The Mechanism of Plastic Deformation of Crystals
Part I. Theoretical
G.I. Taylor, 1934
Prepared for 18.325 Classical Physics Through the Work of
G.I. Taylor
by Catherine Bishop March 16, 2000
INTRODUCTION
It is in this paper [1] that Taylor suggests that the motion of dislocations and their interaction explains the shape of stress strain curves for single crystal metals. He begins by presenting a plot of resolved shear stress vs. shear strain and stating that the form of the plot is similar for may single crystal metals. He comments that the lower limit of strength depends on purity but offers no explanation.
The three types of existing theories to explain why plastic deformation increases the strength of materials are presented. The first supposes that a perfect crystal slips under a small stress but that "locking keys" of material break off and hold up further deformation. The second supposes that perfect crystal slips under a small shear stress but that there is a mosaic structure of planes of misfit present that hold up this slip with increasing degree. The third supposes that perfect crystal is very strong and weakness is imbued by stress concentrations from internal misfits. The strength of the material increases as the concentration of misfits increases with increasing plastic strain. When the number of misfits increases the ratio of maximum stress to mean stress decreases. So the mean stress necessary to cause a given maximum stress in an area of stress concentration increases.
It is suggested that plastic strain occurs when atoms jump from one equilibrium position to another and that for slip to occur the majority of atoms must jump in the slip direction. If all the atoms move at once this would leave the perfect crystal above and below the slip plane unchanged, leave the material un-strengthened, require a stress on the order of the elastic modulus and would not explain the temperature effect.
DISLOCATION STRUCTURE AND ENERGETICS
Taylor suggest two kinds of dislocations, negative and positive, Fig. 1.
[height= 3in] disloc1.eps
Figure 
He suggests that a temperature 
exists analogous to, but probably below, the recrystallization temperature at which a dislocation can move under no applied stress. In an attempt to account for high temperature deformation he suggests that dislocation centers travel across material at low stresses leaving the material a single crystal and still weak.
Atoms sit at sites of minimum potential in the relation to their neighbors. If atoms are spaced a distance 
apart then the potential along a line of atoms due to the lines of atoms either side of it in a plane may be
The potential along a line of atoms centered between the neighboring lines and containing the dislocation center is inferred by the general expression for the potential due to N+2 atoms above and N+1 below each spaced over 
given as
If N is an odd number the shallowest depression in the potential is
at the center and if unoccupied by an atom is considered the
dislocation center. At thermal energy is enough for an atom to
surmount this potential, vacate its site and jump to the central
site. This results in the motion of the dislocation center.
Under the application of an external shear stress the atoms are
displaced a distance 
to the right then the potential becomes
This decreases the height of the barrier to the right of the central position making jumps from the right into the central position easier. So a positive dislocation will migrate to the right under the application of a shear stress to the right.
CALCULATION OF DISLOCATION STRESS INTERACTIONS
Taylor states that the general theory of dislocations in isotropic
elastic solids had been treated by Volterra. When applying the theory of elasticity to an isotropic crystal at least a few lattice
spacings away from the dislocation center for a positive dislocation or
slip of on the positive x axis where 
is the modulus of rigidity, the components of stress are
given by Timpe as
At this point Taylor remarks that the assumption of isotropy appears to be fulfilled
by aluminum.
Taylor states that as a dislocation stress field has a non-negative
shear component like dislocations repel each other and unlike
dislocations attract until they are aligned on slip planes. With a
single dislocation in an otherwise perfect crystal above the
application of a small shear will allow the dislocation to move across
the crystal. If two opposite dislocations are present on parallel slip planes
h apart a small shear will allow them to move in opposite directions
along their respective slip planes until each dislocation reaches a
position of equilibrium where the stress is
where x is the projection of the separation of the dislocations on a
slip plane. The maximum value of stress associated with equilibrium
is 
, so any greater stress will allow the
dislocations to escape their mutual attraction and continue to move
across the crystal. Therefore, the strength of the crystal has increased.
Interpenetrating rectangular nets of positive and negative dislocations can form equilibrium structures as below, Fig. 2.
[height= 2in] disloc2.eps
Figure 
(a) stable (b) unstable.
