We present a nonlinear micropolar continuum model that includes a diffusion–reaction equation for the remodeling of bone with a trabecular lattice microstructure. This micropolar formulation enables the modeling of bone’s adaptive response to mechanical stimuli. The remodeling process is described as a time evolution regulated by a feedback mechanism that adjusts both the orientation of the trabecular lattice microarchitecture, thanks to the micropolar formulation, and the mechanical properties, related to key morphological features such us bone density. The junctions between trabeculae are modeled as nodal points within the micropolar framework, incorporating the following energy densities: mechanical deformation, mechano-biologic contributions, and Rayleigh-type dissipation terms that control the time evolution of remodeling. We present the weak form of the energetic formulation, which includes the diffusion–reaction equation. The cases of isotropic and orthotropic material symmetries class are deduced from the general formulation. This weak formulation is adapted to the development of a computational model based on the use of the finite element method. At the end of this paper, we discuss aspects related to validation, model complexity, and possible clinical applicability.
MartinRBBurrDBSharkeyNA. Skeletal tissue mechanics. New York: Springer, 1998.
2.
HuiskesRWeinansHGrootenboerH, et al. Adaptive bone-remodeling theory applied to prosthetic-design analysis. J Biomech1987; 20(11–12): 1135–1150.
3.
CarterDRHayesWC. Bone compressive strength: the influence of density and strain rate. Science1976; 194(4270): 1174–1176.
4.
CarterDFyhrieDPWhalenR. Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. J Biomech1987; 20(8): 785–794.
5.
GeorgeDAllenaRBourzacC, et al. A new comprehensive approach for bone remodeling under medium and high mechanical load based on cellular activity. Math Mech Complex Syst2020; 8(4): 287–306.
6.
CowinSCSadeghAMLuoGM. An evolutionary Wolff’s law for trabecular architecture. J Biomech Eng1992; 114(1): 129–136.
7.
DoblaréMGarcıaJ. Anisotropic bone remodelling model based on a continuum damage-repair theory. J Biomech2002; 35(1): 1–17.
8.
AllenaRCluzelC. Computational implementation of nonuniform orthotropic directions in the femoral diaphysis based on cortical bone microstructure to build a constitutive model. Math Mech Complex Syst2021; 9(1): 33–58.
9.
TepedinoM. The mechanical role of the periodontal ligament for developing mathematical models in orthodontics. Math Mech Complex Syst2023; 11(4): 525–539.
10.
FrostH. ’mass’ and the ’mechanostat’: a proposal. Anat Rec1987; 219(1): 1–9.
11.
TurnerCH. Homeostatic control of bone structure: an application of feedback theory. Bone1991; 12(3): 203–217.
12.
LekszyckiTDell’IsolaF. A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. Z Angew Math Mech2012; 92(6): 426–444.
13.
GiorgioIDell’IsolaFAndreausU, et al. On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon. Biomech Model Mechanobiol2019; 18: 1639–1663.
14.
CosseratECosseratF. Theorie des corps deformables. Paris: Hermann, 1970.
15.
EremeyevVALebedevLPAltenbachH. Foundations of micropolar mechanics. Berlin and Heidelberg: Springer, 2012.
16.
SteigmannDJ. On the relationship between the Cosserat and Kirchhoff–Love theories of elastic shells. Math Mech Solids1999; 4(3): 275–288.
17.
GrecoLCammarataACastelloD, et al. An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation. Comput Methods Appl Mech Eng2024; 425: 116947.
18.
GrecoLCastelloDCuomoM. An objective minimal constraint formulation for the analysis of elastic articulated structures. Comput Struct2024; 305: 107571.
19.
ShiraniMBîrsanMSteigmannDJ. Quasiconvexity in a model of fiber-reinforced solids based on Cosserat elasticity theory. Math Mech Solids2025; 30(2): 372–380.
20.
