{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T23:07:58Z","timestamp":1792105678256,"version":"4.4.0"},"reference-count":91,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,8,5]],"date-time":"2023-08-05T00:00:00Z","timestamp":1691193600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,5]],"date-time":"2023-08-05T00:00:00Z","timestamp":1691193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006256","name":"Universit\u00e0 degli Studi dell\u2019Aquila","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006256","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Biomech Model Mechanobiol"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifically focus upon the crucial gradual adjustment of bone tissue as a structural material that adapts its mechanical features, such as materials stiffnesses and microstructure, in response to the evolving loading conditions. We postulate that the evolution process relies on a feedback mechanism involving multiple stimulus signals. The mechanical and remodeling behavior of bone tissue is clearly a complex process that is difficult to describe within the framework of classical continuum theories. For this reason, a generalized continuum elastic theory is employed as a proper mathematical context for an adequate description of the examined phenomenon. To simplify the investigation, we considered a two-dimensional problem. Numerical simulations have been performed to illustrate bone evolution in a few significant cases: the bending of a rectangular cantilever plate and a three-point flexure test. The results are encouraging because they can replicate the optimization process observed in bone remodeling. The proposed model provides a likely distribution of stiffnesses and accurately represents the arrangement of trabeculae macroscopically described by the orthotropic symmetry directions, as supported by experimental evidence from the trajectorial theory.<\/jats:p>","DOI":"10.1007\/s10237-023-01755-w","type":"journal-article","created":{"date-parts":[[2023,8,5]],"date-time":"2023-08-05T06:01:30Z","timestamp":1691215290000},"page":"2135-2152","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":82,"title":["An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff\u2019s law"],"prefix":"10.1007","volume":"22","author":[{"given":"Ivan","family":"Giorgio","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francesco","family":"dell\u2019Isola","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ugo","family":"Andreaus","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anil","family":"Misra","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,8,5]]},"reference":[{"issue":"3","key":"1755_CR1","doi-asserted-by":"publisher","first-page":"213","DOI":"10.2140\/memocs.2018.6.213","volume":"6","author":"H Abdoul-Anziz","year":"2018","unstructured":"Abdoul-Anziz H, Seppecher P (2018) Strain gradient and generalized continua obtained by homogenizing frame lattices. Math Mech Complex Syst 6(3):213\u2013250","journal-title":"Math Mech Complex Syst"},{"issue":"12","key":"1755_CR2","doi-asserted-by":"publisher","first-page":"3976","DOI":"10.1177\/1081286519855431","volume":"24","author":"H Abdoul-Anziz","year":"2019","unstructured":"Abdoul-Anziz H, Seppecher P, Bellis C (2019) Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms. Math Mech Solids 24(12):3976\u20133999","journal-title":"Math Mech Solids"},{"issue":"4","key":"1755_CR3","doi-asserted-by":"publisher","first-page":"339","DOI":"10.2140\/memocs.2018.6.339","volume":"6","author":"R Allena","year":"2018","unstructured":"Allena R, Cluzel C (2018) Heterogeneous directions of orthotropy in three-dimensional structures: finite element description based on diffusion equations. Math Mech Complex Syst 6(4):339\u2013351","journal-title":"Math Mech Complex Syst"},{"key":"1755_CR4","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1023\/A:1010658802322","volume":"65","author":"J Andres","year":"2001","unstructured":"Andres J, Bersani AM, Le\u015bniak K (2001) On some almost-periodicity problems in various metrics. Acta Applicandae Mathematica 65:35\u201357","journal-title":"Acta Applicandae Mathematica"},{"issue":"1","key":"1755_CR5","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s00033-021-01668-7","volume":"73","author":"E Barchiesi","year":"2022","unstructured":"Barchiesi E, Hamila N (2022) Maximum mechano-damage power release-based phase-field modeling of mass diffusion in damaging deformable solids. Z Angew Math Phys 73(1):35","journal-title":"Z Angew Math