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A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity

Javier Bonet, Chun Hean Lee, Antonio Gil Orcid Logo, Ataollah Ghavamian

Computer Methods in Applied Mechanics and Engineering, Volume: 373, Start page: 113505

Swansea University Authors: Antonio Gil Orcid Logo, Ataollah Ghavamian

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DOI (Published version): 10.1016/j.cma.2020.113505

Abstract

In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a sy...

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Published in: Computer Methods in Applied Mechanics and Engineering
ISSN: 0045-7825
Published: Elsevier BV 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa55401
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The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie&#x2013;Gr&#xFC;neisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. 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spelling 2020-11-11T18:41:18.6295252 v2 55401 2020-10-12 A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false ea56d8e69b28541a1b2c201f7dc0b6d4 Ataollah Ghavamian Ataollah Ghavamian true false 2020-10-12 CIVL In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie–Grüneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov–Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation. Journal Article Computer Methods in Applied Mechanics and Engineering 373 113505 Elsevier BV 0045-7825 Large strain thermo-elasticity, Petrov–Galerkin, Explicit dynamics, Polyconvexity, Conservation laws, Ballistic energy 1 1 2021 2021-01-01 10.1016/j.cma.2020.113505 COLLEGE NANME Civil Engineering COLLEGE CODE CIVL Swansea University 2020-11-11T18:41:18.6295252 2020-10-12T12:32:33.3039741 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Javier Bonet 1 Chun Hean Lee 2 Antonio Gil 0000-0001-7753-1414 3 Ataollah Ghavamian 4 55401__18409__012c2b3d3656474a9cad18a10dacf095.pdf 55401.pdf 2020-10-12T12:34:55.1243671 Output 18131629 application/pdf Accepted Manuscript true 2021-11-02T00:00:00.0000000 © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license true eng http://creativecommons.org/licenses/by-nc-nd/4.0/
title A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
spellingShingle A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
Antonio Gil
Ataollah Ghavamian
title_short A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
title_full A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
title_fullStr A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
title_full_unstemmed A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
title_sort A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
ea56d8e69b28541a1b2c201f7dc0b6d4
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
ea56d8e69b28541a1b2c201f7dc0b6d4_***_Ataollah Ghavamian
author Antonio Gil
Ataollah Ghavamian
author2 Javier Bonet
Chun Hean Lee
Antonio Gil
Ataollah Ghavamian
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container_title Computer Methods in Applied Mechanics and Engineering
container_volume 373
container_start_page 113505
publishDate 2021
institution Swansea University
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doi_str_mv 10.1016/j.cma.2020.113505
publisher Elsevier BV
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering
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description In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie–Grüneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov–Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation.
published_date 2021-01-01T04:09:35Z
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