Distribution of plastic strain in alloys containing small particles

METALLOGRAPHY 8, 181-202 (1975)

181

INVITED

PAPER

Distribution of Plastic Strain in Alloys Containing Small Particles

ERHARD HORNBOGEN ANDKARL-HEINZ ZUM GAHR Institut fiir Werkstoffe, Ruhr-Universitdt Bochum

This paper discusses the physical origin and metallographic measurements of different distributions of slip steps in an Fe-Ni-Al alloy exhibiting four characteristically different microstructures.

h Introduction

A certain macroscopic amount of plastic strain can be distributed homogeneously or heterogeneously in a crystal volume. The principle geometrical possibilities are indicated in Fig. 1 for the case of slip in one system. The parameters are the height H and the distance A of the slip steps. A third parameter has to be used if slip does not occur in atomically sharp steps but in slip bands of a thickness B > b, where b is the atomic spacing or Burger's vector, and n is the number of glide dislocation in a particular slip plane: Y.H =

EA

-

Y.nb sin ZA

(1)

During investigations of crystal plasticity, observations have been reported, that heterogeneous slip is favored by low stacking fault energy [-1], radiation damage [-2] and second phase particles [-3]. There was, however, little systematic effort on this aspect of crystal plasticity, until it became evident t h a t it m a y be of importance for crack formation and propagation mechanisms in metals [-4, 5]. Slip distribution at small strains is basically determined by the number and the nature of sources, and the ability of the dislocations to leave their glide plane by cross slip. Accordingly in Table 1 several crystal-structural and micro-structural features are listed t h a t are expected to favor heterogeneous or homogeneous distribution of strain. It is evident t h a t there are completely different reasons t h a t can lead to the same effect. This applies, © American Elsevier Publishing Company, Inc., 1975

Erhard Hornbogen and Karl-Heinz Zum Gahr

182

i ..... h-

I r

i

FIG. 1. Geometric possibilities for plastic deformation in one slip system. A - spacing between slip steps; H -- height of slip steps; a = angle between slip plane and crystal surface; B -- thickness of a slip band.

TABLE 1 Factors That Have an Effect on Slip Distribution

Coarse slip is favored by 1. 2. 3. 4. 5. 6. 7.

low stacking fault energy of solid solution sheared precipitate particles short range order radiation damage and holes few slip systems operating emissary dislocations large grain size

Fine slip is favored by 1. 2. 3. 4. 5. 6.

high stacking fault energy by-passed particles dislocation forest dislocation climb many slip systems operating small grain size

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183

for example, to the role of stacking fault energy and finely dispersed particles. Lowering of the stacking fault energy favors heterogeneous slip because there is an increasing stress required to push dissociated dislocations together to allow cross slip. Particles show a corresponding effect on slip, only if they are sheared by dislocations. Then local work softening due to a reduced cross section of the obstacles is the reason for concentration of strain. This mechanism seems to be the one which is very effective and most frequently occuring in metallic materials. Its principle features, supplemented by experimental evidence obtained with fcc Fe-Ni-A1 alloys, will be discussed in the following sections. Ih Critical Particle Sizes

If particles increase their diameter d during a precipitation process, in most cases a critical diameter dc exists, above which the particles are not yet sheared, but bypassed. Dislocations can then be bowed out to semicircles, and a variety of multiplication mechanisms can start to operate, Fig. 2. They all lead to local work hardening of the plane in which one dislocation has moved, and therefore, favor an even distribution of strain.

shearing

by o passing

2

A. J. _ L O T •

T T

I

dc

particle diameter

FIG. 2. Critical shear stress as a function of particle diameter, schematic.

Erhard Hornbogen and Karl-Heinz Zum Gahr

184

The general equation for the increase in yield stress Ar from the interaction of a flexible dislocation line with obstacles of a spacing D is E6]: =

Gb ( K y / '

(2)

From this equation the critical particle size do can be estimated (special geometrical features of the particle are neglected) :

K(d < dc) < Gb~ K (dc) = Gb2

condition for shearing condition for transition shearing by-passing

K is force on the dislocation by the obstacle; G is the shear modulus of the matrix lattice. Three types of obstacles have to be distinguished if values for dc are to be derived for special cases (Fig. 3). (1) Coherent particles: the dislocation can move from the matrix lattice into the particle without or with only small changes of its Burger's vector. (2) Noncoherent particles: the interfacial structure between particle and matrix makes it necessary to form a new dislocation inside the particle in order to shear it. (3) Liquid particles or holes: the critical shear stress of the particle is zero. However, a segment of the dislocation that has moved into the hole has to be reproduced to allow it to teai" loose from its interface as the dislocation moves on.

coherent

non-coherent

-

shearing:coherent and incoherent particles FIG. 3. Interaction between dislocations and particles for shearing of coherent (ba ~ b~) and non-coherent (ba ~ b~) particles.