Phys"},{"issue":"5","key":"1755_CR6","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1002\/jor.1100080506","volume":"8","author":"G Beaupr\u00e9","year":"1990","unstructured":"Beaupr\u00e9 G, Orr T, Carter D (1990) An approach for time-dependent bone modeling and remodeling\u2013theoretical development. J Orthop Res 8(5):651\u2013661","journal-title":"J Orthop Res"},{"key":"1755_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00033-016-0687-2","volume":"67","author":"E Bednarczyk","year":"2016","unstructured":"Bednarczyk E, Lekszycki T (2016) A novel mathematical model for growth of capillaries and nutrient supply with application to prediction of osteophyte onset. Z Angew Math Phys 67:1\u201314","journal-title":"Z Angew Math Phys"},{"issue":"6","key":"1755_CR8","doi-asserted-by":"publisher","first-page":"976","DOI":"10.1177\/10812865211048925","volume":"27","author":"E Bednarczyk","year":"2022","unstructured":"Bednarczyk E, Lekszycki T (2022) Evolution of bone tissue based on angiogenesis as a crucial factor: new mathematical attempt. Math Mech Solids 27(6):976\u2013988","journal-title":"Math Mech Solids"},{"key":"1755_CR9","volume-title":"Topology Optimization: Theory, Methods, and Applications","author":"MP Bendsoe","year":"2003","unstructured":"Bendsoe MP, Sigmund O (2003) Topology Optimization: Theory, Methods, and Applications. Springer, Berlin, Heidelberg"},{"issue":"2","key":"1755_CR10","doi-asserted-by":"publisher","first-page":"59","DOI":"10.3390\/biomimetics7020059","volume":"7","author":"N Branecka","year":"2022","unstructured":"Branecka N, Yildizdag ME, Ciallella A, Giorgio I (2022) Bone remodeling process based on hydrostatic and deviatoric strain mechano-sensing. Biomimetics 7(2):59. https:\/\/doi.org\/10.3390\/biomimetics7020059","journal-title":"Biomimetics"},{"issue":"4270","key":"1755_CR11","doi-asserted-by":"publisher","first-page":"1174","DOI":"10.1126\/science.996549","volume":"194","author":"DR Carter","year":"1976","unstructured":"Carter DR, Hayes WC (1976) Bone compressive strength: the influence of density and strain rate. Science 194(4270):1174\u20131176","journal-title":"Science"},{"issue":"8","key":"1755_CR12","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1016\/0021-9290(87)90058-3","volume":"20","author":"D Carter","year":"1987","unstructured":"Carter D, Fyhrie DP, Whalen R (1987) Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. J Biomech 20(8):785\u2013794","journal-title":"J Biomech"},{"issue":"4","key":"1755_CR13","doi-asserted-by":"publisher","first-page":"353","DOI":"10.2140\/memocs.2018.6.353","volume":"6","author":"C Cluzel","year":"2018","unstructured":"Cluzel C, Allena R (2018) A general method for the determination of the local orthotropic directions of heterogeneous materials: application to bone structures using $$\\mu$$ct images. Math Mech Complex syst 6(4):353\u2013367","journal-title":"Math Mech Complex syst"},{"issue":"1","key":"1755_CR14","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1093\/ejo\/cjaa016","volume":"43","author":"MA Cornelis","year":"2021","unstructured":"Cornelis MA, Tepedino M, Riis NDV, Niu X, Cattaneo PM (2021) Treatment effect of bone-anchored maxillary protraction in growing patients compared to controls: a systematic review with meta-analysis. Eur J Orthod 43(1):51\u201368","journal-title":"Eur J Orthod"},{"issue":"1","key":"1755_CR15","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1115\/1.2895436","volume":"114","author":"S Cowin","year":"1992","unstructured":"Cowin S, Sadegh A, Luo G (1992) An evolutionary Wolff\u2019s law for trabecular architecture. J Biomech Eng 114(1):129\u2013136","journal-title":"J Biomech Eng"},{"key":"1755_CR16","doi-asserted-by":"publisher","first-page":"104492","DOI":"10.1016\/j.euromechsol.2021.104492","volume":"93","author":"M Cuomo","year":"2022","unstructured":"Cuomo M, Boutin C, Contrafatto L, Gazzo S (2022) Effective anisotropic properties of fibre network sheets. Eur J Mech A Sol 93:104492","journal-title":"Eur J Mech A Sol"},{"issue":"2","key":"1755_CR17","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/0021-9290(88)90006-1","volume":"21","author":"JD Currey","year":"1988","unstructured":"Currey JD (1988) The effect of porosity and mineral content on the Young\u2019s modulus of elasticity of compact bone. J Biomech 21(2):131\u2013139","journal-title":"J Biomech"},{"issue":"9","key":"1755_CR18","doi-asserted-by":"publisher","first-page":"1011","DOI":"10.1002\/zamm.201600126","volume":"97","author":"C Cyron","year":"2017","unstructured":"Cyron C, Aydin R (2017) Mechanobiological free energy: a variational approach to tensional homeostasis in tissue equivalents. ZAMM-Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik 97(9):1011\u20131019","journal-title":"ZAMM-Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik"},{"issue":"2","key":"1755_CR19","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1615\/IntJMultCompEng.v5.i2.40","volume":"5","author":"F De Angelis","year":"2007","unstructured":"De Angelis F (2007) A variationally consistent formulation of nonlocal plasticity. Int J Multiscale Comput Eng 5(2):105\u2013116","journal-title":"Int J Multiscale Comput Eng"},{"issue":"2\u20133","key":"1755_CR20","first-page":"127","volume":"74","author":"S De Cicco","year":"2022","unstructured":"De Cicco S (2022) Non-simple elastic materials with double porosity structure. Archives of Mechanics 74(2\u20133):127\u2013142","journal-title":"Archives of Mechanics"},{"issue":"1","key":"1755_CR21","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1177\/1081286519867109","volume":"25","author":"S De Cicco","year":"2020","unstructured":"De Cicco S, De Angelis F (2020) A plane strain problem in the theory of elastic materials with voids. Math Mech Solids 25(1):46\u201359","journal-title":"Math Mech Solids"},{"issue":"12","key":"1755_CR22","doi-asserted-by":"publisher","first-page":"2715","DOI":"10.1177\/10812865221078822","volume":"27","author":"F dell\u2019Isola","year":"2022","unstructured":"dell\u2019Isola F, Eugster SR, Fedele R, Seppecher P (2022) Second-gradient continua: From lagrangian to eulerian and back. Math Mech Solids 27(12):2715\u20132750","journal-title":"Math Mech Solids"},{"issue":"12","key":"1755_CR23","doi-asserted-by":"publisher","first-page":"2715","DOI":"10.1177\/10812865221078822","volume":"27","author":"F dell\u2019Isola","year":"2022","unstructured":"dell\u2019Isola F, Eugster SR, Fedele R, Seppecher P (2022) Second-gradient continua: From Lagrangian to Eulerian and back. Math Mech Solids 27(12):2715\u20132750","journal-title":"Math Mech Solids"},{"issue":"6","key":"1755_CR24","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/S0093-6413(02)00297-5","volume":"29","author":"A DiCarlo","year":"2002","unstructured":"DiCarlo A, Quiligotti S (2002) Growth and balance. Mech Res Commun 29(6):449\u2013456","journal-title":"Mech Res Commun"},{"issue":"1","key":"1755_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0021-9290(01)00178-6","volume":"35","author":"M Doblar\u00e9","year":"2002","unstructured":"Doblar\u00e9 M, Garc\u0131a J (2002) Anisotropic bone remodelling model based on a continuum damage-repair theory. J Biomech 35(1):1\u201317","journal-title":"J Biomech"},{"issue":"14","key":"1755_CR26","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1016\/j.ijsolstr.2012.04.007","volume":"49","author":"VA Eremeyev","year":"2012","unstructured":"Eremeyev VA, Pietraszkiewicz W (2012) Material symmetry group of the non-linear polar-elastic continuum. Int J Solids Struct 49(14):1993\u20132005","journal-title":"Int J Solids Struct"},{"issue":"2","key":"1755_CR27","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1177\/1081286515582862","volume":"21","author":"VA Eremeyev","year":"2016","unstructured":"Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math Mech Solids 21(2):210\u2013221","journal-title":"Math Mech Solids"},{"key":"1755_CR28","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s11223-017-9901-5","volume":"49","author":"V Eremeyev","year":"2017","unstructured":"Eremeyev V, Skrzat A, Stachowicz F (2017) Linear micropolar elasticity analysis of stresses in bones under static loads. Strength Mater 49:575\u2013585","journal-title":"Strength Mater"},{"key":"1755_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00033-020-01395-5","volume":"71","author":"VA Eremeyev","year":"2020","unstructured":"Eremeyev VA, Lurie SA, Solyaev YO, dell\u2019Isola F (2020) On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity. Z Angew Math Phys 71:1\u201316","journal-title":"Z Angew Math Phys"},{"issue":"4","key":"1755_CR30","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1115\/1.1388075","volume":"54","author":"HA Eschenauer","year":"2001","unstructured":"Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331\u2013390","journal-title":"Appl Mech Rev"},{"issue":"5","key":"1755_CR31","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1007\/s00161-022-01100-z","volume":"34","author":"R Fedele","year":"2022","unstructured":"Fedele R (2022) Approach \u00e0 la Piola for the equilibrium problem of bodies with