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The effect of mechanical load-induced intraosseous pressure gradients on bone remodeling. (English) Zbl 1425.74321
Abali, Bilen Emek (ed.) et al., New achievements in continuum mechanics and thermodynamics. A tribute to Wolfgang H. Müller. Cham: Springer. Adv. Struct. Mater. 108, 29-49 (2019).
Summary: It is well established that changes in bone blood and interstitial fluid flows are associated with changes in the bone remodeling process. These flows in bone are a result not only of trans-cortical pressure gradients produced by vascular and hydro-static pressure, but also of mechanical loadings. Mechanical load-induced intraosseous pressure gradients may result in some fluid stimuli effects which, in turn, may enable bone cells to detect external mechanical signals. In this paper, the exploitation of a 2D continuum model based on classical poroelasticity is presented within a variational framework. The investigation is aimed at describing how mechanical actions can affect the remodeling process of a bone tissue. The focus is on the introduction of a physically motivated strain energy contribution aimed to take into account the presence of saturating fluid in the interconnected pores of bone tissue. The interaction with a bio-resorbable organic ceramic material like those used in bone graft implants is also considered in presented model. Numerical results are provided in a relevant exemplary case.
For the entire collection see [Zbl 1411.74006].
MSC:
74L15 Biomechanical solid mechanics
92C10 Biomechanics
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[1] Abali BE, Müller WH, Eremeyev VA (2015) Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mechanics of Advanced Materials and Modern Processes 1(1):4
[2] Abali BE, Müller WH, dell’Isola F (2017) Theory and computation of higher gradient elasticity theories based on action principles. Archive of Applied Mechanics pp 1-16
[3] Abbas IA, Abdalla AEN, Alzahrani FS, Spagnuolo M (2016) Wave propagation in a generalized thermoelastic plate using eigenvalue approach. Journal of Thermal Stresses 39(11):1367-1377
[4] Abd-alla Aen, Alshaikh F, Del Vescovo D, Spagnuolo M (2017) Plane waves and eigenfrequency study in a transversely isotropic magneto-thermoelastic medium under the effect of a constant angular velocity. Journal of Thermal Stresses 40(9):1079-1092
[5] Abd-alladan AenN, Hamdan AM, Almarashi AA, Battista A (2017) The mathematical modeling for bulk acoustic wave propagation velocities in transversely isotropic piezoelectric materials. Mathematics and Mechanics of Solids 22(4):823-836 · Zbl 1371.74138
[6] Abdoul-Anziz H, Seppecher P (2018) Strain gradient and generalized continua obtained by homogenizing frame lattices. Mathematics and mechanics of complex systems 6(3):213-250 · Zbl 1403.35028
[7] Alibert JJ, Seppecher P, dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids 8(1):51-73 · Zbl 1039.74028
[8] Allena R, Cluzel C (2018) Heterogeneous directions of orthotropy in three-dimensional structures: finite element description based on diffusion equations. Mathematics and Mechanics of Complex Systems 6(4):339-351 · Zbl 1453.74021
[9] Altenbach H, Eremeyev V (2009) On the linear theory of micropolar plates. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 89(4):242-256 · Zbl 1160.74030
[10] Altenbach H, Eremeyev V (2015) On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Mathematics and Mechanics of Complex Systems 3(3):273-283 · Zbl 1327.74011
[11] Altenbach J, Altenbach H, Eremeyev VA (2010) On generalized cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics 80(1):73-92 · Zbl 1184.74042
[12] Andreaus U, Placidi L, Rega G (2010) Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator. Communications in Nonlinear Science and Numerical Simulation 15(9):2603-2616 · Zbl 1222.70020
[13] Andreaus U, Spagnuolo M, Lekszycki T, Eugster SR (2018) A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Continuum Mechanics and Thermodynamics pp 1-21 · Zbl 1396.74070
[14] Auffray N, dell’Isola F, Eremeyev V, Madeo A, Rosi G (2015) Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids 20(4):375-417 · Zbl 1327.76008