EremeyevVSkrzatAStachowiczF. Linear micropolar elasticity analysis of stresses in bones under static loads. Strength Mater2017; 49: 575–585.
21.
dell’IsolaFMaierGPeregoU, et al. The complete works of Gabrio Piola: volume I. Cham: Springer, 2014.
22.
AbaliBEMüllerWHEremeyevVA. Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mech Adv Mater Mod Process2015; 1(1): 4.
23.
Dell’IsolaFAndreausUCazzaniA, et al. The complete works of Gabrio Piola: volume II-commented English translation. Cham: Springer, 2019.
24.
EugsterSRDell’IsolaFFedeleR, et al. Piola transformations in second-gradient continua. Mech Res Commun2022; 120: 103836.
25.
FedeleR. Piola’s approach to the equilibrium problem for bodies with second gradient energies. Part I: first gradient theory and differential geometry. Contin Mech Thermodyn2022; 34(2): 445–474.
26.
FedeleR. Approach à la Piola for the equilibrium problem of bodies with second gradient energies. Part II: variational derivation of second gradient equations and their transport. Contin Mech Thermodyn2022; 34(5): 1087–1111.
27.
FedeleRPlacidiLFabbrocinoF. A review of inverse problems for generalized elastic media: formulations, experiments, synthesis. Contin Mech Thermodyn2024; 36(6): 1413–1453.
28.
SteigmannDJ. Constitutively admissible gradients of plastic deformation in isotropic solids. Math Mech Solids2025; 30: 2032–2042.
29.
PlacidiLBarchiesiE. Energy approach to brittle fracture in strain-gradient modelling. Proc R Soc A Math Phys Eng Sci2018; 474(2210): 20170878.
30.
MisraAPlacidiLdell’IsolaF, et al. Identification of a geometrically nonlinear micromorphic continuum via granular micromechanics. Z Angew Math Phys2021; 72: 1–21.
31.
PlacidiLBarchiesiEMisraA, et al. Micromechanics-based elasto-plastic–damage energy formulation for strain gradient solids with granular microstructure. Contin Mech Thermodyn2021; 33(5): 2213–2241.
32.
MisraAPlacidiL. Emergence of bimodulus (tension–compression asymmetric) behavior modeled in granular micromechanics framework. Math Mech Solids2025; 30: 1711–1726.
33.
AlibertJJSeppecherPDell’IsolaF. Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
34.
AydinGYildizdagMEAbaliBE. Strain-gradient modeling and computation of 3-D printed metamaterials for verifying constitutive parameters determined by asymptotic homogenization. In: GiorgioIPlacidiLBarchiesiE, et al. (eds) Theoretical analyses, computations, and experiments of multiscale materials: a tribute to francesco dell’isola. Cham: Springer, 2022, pp. 343–357.
35.
CiallellaAPasqualiDD’AnnibaleF, et al. Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: numerical modeling and experiments. Math Mech Solids2022; 27(10): 2170–2188.
36.
HarschJGanzoschGBarchiesiE, et al. Experimental analysis, discrete modeling and parameter optimization of SLS-printed bi-pantographic structures. Math Mech Solids2022; 27(10): 2201–2217.
37.
CiallellaALa ValleGVintacheA, et al. Deformation mode in 3-point flexure on pantographic block. Int J Solids Struct2023; 265: 112129.
38.
ChekeresOSalnikovVD’annibaleF. From approximation of dissipative systems to representative space-time volume elements for metamaterials. Contin Mech Thermodyn2024; 36(6): 1597–1605.
39.
Murcia TerranovaLCardilloCAretusiG. An enhanced beam model incorporating a hysteresis-based solid friction damping mechanism for cementitious materials. Contin Mech Thermodyn2025; 37(1): 2.
40.
MadeoAGeorgeDLekszyckiT, et al. A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Compt Rendus Méc2012; 340(8): 575–589.
41.
GiorgioIAndreausUDell’IsolaF, et al. Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mech Lett2017; 13: 141–147.