second gradient energies. part II: variational derivation of second gradient equations and their transport. Continuum Mech Thermodyn 34(5):1087\u20131111","journal-title":"Continuum Mech Thermodyn"},{"key":"1755_CR32","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s00161-018-0668-0","volume":"31","author":"P Franciosi","year":"2019","unstructured":"Franciosi P, Spagnuolo M, Salman OU (2019) Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mech Thermodyn 31:101\u2013132","journal-title":"Continuum Mech Thermodyn"},{"issue":"1","key":"1755_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/ar.1092190104","volume":"219","author":"HM Frost","year":"1987","unstructured":"Frost HM (1987) Bone \u201cmass\" and the \u201cmechanostat\": a proposal. Anat Rec 219(1):1\u20139","journal-title":"Anat Rec"},{"key":"1755_CR34","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.mechrescom.2019.02.007","volume":"96","author":"J Ganghoffer","year":"2019","unstructured":"Ganghoffer J, Rahouadj R, Boisse J, Schiavi J (2019) A phase field approach for bone remodeling based on a second-gradient model. Mech Res Commun 96:37\u201344","journal-title":"Mech Res Commun"},{"key":"1755_CR35","doi-asserted-by":"publisher","first-page":"104009","DOI":"10.1016\/j.euromechsol.2020.104009","volume":"82","author":"S Gazzo","year":"2020","unstructured":"Gazzo S, Cuomo M, Boutin C, Contrafatto L (2020) Directional properties of fibre network materials evaluated by means of discrete homogenization. Eur J Mech A Sol 82:104009","journal-title":"Eur J Mech A Sol"},{"issue":"4","key":"1755_CR36","doi-asserted-by":"publisher","first-page":"307","DOI":"10.2140\/memocs.2018.6.307","volume":"6","author":"D George","year":"2018","unstructured":"George D, Allena R, Remond Y (2018) A multiphysics stimulus for continuum mechanics bone remodeling. Math Mech Complex Syst 6(4):307\u2013319","journal-title":"Math Mech Complex Syst"},{"key":"1755_CR37","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1007\/s00161-018-0726-7","volume":"31","author":"D George","year":"2019","unstructured":"George D, Allena R, Remond Y (2019) Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction. Continuum Mech Thermodyn 31:725\u2013740","journal-title":"Continuum Mech Thermodyn"},{"key":"1755_CR38","doi-asserted-by":"publisher","first-page":"1325","DOI":"10.1007\/s10237-016-0765-6","volume":"15","author":"I Giorgio","year":"2016","unstructured":"Giorgio I, Andreaus U, Scerrato D, dell\u2019Isola F (2016) A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomech Model Mechanobiol 15:1325\u20131343","journal-title":"Biomech Model Mechanobiol"},{"issue":"7","key":"1755_CR39","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/j.crme.2016.02.009","volume":"344","author":"I Giorgio","year":"2016","unstructured":"Giorgio I, Della Corte A, dell\u2019Isola F, Steigmann DJ (2016) Buckling modes in pantographic lattices. CR Mec 344(7):487\u2013501","journal-title":"CR Mec"},{"key":"1755_CR40","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.eml.2017.02.008","volume":"13","author":"I Giorgio","year":"2017","unstructured":"Giorgio I, Andreaus U, dell\u2019Isola F, Lekszycki T (2017) Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mech Lett 13:141\u2013147","journal-title":"Extreme Mech Lett"},{"issue":"6","key":"1755_CR41","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1007\/s10237-019-01166-w","volume":"18","author":"I Giorgio","year":"2019","unstructured":"Giorgio I, dell\u2019Isola F, Andreaus U, Alzahrani F, Hayat T, Lekszycki T (2019) On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon. Biomech Model Mechanobiol 18(6):1639\u20131663","journal-title":"Biomech Model Mechanobiol"},{"issue":"7","key":"1755_CR42","doi-asserted-by":"publisher","first-page":"1074","DOI":"10.1177\/1081286520978516","volume":"26","author":"I Giorgio","year":"2021","unstructured":"Giorgio I, Spagnuolo M, Andreaus U, Scerrato D, Bersani AM (2021) In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical metamaterials. Math Mech Solids 26(7):1074\u20131103","journal-title":"Math Mech Solids"},{"key":"1755_CR43","doi-asserted-by":"publisher","first-page":"113476","DOI":"10.1016\/j.cma.2020.113476","volume":"373","author":"L Greco","year":"2021","unstructured":"Greco L, Cuomo M (2021) An implicit G1-conforming bi-cubic interpolation for the analysis of smooth and folded Kirchhoff-Love shell assemblies. Comput Methods Appl Mech Eng 373:113476","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1755_CR44","doi-asserted-by":"publisher","first-page":"114176","DOI":"10.1016\/j.cma.2021.114176","volume":"387","author":"L