. 2018, 43, 171–184.
- 12.
Ciancio, V. Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversibile thermodynamic with internal variables. Int. J. Math. Comput. Eng. 2024, 2, 141–154.
- 13.
Ciancio, A.; Flora, B.F.F. Application of a generalized heat equation to ultra-fast processes in viscoanelastic isotropic medium. Int. J. Math. Comput. Eng. 2025, 3, 389–404.
- 14.
Ciancio, A.; Flora, B.F.F.; Ciancio, V. Derivations of the stress-strain relations for anisotropic viscoanelastic media in irreversible thermodynamics with internal variables. Atti Accad. Pelorit. Pericolanti Cl. Sci. Fis. Mat. Nat. 2025, 103, A1-1.
- 15.
Maxwell, J.C. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 1867, 157, 49–88.
- 16.
Berezovski, A.; V´an, P. Microinertia and internal variables. Contin. Mech. Thermodyn. 2015, 28, 1027–1037.
- 17.
Wang, Y.C.; Ko, C.C.; Chang, K.W. Anomalous effective viscoelastic, thermoelastic, dielectric, and piezoelectric properties of negative-stiffness composites and their stability. Phys. Status Solidi B 2015, 252, 1640–1655.
- 18.
Bulıcek, M.; M´alek, J.; Prusa, V.; et al. On incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion and purely spherical elastic response. SIAM J. Math. Anal. 2021, 53, 3985–4030.
- 19.
Safa, B.N.; Santare, M.H.; Elliott, D.M. A Reactive Inelasticity Theoretical Framework for Modeling Viscoelasticity, Plastic Deformation, and Damage in Fibrous Soft Tissue. J. Biomech Eng. 2019, 141, 021005.
- 20.
Song, R.; Muliana, A.; Rajagopal, K. A thermodynamically consistent model for viscoelastic polymers undergoing microstructural changes. Int. J. Eng. Sci. 2019, 142, 106–124.
- 21.
Berjamin, H.; Destrade, M.; Parnell, W.J. On the thermodynamic consistency of Quasi-linear viscoelastic models for soft solids. Mech. Res. Commun. 2021, 111, 103648.
- 22.
Atanackovic, T.M.; Dolicanin, C.; Kacapor, E. Internal Variable Theory in Viscoelasticity: Fractional Generalizations and Thermodynamical Restrictions. Mathematics 2022, 10, 1708.
- 23.
Ciancio, A.; Ciancio, V.; Flora, B.F.F. A Fractional Rheological Model of Viscoanelastic Media. Axioms 2023, 12, 243.
- 24.
Kluitenberg, G.A. On heat dissipation due to irreversible mechanical phenomena in continuous media. Physica 1967, 35, 177–192.
- 25.
Cattaneo, C. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. C. R. Acad. Sci. 1958, 247, 431–433.
- 26.
Vernotte, P. Les paradoxes de la theorie continue de l’equation de la chaleur. C. R. Acad. Sci. 1958, 246, 3154–3155.
- 27.
Joseph, D.D.; Preziosi, L. Heat waves. Rev. Mod. Phys. 1989, 61, 41–73.
How to Cite
Ciancio, A.; Flora, B. F. F.; Ciancio, V. On the Evolution of Internal Variables for Anisotropic Viscoelastic Media. Nonlinear Analysis and Computer Simulations 2026, 1 (3), 12. https://doi.org/10.53941/nacs.2026.100012.
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Copyright (c) 2026 by the authors.
This work is licensed under a Creative Commons Attribution 4.0 International License.
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