300
Z. Louna et al.
The physical meaning of the introduced growth stress is that of an internal stress,
the counterpart of the back stress in plasticity theories.
16.5 Conclusion
A unifying mechanical constitutive framework for growing solid bodies exhibiting
scale effects is developed, based on the virtual power, energy, and entropy principles.
Different variants of higher gradient theories are exposed, adopting the viewpoint
of the phenomenology. The consideration of strain gradient terms in the constitutive
models developed for biological tissues like bone is motivated by the occurrence of
pronounced microscopic strain gradients within their internal architecture showing a
scale hierarchy and strong contrasts of mechanical properties between the different
phases. A strain gradient model for bone remodeling has been developed based on the
homogenization of the trabecular architecture in order to exemplify the construction
of such strain gradient models starting from the microstructural level.
The consideration of surface growth effects in constitutive models based on the
mechanics of generalized continua shall deserve future contributions. Moreover,
since the present theories are applicable to hard tissues like bone, they restrict to a
small strains situation; it thus remains to extend them to large strains situations as
exhibited by soft biological tissues.
References
Alibert, J.-J., & Della Corte, A. (2015). Second-gradient continua as homogenized limit of pantographic microstructured plates: A rigorous proof. Zeitschrift fur Angewandte Mathematik und
Physik, 66(5), 2855–2870.
Alibert, J. J., 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.
Altenbach, H., & Eremeyev, V. A. (2009). On the linear theory of micropolar plates. Zeitschrift für
Angewandte Mathematik und Mechanik, 89(4), 242–256.
Andreaus, U., Giorgio, I., & Lekszycki, T. (2014). A 2-D continuum model of a mixture of
bone tissue and bio-resorbable material for simulating mass density redistribution under load
slowly variable in time. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für
Angewandte Mathematik und Mechanik, 94(12), 978–1000.
Barchiesi, E., Eugster, S. R., Placidi, L., & dell’Isola, F. (2019). Pantographic beam: A complete
second gradient 1D-continuum in plane. Zeitschrift für Angewandte Mathematik und Physik,
70(5), 135.
Berkache, K., Deogekar, S., Goda, I., Picu, R. C., & Ganghoffer, J.-F. (2017). Construction of second
gradient continuum models for random fibrous networks and analysis of size effects. Composite
Structures, 181, 347–357.
Bowman, S. M., et al. (1998). Creep contributes to the fatigue behavior of bovine trabecular bone.
Journal of Biomechanical Engineering, 120, 647–654.
Buechner, P. M., & Lakes, R. S. (2003). Size effects in the elasticity and viscoelasticity of bone.
Biomechanics and Modeling in Mechanobiology, 1(4), 295–301.
Z. Louna et al.
The physical meaning of the introduced growth stress is that of an internal stress,
the counterpart of the back stress in plasticity theories.
16.5 Conclusion
A unifying mechanical constitutive framework for growing solid bodies exhibiting
scale effects is developed, based on the virtual power, energy, and entropy principles.
Different variants of higher gradient theories are exposed, adopting the viewpoint
of the phenomenology. The consideration of strain gradient terms in the constitutive
models developed for biological tissues like bone is motivated by the occurrence of
pronounced microscopic strain gradients within their internal architecture showing a
scale hierarchy and strong contrasts of mechanical properties between the different
phases. A strain gradient model for bone remodeling has been developed based on the
homogenization of the trabecular architecture in order to exemplify the construction
of such strain gradient models starting from the microstructural level.
The consideration of surface growth effects in constitutive models based on the
mechanics of generalized continua shall deserve future contributions. Moreover,
since the present theories are applicable to hard tissues like bone, they restrict to a
small strains situation; it thus remains to extend them to large strains situations as
exhibited by soft biological tissues.
References
Alibert, J.-J., & Della Corte, A. (2015). Second-gradient continua as homogenized limit of pantographic microstructured plates: A rigorous proof. Zeitschrift fur Angewandte Mathematik und
Physik, 66(5), 2855–2870.
Alibert, J. J., 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.
Altenbach, H., & Eremeyev, V. A. (2009). On the linear theory of micropolar plates. Zeitschrift für
Angewandte Mathematik und Mechanik, 89(4), 242–256.
Andreaus, U., Giorgio, I., & Lekszycki, T. (2014). A 2-D continuum model of a mixture of
bone tissue and bio-resorbable material for simulating mass density redistribution under load
slowly variable in time. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für
Angewandte Mathematik und Mechanik, 94(12), 978–1000.
Barchiesi, E., Eugster, S. R., Placidi, L., & dell’Isola, F. (2019). Pantographic beam: A complete
second gradient 1D-continuum in plane. Zeitschrift für Angewandte Mathematik und Physik,
70(5), 135.
Berkache, K., Deogekar, S., Goda, I., Picu, R. C., & Ganghoffer, J.-F. (2017). Construction of second
gradient continuum models for random fibrous networks and analysis of size effects. Composite
Structures, 181, 347–357.
Bowman, S. M., et al. (1998). Creep contributes to the fatigue behavior of bovine trabecular bone.
Journal of Biomechanical Engineering, 120, 647–654.
Buechner, P. M., & Lakes, R. S. (2003). Size effects in the elasticity and viscoelasticity of bone.
Biomechanics and Modeling in Mechanobiology, 1(4), 295–301.