The shear stress due to a row of positive dislocations regularly spaced a distance d apart in a line perpendicular to the slip plane is given by the sum of the stress fields from Eq. 4
If a net of dislocations as in
Fig. 2 (a) has the positive lattice displaced
relative to the negative then the coordinates of the negative
centers with respect to the positive are 
where n and m are integers and a is the separation
distance on the slip plane of dislocation centers. If S' is the shear
stress at the origin due to all the dislocations then for each
increase in by a, S' changes by 
due
to the polarization of positive and negative dislocations on either
side of the crystal. Therefore is the
increase in the mean shear stress for an increase in of a over the whole crystal due to
dislocations.
When the boundary is taken into account so that the mean stress through
the crystal is equal to the externally applied stress, the stress at
the positive centers of dislocation is 
where 
. If the temperature is above then the
equilibrium condition is 
or
It is noted that F is a function of 
and a/d so for a
given value of a/d, 
is the maximum of F with respect to
. When graphed for configurations (a) and (b) it is seen that (a) is the
stable configuration and the shear strength of the crystal is finite
and given by
. A similar calculation is done for
diagonal lattices of dislocations and was determined.
If a dislocation center travels a distance x from the surface into
the crystal the displacement is 
where L is the
dimension of the crystal. For a row of
dislocations spaced a apart the total shift is 
. In a rectangular arrangement with slip plane spacing
the displacement of two slip planes separated by h is
. So by definition, the shear strain is the
displacement of the slip planes divided by the separation of the
planes or 
.
For the diagonal arrangement with slip plane spacing
, the shear strain is 
. By
taking the ratio of S to 
one gets
Where 
is a constant determined by the arrangement of
dislocations and the (S, s) curve is parabolic. When the
temperature is below the condition for equilibrium becomes 
where 
is the finite stress required
to move a single dislocation in a perfect crystal. The (S, s) curve
is then governed by
Taylor discusses the selection of a value of 
to compare to
experimental results. He suggests that if dislocations are produced
at random at the boundary or in the crystal then a regular lattice
with low is unlikely. He also says that the presence of
dislocations in a region of crystal will tend to prevent the random
generation of future dislocations there. In the end he just chooses a
value of 
for the ``present very imperfect state of the
theory'' and says that this gives the correct order of
magnitude for the strength of crystals.
COMMENTS
At the beginning of the paper Taylor introduces positive and negative
dislocations. Today materials scientists discus dislocations with edge
and screw characteristics. Taylor's dislocations are positive and
negative edge dislocations which have a particular slip plane. Left and right handed screw dislocations
have Burger's vectors parallel to the dislocation line direction and
so have no slip plane.
There are also dislocations of mixed character. (The Burger's vector
is analogous to Taylor's or the size of the step on the
surface of a crystal after a single dislocation passes from the
interior to the boundary).
Taylor compares stress-strain data for
single crystals of materials with different crystal structures,
melting temperatures, and anisotropy ratios [2]: Al
(fcc, 933K, 0.81), Fe (bcc, 1811K, 0.47), Au (fcc,
1337K, 0.35), Cu (fcc, 1357K, 0.31). The homologous temperature,
, where T is the absolute experimental temperature and 
is the absolute melting point of the material, gives
an indication of the likelihood of dynamic recovery in a material. If 
then dynamic
recovery, the activated annihilation of point defects such as
interstitials and vacancies, should be considered which will soften the material. It is
well known that copper undergoes dynamic recovery at room
temperature. The
crystal structure of the material determines the number of slip
systems and the length of the Burger's vector for the material [3]. The larger the
number of slip systems the easier it is for dislocations to glide in a crystal. There at 16 and 12 slip systems for fcc and bcc,
respectively, which are 
and 
. The longer the Burger's vector the higher the self energy of
the dislocation and the higher the chance that it will form partial
dislocations and a Lomer-Cottrell lock which will strengthen the material.
The preparation of the samples is very important. Taylor states that the (S,s) curves are very sensitive to purity. Impurities form a Cottrell atmospheres around dislocations and increasing material strength at low temperatures. A non-equilibrium concentration of point defects such as vacancies and interstitials due to rapid quenching increases the mobility of dislocations in a material. In a single crystal precipitates or other dislocations can pin dislocations in a slip plane in such a way as to create a Frank-Reed dislocation source or a place inside the crystal to generate new dislocations. I've mentioned a few modern ways of thinking of some of the issues that Taylor mentions in his paper. An excellent comprehensive reference is Hirth and Lothe [4].
REFERENCES
