Tresca and von Mises criteria

7.7.1 Tresca and von Mises criteria

In dislocation theory it is usual to consider the flow stress or yield stress of ductile metals under simple conditions of stressing. In practice, the engineer deals with metals under more complex conditions of stress- ing (e.g. during forming operations) and hence needs to correlate yielding under combined stresses with that in uniaxial testing. To achieve such a yield stress criterion it is usually assumed that the metal is mechanically isotropic and deforms plastically at constant volume,

Figure 7.46 Limiting draw ratios (LDR) as a function of

i.e. a hydrostatic state of stress does not affect yield- average values of R and of elongation to fracture measured

ing. In assuming plastic isotropy, macroscopic shear is in tensile tests at 0 ° , 45 ° and 90 ° to the rolling direction

(after Wilson, 1966; courtesy of the Institute of Metals) allowed to take place along lines of maximum shear

stress and crystallographic slip is ignored, and the yield stress in tension is equal to that in compression, i.e.

3 has low through-thickness strength there is no Bauschinger effect. when the basal plane is oriented parallel to the plane of

p Zn with c/a <

A given applied stress state in terms of the princi-

the sheet. In contrast, hexagonal metals with c/a > 3 1 2 3 axes, X which act along three principal 1 ,X 2 and X 3 , may be separated into the hydro- would have a high R for f1 0 10g parallel to the plane

static part (which produces changes in volume) and of the sheet.

the deviatoric components (which produce changes in Texture-hardening is much less in the cubic met-

shape). It is assumed that the hydrostatic component als, but fcc materials with f1 1 1g h110i slip system

has no effect on yielding and hence the more the stress and bcc with f1 1 0g h1 1 1i are expected to increase R

state deviates from pure hydrostatic, the greater the when the texture has component with f1 1 1g and

tendency to produce yield. The stresses may be rep- f1 1 0g parallel to the plane of the sheet. The range of

resented on a stress–space plot (see Figure 7.47a), in values of R encountered in cubic metals is much less.

which a line equidistant from the three stress axes rep- Face-centred cubic metals have R ranging from about

resents a pure hydrostatic stress state. Deviation from

0.3 for cube-texture, f1 0 0g h0 0 1i, to a maximum, this line will cause yielding if the deviation is suf- in textures so far attained, of just over 1.0. Higher

ficiently large, and define a yield surface which has values are sometimes obtained in body-centred cubic

sixfold symmetry about the hydrostatic line. This arises metals. Values of R in the range 1.4 ¾ 1.8 obtained

because the conditions of isotropy imply equal yield in aluminium-killed low-carbon steel are associated

stresses along all three axes, and the absence of the with significant improvements in deep-drawing per-

Bauschinger effect implies equal yield stresses along formance compared with rimming steel, which has

1 1 R . Taking a section through stress space, -values between 1.0 and 1.4. The highest values of

perpendicular to the hydrostatic line gives the two R in steels are associated with texture components

simplest yield criteria satisfying the symmetry require- with f1 1 1g parallel to the surface, while crystals with

ments corresponding to a regular hexagon and a circle. f1 0 0g parallel to the surface have a strongly depress-

The hexagonal form represents the Tresca criterion ing effect on R.

(see Figure 7.47c) which assumes that plastic shear In most cases it is found that the R values vary

takes place when the maximum shear stress attains a with testing direction and this has relevance in rela-

critical value k equal to shear yield stress in uniaxial tion to the strain distribution in sheet metal forming. In

tension. This is expressed by particular, ear formation on pressings generally devel-

ops under a predominant uniaxial compressive stress

at the edge of the pressing. The ear is a direct con- max D (7.32)

2 Dk

sequence of the variation in strain ratio for different directions of uniaxial stressing, and a large variation

1 2 3 . This crite- in R value, where R D ⊲R max

rion is the isotropic equivalent of the law of resolved relates with a tendency to form pronounced ears. On

min ⊳ generally cor-

shear stress in single crystals. The tensile yield stress this basis we could write a simple recipe for good

1 2 3 D 0.

236 Modern Physical Metallurgy and Materials Engineering

1 2 3 , (b) von Mises yield criterion and (c) Tresca yield criterion .

