Numerical simulations for the evolution of elastoviscoplastic materials
The model proposed by Groma, Csikor and Zaiser (see the model) describes the dynamics of
dislocation densities in a bounded 2-dimensional material submitted to an exterior applied stress.
Dislocations are defects in crystals that move when a stress field is applied on
the material. These
defects are one of the main explanations of
the elastoviscoplasticity behavior of metals.
This model has been introduced in order to describe the possible
accumulation of dislocations on the boundary layer of a bounded
channel.
Applying vertically a constant shear stress at the boundary of this channel (say fixed at boundary walls),
we are able to compute (numerically) the displacement inside the material (for a theoretical study,
see part 1 and
part 2).
The simulation below shows the deformation of the material that converges
numerically to some particular deformation where we observe the presence of boundary layers. This effect
is directly related to the introduction of the back stress in the model of Groma, Csikor and Zaiser.
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After long time (see the above simulations), we have removed the shear stress. In this case,
as it is illustrated by the next simulation, the material returns back to its original position (click here for a theoretical study of this model). This
is only maintained at a specific scale. In reality (or say at a smaller scale), the distribution of dislocations
inside the material is altered. In fact, we do not obtain the same dislocation densities
as they were at the initial time.
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Simulations of the transport of interfaces by a "Splitting" type algorithm
In the framework of the project, CERMICS-ENPC (PDE and materials group)
and CEA, we were searching a numerical method in order to propagate interfaces, following a given vector field in a
cartesian grid. In this direction, we have proposed an algorithm of the "splitting" type (see Chapter 6
of my thesis for the detais of the algorithm). The basic idea is to split the propagation of the interface
following the two orthonormal axis of the 2D grid. The splitting algorithm is numerically shown to be
exact for a propagation with a constant vector field a=(a1,a2) (see the simulation below showing the translation of a circle with the constant vector field a=(1,2)).
In the case of variable vector fields, our algorithm is not monotone which creates some instabilities
in fronts propagation (see the simulation below of a rotating square, a=(-y,x)).
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Numerical computation of the effective Hamiltonian in homogenization of straight dislocations
We consider the microscopic model of N straight parallel edge dislocations lines moving in the same glide plane (click here for the details). We can write down the fully overdamped dynamics, where the velocity is proportional to the sum of all forces including the exterior applied shear stress S. A natural question is then: what is the macroscopic behavior of this system? In fact, we expect that this behavior will describe the limit macroscopic plastic strain governed by a Hamilton-Jacobi equation with an effective Hamiltonian f. The below simulations show the numerical computation of f, where we initially put N dislocations in an interval of length l=10 which is repeated periodically.
The level sets of the function f are represented in the first simulation below. Here the abscissas stand for the values of N (the number of particles), and the ordinates stand for the values of the exterior applied shear stress S. The next simulation shows, for N=1, 10, 20, the graph of the map that to each value S of the exterior applied shear stress associates the value f(N/l, S).
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