332
R. dell’Erba
Finite element analysis is a well-established method, but these numerical techniques are usually computationally expensive. The proposed algorithm offers the
advantage of limited computational costs. Since the algorithm is based on a linear
operation, that is, the computation of the barycenter, its computational cost increases
only linearly with the number of particles in the system. In (dell’Erba 2018a, b), it
is shown how the model may exhibit a rich range of behaviors, such as asymptotic convergence to the equilibrium configuration, instabilities of various kinds
and, in well-determined circumstances, spontaneous growth of the crack length
after an almost steady state. So far we have tried to describe the deformation of
a continuum medium by this tool useful for complex microstructures not easily
analyzed by Cauchy continuum theory generating and generating big quantity of
experimental data. Classical Cauchy continua are not able to give accuracy prediction
in highly non-homogeneous microstructure, though generalizations have to be introduced either considering additional degrees of freedom, to account for the kinematics
at the level of the microstructure (Seddik et al. 2008; Pietraszkiewicz and Eremeyev
2009; Altenbach et al. 2010, 2013; Eremeyev et al. 2012; Altenbach et al. 2010),
or including in the deformation energy density higher gradients of the displacement
than the first one (Abali et al. 2017; Cuomo et al. 2017; Turco et al. 2016; Dell’Isola
et al. 2016, 2015; Javili et al. 2013; Seppecher et al. 2011; Forest et al. 2011; Placidi
2015; Rosi et al. 2013). The latter is a particularly relevant topic if you consider the
technological interest in developing exotic mechanical metamaterials able to perform
targeted tasks (Dell’Isola et al. 2015; Bückmann 2012; Dell’Isola 2019; Barchiesi
et al. 2018; Carcaterra et al. 2015; Turco et al. 2017; Dell’Isola et al. 2016; Milton
and Seppecher 2012); therefore, the investigation of new and efficient algorithms is
of great interest at the moment.
This approach seemed particularly promising considering the emerging role of
microstructured continua, manufactured with computer-aided methods, as a technological resource (see Altenbach et al. 2010; Eremeyev et al. 2012; Altenbach
et al. 2013; Dell’Isola et al. 2015; Placidi et al. 2017; Placidi et al. 2010; Altenbach
and Eremeyev 2009; Altenbach and Eremeyev 2013; Eremeyev and Pietraszkiewicz
2012; Forest 2009) because the presence of a complex microstructure often leads to
macroscopic behaviors that require generalized continua for their accurate modeling
(see Abali et al. 2017; Cuomo et al. 2017; Turco et al. 2016; Dell’Isola et al. 2015,
2016; Javili et al. 2013; Forest et al. 2011; Placidi 2015; Rosi et al. 2013; Alibert
et al. 2003) for more details and (Andreaus and Placidi 2013) for a historical survey
on the subject). For interesting results in nth gradient theory, the reader can see
(dell’Isola et al. 2015; Javili et al. 2013; Rosi et al. 2013; Carcaterra et al. 2015;
Alibert et al. 2003; Andreaus and Placidi 2013; Madeo et al. 2013; Madeo et al.
2008; Dell’Isola et al. 2011; Auffray et al. 2015; Dell’Isola et al. 2009; Dell’Isola
and Seppecher 1995; Dell’isola and Seppecher 1997; dell’Isola et al. 2012; Alibert
and Della Corte 2015; Placidi et al. 2013); on the importance of this topic, see (Placidi
et al. 2017), where a general overview of recent results is provided. Note that nth
gradient theories can be contextualized in a more general framework of micromorphic/microstructured continua, which is a very active research field (see, for instance,
Eringen 2012; Germain 1973; Mindlin 1964) for classical references (Seddik et al.
R. dell’Erba
Finite element analysis is a well-established method, but these numerical techniques are usually computationally expensive. The proposed algorithm offers the
advantage of limited computational costs. Since the algorithm is based on a linear
operation, that is, the computation of the barycenter, its computational cost increases
only linearly with the number of particles in the system. In (dell’Erba 2018a, b), it
is shown how the model may exhibit a rich range of behaviors, such as asymptotic convergence to the equilibrium configuration, instabilities of various kinds
and, in well-determined circumstances, spontaneous growth of the crack length
after an almost steady state. So far we have tried to describe the deformation of
a continuum medium by this tool useful for complex microstructures not easily
analyzed by Cauchy continuum theory generating and generating big quantity of
experimental data. Classical Cauchy continua are not able to give accuracy prediction
in highly non-homogeneous microstructure, though generalizations have to be introduced either considering additional degrees of freedom, to account for the kinematics
at the level of the microstructure (Seddik et al. 2008; Pietraszkiewicz and Eremeyev
2009; Altenbach et al. 2010, 2013; Eremeyev et al. 2012; Altenbach et al. 2010),
or including in the deformation energy density higher gradients of the displacement
than the first one (Abali et al. 2017; Cuomo et al. 2017; Turco et al. 2016; Dell’Isola
et al. 2016, 2015; Javili et al. 2013; Seppecher et al. 2011; Forest et al. 2011; Placidi
2015; Rosi et al. 2013). The latter is a particularly relevant topic if you consider the
technological interest in developing exotic mechanical metamaterials able to perform
targeted tasks (Dell’Isola et al. 2015; Bückmann 2012; Dell’Isola 2019; Barchiesi
et al. 2018; Carcaterra et al. 2015; Turco et al. 2017; Dell’Isola et al. 2016; Milton
and Seppecher 2012); therefore, the investigation of new and efficient algorithms is
of great interest at the moment.
This approach seemed particularly promising considering the emerging role of
microstructured continua, manufactured with computer-aided methods, as a technological resource (see Altenbach et al. 2010; Eremeyev et al. 2012; Altenbach
et al. 2013; Dell’Isola et al. 2015; Placidi et al. 2017; Placidi et al. 2010; Altenbach
and Eremeyev 2009; Altenbach and Eremeyev 2013; Eremeyev and Pietraszkiewicz
2012; Forest 2009) because the presence of a complex microstructure often leads to
macroscopic behaviors that require generalized continua for their accurate modeling
(see Abali et al. 2017; Cuomo et al. 2017; Turco et al. 2016; Dell’Isola et al. 2015,
2016; Javili et al. 2013; Forest et al. 2011; Placidi 2015; Rosi et al. 2013; Alibert
et al. 2003) for more details and (Andreaus and Placidi 2013) for a historical survey
on the subject). For interesting results in nth gradient theory, the reader can see
(dell’Isola et al. 2015; Javili et al. 2013; Rosi et al. 2013; Carcaterra et al. 2015;
Alibert et al. 2003; Andreaus and Placidi 2013; Madeo et al. 2013; Madeo et al.
2008; Dell’Isola et al. 2011; Auffray et al. 2015; Dell’Isola et al. 2009; Dell’Isola
and Seppecher 1995; Dell’isola and Seppecher 1997; dell’Isola et al. 2012; Alibert
and Della Corte 2015; Placidi et al. 2013); on the importance of this topic, see (Placidi
et al. 2017), where a general overview of recent results is provided. Note that nth
gradient theories can be contextualized in a more general framework of micromorphic/microstructured continua, which is a very active research field (see, for instance,
Eringen 2012; Germain 1973; Mindlin 1964) for classical references (Seddik et al.