Greco","year":"2021","unstructured":"Greco L, Scrofani A, Cuomo M (2021) A non-linear symmetric G1-conforming B\u00e9zier finite element formulation for the analysis of Kirchhoff beam assemblies. Comput Methods Appl Mech Eng 387:114176","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1755_CR45","doi-asserted-by":"publisher","DOI":"10.1177\/10812865231152228","author":"A Grillo","year":"2023","unstructured":"Grillo A, Di Stefano S (2023) A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint. Math Mech Sol. https:\/\/doi.org\/10.1177\/10812865231152228","journal-title":"Math Mech Sol"},{"key":"1755_CR46","doi-asserted-by":"publisher","DOI":"10.1177\/10812865231178291","author":"A Grillo","year":"2023","unstructured":"Grillo A, Di Stefano S (2023) Addendum to \u201cA formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheononic constraint\u201d. Math Mech Sol. https:\/\/doi.org\/10.1177\/10812865231178291","journal-title":"Math Mech Sol"},{"issue":"9","key":"1755_CR47","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1177\/1081286513515265","volume":"20","author":"A Grillo","year":"2015","unstructured":"Grillo A, Wittum G, Tomic A, Federico S (2015) Remodelling in statistically oriented fibre-reinforced materials and biological tissues. Math Mech Solids 20(9):1107\u20131129","journal-title":"Math Mech Solids"},{"key":"1755_CR48","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s10665-017-9940-8","volume":"109","author":"A Grillo","year":"2018","unstructured":"Grillo A, Carfagna M, Federico S (2018) An Allen-Cahn approach to the remodelling of fibre-reinforced anisotropic materials. J Eng Math 109:139\u2013172","journal-title":"J Eng Math"},{"key":"1755_CR49","doi-asserted-by":"publisher","first-page":"6","DOI":"10.3389\/fbioe.2014.00006","volume":"2","author":"R Hambli","year":"2014","unstructured":"Hambli R (2014) Connecting mechanics and bone cell activities in the bone remodeling process: an integrated finite element modeling. Front Bioeng Biotechnol 2:6","journal-title":"Front Bioeng Biotechnol"},{"issue":"11\u201312","key":"1755_CR50","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1016\/0021-9290(87)90030-3","volume":"20","author":"R Huiskes","year":"1987","unstructured":"Huiskes R, Weinans H, Grootenboer H, Dalstra M, Fudala B, Slooff T (1987) Adaptive bone-remodeling theory applied to prosthetic-design analysis. J Biomech 20(11\u201312):1135\u20131150","journal-title":"J Biomech"},{"issue":"4","key":"1755_CR51","first-page":"1","volume":"35","author":"K Jankowski","year":"2022","unstructured":"Jankowski K, Pawlikowski M, Doma\u0144ski J (2022) Multi-scale constitutive model of human trabecular bone. Continu Mech Thermodyn 35(4):1\u201314","journal-title":"Continu Mech Thermodyn"},{"issue":"4","key":"1755_CR52","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1111\/joa.12446","volume":"228","author":"TL Kivell","year":"2016","unstructured":"Kivell TL (2016) A review of trabecular bone functional adaptation: what have we learned from trabecular analyses in extant hominoids and what can we apply to fossils? J Anat 228(4):569\u2013594","journal-title":"J Anat"},{"issue":"1","key":"1755_CR53","doi-asserted-by":"publisher","first-page":"303","DOI":"10.2140\/jomms.2011.6.303","volume":"6","author":"C Kumar","year":"2011","unstructured":"Kumar C, Jasiuk I, Dantzig J (2011) Dissipation energy as a stimulus for cortical bone adaptation. J Mech Mater Struct 6(1):303\u2013319","journal-title":"J Mech Mater Struct"},{"key":"1755_CR54","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1007\/s00256-015-2203-8","volume":"44","author":"C Lamm","year":"2015","unstructured":"Lamm C, Dockner M, Pospischek B et al (2015) Micro-CT analyses of historical bone samples presenting with osteomyelitis. Skeletal Radiol 44:1507\u20131514","journal-title":"Skeletal Radiol"},{"issue":"1","key":"1755_CR55","first-page":"160","volume":"56","author":"LE Lanyon","year":"1974","unstructured":"Lanyon LE (1974) Experimental support for the trajectorial theory of bone structure. J Bone Joint Surg 56(1):160\u2013166","journal-title":"J Bone Joint Surg"},{"issue":"2","key":"1755_CR56","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1007\/s00033-022-01715-x","volume":"73","author":"G La Valle","year":"2022","unstructured":"La Valle G (2022) A new deformation measure for the nonlinear micropolar continuum. Z Angew Math Phys 73(2):78","journal-title":"Z Angew Math