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185

From these considerations the following examples for estimates of critical particle size can be derived: Coherent and ordered particles with an antiphase boundary energy ~aP~ which exerts a force K = ~'aP~"d on a single dislocation: dc -

Gb~

(3)

~/APB

Coherent particles f~ the critical shear stress of which r~, is different from that of the matrix r,, K = I r, -- r~ I bd:

Gb

dc -

(4)

For non-coherent particles it must be assumed that the theoretical shear stress for plastic shear of a perfect crystal or the stress for spontaneous production of dislocations r~h~ ~ G~/4~ has to be applied to shear the particle:

Gb dc

-

4~G,b -

T thO

(5)

Gfl

where G = G~ is the shear modulus of the matrix, Ga that of the particle. Holes or liquid particles have to be treated similarly to noncoherent particles, dc can therefore not be derived by using Eq. 4 and r~ = 0, but by introducing r,h, into Eq. (5) : do = 4~b (6) Different from Eqs. (3), (4) and (5), this only is the critical size above which dislocations bow to semicircles (i.e., the Orowan stress is reached). Holes and liquid particles are however, sheared up to unlimited sizes, so t h a t no local work hardening can be expected at d > de. The same is true for the special case of coherent ordered particles, if deformed by dislocation pairs t h a t avoid destruction of order [-3]. Correspondingly estimates of the critical particle size can be derived on the basis that the curvature of a matrix dislocation at a particle leads to local stress higher than the external stress. If it is assumed, t h a t at the Orowan stress, the minimum radius of curvature at the particle is pmin = d/2, where d = particle diameter, the critical conditions for shearing can be derived from equalizing that stress with the strength of the particle. In spite of the simplified treatment which neither considers the character of the dislocation, the dependence of line energy on curvature, nor the shape of the particles, Eqs. (3) to (6) seem to be better than an order of magnitude estimate. Values calculated for special alloys are in principle agreements with microscopic observations (Table 2).

186

Erhard Hornbogen and Karl-Heinz Z u m Gahr

TABLE 2 Critical Particle Sizes For Cutting-Process

Precipitation

Shear modulus G~ [ N / m m 2]

Critical Particle diameter d~[nm]

Diamant in a-Fe TiC in a-Fe Cu in a-Fe NisAI in Ni

505000

0,5

210000 46000 --

3 14 100

NisA1 in Ni

--

~

Holes

--

oo

Phase-Interface incoherent incoherent incoherent coherent (single dislocation) coherent (dislocation-pair) --

III. Shearing of Obstacles

Most alloys in which formation and growth of particles can occur, can be obtained in three characteristic conditions with respect to their deformation behavior. (1) Solid solutions: The slip distribution is, for example, determined by the stacking fault energy. (2) d > do: Slip is spread homogeneously over the volume due to preferred work hardening of particular slip planes. (3) d < do: Slip is concentrated on a few planes because of local work softening (Fig. 4). Only case 3 will be discussed in the following. We will attempt to find a parameter by which the tendency to produce coarse slip under these conditions can be characterized. It has to be emphasized, that slip coarsening effects discussed here, occur for all particles whether coherent or noncoherent if d < de. To show the principle features, the case of coherent and ordered particles will be discussed. Using Eq. 2 and K = ~'APB'd, the increase in yield stress due to such particles is obtained as a function of volume portion f, particle diameter d, and the antiphase boundary energy ~'APB if interaction with one flexible dislocation is considered [3]. Ar (0) =

~'A~B3/2f~/8 d 1/2 Gl12b~

(7)

