{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,7]],"date-time":"2025-03-07T07:40:27Z","timestamp":1741333227401,"version":"3.38.0"},"reference-count":53,"publisher":"Tech Science Press","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["CMC"],"published-print":{"date-parts":[[2025]]},"DOI":"10.32604\/cmc.2024.059950","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T06:28:16Z","timestamp":1735885696000},"page":"193-222","source":"Crossref","is-referenced-by-count":0,"title":["Toward Analytical Homogenized Relaxation Modulus for Fibrous Composite Material with Reduced Order Homogenization Method"],"prefix":"10.32604","volume":"82","author":[{"given":"Huilin","family":"Jia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shanqiao","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zifeng","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"17807","published-online":{"date-parts":[[2025]]},"reference":[{"key":"ref1","article-title":"3D printing of curvilinear fiber reinforced variable stiffness composite structures: A review","volume":"272","author":"Guo","year":"2024","journal-title":"Compos. Part B Eng."},{"key":"ref2","doi-asserted-by":"crossref","DOI":"10.1016\/j.conbuildmat.2024.138822","article-title":"Mechanical roperties and durability of carbon fiber reinforced cementitious composites: A review","volume":"452","author":"Aduwenye","year":"2024","journal-title":"Constr. Build. Mater."},{"key":"ref3","doi-asserted-by":"crossref","DOI":"10.1016\/j.compscitech.2024.110982","article-title":"Multi-scale numerical calculations for the interphase mechanical properties of carbon fiber reinforced thermoplastic composites","volume":"260","author":"Li","year":"2025","journal-title":"Compos. Sci. Technol."},{"key":"ref4","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.32604\/jrm.2024.051201","article-title":"A review on the advancement of renewable natural fiber hybrid composites: Prospects, challenges, and industrial applications","volume":"12","author":"Mohammed","year":"2024","journal-title":"J. Renew. Mater."},{"key":"ref5","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcomc.2020.100082","article-title":"Environmental benign natural fibre reinforced thermoplastic composites: A review","volume":"4","author":"Awais","year":"2021","journal-title":"Compos. Part C Open Access"},{"key":"ref6","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1016\/j.ijsolstr.2011.04.007","article-title":"A time dependent model for unidirectional fibre-reinforced composites with viscoelastic matrices","volume":"48","author":"Nedjar","year":"2011","journal-title":"Int. J. Solids Struct."},{"key":"ref7","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.mechmat.2018.08.005","article-title":"Closed-form expressions for effective viscoelastic properties of fiber-reinforced composites considering fractional matrix behavior","volume":"127","author":"Hofer","year":"2018","journal-title":"Mech. Mater."},{"key":"ref8","first-page":"1","article-title":"Extended mean-field homogenization of viscoelastic-viscoplastic polymer composites undergoing hybrid progressive degradation induced by interface debonding and matrix ductile damage","volume":"210\u2013211","author":"Chen","year":"2021","journal-title":"Int. J. Solids Struct."},{"key":"ref9","doi-asserted-by":"crossref","DOI":"10.1016\/j.euromechsol.2022.104859","article-title":"Mean-field descriptions for the viscoelastic response of thermorheologically complex reinforced solids","volume":"98","author":"Suarez-Afanador","year":"2023","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref10","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijsolstr.2023.112642","article-title":"Multiscale simulation of polymer curing of composites combined mean-field homogenisation methods at large strains","volume":"290","author":"Lenz","year":"2024","journal-title":"Int. J. Solids Struct."},{"key":"ref11","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.compstruct.2018.09.056","article-title":"A novel implementation of asymptotic homogenization for viscoelastic composites with periodic microstructures","volume":"208","author":"Li","year":"2019","journal-title":"Compos. Struct."},{"key":"ref12","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1016\/S0020-7683(97)00166-2","article-title":"Asymptotic homogenization of viscoelastic composites with periodic microstructures","volume":"35","author":"Yi","year":"1998","journal-title":"Int. J. Solids Struct."},{"key":"ref13","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1007\/s00419-011-0526-z","article-title":"Homogenization of viscoelastic-matrix fibrous composites with square-lattice reinforcement","volume":"81","author":"Andrianov","year":"2011","journal-title":"Arch. Appl. Mech."},{"key":"ref14","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.euromechsol.2014.07.012","article-title":"Field statistics in linear viscoelastic composites and polycrystals","volume":"49","author":"Badulescu","year":"2015","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref15","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1177\/0021998304042732","article-title":"The response of a fiber-reinforced composite with a viscoelastic matrix phase","volume":"38","author":"Debotton","year":"2004","journal-title":"J. Compos. Mater."