The circular cylinder is described by the equation

1 2 ⊳ 2 C 2 3 ⊳ 2 C 3 1 ⊳ 2 D constant ⊲ 7.33⊳

and is the basis of the von Mises yield criterion (see Figure 7.47b). This criterion implies that yielding will occur when the shear energy per unit volume reaches

a critical value given by the constant. This constant is equal to 6k 2 or 2Y 2 where k is the yield stress in

and Y is the yield stress in uniaxial tension when

2 D 3 D 0. Clearly Y D 3k compared to Y D 2k for the Tresca criterion and, in general, this is found to agree somewhat closer with experiment.

Figure 7.48 The von Mises yield ellipse and Tresca yield hexagon In many practical working processes (e.g. rolling), .

the deformation occurs under approximately plane strain conditions with displacements confined to the

inscribed in the ellipse as shown in Figure 7.48. Thus,

1 2 have opposite signs, the Tresca crite- direction is zero, and, in fact, the deformation condi-

X 1 X 2 plane. It does not follow that the stress in this

the edges of the hexagon CD and FA. When they have dency for one pair of principal stresses to extend the

3 D 1 2 1 C 2 ⊳ so that the ten-

1 2 D 2k D Y and metal along the X 3 axis is balanced by that of the other

defines the hexagon edges AB, BC, DE and EF.

3 from

the von Mises criterion, the yield criterion becomes

7.7.2 Effective stress and strain

1 2 ⊳D 2k For an isotropic material, a knowledge of the uniaxial tensile test behaviour together with the yield func-

tion should enable the stress–strain behaviour to be given when

2 D 0,

predicted for any stress system. This is achieved by p

defining an effective stress–effective strain relation-

3 D 1.15Y n is the uniaxial stress–strain For plane strain conditions, the Tresca and von

1 D 2k D 2Y/

relationship then we may write. Mises criteria are equivalent and two-dimensional flow

(7.35) occurs when the shear stress reaches a critical value. The above condition is thus equally valid when written

for any state of stress. The stress–strain behaviour of

0 1 2 0 3 0 defined

a thin-walled tube with internal pressure is a typi-

0 1 cal example, and it is observed that the flow curves 1 1

obtained in uniaxial tension and in biaxial torsion

coincide when the curves are plotted in terms of effec- surface becomes two-dimensional and the von Mises

3 D 0 and the yield

tive stress and effective strain. These quantities are criterion becomes

defined by:

1 2 D 3k DY

1 2 ⊳ 2 2 3 ⊳ 2 3 1 ⊳ 2 ] 1/2 which describes an ellipse in the stress plane. For the

Tresca criterion the yield surface reduces to a hexagon ⊲ 7.36⊳

Mechanical behaviour of materials 237 and

therefore, to increase the temperature of the deformed p

metal above the strain-ageing temperature before it

[⊲ε 1 2 ⊳ 2 2 εD 2 2 C ⊲ε 1/2 3 ⊳ 3 C ⊲ε 1 ⊳ ]

recovers its original softness and other properties.

3 The removal of the cold-worked condition occurs

by a combination of three processes, namely: where ε 1 ,ε 2 and ε 3 are the principal strains, both

(1) recovery, (2) recrystallization and (3) grain growth. of which reduce to the axial normal components of

These stages have been successfully studied using stress and strain for a tensile test. It should be empha-

light microscopy, transmission electron microscopy, or sized, however, that this generalization holds only for

X-ray diffraction; mechanical property measurements isotropic media and for constant loading paths, i.e.

(e.g. hardness); and physical property measurements

1 2 3 where ˛ and ˇ are constants inde- (e.g. density, electrical resistivity and stored energy).

1 . Figure 7.49 shows the change in some of these prop- erties on annealing. During the recovery stage the decrease in stored energy and electrical resistivity

7.8 Annealing

is accompanied by only a slight lowering of hard- ness, and the greatest simultaneous change in proper-

ties occurs during the primary recrystallization stage. When a metal is cold-worked, by any of the many

