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R. dell’Erba
control algorithm in PBD problems. Note that most PBD methods hide the dynamics
inside their relationship; moreover, they ask for the knowledge of the velocity of
the particles. We try to describe the deformation of a continuum medium by this
tool useful for complex microstructures not easily analysed by Cauchy continuum.
Classical Cauchy continua are not able to give accurate predictions in highly nonhomogeneous microstructure: generalizations have to be introduced either considering additional degrees of freedom, to account for the kinematics at the level of the
microstructure, or including in the deformation energy density higher gradients of
the displacement than the first one. As part of generalized continua theory, second
gradient models allow to study microstructured materials, by taking into account the
gradient of strain instead of adding independent variables. That class of models is thus
able to capture exotic behaviours. (Abali et al. 2015; Andreaus et al. 2018; Auffray
et al. 2015; dell’Isola et al. 2019a; Eremeyev et al. 2018; Misra et al. 2018; Nejadsadeghi et al. 2019; Placidi and El Dhaba 2017; Placidi et al. 2015, 2016, 2017; Turco
et al. 2016b c, 2017). The derivation of such models has raised many mathematical
challenges, such as that of homogenization methods. (Abdoul-Anziz and Seppecher
2018; Alibert et al. 2003). Such frameworks have permitted the development of
generalized continua theory to go beyond Cauchy continua (Toupin 1964; Green
1965; Mindlin 1965; Maugin 2010). (This class of models has been used extensively
in the recent years, for example to study microstructured materials. (Altenbach and
Eremeyev 2009; Altenbach et al. 2013; Carcaterra et al. 2015; dell’Isola et al. 2015a,
b, 2016d, 2017; Eremeyev 2018, 2019a, b; Eremeyev and Sharma 2019; Eremeyev
et al. 2012; Misra and Poorsolhjouy 2015; Misra and Singh 2013; Misra and Singh
2015; Niiranen et al. 2016; Sharma and Eremeyev 2019; Spagnuolo and Andreaus
2019; Yildizdag et al. 2019). Many efforts have been made in the recent years
towards the numerical modelling and simulation of those models (Balobanov et al.
2019; Barchiesi et al. 2019a, b; de Angelo et al. 2019; dell’Isola et al. 2016b, c, 2019b,
c; Golaszewski et al. 2019; Khakalo and Niiranen 2017; Spagnuolo et al. 2017; Turco
and Rizzi 2016; Turco et al. 2016a, 2018; Cazzani et al. 2016a, b; Spagnuolo et al.
2019; Turco et al. 2016b, c; Yang and Müller 2019; Yang et al. 2018). The latter is
a particularly relevant topic if you consider the technological interest in developing
exotic mechanical metamaterials able to perform targeted tasks; therefore, the investigation of new and efficient algorithms is of great interest at the moment. In our tool,
according to our experience in control of robot swarm, the new position of a particle
is determined by the spatial position of its neighbours. The system (dell’Erba 2018a,
b; Battista et al. 2016) exhibits very different behaviours changing lattice type and
its internal parameters. Therefore, the computation of new positions for a particle
set can be considered as a constrained geometrical problem leading to a transformation operator between the matrices representing the particles configuration, C t , for
a discrete set of time steps t1, t2, … tn. In this paper, we summarize some results
and try to examine the behaviour of a two-dimensional beam under bending stress,
which still needs to be improved.
R. dell’Erba
control algorithm in PBD problems. Note that most PBD methods hide the dynamics
inside their relationship; moreover, they ask for the knowledge of the velocity of
the particles. We try to describe the deformation of a continuum medium by this
tool useful for complex microstructures not easily analysed by Cauchy continuum.
Classical Cauchy continua are not able to give accurate predictions in highly nonhomogeneous microstructure: generalizations have to be introduced either considering additional degrees of freedom, to account for the kinematics at the level of the
microstructure, or including in the deformation energy density higher gradients of
the displacement than the first one. As part of generalized continua theory, second
gradient models allow to study microstructured materials, by taking into account the
gradient of strain instead of adding independent variables. That class of models is thus
able to capture exotic behaviours. (Abali et al. 2015; Andreaus et al. 2018; Auffray
et al. 2015; dell’Isola et al. 2019a; Eremeyev et al. 2018; Misra et al. 2018; Nejadsadeghi et al. 2019; Placidi and El Dhaba 2017; Placidi et al. 2015, 2016, 2017; Turco
et al. 2016b c, 2017). The derivation of such models has raised many mathematical
challenges, such as that of homogenization methods. (Abdoul-Anziz and Seppecher
2018; Alibert et al. 2003). Such frameworks have permitted the development of
generalized continua theory to go beyond Cauchy continua (Toupin 1964; Green
1965; Mindlin 1965; Maugin 2010). (This class of models has been used extensively
in the recent years, for example to study microstructured materials. (Altenbach and
Eremeyev 2009; Altenbach et al. 2013; Carcaterra et al. 2015; dell’Isola et al. 2015a,
b, 2016d, 2017; Eremeyev 2018, 2019a, b; Eremeyev and Sharma 2019; Eremeyev
et al. 2012; Misra and Poorsolhjouy 2015; Misra and Singh 2013; Misra and Singh
2015; Niiranen et al. 2016; Sharma and Eremeyev 2019; Spagnuolo and Andreaus
2019; Yildizdag et al. 2019). Many efforts have been made in the recent years
towards the numerical modelling and simulation of those models (Balobanov et al.
2019; Barchiesi et al. 2019a, b; de Angelo et al. 2019; dell’Isola et al. 2016b, c, 2019b,
c; Golaszewski et al. 2019; Khakalo and Niiranen 2017; Spagnuolo et al. 2017; Turco
and Rizzi 2016; Turco et al. 2016a, 2018; Cazzani et al. 2016a, b; Spagnuolo et al.
2019; Turco et al. 2016b, c; Yang and Müller 2019; Yang et al. 2018). The latter is
a particularly relevant topic if you consider the technological interest in developing
exotic mechanical metamaterials able to perform targeted tasks; therefore, the investigation of new and efficient algorithms is of great interest at the moment. In our tool,
according to our experience in control of robot swarm, the new position of a particle
is determined by the spatial position of its neighbours. The system (dell’Erba 2018a,
b; Battista et al. 2016) exhibits very different behaviours changing lattice type and
its internal parameters. Therefore, the computation of new positions for a particle
set can be considered as a constrained geometrical problem leading to a transformation operator between the matrices representing the particles configuration, C t , for
a discrete set of time steps t1, t2, … tn. In this paper, we summarize some results
and try to examine the behaviour of a two-dimensional beam under bending stress,
which still needs to be improved.