[15] Battista A, Cardillo C, Del Vescovo D, Rizzi N, Turco E (2015) Frequency shifts induced by large deformations in planar pantographic continua. Nanomechanics Science and Technology: An International Journal 6(2)
[16] Battista A, Del Vescovo D, Rizzi NL, Turco E (2017a) Frequency shifts in natural vibrations in pantographic metamaterials under biaxial tests. Technische Mechanik 37(1):1-17
[17] Battista A, Rosa L, dell’Erba R, Greco L (2017b) Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena. Mathematics and Mechanics of Solids 22(11):2120-2134 · Zbl 1395.74005
[18] Beaupré GS, Orr TE, Carter DR (1990) An approach for time-dependent bone modeling and remodeling—theoretical development. Journal of Orthopaedic Research 8(5):651-661
[19] Berezovski A, Yildizdag ME, Scerrato D (2018) On the wave dispersion in microstructured solids. Continuum Mechanics and Thermodynamics https://doi.org/10.1007/s00161-018-0683-1:1-20
[20] Bertram A, Glüge R (2016) Gradient materials with internal constraints. Mathematics and Mechanics of Complex Systems 4(1):1-15 · Zbl 1333.74011
[21] Biot MA (1941) General theory of three-dimensional consolidation. Journal of applied physics 12(2):155-164 · JFM 67.0837.01
[22] Camar-Eddine M, Seppecher P (2001) Non-local interactions resulting from the homogenization of a linear diffusive medium. Comptes Rendus de l’Academie des Sciences Series I Mathematics 332(5):485-490 · Zbl 1032.74047
[23] Carinci G, De Masi A, Giardinà C, Presutti E (2014a) Hydrodynamic limit in a particle system with topological interactions. Arabian Journal of Mathematics 3(4):381-417 · Zbl 1323.60125
[24] Carinci G, De Masi A, Giardinà C, Presutti E (2014b) Super-hydrodynamic limit in interacting particle systems. Journal of Statistical Physics 155(5):867-887 · Zbl 1297.82022
[25] Chatzigeorgiou G, Javili A, Steinmann P (2014) Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Mathematics and Mechanics of Solids 19(2):193-211 · Zbl 1355.74065
[26] Cluzel C, Allena R (2018) A general method for the determination of the local orthotropic directions of heterogeneous materials: application to bone structures using μct images. Mathematics and Mechanics of Complex Systems 6(4):353-367 · Zbl 07093773
[27] Coussy O (2004) Poromechanics. John Wiley & Sons
[28] CuomoM(2017) Forms of the dissipation function for a class of viscoplastic models. Mathematics and Mechanics of Complex Systems 5(3):217-237 · Zbl 1386.74007
[29] Cuomo M, dell’Isola F, Greco L, Rizzi N (2016) First versus second gradient energies for planar sheets with two families of inextensible fibres: Investigation on deformation boundary layers, discontinuities and geometrical instabilities. Composites Part B: Engineering
[30] De Masi A, Olla S (2015) Quasi-static hydrodynamic limits. Journal of Statistical Physics 161(5):1037-1058 · Zbl 1332.82070
[31] De Masi A, Galves A, Löcherbach E, Presutti E (2015) Hydrodynamic limit for interacting neurons. Journal of Statistical Physics 158(4):866-902 · Zbl 1315.35222
[32] dell’Isola F, Seppecher P (1997) Edge contact forces and quasi-balanced power. Meccanica 32(1):33-52 · Zbl 0877.73055
[33] dell’Isola F, Steigmann D (2015) A two-dimensional gradient-elasticity theory for woven fabrics. Journal of Elasticity 118(1):113-125 · Zbl 1305.74024
[34] dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”. Zeitschrift für angewandte Mathematik und Physik 63(6):1119-1141 · Zbl 1330.76016
[35] dell’Isola F, Andreaus U, Placidi L (2015a) 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. Mathematics and Mechanics of Solids 20(8):887-928 · Zbl 1330.74006
[36] dell’Isola F, Seppecher P, Della Corte A (2015b) The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc R Soc A 471(2183):20150,415 · Zbl 1371.82032
[37] dell’Isola F, Madeo A, Seppecher P (2016) Cauchy tetrahedron argument applied to higher contact interactions. Archive for Rational Mechanics and Analysis 219(3):1305-1341 · Zbl 1395.74071
[38] Di Nino S, D’Annibale F, Luongo A (2017) A simple model for damage analysis of a framemasonry shear-wall system. International Journal of Solids and Structures 129:119-134
[39] Enakoutsa K, DEl Vescovo D, Scerrato D (2017) Combined polarization field gradient and strain field gradient effects in elastic flexoelectric materials. Mathematics and Mechanics of Solids 22(5):938-951 · Zbl 1371.74098
