18 A Plausible Description of Continuum …
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to avoid board effects. We decide the motions of some points, called leaders, for all
the time windows we are investigating; we can also decide that they will be leader
only for a certain time and late become followers (category change).
Now we can calculate, for each time step, the new configuration of the lattice
in three separate operations. When time increases from t 0 to t 1 , the leaders change
their position from initial configuration according to the prescribed equation. So far
we build a new intermediate lattice where only the leaders have been moved. Now
we take care that the followers are no longer in equilibrium position owing to the
leader’s displacement. How we can calculate it? As an example, if the interactions rule
establishes that a follower has to be in the barycenter of all its neighbors, we calculate
the new position of each follower, taking into account the leader displacement. So far
note as at this stage, only the leader’s neighbors are involved. Finally, we take into
account the rules governing the frame displacement. This is our new configuration at
time t 1 . It is important to note as reached the configuration is not an equilibrium one,
because the three operations must be repeated for many time steps, after the leaders
stop. To be more clear if at time step one the leaders have moved, we calculate the
follower’s displacement. This operation involves only the neighbors of the leaders
and not the other far followers. Later, we calculate the frame displacement to close
the loop. Now there are some followers (the neighbors of the leader’s neighbors) that
there are no longer in equilibrium because there has been the displacement of the
leader’s neighbors. So we need another time step to adjust the configuration and so
on. At a certain time, all the followers are involved in the calculation. The followers
will suffer the leader’s motion after (k − 1) time steps where k is the distance from
the leaders, measured in layers. In this meaning, the leader motion “propagates”
through the lattice to influence the position of all the followers in a time depending
on the lattice dimensions and how many shells of points are being considered in the
neighbor’s definition. In the same way when leaders stop the followers continue to
adjust their position in many time steps.
At this point, we can open a long discussion on the concept of time which is present
here only as “step.” So our question is concerning if we have to consider some virtual
configuration between time t i , and t i+1, until the equilibrium is reached or not. In the
second case, we have used in this work, the second movement of the leaders happens
when the second shell of neighbors is just interested from the first displacements of
the leaders. There are some conceptual differences in the two methods we are still
investigating.
In the next future, we are considering the possibility to discuss the proposed model
in a fully variational setting, which is by no means trivial but would provide clear
methodological advantages (see (Lanczos 2012) for an introduction and (Placidi et al.
2008; Dell’Isola and Placidi 2011; Dell’Isola et al. 2016; Dell’Isola and Gavrilyuk
2012; dell’Isola et al. 2014) for illustrative cases concerning continua with nonclassical properties); therefore, we like to introduce pseudoenergetic considerations
by two formulations PE1 and PE2 to give a contour plot of the strain distribution. The first is the sum, extended to the neighbors, of squares of the differences
between the distances of the point from its neighbors minus the distance in the initial
configuration, i.e.,
341
to avoid board effects. We decide the motions of some points, called leaders, for all
the time windows we are investigating; we can also decide that they will be leader
only for a certain time and late become followers (category change).
Now we can calculate, for each time step, the new configuration of the lattice
in three separate operations. When time increases from t 0 to t 1 , the leaders change
their position from initial configuration according to the prescribed equation. So far
we build a new intermediate lattice where only the leaders have been moved. Now
we take care that the followers are no longer in equilibrium position owing to the
leader’s displacement. How we can calculate it? As an example, if the interactions rule
establishes that a follower has to be in the barycenter of all its neighbors, we calculate
the new position of each follower, taking into account the leader displacement. So far
note as at this stage, only the leader’s neighbors are involved. Finally, we take into
account the rules governing the frame displacement. This is our new configuration at
time t 1 . It is important to note as reached the configuration is not an equilibrium one,
because the three operations must be repeated for many time steps, after the leaders
stop. To be more clear if at time step one the leaders have moved, we calculate the
follower’s displacement. This operation involves only the neighbors of the leaders
and not the other far followers. Later, we calculate the frame displacement to close
the loop. Now there are some followers (the neighbors of the leader’s neighbors) that
there are no longer in equilibrium because there has been the displacement of the
leader’s neighbors. So we need another time step to adjust the configuration and so
on. At a certain time, all the followers are involved in the calculation. The followers
will suffer the leader’s motion after (k − 1) time steps where k is the distance from
the leaders, measured in layers. In this meaning, the leader motion “propagates”
through the lattice to influence the position of all the followers in a time depending
on the lattice dimensions and how many shells of points are being considered in the
neighbor’s definition. In the same way when leaders stop the followers continue to
adjust their position in many time steps.
At this point, we can open a long discussion on the concept of time which is present
here only as “step.” So our question is concerning if we have to consider some virtual
configuration between time t i , and t i+1, until the equilibrium is reached or not. In the
second case, we have used in this work, the second movement of the leaders happens
when the second shell of neighbors is just interested from the first displacements of
the leaders. There are some conceptual differences in the two methods we are still
investigating.
In the next future, we are considering the possibility to discuss the proposed model
in a fully variational setting, which is by no means trivial but would provide clear
methodological advantages (see (Lanczos 2012) for an introduction and (Placidi et al.
2008; Dell’Isola and Placidi 2011; Dell’Isola et al. 2016; Dell’Isola and Gavrilyuk
2012; dell’Isola et al. 2014) for illustrative cases concerning continua with nonclassical properties); therefore, we like to introduce pseudoenergetic considerations
by two formulations PE1 and PE2 to give a contour plot of the strain distribution. The first is the sum, extended to the neighbors, of squares of the differences
between the distances of the point from its neighbors minus the distance in the initial
configuration, i.e.,