Phys"},{"issue":"6","key":"1755_CR57","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1002\/zamm.201100082","volume":"92","author":"T Lekszycki","year":"2012","unstructured":"Lekszycki T, dell\u2019Isola F (2012) A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM J Appl Math Mech 92(6):426\u2013444","journal-title":"ZAMM J Appl Math Mech"},{"issue":"8","key":"1755_CR58","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1016\/j.crme.2012.05.003","volume":"340","author":"A Madeo","year":"2012","unstructured":"Madeo A, George D, Lekszycki T, Nierenberger M, R\u00e9mond Y (2012) A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Comptes Rendus M\u00e9canique 340(8):575\u2013589","journal-title":"Comptes Rendus M\u00e9canique"},{"issue":"1","key":"1755_CR59","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1038\/s41413-019-0061-z","volume":"7","author":"EA Masters","year":"2019","unstructured":"Masters EA, Trombetta RP, MesyBentley KL et al (2019) Evolving concepts in bone infection: redefining \u201cbiofilm\u2019\u2019, \u201cacute vs. chronic osteomyelitis\u2019\u2019, \u201cthe immune proteome\u2019\u2019 and \u201clocal antibiotic therapy\u2019\u2019. Bone Res 7(1):20","journal-title":"Bone Res"},{"issue":"4","key":"1755_CR60","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1002\/jor.1100130405","volume":"13","author":"M Mullender","year":"1995","unstructured":"Mullender M, Huiskes R (1995) Proposal for the regulatory mechanism of Wolff\u2019s law. J Orthop Res 13(4):503\u2013512","journal-title":"J Orthop Res"},{"issue":"2","key":"1755_CR61","first-page":"49","volume":"12","author":"M Nowak","year":"2010","unstructured":"Nowak M (2010) On some properties of bone functional adaptation phenomenon useful in mechanical design. Acta Bioeng Biomech 12(2):49\u201354","journal-title":"Acta Bioeng Biomech"},{"key":"1755_CR62","first-page":"97","volume-title":"Developments and Novel Approaches in Biomechanics and Metamaterials. Advanced Structured Materials","author":"M Nowak","year":"2020","unstructured":"Nowak M (2020) New aspects of the trabecular bone remodeling regulatory model\u2013Two postulates based on shape optimization studies. In: Abali B, Giorgio I (eds) Developments and Novel Approaches in Biomechanics and Metamaterials. Advanced Structured Materials, vol 132. Springer, Cham, pp 97\u2013105"},{"issue":"5","key":"1755_CR63","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1016\/0021-9290(86)90015-1","volume":"19","author":"HC Park","year":"1986","unstructured":"Park HC, Lakes RS (1986) Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent. J Biomech 19(5):385\u2013397","journal-title":"J Biomech"},{"key":"1755_CR64","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00033-018-0947-4","volume":"69","author":"L Placidi","year":"2018","unstructured":"Placidi L, Misra A, Barchiesi E (2018) Two-dimensional strain gradient damage modeling: a variational approach. Z Angew Math Phys 69:1\u201319","journal-title":"Z Angew Math Phys"},{"issue":"2","key":"1755_CR65","doi-asserted-by":"publisher","first-page":"77","DOI":"10.2140\/memocs.2018.6.77","volume":"6","author":"L Placidi","year":"2018","unstructured":"Placidi L, Barchiesi E, Misra A (2018) A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math Mech Complex Syst 6(2):77\u2013100","journal-title":"Math Mech Complex Syst"},{"key":"1755_CR66","doi-asserted-by":"publisher","first-page":"1143","DOI":"10.1007\/s00161-018-0693-z","volume":"31","author":"L Placidi","year":"2019","unstructured":"Placidi L, Misra A, Barchiesi E (2019) Simulation results for damage with evolving microstructure and growing strain gradient moduli. Continuum Mech Thermodyn 31:1143\u20131163","journal-title":"Continuum Mech Thermodyn"},{"issue":"8","key":"1755_CR67","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1016\/0021-9290(94)90223-2","volume":"27","author":"P Prendergast","year":"1994","unstructured":"Prendergast P, Taylor D (1994) Prediction of bone adaptation using damage accumulation. J Biomech 27(8):1067\u20131076","journal-title":"J Biomech"},{"key":"1755_CR68","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1007\/978-3-031-26186-2_39","volume-title":"Sixty Shades of Generalized Continua","author":"BC Sarar","year":"2023","unstructured":"Sarar BC, Yildizdag ME, Abali BE (2023) Comparison of homogenization techniques in strain gradient elasticity for determining material parameters. Advanced structured materials. Sixty Shades of Generalized Continua. Springer, Cham, pp 631\u2013644"},{"issue":"4","key":"1755_CR69","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1177\/09544119JEIM895","volume":"225","author":"M