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Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity

Zeng Liu, Rogelio Ortigosa Orcid Logo, Antonio Gil Orcid Logo, Javier Bonet Orcid Logo

Journal of the Mechanics and Physics of Solids, Volume: 203, Start page: 106194

Swansea University Author: Antonio Gil Orcid Logo

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DOI (Published version): 10.1016/j.jmps.2025.106194

Abstract

This paper discusses a new phenomenological continuum formulation for the constitutive modelling of viscoelastic materials at large strains. Following pioneering works in [1, 2, 3, 4], the formulation shares some common ingredients with other phenomenological approaches, including the multiplicative...

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Published in: Journal of the Mechanics and Physics of Solids
ISSN: 0022-5096 1873-4782
Published: Elsevier Ltd 2025
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Very importantly, the evolution law is shown to be compatible with the second law of thermodynamics by construction and have a closed-form solution in the case of incompressible viscoelasticity when using a prototypical neo-Hookean type of non-equilibrium strain energy density. Second, to facilitate the joint consideration of anisotropy and thermodynamic equilibrium, a frame indifferent stress free configuration is introduced which facilitates the definition of objective strain measures. Third, the methodology is extended from isotropy to transverse isotropy via the consideration of the appropriate structural tensor. The formulation is first displayed for the simple case of a single transversely isotropic invariant contribution with corresponding closed-form solution, and then straightforwardly extended to the consideration of the second transversely isotropic invariant, multiple families of fibres, or even more complex symmetry groups. To demonstrate the capability of the new framework, a specialised form of the eight-chain long-term strain energy (long term) and a neo-Hookean strain energy (non-equilibrium) have been adopted for the description of the mechanical behaviour of VHB 4910 polymer, due to its use in current Electro-Active Polymers based soft robotics. Good agreement is found between in silico predictions and available experimental data on various tests, including loading-unloading cyclic tests, single-step relaxation tests and a multi-step relaxation test. 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spelling 2025-06-10T13:14:21.8984289 v2 69506 2025-05-14 Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false 2025-05-14 ACEM This paper discusses a new phenomenological continuum formulation for the constitutive modelling of viscoelastic materials at large strains. Following pioneering works in [1, 2, 3, 4], the formulation shares some common ingredients with other phenomenological approaches, including the multiplicative decomposition of the deformation gradient into viscous and elastic contributions, the additive Maxwell-type decomposition of the strain energy density, and the definition of a set of kinematic internal state variables with their associated evolution laws. Our formulation departs from other state-of-the-art methodologies via three distinct novelties. First, and revisiting previous work by Bonet [5], the paper introduces a thermodynamically consistent linear rate type evolution law in terms of stress-type variables, which resembles the return mapping algorithm typically used in elastoplasticity, facilitating the modelling link between both inelastic constitutive models. In this sense, the proposed viscoelastic evolution law can be identified with a classical plastic flow rule. Very importantly, the evolution law is shown to be compatible with the second law of thermodynamics by construction and have a closed-form solution in the case of incompressible viscoelasticity when using a prototypical neo-Hookean type of non-equilibrium strain energy density. Second, to facilitate the joint consideration of anisotropy and thermodynamic equilibrium, a frame indifferent stress free configuration is introduced which facilitates the definition of objective strain measures. Third, the methodology is extended from isotropy to transverse isotropy via the consideration of the appropriate structural tensor. The formulation is first displayed for the simple case of a single transversely isotropic invariant contribution with corresponding closed-form solution, and then straightforwardly extended to the consideration of the second transversely isotropic invariant, multiple families of fibres, or even more complex symmetry groups. To demonstrate the capability of the new framework, a specialised form of the eight-chain long-term strain energy (long term) and a neo-Hookean strain energy (non-equilibrium) have been adopted for the description of the mechanical behaviour of VHB 4910 polymer, due to its use in current Electro-Active Polymers based soft robotics. Good agreement is found between in silico predictions and available experimental data on various tests, including loading-unloading cyclic tests, single-step relaxation tests and a multi-step relaxation test. Finally, biaxial loading-unloading cyclic and relaxation tests are presented to further showcase performance in anisotropic scenarios. Journal