},{"key":"ref16","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1016\/j.euromechsol.2011.04.002","article-title":"Sequential linearization method for viscous\/elastic heterogeneous materials","volume":"30","author":"Kowalczyk-Gajewska","year":"2011","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref17","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1016\/S0997-7538(02)01247-0","article-title":"A quasi-elastic affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites","volume":"21","author":"Brenner","year":"2002","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref18","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcomc.2022.100248","article-title":"Time-domain asymptotic homogenization for linear-viscoelastic composites: Mathematical formulae and finite element implementation","volume":"8","author":"Zhai","year":"2022","journal-title":"Compos. Part C Open Access"},{"key":"ref19","doi-asserted-by":"crossref","DOI":"10.1016\/j.compstruct.2020.112690","article-title":"A numerical homogenization scheme used for derivation of a homogenized viscoelastic-viscoplastic model for the transverse response of fiber-reinforced polymer composites","volume":"252","author":"Liu","year":"2020","journal-title":"Compos. Struct."},{"key":"ref20","doi-asserted-by":"crossref","DOI":"10.1016\/j.euromechsol.2023.105108","article-title":"Time domain analytical bounds to the homogenized viscous kernels of linear viscoelastic composites","volume":"102","author":"Carini","year":"2023","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref21","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.ijplas.2011.12.004","article-title":"Micromechanical modeling of the interactions between the microstructure and the dissipative deformation mechanisms in steels under cyclic loading","volume":"32\u201333","author":"Mareau","year":"2012","journal-title":"Int. J. Plast."},{"key":"ref22","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.ijplas.2018.02.007","article-title":"Latent hardening within the elasto-plastic self-consistent polycrystal homogenization to enable the prediction of anisotropy of AA6022-T4 sheets","volume":"105","author":"Zecevic","year":"2018","journal-title":"Int. J. Plast."},{"key":"ref23","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/j.compscitech.2004.09.010","article-title":"Constitutive modelling of viscoelastic unloading of short glass fibre-reinforced polyethylene","volume":"65","author":"R\u00e9mond","year":"2005","journal-title":"Compos. Sci. Technol."},{"key":"ref24","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1615\/IntJMultCompEng.v4.i4.80","article-title":"Study of various estimates of the macroscopic tangent operator in the incremental homogenization of elastoplastic composites","volume":"4","author":"Pierard","year":"2006","journal-title":"Int. J. Multiscale Comput. Eng."},{"key":"ref25","series-title":"Comptes Rendus Acad\u00e9mie Sci. S\u00e9r. II M\u00e9canique Phys. Chim. Astron.","first-page":"563","article-title":"Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Casta\u00f1eda\u2019s nonlinear variational procedure","volume":"320","author":"Suquet","year":"1995"},{"key":"ref26","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmps.2020.104251","article-title":"Tangent second-order homogenisation estimates for incompressible hyperelastic composites with fibrous microstructures and anisotropic phases","volume":"147","author":"Bleiler","year":"2021","journal-title":"J. Mech. Phys. Solids"},{"key":"ref27","doi-asserted-by":"crossref","first-page":"102632","DOI":"10.1016\/j.ijplas.2019.11.011","article-title":"Homogenization using modified Mori-Tanaka and TFA framework for elastoplastic-viscoelastic-viscoplastic composites: Theory and numerical validation","volume":"127","author":"Barral","year":"2020","journal-title":"Int. J. Plast."},{"key":"ref28","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1098\/rspa.1992.0062","article-title":"On transformation strains and uniform fields in multiphase elastic media","volume":"437","author":"Dvorak","year":"1992","journal-title":"Proc. R Soc. Lond A"},{"key":"ref29","doi-asserted-by":"crossref","first-page":"6937","DOI":"10.1016\/S0020-7683(03)00346-9","article-title":"Nonuniform transformation field analysis","volume":"40","author":"Michel","year":"2003","journal-title":"Int. J. Solids Struct."},{"key":"ref30","doi-asserted-by":"crossref","first-page":"5477","DOI":"10.1016\/j.cma.2003.12.071","article-title":"Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis","volume":"193","author":"Michel","year":"2004","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/nme.4473","article-title":"Hybrid impotent-incompatible eigenstrain based homogenization","volume":"95","author":"Fish","year":"2013","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref32","volume":"127","author":"Sanchez-Palencia","year":"1980","journal-title":"Non-Homogeneous Media and Vibration Theory"},{"journal-title":"Asymptotic Analysis for Periodic Structures","year":"1978","author":"Papanicolau","key":"ref33"},{"key":"ref34","doi-asserted-by":"crossref","first-page":"6429","DOI":"10.1016\/S0020-7683(02)00255-X","article-title":"Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: A coupled thermo-viscoelastic example problem","volume":"39","author":"Yu","year":"2002","journal-title":"Int. J. Solids Struct."