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187

If n dislocations have moved in a slip plane, the average particle is completely sheared off and hr = 0 if nb= d (where d is the particle diameter in direction of b). The exact function At(n) depends in a complicated manner on shape, size distribution, volume fraction and properties of the particles. The principle features of this function can, however, be shown in a simplified calculation for cube shaped, ordered, coherent particles in interaction with a straight dislocation line [71: /,r ( 0 )

-

v'~ ")'ApBd 4 bD

(8)

If cube shaped particles are sheared in [111} planes (as is the case in m a n y Ni-base superalloys) the effective diameter d decreases linearly with the number of passing dislocations (Fig. 5) : At(n)

=

g3 "Y (d-- nb)

=

g3"yd (1 b--5

-

n@)

(9)

The tendency to produce inhomogeneous slip will be more pronounced the more Ar decreases b y the passage of one dislocation. Therefore the

'Z'y = TO, Zl T

!

1 ! ! L

d n=--ff

n

Fie. 4. Local critical shear stress as a function of the number of dislocations that have passed a slip plane (d > de: local work hardening; d < d~: local work softening): n -- number of dislocations; d -- particle diameter, b = Burgersvector.

Erhard Hornbogen and Karl-Heinz Zum Gahr

188

I

I

~

.

I

.

.

.

.

s

°

I

:

.

I

•

I

FIG. 5. Shearing of a cube-shaped particle in a {111} plane: n --- number of shearing dislocations; b = Burgersvector. amount of dhr/dn can be expected to be a property that characterizes the tendency for coarsening of slip due to work softening: dAy(n)dn = ---.--=V34 ~/ D - C d f'13

(10)

C is a geometrical constant, which depends on shape and distribution of the particles. C = ¼V3/0.82 for randomly distributed cubes because f1/s = 0.82 d / D . This indicates that the tendency for coarse ~ip is more pronounced the higher the volume portion of the particles is, and the smaller the particles are. During an aging cycle this is the condition at which precipitation is replaced by particle coarsening. A maximum possible volume portion is precipitated as particles of minimum size, and the particles are dispersed at a minimum spacing Drain. The yield stress ~ of an alloy is composed of the contribution of the dispersed obstacles Ar and that of the matrix crystal ~o, which usually is small as compared to AT: T~

=

1"o -'[- AT.

(11)

Because ro may become large in certain cases, for example, in bcc transition metals, it seems reasonable to relate the work softening rate [Eq. (10)"] to

189

Distribution of S t r a i n - - A n Invited Paper

the yield stress of the matrix and use a parameter which characterizes the tendency to form high, sharp slip steps in alloys with particles (Fig. 6) : 1 -t- to~AT

= S ['N/mm2]

-- dn

O
-

-

dn

(12)

"

For bcc alloys, where AT is small, the factor S--~ 0. Therefore we expect fine slip distribution and only light softening. But for the fcc alloys AT is much greater than r0. Then S ~ I - d A r / d n [ which produces strong local softening and at the crystal's surface high slip steps (coarse slip distribution). Therefore strain distribution in alloys with small particles should depend predominently on the two parameters dc and S.

ry(O)

,dr

TO

ry (n) I

O"C

L

T

FIG. 6. Temperature dependence of the critical shear stress for bcc crystals as a function of the number of shearing dislocations (schematic): Ar ----precipitation hardening; ro -- matrix shear stress (to < solid solution shear stress); r~(n) -- local critical shear stress after shearing of n dislocations,

Erhard Hornbogen and Karl-Heinz Zum Gahr

190

IV. Materials and Experimental Methods The general phenomena discussed above, should be tested with a material t h a t can be obtained in the following microstructural conditions: (a) Homogeneous Solid Solution (b) Coherent Phase Mixture (c) Noncoherent Phase Mixture. An alloy of Fe with 36 a t % Ni and 12 a t % A1 fulfills this requirement. The exact analysis was (in w t % ) 0.004 C; 0.30 Si; 0.02 Mn; 0.03 Cr; 38.60 Ni; 5.96 A1; bal Fe. The specimens were prepared from the vacuum cast rectangular (80 × 18 × 75 mm) ingot b y following procedure: (1) Homogenization by annealing 10 min at 1300°C and water quenching. (2) The amount of cold-rolling was increased from about 36% reduction pro pass to 68% for the final pass. Surface cracks were removed b y grinding between the different passes. (3) The shaped material was homogenized again. The average grain diameter of the ~,-solid solution was 185 ~m. TABLE 3 Fe--36 at% Ni-12 at% A1 Alloy

