SIMULATION OF PLASTIC DEFORMATION, RECRYSTALLISATION
AND GRAIN COARSENING BY CELLULAR AUTOMATON
András Roósz*, Péter Barkóczy*, János Geiger**
*University of Miskolc, Faculty of Material Science and Metallurgical Engineering, Department of
Physical Metallurgy
**University of Miskolc, Faculty of Mechanical Engineering, Department of Descriptive Geometry
Abstract
A complex plastic deformation, recrystallisation and grain coarsening simulation method is
developed by cellular automaton (CA) algorithm instead of the more complicat traditional analytical
method. The result of the CA method is a grain structure from which the mechanical properties of
the metals or alloys can be determined. The simulation of plastic deformation is investigated on
geometrical aspects. This method is usable for different deformations e.g. drawing, rolling etc.. The
simulation of recrystallisation and grain coarsening take into consideration the different enthalpies
e.g. stored enthalpy, activation enthalpies etc.. The method can examine the process separately or
entirely.
Keywords
simulation, cellular automaton, plastic deformation, grain coarsening, recrystallisation
Introduction
One of the most important technological processes of producing metals and alloys is their plastic
deformation (hot- and cold deformation) and the recrystallisation following it. Combining the
plastic deformation with recrystallisation the mechanical properties of materials can be changed in a
wide range. The recrystallisation and precipitation can be taken place at the same time (e.g. Alalloys), and the recrystallisation can be followed by other phase-transformation as well (e.g. in case
of the hot deformation of steels). The planning of technologies is highly promoted by the simulation
of the processes. The simulation of the previously mentioned processes is too difficult by traditional
analytical methods, because the differential equations, which describe the processes, cannot be
solved generally. The Cellular Automaton (CA) is a very usable method for modelling these
consecutive processes. The grain structures simulated by CA are very similar to the real structures
observed by microscope, so the transformation of structures can be well followed visually.
Simulation Procedure
In the three dimensions CA method used in the simulation the discrete space filling volume element
(cell) is cube, which represents an atomic group. Discrete space and state variables are corresponded
to each cells. The states are represented by a function as follows (Eq. 1):
(1),
where
u: is a vector, determining the position of a cell,
q: crystallographic orientation of grain containing the cell (1<q<qmax, where qmax is the maximum
orientation number of the grains),
Gb: boundary free enthalpy of a cell (definitely zero if the cell is not on the grain boundary)
Gst: stored free enthalpy of a cell coming from the plastic deformation,
Gt: thermal free enthalpy of a cell.
The size of cells is not determined in the beginning of simulation, but they get values from
measurements at the end of process. During the simulation the Neumann neighbourhood was used.
The transition rules of individual processes based on free enthalpy.
The simulation of plastic deformation
During the plastic deformation the volume of the grains are constant while the shape of grain
changes, the length of edges and grain surfaces per unit volume increase. During the
recrystallisation the nucleation takes place at the grain boundary usually (heterogenous nucleation)
and then it is very important to know these quantities. Recently a solution has been published by
Réti and Czinege [1] as well as by Singh and Badhesia [2] for determining the topologic parameters
of structures developing during the deformation. These methods do not take into consideration that
the grain surfaces generally are not planes, i.e. the grain boundaries on the metallographic sections
are not straight lines. At the simulation of recrystallisation it is necessary to know the new coordinates of all cells, and then those were calculated as follows.
Each cell can be characterised by a u vector before deformation (Fig. 1). The size of a cell
represents a unit. The coordinates of vectors can be only integer number. In the deformed grain
structure the vectors of cells (v) can be calculated from the product of the deformation tensor (F)
and the original vectors (u):
Fig. 1. The sketch of simulation of plastic deformation.
or
(2)
where: ui, vi are the coordinates of the original and the deformed vector of cells:
a=h0/(h0-h), b=w0/(w0-w) and c=l0/(l0-l) are the rate of deformation in the x, y, z directions,
h0, w0, l0 and h, w, l are the measures of specimen before and after the deformation (in the
figure w=w0),
abc=1 because of the mass conservation law.
The vi coordinates of cells must be integer numbers after the deformation. Results of the Eq. 2. must
be rounded off, so there will be cells whose state are undefined. These cells take the q value of
defined cells being in their neighbourhood. In Fig. 3. the result of rolling and in Fig 4. the result of
drawing are shown.