[40] Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Mathematics and Mechanics of Complex Systems 3(1):43-82 · Zbl 1309.74008
[41] Eremeyev VA (2018) On the material symmetry group for micromorphic media with applications to granular materials. Mechanics Research Communications 94:8-12
[42] Eremeyev VA, dell’Isola F, Boutin C, Steigmann D (2018a) Linear pantographic sheets: existence and uniqueness of weak solutions. Journal of Elasticity 132(2):175-196 · Zbl 1398.74011
[43] Eremeyev VA, Rosi G, Naili S (2018b) Comparison of anti-plane surface waves in straingradient materials and materials with surface stresses. Mathematics and Mechanics of Solids p 1081286518769960
[44] Eugster SR, Glocker C (2017) On the notion of stress in classical continuum mechanics. Mathematics and Mechanics of Complex Systems p 299 · Zbl 1386.70004
[45] Ferretti M, Piccardo G, Luongo A (2017) Weakly nonlinear dynamics of taut strings traveled by a single moving force. Meccanica 52(13):3087-3099 · Zbl 1380.74063
[46] Franciosi P, Spagnuolo M, Salman OU (2018) Mean green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mechanics and Thermodynamics https://doi.org/10.1007/s00161-018-0668-0:1-32
[47] Ganghoffer JF (2012) A contribution to the mechanics and thermodynamics of surface growth. application to bone external remodeling. International Journal of Engineering Science 50(1):166-191 · Zbl 1423.74613
[48] Ganghoffer JF (2016) Spatial and material stress tensors in continuum mechanics of growing solid bodies. Mathematics and Mechanics of Complex Systems 3(4):341-363 · Zbl 1381.74024
[49] George D, Allena R, Remond Y (2018a) Cell nutriments and motility for mechanobiological bone remodeling in the context of orthodontic periodontal ligament deformation. Journal of Cellular Immunotherapy
[50] George D, Allena R, Remond Y (2018b) Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction. Continuum Mechanics and Thermodynamics pp 1-16
[51] George D, Allena R, Remond Y (2018c) A multiphysics stimulus for continuum mechanics bone remodeling. Mathematics and Mechanics of Complex Systems 6(4):307-319 · Zbl 1422.65295
[52] Giorgio I, Andreaus U, Scerrato D, dell’Isola F (2016) A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomechanics and modeling in mechanobiology 15(5):1325-1343
[53] Giorgio I, Andreaus U, Scerrato D, Braidotti P (2017) Modeling of a non-local stimulus for bone remodeling process under cyclic load: Application to a dental implant using a bioresorbable porous material. Mathematics and Mechanics of Solids 22(9):1790-1805 · Zbl 1391.74156
[54] Goda I, Ganghoffer JF (2015) 3d plastic collapse and brittle fracture surface models of trabecular bone from asymptotic homogenization method. International Journal of Engineering Science 87:58-82 · Zbl 1423.74617
[55] Goda I, Assidi M, Belouettar S, Ganghoffer JF (2012) A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. Journal of the mechanical behavior of biomedical materials 16:87-108
[56] Goda I, Assidi M, Ganghoffer JF (2014) A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech Model Mechanobiol 13:53-83
[57] Gusev AA, Lurie SA (2017) Symmetry conditions in strain gradient elasticity. Mathematics and Mechanics of Solids 22(4):683-691 · Zbl 1371.74049
[58] Hillsley MV, Frangos JA (1994) Bone tissue engineering: the role of interstitial fluid flow. Biotechnology and bioengineering 43(7):573-581
[59] Javili A, Chatzigeorgiou G, Steinmann P (2013) Computational homogenization in magnetomechanics. International Journal of Solids and Structures 50(25):4197-4216
[60] Lekszycki T, dell’Isola F (2012) A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMMZeitschrift für Angewandte Mathematik und Mechanik 92(6):426-444 · Zbl 1241.92010
[61] Lekszycki T, Bucci S, Del Vescovo D, Turco E, Rizzi NL (2017) A comparison between different approaches for modelling media with viscoelastic properties via optimization analyses. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 97(5):515-531
[62] Luongo A, D’Annibale F (2017) Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic beck’s beam. Mathematics and Mechanics of Solids 22(6):1347-1365 · Zbl 1371.74134
[63] Madeo A, dell’Isola F, Darve F (2013) A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. Journal of the Mechanics and Physics of Solids 61(11):2196-2211
