282
Z. Louna et al.
in particular that strain gradient models allow for the description of microstructurerelated size effects which are known to be important in hierarchically heterogeneous
materials like trabecular bones, but more generally in many biological tissues, soft
or hard, which exhibit the same features motivating the recourse to generalized
continuum mechanics.
In terms of mass change (growth), a clear distinction can be made between volumetric and surface growth, as emphasized in (Skalak et al. 1997; Epstein and Maugin
2000; Ganghoffer et al. 2014; Goda et al. 2016). Volumetric growth refers to the
processes taking place in the bulk of the material, while surface growth involves
deposition of mass at a surface, mostly occurring in hard tissues. The structural hierarchy of living tissues with microstructures, either soft or hard, such as trabecular
bones (in the hard category) or tendons (a soft tissue) plays an important role in
determining their macroscopic mechanical behavior as well as the stress and strain
distribution at the macroscopic scale. Such microstructural effects become especially
pronounced near the bone–implant interfaces and more generally in zones with high
strain gradients, like in any heterogeneous material. This issue can be investigated
using generalized continuum mechanics theories.
To incorporate the scale of the microstructure of a heterogeneous material within
the continuum framework, a number of phenomenological remedies to the lack of
microstructural features of classical continuum mechanics (first gradient) have been
proposed in the past decades requiring to abandon the local action hypothesis of
classical continuum mechanics. Such enhanced continuum models aim to incorporate
information on the microstructure and can be categorized into three main classes: (i)
non-local integral models (Kröner 1976; Eringen and Edelen 1972) (ii) higher-order
gradient models (Alibert et al. 2003; dell’Isola et al. 2015a, b; Madeo et al. 2011,
2012; Alibert and Della Corte 2015; Goda and Ganghoffer 2016; Berkache et al. 2017;
Giorgio et al. 2017; Reda et al. 2017; dell’Isola et al. 2019; Barchiesi et al. 2019) and
(iii) Cosserat theories and variants of it (Cosserat and Cosserat 1909; Altenbach and
Eremeyev 2009; Goda et al. 2012; Goda and Ganghoffer 2015a; Eremeyev 2019;
Giorgio et al. 2019).
Evolution laws for growing solid bodies incorporating strain gradients are
developed along the line of the phenomenology in the next section.
16.2 Evolution Laws for Growing Solid Bodies with Strain
Gradient Effects
Solid bodies experience growth due to new mass being produced in their bulk or mass
is being exchanged by a convective flux across their external boundary. In order to set
the stage, we recall the balance laws of mass and momentum written in (Ganghoffer
2010). The general form of the mass balance in Eulerian format for an evolving
domain t is given in terms of the actual density ρ as
Z. Louna et al.
in particular that strain gradient models allow for the description of microstructurerelated size effects which are known to be important in hierarchically heterogeneous
materials like trabecular bones, but more generally in many biological tissues, soft
or hard, which exhibit the same features motivating the recourse to generalized
continuum mechanics.
In terms of mass change (growth), a clear distinction can be made between volumetric and surface growth, as emphasized in (Skalak et al. 1997; Epstein and Maugin
2000; Ganghoffer et al. 2014; Goda et al. 2016). Volumetric growth refers to the
processes taking place in the bulk of the material, while surface growth involves
deposition of mass at a surface, mostly occurring in hard tissues. The structural hierarchy of living tissues with microstructures, either soft or hard, such as trabecular
bones (in the hard category) or tendons (a soft tissue) plays an important role in
determining their macroscopic mechanical behavior as well as the stress and strain
distribution at the macroscopic scale. Such microstructural effects become especially
pronounced near the bone–implant interfaces and more generally in zones with high
strain gradients, like in any heterogeneous material. This issue can be investigated
using generalized continuum mechanics theories.
To incorporate the scale of the microstructure of a heterogeneous material within
the continuum framework, a number of phenomenological remedies to the lack of
microstructural features of classical continuum mechanics (first gradient) have been
proposed in the past decades requiring to abandon the local action hypothesis of
classical continuum mechanics. Such enhanced continuum models aim to incorporate
information on the microstructure and can be categorized into three main classes: (i)
non-local integral models (Kröner 1976; Eringen and Edelen 1972) (ii) higher-order
gradient models (Alibert et al. 2003; dell’Isola et al. 2015a, b; Madeo et al. 2011,
2012; Alibert and Della Corte 2015; Goda and Ganghoffer 2016; Berkache et al. 2017;
Giorgio et al. 2017; Reda et al. 2017; dell’Isola et al. 2019; Barchiesi et al. 2019) and
(iii) Cosserat theories and variants of it (Cosserat and Cosserat 1909; Altenbach and
Eremeyev 2009; Goda et al. 2012; Goda and Ganghoffer 2015a; Eremeyev 2019;
Giorgio et al. 2019).
Evolution laws for growing solid bodies incorporating strain gradients are
developed along the line of the phenomenology in the next section.
16.2 Evolution Laws for Growing Solid Bodies with Strain
Gradient Effects
Solid bodies experience growth due to new mass being produced in their bulk or mass
is being exchanged by a convective flux across their external boundary. In order to set
the stage, we recall the balance laws of mass and momentum written in (Ganghoffer
2010). The general form of the mass balance in Eulerian format for an evolving
domain t is given in terms of the actual density ρ as