Maticové rozklady v teorii konstitutivních vztahů pro spojité prostředí

• Thesis title in Czech: Maticové rozklady v teorii konstitutivních vztahů pro spojité prostředí
• Thesis title in English: Matrix decompositions in constitutive relations for continuous medium
• Key words: mechanika kontinua|maticový rozklad|konstitutivní vztahy
• English key words: continuum mechanics|matrix decomposition|constitutive relations
• Academic year of topic announcement: 2021/2022
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: Mgr. Vít Průša, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 01.11.2021
• Date of assignment: 11.11.2021
• Confirmed by Study dept. on: 22.05.2023
• Date and time of defence: 07.06.2023 09:15
• Date of electronic submission: 04.05.2023
• Date of submission of printed version: 09.05.2023
• Date of proceeded defence: 07.06.2023
• Opponents: prof. RNDr. Martin Kružík, Ph.D., DSc.

Guidelines

The standard approach to the Green elastic materials, see Truesdell & Noll (2004), is based on the assumption that the Helmholtz free energy depends on the invariants of a strain tensor, while the set of invariants differs depending on the material symmetries of the given material. Furthermore the whole elasticity theory and the set of invariants being used is usually based on the polar decomposition of the deformation gradient. Srinivasa (2012) proposed to use another matrix decomposition, namely the QR decomposition. From a strict theoretical point of view both approaches are equivalent, but it is claimed that the alternative decomposition has certain features that make it more useful in practical applications such as experimental data fitting. The reason is that the alternative set of invariants allows one---amongst others---handle more transparently the experimental error.

The objective of the thesis is to track the relevant literature on the subject matter, and investigate various question related to the constitutive relations specified in terms of the new set of invariants, especially their applicability in the case of anisotropic solids. Further developments might include either analytical of numerical solution of simple boundary value problems for the newly established constitutive relations, and alternatively investigation of the applicability of the new set of invariants in theories regarding the inelastic response of solids or even the response of fluids.

References

Annin, B. D. and K. V. Bagrov (2021). Numerical simulation of the hyperelastic material using new strain measure. Acta Mech. 232 (5), 1809–1813.
Clayton, J. and A. Freed (2020). A constitutive framework for finite viscoelasticity and damage based on the Gram–Schmidt decomposition. Acta Mech. 231 (8), 3319–3362.
Freed, A., J.-B. le Graverend, and K. Rajagopal (2019). A decomposition of Laplace stretch with applications in inelasticity. Acta Mech. 230 (9), 3423–3429.
Freed, A. and S. Zamani (2019). Elastic Kelvin–Poisson–Poynting solids described through scalar conjugate stress/strain pairs derived from a QR factorization of F. J. Mech. Phys. Solids 129, 278–293.
Freed, A. D. (2017). A note on stress/strain conjugate pairs: Explicit and implicit theories of thermoelasticity for anisotropic materials. Int. J. Eng. Sci. 120, 155–171.
Freed, A. D. and A. R. Srinivasa (2015). Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient. Acta Mech. 226 (8), 2645–2670.
Gao, X.-L. and Y. Li (2018). The upper triangular decomposition of the deformation gradient: possible decompositions of the distortion tensor. Acta Mech. 229 (5), 1927–1948.
Holzapfel, G. A. (2000). Nonlinear solid mechanics: A continuum approach for engineering. Chichester: Wiley.
Li, Y. and X.-L. Gao (2019). Constitutive equations for hyperelastic materials based on the upper triangular decomposition of the deformation gradient. Math. Mech. Solids 24 (6), 1785–1799.
Paul, S. and A. Freed (2020). A simple and practical representation of compatibility condition derived using a QR decomposition of the deformation gradient. Acta Mech. 231 (8), 3289–3304.
Paul, S. and A. Freed (2021). A constitutive model for elastic–plastic materials using scalar conjugate stress/strain base pairs. J. Mech. Phys. Solids 155, 104535.
Salamatova, V. Y. and Y. V. Vasilevskii (2021). On ellipticity of hyperelastic models restored by experimental data. J. Math. Sci. 253 (5), 720–729.
Salamatova, V. Y., Y. V. Vassilevski, and L. Wang (2018). Finite element models of hyperelastic materials based on a new strain measure. Diff. Equat. 54 (7), 971–978.
Srinivasa, A. R. (2012). On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials. Int. J. Eng. Sci. 60, 1–12.
Truesdell, C. and W. Noll (2004). The non-linear field theories of mechanics (3rd ed.). Berlin: Springer.