Article Journal of the Mechanics and Physics of Solids 203 106194 Elsevier Ltd 0022-5096 1873-4782 Large strain; Finite viscoelasticity; Maxwell rheological model; Multiplicative decomposition 1 10 2025 2025-10-01 10.1016/j.jmps.2025.106194 COLLEGE NANME Aerospace, Civil, Electrical, and Mechanical Engineering COLLEGE CODE ACEM Swansea University Another institution paid the OA fee The authors acknowledge funding received from grants PID2022-141957OB-C21 and PID2022-141957OA-C22 financed by MCIN/AEI /10.13039/501100011033/ FEDER, UE. R. Ortigosa also acknowledges the support provided by the Autonomous Community of the Region of Murcia, Spain, through the programme for the development of scientific and technical research by competitive groups (21996/PI/22), included in the Regional Program for the Promotion of Scientific and Technical Research of Fundación Séneca - Agencia de Ciencia y Tecnología de la Región de Murcia. A. J. Gil wishes to acknowledge the support provided by the Defence, Science and Technology Laboratory (Dstl) and The Leverhulme Trust Foundation (UK) through a Leverhulme Fellowship. 2025-06-10T13:14:21.8984289 2025-05-14T11:03:36.5880910 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Zeng Liu 1 Rogelio Ortigosa 0000-0002-4542-2237 2 Antonio Gil 0000-0001-7753-1414 3 Javier Bonet 0000-0002-0430-5181 4 69506__34450__a0683c200a7a4eaf96a8adb4be144fc0.pdf 69506.VOR.pdf 2025-06-10T13:08:01.9891751 Output 3833738 application/pdf Version of Record true © 2025 The Authors. This is an open access article under the CC BY-NC-ND license. true eng http://creativecommons.org/licenses/by-nc-nd/4.0/
title Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
spellingShingle Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
Antonio Gil
title_short Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
title_full Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
title_fullStr Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
title_full_unstemmed Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
title_sort Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
author Antonio Gil
author2 Zeng Liu
Rogelio Ortigosa
Antonio Gil
Javier Bonet
format Journal article
container_title Journal of the Mechanics and Physics of Solids
container_volume 203
container_start_page 106194
publishDate 2025
institution Swansea University
issn 0022-5096
1873-4782
doi_str_mv 10.1016/j.jmps.2025.106194
publisher Elsevier Ltd
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
document_store_str 1
active_str 0
description This paper discusses a new phenomenological continuum formulation for the constitutive modelling of viscoelastic materials at large strains. Following pioneering works in [1, 2, 3, 4], the formulation shares some common ingredients with other phenomenological approaches, including the multiplicative decomposition of the deformation gradient into viscous and elastic contributions, the additive Maxwell-type decomposition of the strain energy density, and the definition of a set of kinematic internal state variables with their associated evolution laws. Our formulation departs from other state-of-the-art methodologies via three distinct novelties. First, and revisiting previous work by Bonet [5], the paper introduces a thermodynamically consistent linear rate type evolution law in terms of stress-type variables, which resembles the return mapping algorithm typically used in elastoplasticity, facilitating the modelling link between both inelastic constitutive models. In this sense, the proposed viscoelastic evolution law can be identified with a classical plastic flow rule. Very importantly, the evolution law is shown to be compatible with the second law of thermodynamics by construction and have a closed-form solution in the case of incompressible viscoelasticity when using a prototypical neo-Hookean type of non-equilibrium strain energy density. Second, to facilitate the joint consideration of anisotropy and thermodynamic equilibrium, a frame indifferent stress free configuration is introduced which facilitates the definition of objective strain measures. Third, the methodology is extended from isotropy to transverse isotropy via the consideration of the appropriate structural tensor. The formulation is first displayed for the simple case of a single transversely isotropic invariant contribution with corresponding closed-form solution, and then straightforwardly extended to the consideration of the second transversely isotropic invariant, multiple families of fibres, or even more complex symmetry groups. To demonstrate the capability of the new framework, a specialised form of the eight-chain long-term strain energy (long term) and a neo-Hookean strain energy (non-equilibrium) have been adopted for the description of the mechanical behaviour of VHB 4910 polymer, due to its use in current Electro-Active Polymers based soft robotics. Good agreement is found between in silico predictions and available experimental data on various tests, including loading-unloading cyclic tests, single-step relaxation tests and a multi-step relaxation test. Finally, biaxial loading-unloading cyclic and relaxation tests are presented to further showcase performance in anisotropic scenarios.
published_date 2025-10-01T05:28:22Z
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