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Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity

Roman Poya Orcid Logo, Rogelio Ortigosa, Antonio Gil Orcid Logo

International Journal for Numerical Methods in Engineering, Volume: 124, Issue: 16, Pages: 3436 - 3493

Swansea University Author: Antonio Gil Orcid Logo

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DOI (Published version): 10.1002/nme.7254

Abstract

A new computational framework for large strain elasticity in principal stretches is presented. Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares i.e. eigenvalues of Cauchy-Green strain tensor. The proposed framework has thre...

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Published in: International Journal for Numerical Methods in Engineering
ISSN: 0029-5981 1097-0207
Published: Wiley 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63263
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Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares i.e. eigenvalues of Cauchy-Green strain tensor. The proposed framework has three key features. First, the eigendecomposition of the tangent elasticity and initial (geometric) stiffness operators is obtained in closed-form from principal information alone. Crucially, these newly found eigenvalues describe the general convexity conditions of isotropic hyperelastic energies. In other words, convexity is postulated concisely through tangent eigenvalues supplementing the original work of J. M. Ball 1. Consequently, this novel finding opens the door for designing efficient automated Newton-style algorithms with stabilised tangents via closed-form semi-positive definite projection or spectral shifting that converge irrespective of mesh resolution, quality, loading scenario and without relying on path-following techniques. A critical study of closed-form tangent stabilisation in the context of isotropic hyperelasticity is therefore undertaken in this work. Second, in addition to high order displacement-based formulation, mixed Hu-Washizu variational principles are formulated in terms of principal stretches by introducing stretch work conjugate Lagrange multipliers that enforce principal stretch-stress compatibility. This is similar to enhanced strain methods. However, the resulting mixed finite element scheme is cost-efficient, specially compared to approximating the entire strain tensors since the formulation is in the scalar space of singular values. 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The second author acknowledges the financial support through the contract 21132/SF/19, Fundaciòn Sèneca, Regiòn de Murcia (Spain), through the program Saavedra Fajardo. Second author is funded by Fundaciòn Sèneca (Murcia, Spain) through grant 20911/PI/18. 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spelling v2 63263 2023-04-27 Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false 2023-04-27 ACEM A new computational framework for large strain elasticity in principal stretches is presented. Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares i.e. eigenvalues of Cauchy-Green strain tensor. The proposed framework has three key features. First, the eigendecomposition of the tangent elasticity and initial (geometric) stiffness operators is obtained in closed-form from principal information alone. Crucially, these newly found eigenvalues describe the general convexity conditions of isotropic hyperelastic energies. In other words, convexity is postulated concisely through tangent eigenvalues supplementing the original work of J. M. Ball 1. Consequently, this novel finding opens the door for designing efficient automated Newton-style algorithms with stabilised tangents via closed-form semi-positive definite projection or spectral shifting that converge irrespective of mesh resolution, quality, loading scenario and without relying on path-following techniques. A critical study of closed-form tangent stabilisation in the context of isotropic hyperelasticity is therefore undertaken in this work. Second, in addition to high order displacement-based formulation, mixed Hu-Washizu variational principles are formulated in terms of principal stretches by introducing stretch work conjugate Lagrange multipliers that enforce principal stretch-stress compatibility. This is similar to enhanced strain methods. However, the resulting mixed finite element scheme is cost-efficient, specially compared to approximating the entire strain tensors since the formulation is in the scalar space of singular values. Third, the proposed framework facilitates simulating rigid and stiff systems and those that are nearly-inextensible in principal directions, a constituent of elasticity that cannot be easily studied using standard formulations. Journal Article International Journal for Numerical Methods in Engineering 124 16 3436 3493 Wiley 0029-5981 1097-0207 convexity conditions; large strain elasticity; mixed finite elements; principal stretches 30 8 2023 2023-08-30 10.1002/nme.7254 COLLEGE NANME Aerospace, Civil, Electrical, and Mechanical Engineering COLLEGE CODE ACEM Swansea University The first author thanks Brent Meranda manager of the Meshing & Abstraction Group, Simcenter 3D, Siemens Digital Industries Software. The second author acknowledges the financial support through the contract 21132/SF/19, Fundaciòn Sèneca, Regiòn de Murcia (Spain), through the program Saavedra Fajardo. Second author is funded by Fundaciòn Sèneca (Murcia, Spain) through grant 20911/PI/18. The fourth author acknowledges the financial support received through the European Training Network ProTechtion (Project ID: 764636). 2024-07-29T14:24:37.9659320 2023-04-27T15:01:36.6261415 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Roman Poya 0000-0003-2350-4933 1 Rogelio Ortigosa 2 Antonio Gil 0000-0001-7753-1414 3 63263__27246__1a720e943f034de3b3ef12097f6b2bca.pdf 63263.pdf 2023-04-27T15:10:47.6564007 Output 46424372 application/pdf Accepted Manuscript true 2024-05-07T00:00:00.0000000 true eng
title Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
spellingShingle Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
Antonio Gil
title_short Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
title_full Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
title_fullStr Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
title_full_unstemmed Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
title_sort Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
author Antonio Gil
author2 Roman Poya
Rogelio Ortigosa
Antonio Gil
format Journal article
container_title International Journal for Numerical Methods in Engineering
container_volume 124
container_issue 16
container_start_page 3436
publishDate 2023
institution Swansea University
issn 0029-5981
1097-0207
doi_str_mv 10.1002/nme.7254
publisher Wiley
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 A new computational framework for large strain elasticity in principal stretches is presented. Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares i.e. eigenvalues of Cauchy-Green strain tensor. The proposed framework has three key features. First, the eigendecomposition of the tangent elasticity and initial (geometric) stiffness operators is obtained in closed-form from principal information alone. Crucially, these newly found eigenvalues describe the general convexity conditions of isotropic hyperelastic energies. In other words, convexity is postulated concisely through tangent eigenvalues supplementing the original work of J. M. Ball 1. Consequently, this novel finding opens the door for designing efficient automated Newton-style algorithms with stabilised tangents via closed-form semi-positive definite projection or spectral shifting that converge irrespective of mesh resolution, quality, loading scenario and without relying on path-following techniques. A critical study of closed-form tangent stabilisation in the context of isotropic hyperelasticity is therefore undertaken in this work. Second, in addition to high order displacement-based formulation, mixed Hu-Washizu variational principles are formulated in terms of principal stretches by introducing stretch work conjugate Lagrange multipliers that enforce principal stretch-stress compatibility. This is similar to enhanced strain methods. However, the resulting mixed finite element scheme is cost-efficient, specially compared to approximating the entire strain tensors since the formulation is in the scalar space of singular values. Third, the proposed framework facilitates simulating rigid and stiff systems and those that are nearly-inextensible in principal directions, a constituent of elasticity that cannot be easily studied using standard formulations.
published_date 2023-08-30T14:24:36Z
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