[64] Misra A, Poorsolhjouy P (2015a) Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Mathematics and Mechanics of Solids https://doi.org/10.1177/1081286515576821
[65] Misra A, Poorsolhjouy P (2015b) Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Mathematics and Mechanics of Complex Systems 3(3):285-308 · Zbl 1329.74225
[66] Misra A, Singh V (2013) Micromechanical model for viscoelastic materials undergoing damage. Continuum Mechanics and Thermodynamics 25(2-4):343-358 · Zbl 1343.74039
[67] Misra A, Singh V (2015) Thermomechanics-based nonlinear rate-dependent coupled damageplasticity granular micromechanics model. Continuum Mechanics and Thermodynamics 27(4-5):787
[68] Pagnini LC, Piccardo G (2016) The three-hinged arch as an example of piezomechanic passive controlled structure. Continuum Mechanics and Thermodynamics 28(5):1247-1262 · Zbl 1355.74058
[69] Pideri C, Seppecher P (1997) A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mechanics and Thermodynamics 9(5):241-257 · Zbl 0893.73006
[70] Pietraszkiewicz W, Eremeyev V (2009) On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures 46(3):774-787 · Zbl 1215.74004
[71] Placidi L (2015) A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Continuum Mechanics and Thermodynamics 27(4-5):623 · Zbl 1341.74016
[72] Placidi L, dell’Isola F, Ianiro N, Sciarra G (2008) Variational formulation of pre-stressed solid– fluid mixture theory, with an application to wave phenomena. European Journal of Mechanics- A/Solids 27(4):582-606 · Zbl 1146.74012
[73] Placidi L, Greco L, Bucci S, Turco E, Rizzi N (2016) A second gradient formulation for a 2d fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5)(114) · Zbl 1432.74034
[74] Placidi L, Barchiesi E, Misra A (2018) A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Mathematics and Mechanics of Complex Systems 6(2):77-100 · Zbl 1452.74019
[75] Rinaldi A, Placidi L (2014) A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. ZAMM-Journal of Applied Mathematics and Mechanics /Zeitschrift für Angewandte Mathematik und Mechanik 94(10):862-877 · Zbl 1301.74042
[76] Rosi G, Placidi L, Auffray N (2018) On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure. European Journal of Mechanics-A/Solids 69:179-191 · Zbl 1406.74092
[77] Saeb S, Steinmann P, Javili A (2016) Aspects of computational homogenization at finite deformations: A unifying review from Reuss’ to Voigt’s bound. Applied Mechanics Reviews 68(5):050,801
[78] Sciarra G, dell’Isola F, Coussy O (2007) Second gradient poromechanics. International Journal of Solids and Structures 44(20):6607-6629 · Zbl 1166.74341
[79] Seppecher P (1993) Equilibrium of a Cahn-Hilliard fluid on a wall: influence of the wetting properties of the fluid upon the stability of a thin liquid film. European journal of mechanics series B fluids 12:69-69 · Zbl 0766.76034
[80] Seppecher P (2000) Second-gradient theory: application to Cahn-Hilliard fluids. In: Continuum thermomechanics, Springer, pp 379-388
[81] Seppecher P, Alibert JJ, dell’Isola F (2011) Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, IOP Publishing, vol 319, p 012018
[82] Shirani M, Andani MT, Kadkhodaei M, Elahinia M (2017) Effect of loading history on phase transition and martensitic detwinning in shape memory alloys: Limitations of current approaches and development of a 1d constitutive model. Journal of Alloys and Compounds 729:390-406
[83] Spagnuolo M, Andreaus U (2018) A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Mathematics and Mechanics of Solids p 1081286517737000 · Zbl 1425.74267
[84] Spagnuolo M, Barcz K, Pfaff A, dell’Isola F, Franciosi P (2017) Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mechanics Research Communications
[85] Spingarn C,Wagner D, Remond Y, George D (2018) Theoretical numerical modeling of the oxygen diffusion effects within the periodontal ligament for orthodontic tooth displacement. Journal of Cellular Immunotherapy
[86] Steigmann D, Agrawal A (2016) Electromechanics of polarized lipid bilayers. Mathematics and Mechanics of Complex Systems 4(1):31-54 · Zbl 1333.74067
[87] Wilmanski K (1998) A thermodynamic model of compressible porous materials with the balance equation of porosity. Transport in Porous Media 32(1):21-47
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