Sarikanat","year":"2011","unstructured":"Sarikanat M, Yildiz H (2011) Determination of bone density distribution in proximal femur by using the 3D orthotropic bone adaptation model. Proc Inst Mech Eng 225(4):365\u2013375","journal-title":"Proc Inst Mech Eng"},{"issue":"11","key":"1755_CR70","doi-asserted-by":"publisher","first-page":"2436","DOI":"10.3390\/sym14112436","volume":"14","author":"D Scerrato","year":"2022","unstructured":"Scerrato D, Giorgio I, Bersani AM, Andreucci D (2022) A proposal for a novel formulation based on the hyperbolic Cattaneo\u2019s equation to describe the mechano-transduction process occurring in bone remodeling. Symmetry 14(11):2436","journal-title":"Symmetry"},{"key":"1755_CR71","doi-asserted-by":"publisher","first-page":"110144","DOI":"10.1016\/j.ymssp.2023.110144","volume":"190","author":"S Sessa","year":"2023","unstructured":"Sessa S (2023) An inverse strategy for identifying the mechanical parameters of an asymmetric hysteretic constitutive model. Mech Syst Signal Process 190:110144","journal-title":"Mech Syst Signal Process"},{"issue":"7","key":"1755_CR72","doi-asserted-by":"publisher","first-page":"1133","DOI":"10.3390\/sym12071133","volume":"12","author":"M Shirani","year":"2020","unstructured":"Shirani M, Steigmann DJ (2020) A Cosserat model of elastic solids reinforced by a family of curved and twisted fibers. Symmetry 12(7):1133","journal-title":"Symmetry"},{"issue":"1","key":"1755_CR73","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1177\/1081286519861827","volume":"25","author":"Y Solyaev","year":"2020","unstructured":"Solyaev Y, Lurie S, Barchiesi E, Placidi L (2020) On the dependence of standard and gradient elastic material constants on a field of defects. Math Mech Solids 25(1):35\u201345","journal-title":"Math Mech Solids"},{"issue":"12","key":"1755_CR74","doi-asserted-by":"publisher","first-page":"2660","DOI":"10.3390\/sym14122660","volume":"14","author":"M Spagnuolo","year":"2022","unstructured":"Spagnuolo M (2022) Symmetrization of mechanical response in fibrous metamaterials through micro-shear deformability. Symmetry 14(12):2660","journal-title":"Symmetry"},{"key":"1755_CR75","doi-asserted-by":"crossref","unstructured":"Spencer A (1982) The formulation of constitutive equation for anisotropic solids. In: Mechanical Behavior of Anisotropic Solids\/Comportment M\u00e9chanique des Solides Anisotropes: Proceedings of the Euromech Colloquium 115 Villard-de-Lans, June 19\u201322, 1979\/Colloque Euromech 115 Villard-de-Lans, 19\u201322 Juin 1979. Springer, pp 3\u201326","DOI":"10.1007\/978-94-009-6827-1_1"},{"key":"1755_CR76","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-7091-4336-0","volume-title":"Continuum Theory of the Mechanics of Fibre-reinforced Composites","author":"AJM Spencer","year":"1984","unstructured":"Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. Continuum Theory of the Mechanics of Fibre-reinforced Composites. Springer, Vienna, pp 1\u201332"},{"issue":"2","key":"1755_CR77","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.ijnonlinmec.2007.02.015","volume":"42","author":"AJM Spencer","year":"2007","unstructured":"Spencer AJM, Soldatos KP (2007) Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int J Non-Linear Mech 42(2):355\u2013368","journal-title":"Int J Non-Linear Mech"},{"issue":"7","key":"1755_CR78","doi-asserted-by":"publisher","first-page":"734","DOI":"10.1016\/j.ijnonlinmec.2012.04.007","volume":"47","author":"DJ Steigmann","year":"2012","unstructured":"Steigmann DJ (2012) Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int J Non-Linear Mech 47(7):734\u2013742","journal-title":"Int J Non-Linear Mech"},{"issue":"3","key":"1755_CR79","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s10409-015-0413-x","volume":"31","author":"DJ Steigmann","year":"2015","unstructured":"Steigmann DJ, dell\u2019Isola F (2015) Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech Sin 31(3):373\u2013382","journal-title":"Acta Mech Sin"},{"issue":"5","key":"1755_CR80","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1177\/1081286520968149","volume":"26","author":"D Timofeev","year":"2021","unstructured":"Timofeev D, Barchiesi E, Misra A, Placidi L (2021) Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math Mech Solids 26(5):738\u2013770","journal-title":"Math Mech Solids"},{"issue":"5","key":"1755_CR81","doi-asserted-by":"publisher","first-page":"1027","DOI":"10.1093\/imamat\/hxu039","volume":"79","author":"A