42.
GanghofferJRahouadjRBoisseJ, et al. A phase field approach for bone remodeling based on a second-gradient model. Mech Res Commun2019; 96: 37–44.
43.
KrönerE. Elasticity theory of materials with long range cohesive forces. Int J Solids Struct1967; 3(5): 731–742.
44.
EringenACSuhubiES. Nonlinear theory of simple micro-elastic solids–I. Int J Eng Sci1964; 2(2): 189–203.
45.
PolizzottoCFuschiPPisanoAA. Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions, I. Math Mech Complex Syst2022; 10(3): 205–231.
46.
dell’IsolaFAndreausUPlacidiL. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20(8): 887–928.
47.
La ValleGSoizeC. A higher-order nonlocal elasticity continuum model for deterministic and stochastic particle-based materials. Z Angew Math Phys2024; 75(2): 49.
48.
La ValleGSoizeC. Stochastic second-gradient continuum theory for particle-based materials: part II. Z Angew Math Phys2024; 75(3): 93.
49.
La ValleGSoizeC. Identifying second-gradient continuum models in particle-based materials with pairwise interactions using acoustic tensor methodology. J Elast2024; 156(2): 623–639.
50.
La ValleGSoizeC. From discrete to continuum: a generalized Euler–Maclaurin framework for scale effects in nanomechanics. J Elast2025; 157(3): 67.
51.
AllenaRRémondY. Theramechanics: how acting on mechanics will help conceive new medical treatments. Math Mech Complex Syst2023; 11(4): 541–566.
52.
AllenaRRémondY. How can theramechanics drive bone remodeling?Math Mech Complex Syst2025; 13(1): 55–71.
53.
GiorgioIDell’IsolaFAndreausU, et al. An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law. Biomech Model Mechanobiol2023; 22(6): 2135–2152.
54.
CyronCJAydinR. Mechanobiological free energy: a variational approach to tensional homeostasis in tissue equivalents. J Appl Math Mech/Z Angew Math Mech2017; 97(9): 1011–1019.
55.
EremeyevVAPietraszkiewiczW. Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math Mech Solids2016; 21(2): 210–221.
56.
GrilloADi StefanoS. Comparison between different viewpoints on bulk growth mechanics. Math Mech Complex Syst2023; 11(2): 287–311.
57.
GrilloADi StefanoS. An a posteriori approach to the mechanics of volumetric growth. Math Mech Complex Syst2023; 11(1): 57–86.
58.
La ValleG. A new deformation measure for the nonlinear micropolar continuum. Z Angew Math Phys2022; 73(2): 78.
59.
dell’IsolaFMisraA. Principle of virtual work as foundational framework for metamaterial discovery and rational design. Compt Rendus Méc2023; 351(Suppl. 3): 1–25.
60.
EremeyevVALebedevLP. Existence theorems in the linear theory of micropolar shells. J Appl Math Mech/Z Angew Math Mech2011; 91(6): 468–476.
61.
NeffPBîrsanMOsterbrinkF. Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements. J Elast2015; 121(1): 119–141.
62.
ShiraniMSteigmannDJ. Quasiconvexity in a model of fiber-reinforced solids based on Cosserat elasticity theory. In: GiorgioIPlacidiLBarchiesiE, et al. (eds) Theoretical analyses, computations, and experiments of multiscale materials: a tribute to francesco dell’isola. Cham: Springer, 2022, pp. 273–283.
63.
ShiraniMGiorgioIAstoriD, et al. A mixed Cosserat and higher gradient formulation for fibrous tissues and biomaterials. Int J Solids Struct2025; 324: 113671.
SoizeCFarhatC. A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and high-dimensional nonlinear models. Int J Numer Method Eng2017; 109(6): 837–888.
66.
FarhatCBosAAveryP, et al. Modeling and quantification of model-form uncertainties in eigenvalue computations using a stochastic reduced model. AIAA J2018; 56(3): 1198–1210.
