{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T03:35:34Z","timestamp":1782272134308,"version":"3.54.5"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T00:00:00Z","timestamp":1677715200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>To date, the mechanical models of magnetoelectric couplings at finite strains have mainly been limited to time-independent constitutive equations. This paper enhances the literature by developing a time-dependent electromagnetic constitutive equation to characterise the mechanical behaviour of soft solids at finite strains and take into account the full form of the Maxwell equations. Our formulation introduces a symmetrical total stress and uses recently developed spectral invariants in the amended energy function; as a result, the proposed constitutive equation is relatively simple and is amenable to a finite-element formulation.<\/jats:p>","DOI":"10.3390\/sym15030628","type":"journal-article","created":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T04:22:28Z","timestamp":1677730948000},"page":"628","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Generalised Time-Dependent Mathematical Formulation for Magnetoelectrically Coupled Soft Solids at Finite Strains"],"prefix":"10.3390","volume":"15","author":[{"given":"Mohd Halim Bin Mohd","family":"Shariff","sequence":"first","affiliation":[{"name":"Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roger","family":"Bustamante","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda Mec\u00e1nica, Universidad de Chile, Santiago 837-0456, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4616-1104","authenticated-orcid":false,"given":"Mokarram","family":"Hossain","sequence":"additional","affiliation":[{"name":"Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Swansea SA1 8EN, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"11182","DOI":"10.1002\/anie.201813402","article-title":"Bio-inspired Design and Additive Manufacturing of Soft Materials, Machines, Robots, and Haptic Interfaces","volume":"58","author":"Li","year":"2019","journal-title":"Angew. Chem. Int. Ed."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2001000","DOI":"10.1002\/advs.202001000","article-title":"Direct Ink Writing Based 4D Printing of Materials and Their Applications","volume":"7","author":"Wan","year":"2020","journal-title":"Adv. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5154","DOI":"10.1021\/nn501204t","article-title":"Highly stretchable and sensitive strain sensor based on silver nanowire-elastomer nanocomposite","volume":"8","author":"Amjadi","year":"2012","journal-title":"ACS Nano"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s003390050854","article-title":"The giant magnetostriction in ferromagnetic composites within an elastomer matrix","volume":"68","author":"Bednarek","year":"1999","journal-title":"Appl. Phys. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1038\/nature25443","article-title":"Small-scale soft-bodied robot with multimodal locomotion","volume":"554","author":"Hu","year":"2018","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"108348","DOI":"10.1016\/j.compositesb.2020.108348","article-title":"A review on magneto-mechanical characterizations of magnetorheological elastomers","volume":"200","author":"Bastola","year":"2020","journal-title":"Compos. Part B Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1038\/nature05023","article-title":"Multiferroic and magnetoelectric materials","volume":"442","author":"Eerenstein","year":"2006","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1109\/JSEN.2006.877989","article-title":"Ferritepiezoelectric multilayers for magnetic field sensors","volume":"6","author":"Fetisov","year":"2006","journal-title":"IEEE Sensors J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"040601","DOI":"10.1103\/PhysRevE.88.040601","article-title":"Giant and universal magnetoelectric coupling in soft materials and concomitant ramifications for materials science and biology","volume":"88","author":"Liu","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/j.jmps.2013.08.001","article-title":"An energy formulation of continuum magneto-electro-elasticity with applications","volume":"63","author":"Liu","year":"2014","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103429","DOI":"10.1016\/j.ijengsci.2020.103429","article-title":"Mathematical formulations for elastic magneto-electrically coupled soft materials at finite strains: Time-independent processes","volume":"159","author":"Bustamante","year":"2021","journal-title":"Int. J. Eng. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1177\/10812865221082521","article-title":"Anisotropic stress softening of electromagnetic Mullins materials","volume":"28","author":"Shariff","year":"2023","journal-title":"Math. Mech. Solids"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/B978-0-08-021792-5.50012-4","article-title":"Electromagnetic Forces in Deformable Continua","volume":"4","author":"Pao","year":"1978","journal-title":"Mech. Today"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Eringen, A.C., and Maugin, G.A. (1990). Electrodynamics of Continua, Springer.","DOI":"10.1007\/978-1-4612-3236-0"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kovetz, A. (2000). Electromagnetic Theory, University Press.","DOI":"10.1093\/oso\/9780198506041.001.0001"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shariff, M.H.B.M. (2023). On the Smallest Number of Functions Representing Isotropic Functions of Scalars, Vectors and Tensors. arXiv.","DOI":"10.1093\/qjmam\/hbac022"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.euromechsol.2016.01.006","article-title":"New sets of spectral invariants for electro-elastic bodies with one and two families of fibres","volume":"58","author":"Bustamante","year":"2016","journal-title":"Eur. J. Mech.-A\/Solids"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1093\/qjmam\/hbx014","article-title":"Spectral derivatives in continuum mechanics","volume":"70","author":"Shariff","year":"2017","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_19","unstructured":"Ogden, R.W. (1997). Non-Linear Elastic Deformations, Dover."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/S0997-7538(03)00067-6","article-title":"Magnetoelastic modelling of elastomers","volume":"22","author":"Dorfmann","year":"2003","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s00707-004-0202-2","article-title":"Nonlinear electroelasticity","volume":"174","author":"Dorfmann","year":"2005","journal-title":"Acta Mech."},{"key":"ref_22","first-page":"239","article-title":"Theory of invariants","volume":"Volume 1","author":"Eringen","year":"1971","journal-title":"Continuum Physics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1115\/1.3111066","article-title":"Theory of representation for tensor function. A unified invariant approach to constitutive equations","volume":"47","author":"Zheng","year":"1994","journal-title":"Appl. Mech. Rev."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/0020-7225(71)90023-1","article-title":"On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors","volume":"9","author":"Smith","year":"1971","journal-title":"Int. J. Eng.Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1177\/1081286505055472","article-title":"On Constitutive Equations For Anisotropic Nonlinearly Viscoelastic Solids","volume":"12","author":"Merodio","year":"2007","journal-title":"Math. Mech. Solids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1177\/1081286507084411","article-title":"On planar biaxial tests for anisotropic nonlinearly elastic solids: A continuum mechanical framework","volume":"14","author":"Holzapfel","year":"2009","journal-title":"Math. Mech. Solids"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1093\/qjmam\/hbm028","article-title":"Nonlinear transversely isotropic elastic solids: An alternative representation","volume":"61","author":"Shariff","year":"2008","journal-title":"Q. J. Mech. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/628\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:46:03Z","timestamp":1760121963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/628"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,2]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030628"],"URL":"https:\/\/doi.org\/10.3390\/sym15030628","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,2]]}}}