7.8.1 General effects of annealing

However, while these measurements are no doubt industrial shaping operations, changes occur in both

striking and extremely useful, it is necessary to under- its physical and mechanical properties. While the

stand them to correlate such studies with the structural increased hardness and strength which result from the

changes by which they are accompanied. working treatment may be of importance in certain

applications, it is frequently necessary to return the

7.8.2 Recovery

metal to its original condition to allow further forming This process describes the changes in the distribution operations (e.g. deep drawing) to be carried out of for

and density of defects with associated changes in phys- applications where optimum physical properties, such

ical and mechanical properties which take place in as electrical conductivity, are essential. The treatment

worked crystals before recrystallization or alteration given to the metal to bring about a decrease of the

of orientation occurs. It will be remembered that the hardness and an increase in the ductility is known as

structure of a cold-worked metal consists of dense dis- annealing. This usually means keeping the deformed

location networks, formed by the glide and interaction metal for a certain time at a temperature higher than

of dislocations, and, consequently, the recovery stage about one-third the absolute melting point.

of annealing is chiefly concerned with the rearrange- Cold working produces an increase in dislocation

ment of these dislocations to reduce the lattice energy

10 of 10 and does not involve the migration of large-angle –10 12 lines m typical of the annealed state, to 10 12

boundaries. This rearrangement of the dislocations is –10 13 after a few per cent deformation, and up to 10 15 –10 16 lines m in the heavily deformed state. Such an array of dislocations gives rise to a substantial strain energy stored in the lattice, so that the cold-worked condition is thermodynamically unstable relative to the undeformed one. Consequently, the deformed metal will try to return to a state of lower free energy, i.e. a more perfect state. In general, this return to a more equilibrium structure cannot occur spontaneously but only at elevated temperatures where thermally activated processes such as diffusion, cross- slip and climb take place. Like all non-equilibrium processes the rate of approach to equilibrium will be governed by an Arrhenius equation of the form

where the activation energy Q depends on impurity content, strain, etc.

The formation of atmospheres by strain-ageing is one method whereby the metal reduces its excess lattice energy but this process is unique in that it

Figure 7.49 (a) Rate of release of stored energy ⊲P⊳, usually leads to a further increase in the structure-

sensitive properties rather than a reduction to the value for specimens of nickel deformed in torsion and heated at 6 characteristic of the annealed condition. It is necessary,

k/min (Clareborough, Hargreaves and West, 1955) .

238 Modern Physical Metallurgy and Materials Engineering assisted by thermal activation. Mutual annihilation of

removal of vacancies from the lattice, together with the dislocations is one process.

reduced strain energy of dislocations which results, can When the two dislocations are on the same slip

account for the large change in both electrical resis- plane, it is possible that as they run together and

tivity and stored energy observed during this stage, annihilate they will have to cut through intersecting

while the change in hardness can be attributed to the dislocations on other planes, i.e. ‘forest’ dislocations.

rearrangement of dislocations and to the reduction in This recovery process will, therefore, be aided by ther-

the density of dislocations.

mal fluctuations since the activation energy for such a The process of polygonization can be demonstrated cutting process is small. When the two dislocations of

using the Laue method of X-ray diffraction. Diffrac- opposite sign are not on the same slip plane, climb or

tion from a bent single crystal of zinc takes the form cross-slip must first occur, and both processes require

of continuous radial streaks. On annealing, these aster- thermal activation.

isms (see Figure 5.10) break up into spots as shown One of the most important recovery processes which

in Figure 7.50c, where each diffraction spot originates leads to a resultant lowering of the lattice strain

from a perfect polygonized sub-grain, and the distance energy is rearrangement of the dislocations into cell

between the spots represents the angular misorienta- walls. This process in its simplest form was originally

tion across the sub-grain boundary. Direct evidence termed polygonization and is illustrated schematically

in Figure 7.50, whereby dislocations all of one sign for this process is observed in the electron microscope, align themselves into walls to form small-angle or sub-

where, in heavily deformed polycrystalline aggregates grain boundaries. During deformation a region of the

at least, recovery is associated with the formation of lattice is curved, as shown in Figure 7.50a, and the

sub-grains out of complex dislocation networks by observed curvature can be attributed to the formation

a process of dislocation annihilation and rearrange- of excess edge dislocations parallel to the axis of bend-

ment. In some deformed metals and alloys the disloca- ing. On heating, the dislocations form a sub-boundary

tions are already partially arranged in sub-boundaries by a process of annihilation and rearrangement. This is

forming diffuse cell structures by dynamical recovery shown in Figure 7.50b, from which it can be seen that

(see Figure 7.41). The conventional recovery process it is the excess dislocations of one sign which remain

is then one in which these cells sharpen and grow. after the annihilation process that align themselves into

In other metals, dislocations are more uniformly dis- walls.

tributed after deformation, with hardly any cell struc- Polygonization is a simple form of sub-boundary

ture discernible, and the recovery process then involves formation and the basic movement is climb whereby

formation, sharpening and growth of sub-boundaries. the edge dislocations change their arrangement from a

The sharpness of the cell structure formed by defor- horizontal to a vertical grouping. This process involves

mation depends on the stacking fault energy of the the migration of vacancies to or from the edge of the

metal, the deformation temperature and the extent of half-planes of the dislocations (see Section 4.3.4). The

deformation (see Figure 7.42).