},{"key":"ref35","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.compositesb.2016.04.066","article-title":"Computation of the effective mechanical properties including nonclassical moduli of 2.5D and 3D interlocks by micromechanical approaches","volume":"98","author":"Rahali","year":"2016","journal-title":"Compos. Part B Eng."},{"key":"ref36","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","article-title":"Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement","volume":"194","author":"Hughes","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref37","doi-asserted-by":"crossref","DOI":"10.1002\/9780470749081","author":"Cottrell","year":"2009","journal-title":"Isogeometric Analysis: Toward Integration of CAD and FEA"},{"key":"ref38","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1002\/nme.5328","article-title":"On the treatments of heterogeneities and periodic boundary conditions for isogeometric homogenization analysis","volume":"109","author":"Matsubara","year":"2017","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref39","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1002\/nme.5775","article-title":"A framework for implementation of RVE-based multiscale models in computational homogenization using isogeometric analysis","volume":"114","author":"Alberdi","year":"2018","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref40","doi-asserted-by":"crossref","first-page":"2499","DOI":"10.1002\/nme.6948","article-title":"Computational homogenization of higher-order continua","volume":"123","author":"Schmidt","year":"2022","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref41","doi-asserted-by":"crossref","DOI":"10.1016\/j.compscitech.2024.110787","article-title":"Nitsche\u2019s method enhanced isogeometric homogenization of unidirectional composites with cylindrically orthotropic carbon\/graphite fibers","volume":"256","author":"Du","year":"2024","journal-title":"Compos. Sci. Technol."},{"key":"ref42","doi-asserted-by":"crossref","DOI":"10.1016\/j.compstruct.2023.117475","article-title":"Isogeometric homogenization of viscoelastic polymer composites via correspondence principle","volume":"323","author":"Chen","year":"2023","journal-title":"Compos. Struct."},{"key":"ref43","doi-asserted-by":"crossref","first-page":"1216","DOI":"10.1016\/j.cma.2006.08.015","article-title":"Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials","volume":"196","author":"Oskay","year":"2007","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref44","unstructured":"J. Fish, Practical Multiscaling. The Atrium, Southern Gate, Chichester, West Sussex, UK: John Wiley & Sons, 2014."},{"key":"ref45","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/S0045-7825(97)00030-3","article-title":"Computational plasticity for composite structures based on mathematical homogenization: Theory and practice","volume":"148","author":"Fish","year":"1997","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref46","doi-asserted-by":"crossref","first-page":"2016","DOI":"10.1016\/j.cma.2008.12.038","article-title":"Multiple scale eigendeformation-based reduced order homogenization","volume":"198","author":"Yuan","year":"2009","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref47","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1002\/nme.6058","article-title":"A second-order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials","volume":"119","author":"Fish","year":"2019","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref48","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113126","article-title":"A novel second-order reduced homogenization approach for nonlinear thermo-mechanical problems of axisymmetric structures with periodic micro-configurations","volume":"368","author":"Yang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref49","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1016\/j.apm.2020.07.009","article-title":"A second-order reduced multiscale approach for non-linear axisymmetric structures with periodic configurations","volume":"88","author":"Yang","year":"2020","journal-title":"Appl. Math. Model."},{"journal-title":"Computational Inelasticity","year":"2006","author":"Simo","key":"ref50"},{"journal-title":"Micromechanics of Composite Materials","year":"2012","author":"Dvorak","key":"ref51"},{"key":"ref52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0266-3538(00)00133-0","article-title":"Models for the strain-rate-dependent behavior of polymer composites","volume":"61","author":"Thiruppukuzhi","year":"2001","journal-title":"Compos Sci. Technol."},{"key":"ref53","doi-asserted-by":"crossref","DOI":"10.1016\/j.compstruct.2021.113679","article-title":"Constitutive modeling for linear viscoelastic fiber-reinforced composites","volume":"263","author":"Chen","year":"2021","journal-title":"Compos. Struct."}],"container-title":["Computers, Materials & Continua"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/file.techscience.com\/files\/cmc\/2025\/TSP_CMC-82-1\/TSP_CMC_59950\/TSP_CMC_59950.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,7]],"date-time":"2025-03-07T07:10:59Z","timestamp":1741331459000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.techscience.com\/cmc\/v82n1\/59268"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":53,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025]]},"published-print":{"date-parts":[[2025]]}},"URL":"https:\/\/doi.org\/10.32604\/cmc.2024.059950","relation":{},"ISSN":["1546-2226"],"issn-type":[{"type":"electronic","value":"1546-2226"}],"subject":[],"published":{"date-parts":[[2025]]}}}