18 A Plausible Description of Continuum …
339
Starting from the leader’s motion, each time step the displacement propagates of
one shell, determined by the neighbors up to involve all the particles.
To avoid edge effects, we build a frame surrounding the body by an external
shell of point, so that any follower interacts with the same number of elements. Our
objective is homogeneity of the boundary conditions for all the followers. Without
the frame, a corner point has fewer neighbors, with respect to an internal point; so
far if its coordinates are determined, as example, by barycenter of its neighbors, this
point will be attracted toward the inside and the lattice and will collapse on the other
points.
The motion of the frame is simple: It only has to follow the motion of an assigned
follower of its competence; in case the assigned followers are more (i.e., in a corner),
then an average displacement, or a more generic complex rule (Choice 4), is considered as can be seen in Fig. 18.5. In this case, you have more than a possibility, and
the frame can be something more complex than a single shell; as an example, if we
are considering second gradient interaction, we need a double shell to reach. Later
we shall see as in hexagonal lattice you can choose more than one kind of frame, and
the obtained results are completely different.
The process stops when all the elements of the system have moved, and then
restarts at every following time step (for a more detailed description the reader is
referred to dell’Erba 2018a, c).
The model exhibits pronounced nonlinear behaviors, as shown in (Dell’isola and
Seppecher 1997), since composing motions for the leader does not lead to a simple
superposition of effects in the configuration of the system.
To manage fracture phenomena, we assume the interactions are decreasing with
increasing distance between particles. Therefore, when Euclidean distance between
points is “great,” they lose their interaction. To address the problem, we start simply
considering a threshold effect between neighbor elements, so that when the distance
overcomes the threshold, these elements stop to influence each other so they are
no longer taken into account in the calculation of the follower position. To preserve
symmetry of the Lagrangian neighbors, we introduce ghost points with the purpose of
balancing the calculations of the point’s displacements, just to balance the equations.
They have the purpose of balancing the calculations of the point’s displacements.
Where are these ghost elements posed? Typical position, where we put ghost points
(Choice 6), is that is able to recover the original shape of the lattice (see Fig. 18.6).
Anyway other choices lead to different results. All the properties of these ghost
elements are the same of the followers, but their motion is not considered, because
they are not in the list of the followers. They are just in the right position to balance the
cell. As we have seen (Battista et al. 2016), a change in their position produces effects
such as the contraction or loosening of the lattice in the deformed configuration. In
fact, varying the distances of the ghost elements after fracture from the true elements,
plastic-like and elastic-like behaviors can be obtained. As elastic behavior in fracture,
we mean the property of the fracture edges or of the disconnected pieces originated
after fracture has occurred, to recover its original shape. The algorithm can be easily
generalized to second gradient by introducing two different thresholds for the two
shells of neighbors.
339
Starting from the leader’s motion, each time step the displacement propagates of
one shell, determined by the neighbors up to involve all the particles.
To avoid edge effects, we build a frame surrounding the body by an external
shell of point, so that any follower interacts with the same number of elements. Our
objective is homogeneity of the boundary conditions for all the followers. Without
the frame, a corner point has fewer neighbors, with respect to an internal point; so
far if its coordinates are determined, as example, by barycenter of its neighbors, this
point will be attracted toward the inside and the lattice and will collapse on the other
points.
The motion of the frame is simple: It only has to follow the motion of an assigned
follower of its competence; in case the assigned followers are more (i.e., in a corner),
then an average displacement, or a more generic complex rule (Choice 4), is considered as can be seen in Fig. 18.5. In this case, you have more than a possibility, and
the frame can be something more complex than a single shell; as an example, if we
are considering second gradient interaction, we need a double shell to reach. Later
we shall see as in hexagonal lattice you can choose more than one kind of frame, and
the obtained results are completely different.
The process stops when all the elements of the system have moved, and then
restarts at every following time step (for a more detailed description the reader is
referred to dell’Erba 2018a, c).
The model exhibits pronounced nonlinear behaviors, as shown in (Dell’isola and
Seppecher 1997), since composing motions for the leader does not lead to a simple
superposition of effects in the configuration of the system.
To manage fracture phenomena, we assume the interactions are decreasing with
increasing distance between particles. Therefore, when Euclidean distance between
points is “great,” they lose their interaction. To address the problem, we start simply
considering a threshold effect between neighbor elements, so that when the distance
overcomes the threshold, these elements stop to influence each other so they are
no longer taken into account in the calculation of the follower position. To preserve
symmetry of the Lagrangian neighbors, we introduce ghost points with the purpose of
balancing the calculations of the point’s displacements, just to balance the equations.
They have the purpose of balancing the calculations of the point’s displacements.
Where are these ghost elements posed? Typical position, where we put ghost points
(Choice 6), is that is able to recover the original shape of the lattice (see Fig. 18.6).
Anyway other choices lead to different results. All the properties of these ghost
elements are the same of the followers, but their motion is not considered, because
they are not in the list of the followers. They are just in the right position to balance the
cell. As we have seen (Battista et al. 2016), a change in their position produces effects
such as the contraction or loosening of the lattice in the deformed configuration. In
fact, varying the distances of the ghost elements after fracture from the true elements,
plastic-like and elastic-like behaviors can be obtained. As elastic behavior in fracture,
we mean the property of the fracture edges or of the disconnected pieces originated
after fracture has occurred, to recover its original shape. The algorithm can be easily
generalized to second gradient by introducing two different thresholds for the two
shells of neighbors.