342
R. dell’Erba
P E1(t, j) =
all neighbours of j
k=1
(dis(t, k, j) − dis(t 0 k, j))
2
Where dis(t, k, j) is the Euclidean distance between points k and j at time t. The
reason for this choice lies in the attempt to emulate potential energy of material point
subject to Hook’s law. It is a sort of square distance between the actual configuration
C t and the reference configuration C 0 .
To compare time contiguous configuration C t and C t−1 , we define for each point
j and each time t
P E2(t, j) = ||C t − C t−1 ||
where || is the norm of the vector defined by the point j at time t and t − 1.
It must be underlined that this artifice has no direct connection with the usual
energy definition (this is the reason we use the term pseudoenergy) but could be
useful to understand deformation.
Moreover, an algorithm based on the geometric barycenter of the neighbors of
a given particle is consistent with the idea of locally minimizing an elastic potential, as the centroid has the well-known properties of minimizing the sum of the
squared distances from a set of given points in an Euclidean space. Therefore, the
proposed algorithm seems a natural discrete approach from the variational point of
view. Another possibility, close to these concepts, we are considering is to substitute
Eqs. 18.1–18.4 by potential field able to determine particle displacements and avoid
collisions.
18.4 Some Examples
Aim of this section is to show the coherence of the model and its adaptability in
showing different physical phenomena by changing some parameters. Therefore, we
approach some bidimensional problems relative to simple shape object subject to
imposed strain of some leaders which are significant; for some of the tests, we shall
show and discuss the movement of the particles, the XY movement of a significant
particle (if present) and some pseudoenergetic considerations by PE1 or PE2. See
preceding works (dell’Erba 2018a, c; Battista et al. 2016). The behavior of some
more complex ASTM samples and the respect of Saint Venant principle have also
been described; moreover, looking for the limits of the tool, we shall discuss some
not satisfactory results.
R. dell’Erba
P E1(t, j) =
all neighbours of j
k=1
(dis(t, k, j) − dis(t 0 k, j))
2
Where dis(t, k, j) is the Euclidean distance between points k and j at time t. The
reason for this choice lies in the attempt to emulate potential energy of material point
subject to Hook’s law. It is a sort of square distance between the actual configuration
C t and the reference configuration C 0 .
To compare time contiguous configuration C t and C t−1 , we define for each point
j and each time t
P E2(t, j) = ||C t − C t−1 ||
where || is the norm of the vector defined by the point j at time t and t − 1.
It must be underlined that this artifice has no direct connection with the usual
energy definition (this is the reason we use the term pseudoenergy) but could be
useful to understand deformation.
Moreover, an algorithm based on the geometric barycenter of the neighbors of
a given particle is consistent with the idea of locally minimizing an elastic potential, as the centroid has the well-known properties of minimizing the sum of the
squared distances from a set of given points in an Euclidean space. Therefore, the
proposed algorithm seems a natural discrete approach from the variational point of
view. Another possibility, close to these concepts, we are considering is to substitute
Eqs. 18.1–18.4 by potential field able to determine particle displacements and avoid
collisions.
18.4 Some Examples
Aim of this section is to show the coherence of the model and its adaptability in
showing different physical phenomena by changing some parameters. Therefore, we
approach some bidimensional problems relative to simple shape object subject to
imposed strain of some leaders which are significant; for some of the tests, we shall
show and discuss the movement of the particles, the XY movement of a significant
particle (if present) and some pseudoenergetic considerations by PE1 or PE2. See
preceding works (dell’Erba 2018a, c; Battista et al. 2016). The behavior of some
more complex ASTM samples and the respect of Saint Venant principle have also
been described; moreover, looking for the limits of the tool, we shall discuss some
not satisfactory results.