Obstacle for dislocation motion

Interaction dislocationparticles

--

--

Sample

Heat treatment

solid solution Mk

10 min at 1300 °C water quenched

aged 75 h/640 °C

a) 10 min at 1300 °C spherical, coherent, b) 75 h at 640 °C ordered particles water quenched (~', about 13 nm)

overaged 75 h/720 °C

a) 10 min at 1300 °C spherical, coherent, cutting b) 75 h at 720 °C ordered particles and and bypassing water quenched noncoherent rhombical or rod-shaped particles

thermo-mechanical a) 10 min at 1300 °C 17% 192 h/640 °C b) 17% cold rolling c) 192 h at 640 °C water quenched

cutting

spherical, coherent, cutting ordered particles and and bypassing disc-shaped semicoherent particles

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191

The following precipitate phases can form in this supersaturated solution: fcc. Ordered ~/-phase (Ni3A1), which is coherent but metastable; incoherent phases are the bcc d - or a'-phases (Fe, Ni)3A1 and (Fe, Ni)A1. There is also a possibility that a sere;coherent phase forms as a metastable phase by nucleation at dislocations. Sheet tensile test specimens (1.0 X 10 mm 2 cross-section) were prepared after the material was given three different treatments in addition to homogenization. The details and characteristic micromechanical properties are listed in Table 3. After heat treatment the fiat surfaces were polished electrolytically for the investigation of slip morphology. Slip lines were produced under tensile load by 2% strain at a strain rate of 2.10 -4 s-1 at room temperature. The strain distribution was investigated by transmission electron microscopy in the interior and surface replicas on the surface of the material. The latter method proved to yield hatex spheres

shodowed steps .~

.

~

,, <<

. . . . . --~

+7' 7

•

h:l.to,~

.... lightly shodowild ....... shodowed

"'"°" (a)

FIG. 7. Experimental methods for determination of slip step heights: (a) schematic; (b) latex spheres with shadows and slip steps.

Erhard Hornbogen and Karl-Heinz Zum Gahr

192

t,4k

(b) FIG. 7(b).

better resolution compared to scanning electron microscopy. The resolution was between 4 and 5 nm for Formvar replicas with simultaneous oblique shadowing. The strain distribution was obtained from measurements of slip line spacings A and heights H of slip steps. For height determination a parallaxmethod or latex spheres with a standard size of 88 nm was used (Fig. 7). V. Experimental Results It could be established that the slip distribution is principally the same in mono- and polycrystals with a certain microstructure. Only in the close environment of grain boundaries there is usually a somewhat finer distribution of strain as compared to the interior. Figure 8 shows the results of statistical evaluations of the parameters that have been discussed in Eq. 1. The corresponding micrographs show in addition the typical dislocation distributions that have led to the surface morphology of the deformed material, Figs. 9, 10, 11. In the homogeneous solid solution slip disfribution is determined by the number of sources and by the stacking fault energy. For a given grain size stacking fault energy is the only parameter left. The dislocation arrangement is characteristic for fcc metals with a SFE of about 20 erg cm-~ = 20.10 -3 J m -~ (Figs. 9a, 10a, l l a ) .

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193

In the aged condition there is a very pronounced effect due to the cutting of the small ~/-particles (Figs. 9b, 10b, l l b ) . The observations and measurements of surface morphology are in agreement with the observations by transmission electron microscopy. Dislocations are concentrated on very few planes of the primary slip planes of the particular crystallites and piled up at grain boundaries in the interior of the material. In the ovcraged condition (Figs. 9c, 10c, llc) the microstructure contains the incoherent phase ( d or d ' ) in addition to "~'. Because the particles of this phase cannot be sheared, slip becomes much finer. Transmission

slip line distance Fe+3~iS,'~Ni+12%Ai experimental method: replica deformation # 2 %

60 50 ,C ~0

~3o

slip distonce in/~m

.c,~o

75hl6,(0°C

~° ~

70

~6 2,0 2,4 2,~ J2

,Yf 4,0 4.4 48 ~2 ~ slip distance in /am

-'~ 50 I

75hl720°C

t

slip distance in /zm

~2~t ~

17%'92h16400C

0,4 0.~ 12 ~6 2,0 2.4 ,~8 3,2 .~6 ~0 ~/; 4,8 5,2 slip distance in /am

(a) Fro. 8.