Fig 2. The initial grain structure before plastic deformation.
a)
b)
c)
d)
Fig 3. The grain structure after rolling
a) parallel section, b) cross section, (a=1.43, b=0.70, c=1.00)
c) parallel section d) cross section, (a=2.04, b=0.49, c=1.00)
a)
c)
b)
d)
Fig 4. the grain structure after drawing
a) parallel section, b) cross section, (a=1.43, b=0.84, c=0.84)
c) parallel section, d) cross section, (a=2.04, b=0.70, c=0.70)
The simulation of recrystallisation and grain coarsening
There are two part-processes of recrystallisation i.e. nucleation and grain growth. In the process of
nucleation new grains (nuclei) develop which are free from the effects of deformation. These nuclei
grow while the deformed part disappears. The recrystallisation can be followed by grain coarsening.
The rules of state-changes has been developed for each part-process separately on the basis of free
enthalpy of cells.
The free enthalpy of cell
As the result of deformation the free enthalpy of the cells increases (Gst, stored free enthalpy). This
free enthalpy depends on the rate of deformation. The deformation of each grain is equal, i.e. the
value of stored free enthalpy does not depend on the location of a cell. The second free enthalpycomponent of the cell is the boundary free enthalpy (Gh). A boundary exists between two cells with
different orientation (different q). The boundary free enthalpy depends on the orientation difference
between two neighbour cells according to Eq. 3 [4] where G0 is the maximum energy of a cell being
on the boundary, qi and qj are the orientations of the neighbour cells and qmax is the maximum
orientation number.
(3)
The third component is the thermal free enthalpy. The thermal free enthalpy of atoms can be
approximated by using the Maxwell-Boltzmann distribution. It is supposed, that the thermal free
enthalpy of cells has also Maxwell-Boltzmann distribution. According to this distribution the ratio
of cells 0<p<1 having higher free enthalpy than Gt can be calculated by using of the Eq. 4:
(4),
where
T is the absolute temperature, R is the gas constant.
Generating a random value p from the Eq 4. the thermal free enthalpy of a cell:
(5)
The total free enthalpy of a cell is the sum of these three free enthalpy parts. If the free enthalpy of
the cells higher than the activation free enthalpy (G) the process takes place.
Nucleation
A nucleus (in this model is one cell) is a separated grain with zero Gst having new crystallographic
orientation. The orientation of a nucleus is a random value independent of the orientation of the
grain which containing of it. If the free enthalpy of a cell higher than the activation free enthalpy of
nucleation it can become to nucleus (Fig. 5.a). Nucleus can develop only in the deformed area.
Fig. 5. Change of the free enthalpy during nucleation (a) grain growth (b) and grain coarsening (c)
Nucleus growth
The growth of a nucleus takes place, if the free enthalpy of a deformed cell which is on the
deformed-undeformed boundary higher than the activation free enthalpy of grain growth (Fig. 5.b).
This cell takes the crystallographic orientation of the nucleus.
Grain coarsening
The free enthalpy of atoms (cells) is higher in the boundary with greater curvature, as in the lower
one. This is the driving force of grain coarsening. A cell takes part in coarsening if it is on the
boundaries of recrystallised grains, and its free enthalpy higher than the activation energy of grain
coarsening (Fig. 5.c).
Results
The effect of temperature and the rate of deformation on the kinetics of recrystallisation
The recrystallisation kinetics can be described by the Avrami (Johnson-Mehl) equation:
(6)
or in another form:
(7)
where F is the recrystallized volume part, B and n are constants, t is the time measured in CA step
(CAs).
The value of Avrami exponent n is determined by the topology and the rate of nucleation and
growth. If the nucleation and growth rate are constant, the theoretical value of n is 4. If the
nucleation rate is zero (place saturated nucleation) and the rate of growth is constant n is 3.
a)
b)
Fig 6. The effect of temperature (a) and rate of deformation (b) on the recrystallisation.
a)
b)
c)
Fig 7. One example series of grain structure under recrystallisation, a) 20 CAs, b) 50 CAs,
c) 100 CAs.