Figure 7.50 (a) Random arrangement of excess parallel edge dislocations and (b) alignment into dislocation walls; (c) Laue photograph of polygonized zinc (after Cahn, 1949) .

Mechanical behaviour of materials 239

susceptible to the segregation of impurities, low con- The most significant changes in the structure-sensitive

7.8.3 Recrystallization

centrations of which can reduce the boundary mobility properties occur during the primary recrystallization

by orders of magnitude. In contrast, special bound- stage. In this stage the deformed lattice is completely

aries which are close to a CSL are much less affected replaced by a new unstrained one by means of a nucle-

by impurity segregation and hence can lead to higher ation and growth process, in which practically stress-

relative mobility.

free grains grow from nuclei formed in the deformed It is well known that the rate of recrystallization matrix. The orientation of the new grains differs con-

depends on several important factors, namely: (1) the siderably from that of the crystals they consume, so

amount of prior deformation (the greater the degree that the growth process must be regarded as incoherent,

of cold work, the lower the recrystallization tempera-

i.e. it takes place by the advance of large-angle bound- ture and the smaller the grain size), (2) the tempera- aries separating the new crystals from the strained

ture of the anneal (as the temperature is lowered the matrix.

time to attain a constant grain size increases exponen- During the growth of grains, atoms get transferred 1 tially ) and (3) the purity of the sample (e.g. zone-

from one grain to another across the boundary. Such a refined aluminium recrystallizes below room tempera- process is thermally activated as shown in Figure 7.51,

ture, whereas aluminium of commercial purity must be and by the usual reaction-rate theory the frequency of

heated several hundred degrees). The role these vari- atomic transfer one way is

ables play in recrystallization will be evident once the mechanism of recrystallization is known. This mecha-

F

nism will now be outlined.

kT Measurements, using the light microscope, of the increase in diameter of a new grain as a function of and in the reverse direction

exp s

time at any given temperature can be expressed as F Ł

shown in Figure 7.52. The diameter increases linearly exp

with time until the growing grains begin to impinge on kT

C F s

one another, after which the rate necessarily decreases. The classical interpretation of these observations is

where F is the difference in free energy per atom that nuclei form spontaneously in the matrix after a between the two grains, i.e. supplying the driving

force for migration, and F Ł

so-called nucleation time, t

0 , and these nuclei then proceed to grow steadily as shown by the linear rela-

is an activation energy.

For each net transfer the boundary moves forward a tionship. The driving force for the process is provided by the stored energy of cold work contained in the

strained grain on one side of the boundary relative to that on the other side. Such an interpretation would

where M is the mobility of the boundary, i.e. the suggest that the recrystallization process occurs in two velocity for unit driving force, and is thus

distinct stages, i.e. first nucleation and then growth.

During the linear growth period the radius of a S

E

MD exp exp

0 ⊳ , where G, the growth rate, is kT

kT

Generally, the open structure of high-angle bound- aries should lead to a high mobility. However they are

Figure 7.52 Variation of grain diameter with time at a constant temperature .

1 The velocity of linear growth of new crystals usually obeys an exponential relationship of the form

Figure 7.51 Variation in free energy during grain growth .

240 Modern Physical Metallurgy and Materials Engineering dR/dt and, assuming the nucleus is spherical, the vol-

will grow to such a size that the boundary mobility

begins to increase with increasing angle. A large angle If the number of nuclei that form in a time increment

° , has a high mobility because dt is N dt per unit volume of unrecrystallized matrix,

of the large lattice irregularities or ‘gaps’ which exist and if the nuclei do not impinge on one another, then

in the boundary transition layer. The atoms on such for unit total volume

a boundary can easily transfer their allegiance from

4 t one crystal to the other. This sub-grain is then able

3 0 ⊳ fD 3 dt

to grow at a much faster rate than the other sub-

3 0 grains which surround it and so acts as the nucleus or

of a recrystallized grain. The further it grows, the

greater will be the difference in orientation between

D (7.42)

the nucleus and the matrix it meets and consumes, until

3 it finally becomes recognizable as a new strain-free This equation is valid in the initial stages when

crystal separated from its surroundings by a large-angle

f − 1. When the nuclei impinge on one another the

boundary.