S t a t i s t i c a l e v a l u a t i o n of m e a s u r e m e n t s of t h e slip d i s t r i b u t i o n .

Erhard Hornbogen and Karl-Heinz Zum Gahr

194

slip s t e p height Fe+36%Ni+I2%AI experimental method: latex spheres,poratlcu(

deformation z 2 % 5o

t ~ g ~0~

-:,oi "; sot

j2° I 70, 0

5

is

2s

~5

Ls

. ~ ,5

.~,: ~5

~ ~b5

1is "s t e p height in nm

7Shl6~0°¢ I0 ~$

25

35 45 55 65 75

95 tO$ fl$ step height in nm

85

:~,JO t

75hl720°C lo

V-7 step height in nm

!,o

17% 192h~>~0°C

To

$

~ 15

25

33

45

33

65

75

aS

step height in nm

(b) Fio. 8(b). electron microscopy indicates an even distribution of dislocations. The step morphology is finer than that of the homogeneous solid solution. In the alloy which was treated thermomechanically, additional to the particles, dislocations were introduced by cold deformation. These dislocations affect the subsequent precipitation process by serving as nucleation sites. The phase which nucleated there, is a tetragonal, ordered phase which can form at dislocations of the fcc lattice. For our purpose it is only important that relatively large particles form which are not fully coherent with the matrix. Therefore two types of additional obstacles are introduced that have to be bypassed by dislocations. The pinned dislocations also can

Distribution of Strain--An Invited Paper

(a)

~

195

(b)

, 40,"" ,1~/,192/6#0

~ ~ ,~"",75h/720"C

(c)

(d)

FIG. 9. Slip lines at the crystal surface (light microscopy, see Table 3): (a) solid solution; (b) aged; (c) overaged; (d) thermo-mecbanical.

196

Erhard Hornbogen and Karl-Heinz Zum Gahr

act as sources so that a further refinement of strain should be expected. This is shown for the different microscopic methods in Figs. 9d, 10d, 11d. Therefore, all experimental results are in basic agreement with the considerations made in the first chapters. The method used is rather accurate if one has to deal with atomically sharp slip steps. If slip bands

(a)

(b) FIG. 10. Surface replicas (see Table 3): (a) solid solution; (b) aged; (c) overaged; (d) thermo-mechanical.

Distribution of Strain--An Invited Paper

197

(c)

(d) FIG. IO(c) and (d). have to be analyzed the thickness of the band has to be measured as the third parameter. Further difficulties occur if metals have to be investigated in which wavy slip occurs as it is common in bcc metals at room temperature and above. For the analysis of fcc metals, the method discussed here appears to be sufficient to characterize all important parameters of slip morphology.

198

Erhard Hornbogen and Karl-Heinz Zum Gahr

VI. Discussion

There is evidence that the phenomena reported above occur in all metallic materials. The experimental results of this report were obtained with an fcc alloy of iron. Investigations with Ni- ['8] and Al-base [-9]

Ca)

(b) Fro. 11. Transmission micrographs (see Table 3): (a) solid solution; (b) aged; (c) overaged; (d) thermo-mechanical.

Distribution of Strain--An Invited Paper

199

(c)

(d) FIG. ll(c) and (d).

precipitation, hardening alloys indicate a corresponding behavior. The same is true for several microscopic analyses-of dislocation-particle interactions in hcp and in bcc Ti-alloys [10, 111. In a-Fe substitutional alloys coarsening of slip was observed in aged Fe-Cu alloys [-121. Here the additional effect of straightening of the originally vary slip lines to {110} slip occurs

200

Erhard Hornbogen and Karl-Heinz Zum Gahr

[12-]. This can be explained by the fact that the stacking fault energy of the a-Fe matrix is extremely high and cross slip correspondingly easy. Therefore the work softening effect must be stronger in a-iron to produce the equivalent distribution of strain as compared to the fcc alloys which usually possess a matrix with a lower stacking fault energy. Recently developed a - F e Si-Ti alloys which contain coherent d-particles (Fe3Al-structure) in large volume portions can produce a slip distribution that is characterized by slip steps as high as they can be produced in fcc alloys under corresponding conditions [13, 14-]. In addition to the differences in stacking fault energy, the strong temperature dependence of the yield stress of the matrix lattice ro must be considered in all bcc transition metals. The relative amount of local work softening is smaller, if r0 is large as compared to ~r [-Eq. (12)2. If the slip behavior of an alloy is considered during an isothermal aging sequence with formation and growth of particles, three distinct conditions are found (Fig. 12): the condition for maximum coarseness of slip at a particle diameter dc' [Eq. (10)2, the critical particle size for transition shearing --* by-passing dc [Eqs. (3-5)-], and eventually a transition of a

