Chapter 16
Strain Gradient Models for Growing
Solid Bodies
Zineeddine Louna, Ibrahim Goda, and Jean-François Ganghoffer
Abstract A unifying mechanical constitutive framework for growing solid bodies
exhibiting scale effects is presented, relying on the principle of virtual power, and
energy, and entropy principles. We focus in this chapter on strain gradient constitutive models and expose different variants of higher gradient theories, adopting a
phenomenological viewpoint. Incorporation of strain gradient terms in the constitutive models developed for biological tissues is motivated by the occurrence of
pronounced microscopic strain gradients within their internal architecture showing
scale hierarchy and strong contrasts of mechanical properties between different
phases. A strain gradient model for bone remodeling is developed following a
micromechanical approach in order to highlight how such models can be constructed
starting from the microstructural level.
16.1 Introduction
The focus of this contribution is the setting up of a modeling framework based on
strain gradient materials for growing solid bodies experiencing a change of mass
due to mass production and an irreversible mass flux across their boundary. Recent
contributions of the authors in the field of bone remodeling account for both first- and
second-order deformation gradients, relying on the thermodynamics of surfaces and
configurational forces for the simulation of the evolution of the external bone surface
induced by mechanical stimulations (Louna et al. 2018, 2019). It has been shown
Z. Louna
LMFTA, Faculté de Physique, USTHB, BP 32 El Alia, 16111 Bab Ezzouar, Algiers, Algeria
I. Goda
Arts et Métiers ParisTech, I2M Bordeaux, UMR CNRS 5295, 33400 Talence, France
J.-F. Ganghoffer (B)
LEM3—Université de Lorraine, CNRS, 7 Rue Félix Savart, 57073 Metz, France
e-mail: jean-francois.ganghoffer@univ-lorraine.fr
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_16
281
Strain Gradient Models for Growing
Solid Bodies
Zineeddine Louna, Ibrahim Goda, and Jean-François Ganghoffer
Abstract A unifying mechanical constitutive framework for growing solid bodies
exhibiting scale effects is presented, relying on the principle of virtual power, and
energy, and entropy principles. We focus in this chapter on strain gradient constitutive models and expose different variants of higher gradient theories, adopting a
phenomenological viewpoint. Incorporation of strain gradient terms in the constitutive models developed for biological tissues is motivated by the occurrence of
pronounced microscopic strain gradients within their internal architecture showing
scale hierarchy and strong contrasts of mechanical properties between different
phases. A strain gradient model for bone remodeling is developed following a
micromechanical approach in order to highlight how such models can be constructed
starting from the microstructural level.
16.1 Introduction
The focus of this contribution is the setting up of a modeling framework based on
strain gradient materials for growing solid bodies experiencing a change of mass
due to mass production and an irreversible mass flux across their boundary. Recent
contributions of the authors in the field of bone remodeling account for both first- and
second-order deformation gradients, relying on the thermodynamics of surfaces and
configurational forces for the simulation of the evolution of the external bone surface
induced by mechanical stimulations (Louna et al. 2018, 2019). It has been shown
Z. Louna
LMFTA, Faculté de Physique, USTHB, BP 32 El Alia, 16111 Bab Ezzouar, Algiers, Algeria
I. Goda
Arts et Métiers ParisTech, I2M Bordeaux, UMR CNRS 5295, 33400 Talence, France
J.-F. Ganghoffer (B)
LEM3—Université de Lorraine, CNRS, 7 Rue Félix Savart, 57073 Metz, France
e-mail: jean-francois.ganghoffer@univ-lorraine.fr
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_16
281