rate of recrystallization decreases and is related to the The recrystallization nucleus therefore has its origin as a sub-grain in the deformed microstructure. Whether it grows to become a strain-free grain depends on three

NG 3 t 4 (7.43)

factors: (1) the stored energy of cold work must be

3 sufficiently high to provide the required driving force, where, for short times, equation (7.43) reduces to

(2) the potential nucleus should have a size advantage equation (7.42). This Johnson–Mehl equation is

over its neighbours, and (3) it must be capable of con- expected to apply to any phase transformation where

tinued growth by existing in a region of high lattice there is random nucleation, constant N and G and small

curvature (e.g. transition band) so that the growing t 0 . In practice, nucleation is not random and the rate

nucleus can quickly achieve a high-angle boundary. In not constant so that equation (7.43) will not strictly

situ experiments in the HVEM have confirmed these apply. For the case where the nucleation rate decreases

factors. Figure 7.53a shows the as-deformed substruc- exponentially, Avrami developed the equation

ture in the transverse section of rolled copper, together n ⊳

with the orientations of some selected areas. The sub-

grains are observed to vary in width from 50 to 500 where k and n are constants, with n ³ 3 for a fast and

nm, and exist between regions 1 and 8 as a transition n ³ 4 for a slow, decrease of nucleation rate. Provided

band across which the orientation changes sharply. On there is no change in the nucleation mechanism, n

C, the sub-grain region 2 grows into the is independent of temperature but k is very sensitive

heating to 200 °

transition region (Figure 7.53b) and the orientation of the new grain well developed at 300 to temperature T; clearly from equation (7.43), k D °

C is identical to

3 / 3 and both N and G depend on T. the original sub-grain (Figure 7.53c). An alternative interpretation is that the so-called

With this knowledge of recrystallization the influ- incubation time t 0 represents a period during which

ence of several variables known to affect the recrys- small nuclei, of a size too small to be observed in the

tallization behaviour of a metal can now be under- light microscope, are growing very slowly. This lat-

stood. Prior deformation, for example, will control the ter interpretation follows from the recovery stage of

extent to which a region of the lattice is curved. The annealing. Thus, the structure of a recovered metal

larger the deformation, the more severely will the lat- consists of sub-grain regions of practically perfect

tice be curved and, consequently, the smaller will be crystal and, thus, one might expect the ‘active’ recrys-

the size of a growing sub-grain when it acquires a tallization nuclei to be formed by the growth of certain

large-angle boundary. This must mean that a shorter sub-grains at the expense of others.

time is necessary at any given temperature for the sub- The process of recrystallization may be pictured as

grain to become an ‘active’ nucleus, or conversely, follows. After deformation, polygonization of the bent

that the higher the annealing temperature, the quicker lattice regions on a fine scale occurs and this results in

will this stage be reached. In some instances, heavily the formation of several regions in the lattice where the

cold-worked metals recrystallize without any signif- strain energy is lower than in the surrounding matrix;

icant recovery owing to the formation of strain-free this is a necessary primary condition for nucleation.

cells during deformation. The importance of impurity During this initial period when the angles between

content on recrystallization temperature is also evident the sub-grains are small and less than one degree,

from the effect impurities have on obstructing disloca- the sub-grains form and grow quite rapidly. However,

tion sub-boundary and grain boundary mobility. as the sub-grains grow to such a size that the angles

The intragranular nucleation of strain-free grains, between them become of the order of a few degrees,

as discussed above, is considered as abnormal sub- the growth of any given sub-grain at the expense of the

grain growth, in which it is necessary to specify others is very slow. Eventually one of the sub-grains

that some sub-grains acquire a size advantage and

Mechanical behaviour of materials 241

1 µ Figure 7.53 Electron micrographs of copper. (a) Cold-rolled 95% at room temperature, transverse section, (b) heated to

300 ° C in the HVEM .