I I ~T

d~fl" - i 'dn < O ~

J-~d-~jmax

d,dT ^ d---n- - > u

I

I

c°herent ~ : i b i i ~ : e $

d¢ ~ dc

•-= ~, ===

J

i

I

dc'

dc

dc"

porticle diometer

FZG. 12. Different critical particle diameters that can occur during particle growth by isothermal aging (schematic): de': condition for highest concentration of slip; de: transition from shearing to by-passing; de": transition from coherency to non-coherency.

Distribution of Strain--An Invited Paper

201

coherent metastable to a noncohcrent more stable particle at a particles size d/'. If d/' ~ d/, the transition from the coarsest to finer slip will be associated with the formation to the noncoherent particle; if d/ ~ dc ~ d/', the condition of coarsest slip will be found at a smaller particle size than that required for the transition to bypassing. The most critical condition is therefore not necessarily that of this transition. It follows from Eqs. (3), (4) and (5) that d~ for noncoherent particles usually can be expected to be much smaller than that for coherent ones. This, however, does not mean that noncoherent particles cannot be sheared and produce coarse slip. The condition under which this is expected is when particles of a size d ~ d~ precipitate in such a high volume portion that considerable precipitation hardening takes place. This will occur, for example, in micro-alloy steels, hardened by small particles of TiC (compare Table 2). In the alloy reported here coarse slip was always produced by coherent particles. Incoherent d-particles led to evenly distributed strains, because they always have to be bypassed at a much smaller size than the ~/-particles which coexisted with them in the microstructure.

VII. Summary The effect of three dimensional obstacles (particles, holes) on the distribution of plastic strain in crystals is discussed. Two parameters determine the morphology of slip: (1) the critical obstacle size dc at which shearing changes to bypassing and consequently local work softening to work hardening; (2) for d ~ dc the parameter dAr/dn characterizes the work softening prodislocation and therefore the tendency for concentration of strain. The metallographic methods for measuring the strain distribution were demonstrated with an Fe-Ni-A1 alloy with four characteristically different microstructures. Our thanks are due to the Research Foundation of VDEh (Forschungsfond des Vereins Deutscher Eisenhi~ttenleute) for support of this work, and to Edelstahlwerke Witten who provided the alloys.

References 1. 2. 3. 4. 5.

S. Mader, Z. Phy. 143, 73 (1957). J. Diehl, Mod. Probl. MetaUphys. l, Springer Berlin 73 (1965). H. Gleiter, E. Hornbogen, Phys. Status Solidi. 12, 235, 251 (1965). E. Hornbogen, Z. Metallic. 58, 31 (1967). G. Liitjering and S. Weissmann, Met. Trans. 1, 1641 (1970), 2, 2599 (1971).

202

Erhard Hornbogen and Karl-Heinz Z u m Gahr

J. Friedel, Dislocations. Pergamon, London 371 (1964). A. Kelly and R. B. Nicholson, Progr. Mater. Sci. 10, 331 (1963). H. P. Klein, Z. Metallk. 61, 564 (1970). N. Ryum, Z. Metallk. 58, 26 (1967). G. Lfitjering, Mater. Sci. Eng. to be published. A. Gysler, G. Lfitjering and V. Gerold, Acta. Met. 22, to be published (1974). U. Bruch, K. H. Zum Gahr and E. Hornbogen, Germ. Met. Soc. Annual Meeting, Bonn (1974). 13. H. Michel and M. Gantois, M~m. Sci. Rev. M~t. 70, 669 (1973). 14. T. Yasunaka, T. Araki, Trans. Nat. Res. Inst. Metals. (Japan) 15, 217 (1973).

6. 7. 8. 9. 10. 11. 12.

Received 4 July 1974