The initial grain structures have been cold-rolled are in different rate and then they have been
recrystallized at different temperatures. The results can be seen in Fig. 6. By the decrease of the rate
of deformation the time of total recrystallisation will increase. It can be seen, that the simulator and
the theory are in correspondence.
Grain coarsening
The kinetics of grain coarsening is described by the equation:
(8)
where A is the average grain area at 0 and t are time parameters, k is a constant at a temperature.
The recrystallisation diagrams (Chohralski diagrams) contain the average grain size as a function of
the rate of deformation following the same deformation and the temperature after constant heat
treatment (e.g. 1 hour). In Fig 10.a, Fig 10.b show the sections of a recrystallisation diagram.
a)
b)
Fig 8. The effect of temperature (a) and the boundary free enthalpy (b) on the average grain area
under gain coarsening.
a)
b)
c)
d)
Fig 9. One example series of grain structure under coarsening, a) initial structure, b) 2500 CAs, c)
5000 CAs, d) 7500 CAs.
a)
b)
Fig 10. The sections of the recrystallisation diagram at constant rate of deformation (a) and a
constant temperature (b) after rolling and 500 CAs.
Simulation of complicate deformation and recrystallisation processes
The real deformation, recrystallisation, coarsening processes would be very complicate. It is true
especially in case of hot deformation of steels, when these processes may follow each other more
than one time. The application of simulation method is shown by an example. The type of
deformation is rolling. The simulated processes: 30% deformation, complete recrystallisation, grain
coarsening two times.
In Fig 11. can be followed the effect of recrystallisation and grain coarsening on the average grain
area, while it does not change under deformation. At the beginning of process of recrystallisation
the average grain area decreases dramatically because there are many small new grains. After that
when the new grains growths the average grain area increases. After the recrystallisation during the
coarsening the average grain area changes linearly.
Fig 11. An example for complicate deformation and recrystallisation process.
a)
b)
c)
d)
e)
f)
g)
h)
Fig 12. Change of grain structure during complicate process shown in Fig. 11
a) initial structure after deformation, b) 40 CAs, c) 100 CAs, d) a 500 CAs,
e) second deformed structure, f) 1040 CAs, g) 1080 CAs, h) 1500 CAs
Conclusions
A CA method has been developed in order to calculate the change of grain structure after plastic
deformation and the recrystallisation following it. The rules of the individual processes has been
determined on an energetic base. The developed method is suitable for the simulation of
complicated, overlapping deformation and recrystallisation processes.
References:
1) R. J. GAYLORD, K. NISHIDATE: Modelling Nature, Springer Verlag, 1994.
2) I. CZINEGE, T. RÉTI: Determination of Local Deformations in Cold Formed Products by a
Measurement of the Geometric Characteristics of the Crystallites, Proceedings of Eighteens
International Machine Tool Design and Research Conference, London 1977, Vol I, Forming p. 159163.
3) S. B. SINGH, H. K. D. H. BHADESHIA: Topology of Grain Deformation, Material Science and
Technology 1998, Vol 14. pp 832-834.
4) J. GEIGER, A. ROÓSZ and P. BARKÓCZY: Simulation of Grain Coarsening in Two
Dimensions by Cellular Automaton, Acta metall., 2001, Vol. 49/4, 623-629.
András Roósz:
Adress:
Tel:
Fax:
E-mail:
H-3515, Miskolc-Egyetemváros, University of Miskolc, Faculty of Material
Science and Metallurgical Engineering, Department of Physical Metallurgy
+00-36-46-565-111/15-43
+00-36-46-365-924
femroosz@gold.uni-miskolc.hu
Péter Barkóczy
Adress:
Tel:
Fax:
E-mail:
H-3515, Miskolc-Egyetemváros, University of Miskolc, Faculty of Material
Science and Metallurgical Engineering, Department of Physical Metallurgy
+00-36-46-565-111/15-43
+00-36-46-365-924
fembarki@gold.uni-miskolc.hu
János Geiger
Adress:
Mechanical
Tel:
Fax:
E-mail:
H-3515, Miskolc-Egyetemváros, University of Miskolc, Faculty of
Engineering, Department of Descriptive Geometry
+00-36-46-565-111/18-09
+00-36-46-365-924
geiger@abrindy.abgr.uni-miskolc.hu