are able to grow at the expense of the normal sub- Segregation of solute atoms to, and precipitation grains. It has been suggested that nuclei may also

on, the grain boundary tends to inhibit intergranu-

be formed by a process involving the rotation of lar nucleation and gives an advantage to intragran- individual cells so that they coalesce with neighbouring

ular nucleation, provided the dispersion is not too cells to produce larger cells by volume diffusion and

fine. In general, the recrystallization behaviour of two- dislocation rearrangement.

phase alloys is extremely sensitive to the dispersion In some circumstances, intergranular nucleation is

of the second phase. Small, finely dispersed particles observed in which an existing grain boundary bows

retard recrystallization by reducing both the nucleation out under an initial driving force equal to the difference

rate and the grain boundary mobility, whereas large in free energy across the grain boundary. This strain-

coarsely dispersed particles enhance recrystallization induced boundary migration is irregular and is from

by increasing the nucleation rate. During deformation,

a grain with low strain (i.e. large cell size) to one zones of high dislocation density and large misorien- of larger strain and smaller cell size. For a boundary

tations are formed around non-deformable particles, to grow in this way the strain energy difference per

and on annealing, recrystallization nuclei are created unit volume across the boundary must be sufficient

within these zones by a process of polygonization by to supply the energy increase to bow out a length of

sub-boundary migration. Particle-stimulated nucleation boundary ³1 µ m.

occurs above a critical particle size which decreases

242 Modern Physical Metallurgy and Materials Engineering with increasing deformation. The finer dispersions tend

of uniform size, and it can be seen that the equilibrium to homogenize the microstructure (i.e. dislocation dis-

grain shape takes the form of a polygon of six sides tribution) thereby minimizing local lattice curvature

with 120 ° inclusive angles. All polygons with either and reducing nucleation.

more or less than this number of sides cannot be in The formation of nuclei becomes very difficult when

equilibrium. At high temperatures where the atoms are the spacing of second-phase particles is so small that

mobile, a grain with fewer sides will tend to become each developing sub-grain interacts with a particle

smaller, under the action of the grain boundary surface before it becomes a viable nucleus. The extreme case

tension forces, while one with more sides will tend to of this is SAP (sintered aluminium powder) which con-

grow.

tains very stable, close-spaced oxide particles. These Second-phase particles have a major inhibiting particles prevent the rearrangement of dislocations into

effect on boundary migration and are particularly cell walls and their movement to form high-angle

effective in the control of grain size. The pinning boundaries, and hence SAP must be heated to a tem-

process arises from surface tension forces exerted by perature very close to the melting point before it

the particle–matrix interface on the grain boundary as recrystallizes.

it migrates past the particle. Figure 7.55 shows that the drag exerted by the particle on the boundary, resolved

7.8.4 Grain growth

in the forward direction, is

When primary recrystallization is complete (i.e. when the growing crystals have consumed all the strained material) the material can lower its energy further by

where is the specific interfacial energy of the reducing its total area of grain surface. With exten-

° . Now if sive annealing it is often found that grain boundaries

boundary; F D F max

there are N particles per unit volume, the volume straighten, small grains shrink and larger ones grow.

3 N/ 3 and the number n intersecting unit The general phenomenon is known as grain growth,

area of boundary is given by and the most important factor governing the process is

the surface tension of the grain boundaries. A grain 2 (7.46) boundary has a surface tension, T (D surface-free

energy per unit area) because its atoms have a higher For a grain boundary migrating under the influence free energy than those within the grains. Consequently,

of its own surface tension the driving force is 2 /R, to reduce this energy a polycrystal will tend to min-

where R is the minimum radius of curvature and as imize the area of its grain boundaries and when this

the grains grow, R increases and the driving force occurs the configuration taken up by any set of grain

decreases until it is balanced by the particle-drag, when boundaries (see Figure 7.54) will be governed by the

growth stops. If R ¾ d the mean grain diameter, then condition that

the critical grain diameter is given by the condition

T A / sin A D T B / sin B D T C / sin C

nF ³ 2 /d crit

Most grain boundaries are of the large-angle type

or

with their energies approximately independent of ori-

(7.47) entation, so that for a random aggregate of grains

d crit ³

T A DT B DT C and the equilibrium grain boundary This Zener drag equation overestimates the driving angles are each equal to 120 ° . Figure 7.54b shows an

force for grain growth by considering an isolated idealized grain in two dimensions surrounded by others

Figure 7.54 (a) Relation between angles and surface tensions at a grain boundary triple point; (b) idealized

Figure 7.55 Diagram showing the drag exerted on a polygonal grain structure .

boundary by a particle .