Tomic","year":"2014","unstructured":"Tomic A, Grillo A, Federico S (2014) Poroelastic materials reinforced by statistically oriented fibres\u2013numerical implementation and application to articular cartilage. The IMA Journal of Applied Mathematics 79(5):1027\u20131059","journal-title":"The IMA Journal of Applied Mathematics"},{"key":"1755_CR82","doi-asserted-by":"publisher","first-page":"110224","DOI":"10.1016\/j.ymssp.2023.110224","volume":"192","author":"J Torabi","year":"2023","unstructured":"Torabi J, Niiranen J (2023) Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach. Mech Syst Signal Process 192:110224","journal-title":"Mech Syst Signal Process"},{"issue":"5","key":"1755_CR83","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1007\/s00161-018-0656-4","volume":"30","author":"E Turco","year":"2018","unstructured":"Turco E (2018) Discrete is it enough? The revival of Piola-Hencky keynotes to analyze three-dimensional Elastica. Continuum Mech Thermodyn 30(5):1039\u20131057","journal-title":"Continuum Mech Thermodyn"},{"issue":"10","key":"1755_CR84","doi-asserted-by":"publisher","first-page":"1950","DOI":"10.1177\/10812865211067987","volume":"27","author":"E Turco","year":"2022","unstructured":"Turco E (2022) Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky\u2019s ideas. Math Mech Solids 27(10):1950\u20131973","journal-title":"Math Mech Solids"},{"key":"1755_CR85","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00033-016-0681-8","volume":"67","author":"E Turco","year":"2016","unstructured":"Turco E, dell\u2019Isola F, Cazzani A, Rizzi NL (2016) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 67:1\u201328","journal-title":"Z Angew Math Phys"},{"issue":"3","key":"1755_CR86","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/8756-3282(91)90043-I","volume":"12","author":"CH Turner","year":"1991","unstructured":"Turner CH (1991) Homeostatic control of bone structure: an application of feedback theory. Bone 12(3):203\u2013217","journal-title":"Bone"},{"issue":"5","key":"1755_CR87","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/S8756-3282(98)00118-5","volume":"23","author":"CH Turner","year":"1998","unstructured":"Turner CH (1998) Three rules for bone adaptation to mechanical stimuli. Bone 23(5):399\u2013407","journal-title":"Bone"},{"key":"1755_CR88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1108\/EC-02-2020-0073","volume":"38","author":"B Vazic","year":"2021","unstructured":"Vazic B, Abali BE, Yang H, Newell P (2021) Mechanical analysis of heterogeneous materials with higher-order parameters. Eng Comput 38:1\u201317","journal-title":"Eng Comput"},{"issue":"1","key":"1755_CR89","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s00161-022-01171-y","volume":"35","author":"B Vazic","year":"2023","unstructured":"Vazic B, Abali BE, Newell P (2023) Generalized thermo-mechanical framework for heterogeneous materials through asymptotic homogenization. Continuum Mech Thermodyn 35(1):159\u2013181","journal-title":"Continuum Mech Thermodyn"},{"key":"1755_CR90","volume-title":"Das Gesetz der Transformation der Knochen (The Law of Bone Remodeling)","author":"J Wolff","year":"1892","unstructured":"Wolff J (1892) Das Gesetz der Transformation der Knochen (The Law of Bone Remodeling). Verlag von August Hirschwald, Berlin"},{"key":"1755_CR91","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.bone.2018.03.003","volume":"111","author":"F Zhao","year":"2018","unstructured":"Zhao F, Kirby M, Roy A, Hu Y, Guo XE, Wang X (2018) Commonality in the microarchitecture of trabecular bone: a preliminary study. Bone 111:59\u201370","journal-title":"Bone"}],"container-title":["Biomechanics and Modeling in Mechanobiology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10237-023-01755-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10237-023-01755-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10237-023-01755-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T09:03:23Z","timestamp":1698483803000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10237-023-01755-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,5]]},"references-count":91,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["1755"],"URL":"https:\/\/doi.org\/10.1007\/s10237-023-01755-w","relation":{},"ISSN":["1617-7959","1617-7940"],"issn-type":[{"value":"1617-7959","type":"print"},{"value":"1617-7940","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,5]]},"assertion":[{"value":"11 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 August 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}