67.
SoizeCFarhatC. Probabilistic learning for modeling and quantifying model-form uncertainties in nonlinear computational mechanics. Int J Numer Method Eng2019; 117(7): 819–843.
68.
JewellEFarhatCSoizeC. A tractable nonparametric probabilistic approach for modeling and quantifying model-form uncertainty in turbulent CFD. J Comput Phys2025; 536: 114067.
69.
HildFRouxS. Digital image correlation: from displacement measurement to identification of elastic properties–a review. Strain2006; 42(2): 69–80.
70.
HaideRFest-SantiniSSantiniM. Use of X-ray micro-computed tomography for the investigation of drying processes in porous media: a review. Dry Technol2022; 40(9): 1731–1744.
71.
SoizeC. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput Methods Appl Mech Eng2006; 195(1–3): 26–64.
72.
NguyenMTDesceliersCSoizeC, et al. Multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations. Int J Multiscale Comput Eng2015; 13(4): 281–295.
73.
ZhangTPledFDesceliersC. Robust multiscale identification of apparent elastic properties at mesoscale for random heterogeneous materials with multiscale field measurements. Materials2020; 13(12): 2826.
74.
LazarusADaltonDHusmeierD, et al. Sensitivity analysis and inverse uncertainty quantification for the left ventricular passive mechanics. Biomech Model Mechanobiol2022; 21(3): 953–982.
75.
MartinMSansaloneVCooperDM, et al. Mechanobiological osteocyte feedback drives mechanostat regulation of bone in a multiscale computational model. Biomech Model Mechanobiol2019; 18(5): 1475–1496.
76.
MartinMLemaireTHaiatG, et al. Bone orthotropic remodeling as a thermodynamically-driven evolution. J Mech Med Biol2020; 20(4): 1950084.
77.
AddessiDD’AnnibaleFPlacidiL, et al. A bone remodeling approach encoding the effect of damage and a diffusive bio-mechanical stimulus. Contin Mech Thermodyn2024; 36(4): 993–1012.
78.
GrecoLCastelloDCuomoM. An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods. Math Mech Solids2024; 29(4): 645–685.
79.
SilvaMJKeavenyTMHayesWC. Computed tomography-based finite element analysis predicts failure loads and fracture patterns for vertebral sections. J Orthop Res1998; 16(3): 300–308.
80.
ZhouBWangJYuYE, et al. High-resolution peripheral quantitative computed tomography (HR-pQCT) can assess microstructural and biomechanical properties of both human distal radius and tibia: ex vivo computational and experimental validations. Bone2016; 86: 58–67.
81.
MartinMLemaireTHaiatG, et al. A thermodynamically consistent model of bone rotary remodeling: a 2D study. Comput Methods Biomech Biomed Eng2017; 20(Suppl. 1): S127–S128.
82.
SansaloneVMartinMHaïatG, et al. A new model of bone remodeling and turnover set up in the framework of generalized continuum mechanics. Math Mech Solids2021; 26(9): 1376–1393.
83.
BersaniAMdell’IsolaFSeppecherP. Lagrange multipliers in infinite dimensional spaces, examples of application. In: AltenbachHÖchsnerA (eds) Encyclopedia of continuum mechanics. Berlin and Heidelberg: Springer, 2020, pp. 1–8.
84.
ForestS. Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media. Philos Trans R Soc A2020; 378(2170): 20190169.
85.
GiorgioIDell’IsolaFMisraA. Chirality in 2D Cosserat media related to stretch-micro-rotation coupling with links to granular micromechanics. Int J Solids Struct2020; 202: 28–38.
86.
DesmoratB. Energy theorems for micromorphic media. Math Mech Complex Syst2025; 14(1): 1–10.
87.
La ValleGSoizeCSteigmannDJ. A Cauchy–Piola framework for micro-based micromorphic continua. Math Mech Complex Syst2025; 14(1): 71–94.