Mechanical behaviour of materials 243 spherical grain. A heterogeneity in grain size is

boundary junction angles between the large grain necessary for grain growth and taking this into account

and the small ones that surround it will not satisfy gives a revised equation:

the condition of equilibrium discussed above. As a

3 2 consequence, further grain boundary movement to

d crit ³

achieve 120 ° angles will occur, and the accompanying 3f 2 Z

movement of a triple junction point will be as shown where Z is the ratio of the diameters of growing grains

in Figure 7.56b. However, when the dihedral angles to the surrounding grains. This treatment explains the

a severe successful use of small particles in refining the grain

at each junction are approximately 120 °

curvature in the grain boundary segments between size of commercial alloys.

the junctions will arise, and this leads to an increase During the above process growth is continuous and

in grain boundary area. Movement of these curved

a uniform coarsening of the polycrystalline aggregate boundary segments towards their centres of curvature usually occurs. Nevertheless, even after growth has

must then take place and this will give rise to the finished the grain size in a specimen which was

configuration shown in Figure 7.56c. Clearly, this previously severely cold-worked remains relatively

sequence of events can be repeated and continued small, because of the large number of nuclei produced

growth of the large grains will result. by the working treatment. Exaggerated grain growth

The behaviour of the dispersed phase is extremely can often be induced, however, in one of two

important in secondary recrystallization and there are ways, namely: (1) by subjecting the specimen to a

many examples in metallurgical practice where the critical strain-anneal treatment or (2) by a process

control of secondary recrystallization with dispersed of secondary recrystallization. By applying a critical

particles has been used to advantage. One example deformation (usually a few per cent strain) to the

is in the use of Fe–3% Si in the production of strip specimen the number of nuclei will be kept to

for transformer laminations. This material is required

a minimum, and if this strain is followed by a with ⊲1 1 0⊳ [0 0 1] ‘Goss’ texture because of the [0 0 1] high-temperature anneal in a thermal gradient some

easy direction of magnetization, and it is found that of these nuclei will be made more favourable for

the presence of MnS particles favours the growth of rapid growth than others. With this technique, if the

secondary grains with the appropriate Goss texture. conditions are carefully controlled, the whole of the

Another example is in the removal of the pores during specimen may be turned into one crystal, i.e. a single

the sintering of metal and ceramic powders, such as crystal. The term secondary recrystallization describes

alumina and metallic carbides. The sintering process the process whereby a specimen which has been

is essentially one of vacancy creep involving the given a primary recrystallization treatment at a low

diffusion of vacancies from the pore of radius r to temperature is taken to a higher temperature to enable

a neighbouring grain boundary, under a driving force the abnormally rapid growth of a few grains to occur.

2 s /r where s is the surface energy. In practice, The only driving force for secondary recrystallization

sintering occurs fairly rapidly up to about 95% full is the reduction of grain boundary-free energy, as

density because there is a plentiful association of in normal grain growth, and consequently, certain

boundaries and pores. When the pores become very special conditions are necessary for its occurrence.

small, however, they are no longer able to anchor One condition for this ‘abnormal’ growth is that

the grain boundaries against the grain growth forces, normal continuous growth is impeded by the presence

and hence the pores sinter very slowly, since they of inclusions, as is indicated by the exaggerated

are stranded within the grains some distance from grain growth of tungsten wire containing thoria, or

any boundary. To promote total sintering, an effective the sudden coarsening of deoxidized steel at about

dispersion is added. The dispersion is critical, however, 1000 °

C. A possible explanation for the phenomenon since it must produce sufficient drag to slow down is that in some regions the grain boundaries become

grain growth, during which a particular pore is crossed free (e.g. if the inclusions slowly dissolve or the

by several migrating boundaries, but not sufficiently boundary tears away) and as a result the grain size

large to give rise to secondary recrystallization when a in such regions becomes appreciably larger than the

given pore would be stranded far from any boundary. average (Figure 7.56a). It then follows that the grain

The relation between grain-size, temperature and strain is shown in Figure 7.57 for commercially pure aluminium. From this diagram it is clear that either a critical strain-anneal treatment or a secondary recrystallization process may be used for the preparation of perfect strain-free single crystals.