Implicitní konstitutivní vztahy na úrovni termodynamických potenciálů

• Thesis title in Czech: Implicitní konstitutivní vztahy na úrovni termodynamických potenciálů
• Thesis title in English: Implicit constitutive relations on the level of thermodynamic potentials
• Key words: termodynamika|konstitutivní vztahy|mechanika kontinua
• English key words: thermodynamics|continuum mechanics|constitutive relations
• Academic year of topic announcement: 2022/2023
• Thesis type: dissertation
• Thesis language: čeština
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. Mgr. Vít Průša, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 07.09.2023
• Date of assignment: 07.09.2023
• Confirmed by Study dept. on: 27.09.2023

Guidelines

Following the pioneering paper by Rajagopal (2003), much research effort has been invested into the development of so-called implicit constitutive relations. In most cases the research community has focused on a special type of implicit constitutive relations. For example, in the theory of elastic solids, the research community focused on constitutive relations wherein the strain is expressed in terms of the stress, which is exactly the opposite of the traditional approach. However, even in this setting one of the variables (strain, stress) is still a privileged one in the sense that the complementary variable (stress, strain) is given by an explicit formula in terms of the primary variable. But the implicit constitutive theory admits more complex constitutive relations wherein neither variable is privileged, and wherein the constitutive relation can not be converted into a form where one of the variables is given as an explicit function of the other. The thesis should explore this complex type of constitutive relations. In particular the thesis should focus on thermodynamic background and solution of simple boundary value problems demonstrating the features of the given constitutive relations.

The key idea behind the thermodynamic approach should be the application of "implicit constitutive relations" already on the level of thermodynamic potentials. For example, one can think of the specification of the Helmholtz free energy as an implicit function of the strain, Helmholtz free energy and the derivative of Helmholtz free energy. In this sense the Helmholtz free energy would be given by a solution to a given partial differential equation, and it will not be a priori given in an explicit form. Same concept might be applied for other potentials.

Literature review should be based on the papers listed below, in particular Cichra, Gazca-Orozco, Průša, Tůma (2023) articulates some open problems.

References

Rajagopal, K. R. (2003). On implicit constitutive theories. Appl. Math. 48 (4), 279–319.
Gokulnath, C., U. Saravanan, and K. R. Rajagopal (2017). Representations for implicit constitutive relations describing non-dissipative response of isotropic materials. Z. angew. Math. Phys. 68 (6), 129.
Cichra, D., Alexei Gazca-Orozco, P., Průša, V., & Tůma, K. (2023). A thermodynamic framework for non-isothermal phenomenological models of isotropic Mullins effect. Proceedings of the Royal Society A, 479(2272), 20220614.
Cichra, D., & Průša, V. (2020). A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation. Mathematics and Mechanics of Solids, 25(12), 2222-2230.
Rajagopal, K. R. and A. R. Srinivasa (2011). A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials. Proc. R. Soc. A-Math. Phys. Eng. Sci. 467 (2125), 39–58.
Rajagopal, K. R. and A. R. Srinivasa (2013). An implicit thermomechanical theory based on a Gibbs potential formulation for describing the response of thermoviscoelastic solids. Int. J. Eng. Sci. 70 (0), 15–28.
Rajagopal, K. R. and A. R. Srinivasa (2015). Inelastic response of solids described by implicit constitutive relations with nonlinear small strain elastic response. Int. J. Plast. 71, 1–9.
Rajagopal, K. R. and A. R. Srinivasa (2016). An implicit three-dimensional model for describing the inelastic response of solids undergoing finite deformation. Z. angew. Math. Phys. 67 (4), 86.
Srinivasa, A. R. (2015). On a class of Gibbs potential-based nonlinear elastic models with small strains. Acta Mech. 226 (2), 571–583.
Giorgi, C., & Morro, A. (2021). A thermodynamic approach to rate-type models of elastic-plastic materials. Journal of Elasticity, 147(1-2), 113-148.
Rajagopal, K. R., & Saccomandi, G. (2022). Implicit nonlinear elastic bodies with density dependent material moduli and its linearization. International Journal of Solids and Structures, 234, 111255.