Characteristics of the relaxation to steady-state deformation in solids
S. G. Psakh’e, A. Yu. Smolin, E. V. Shil’ko, S. Yu. Korostelev, A. I. Dmitriev,
and S. V. Alekseev
Institute of Physics of Strength and Materials Science, Siberian Branch, Russian Academy of Sciences,
634055 Tomsk, Russia
Submitted March 11, 1996
Zh. Tekh. Fiz. 67, 34–37 September 1997
The characteristics of the transient process of relaxation to steady-state deformation in solids are
investigated theoretically. Various loading regimes are modeled by the method of mobile
cellular automata. It is shown that the stressed state in a material is highly inhomogeneous in the
relaxation stage; this property, in turn, can produce stable structures in the velocity field of
the material particles and influence the evolution of deformation in later stages. © 1997 American
Institute of Physics. S1063-78429700709-5
The behavior of materials under mechanical loading is
usually investigated, both experimentally and theoretically,
in the stages of steady-state deformation and at prefracture
strain levels.
1,2
On the other hand, scarcely any attention has
been given to the specific attributes of the initial stage of
deformation. The only way this problem can be studied ex-
perimentally at the present time is on the basis of global
characteristics of the response of materials, such characteris-
tics as the total strain of the sample, acoustic-emission spec-
tra, etc. Even in this case the instrumentation must have high
resolution in the range 10
-6
–10
-9
s. The spatial distribu-
tion of strains can be measured very accurately on the basis
of the special television-optical system described in Refs. 3
and 4. This device is capable of measuring displacement vec-
tors on the surface of a deformable material with a resolution
of 1500 points/mm
2
. In combination with sufficiently high-
performance electronics, such instruments can be used to in-
vestigate the characteristics of the transient process leading
to steady-state deformation. Methods based on continuum
mechanics are currently used for theoretical investigations of
the response of materials at the mesoscopic level. At the
same time, discrete approaches are beginning to make
greater inroads into modeling.
5–12
Apart from their other ad-
vantages, these approaches are far less demanding in com-
puter resources, an asset that is especially useful in the solu-
tion of problems requiring high spatial and temporal
resolution. For this reason we have developed a mobile ki-
netic cellular automata method
8–12
as an elaboration of the
particle method
5
and element dynamics.
8,9
FORMALISM
The material modeled in the given method is represented
by an ensemble of discrete elements cellular automata,
which interact with each other by certain rules, relations, and
laws. The automata, in turn, comprise elements of heteroge-
neous media, in particular, individual grains of a polycrys-
talline material, individual particles of a powder mixture, etc.
Of course, the dimensions of the automata depend on the
conditions of the specific problem.
Since the individual automata are mobile, this approach
can be used to model various processes encountered in a real
material, including permeation effects, mass transfer, frag-
mentation effects, and the formation of defects, cracks, and
voids. Various mechanical loading regimes compression,
tension, shear deformation, etc. can be simulated by impos-
ing additional conditions on the boundary of the modeled
sample.
The modeled system is characterized by the following
quantities in the method of mobile cellular automata: the
radius vectors of the centers of the automata R
i
, the trans-
lational velocities of the automata V
i
, their angles of rota-
tion
i
, and their angular velocities
i
. In addition, each
cellular automaton is characterized by a dimensional param-
eter d
i
, a mass m
i
, and an inertia tensor J
ˆ i
. Interaction be-
tween the automata is analyzed in the pair approximation and
depends on the central forces, the forces of viscous and dry
friction, and the forces of resistance to shear deformation.
The interacting pairs of automata are divided into two types:
bound, when chemical bonds are present elements of one
particle, and independent, when such chemical bonds are
not present elements of different particles.
In the simplest case a cellular automaton contains one
type of material — an initial component of a mixture or a
reaction product, and in the general case composite cellular
automaton it contains a set of several different kinds of ma-
terials, whose characteristics govern its state. In describing
the properties of a composite cellular automaton, we use an
analog of the virtual crystal model, which stipulates that all
the specific characteristics of the automaton and the param-
eters of interaction with its neighbors are determined by av-
eraging the constituents of its composition over the number
of atoms.
A cellular automaton can change its state both as a result
of internal transformations phase transitions and in the
course of chemical reaction with its neighbors.
The evolution of an ensemble of cellular automata is
determined by solving numerically the system of equations
of motion
m
i
d
2
R
i
dt
2
=
j
F
ij
,
J
ˆ
i
d
2
i
dt
2
=
j
K
ij
,
1016 1016 Tech. Phys. 42 (9), September 1997 1063-7842/97/091016-03$10.00 © 1997 American Institute of Physics