strength of alloys with heterogeneous microstructures

S T R E N G T H OF ALLOYS WITH H E T E R O G E N E O U S M I C R O S T R U C T U R E S

E. Hornbogen

lnstitttt fiir Werkstoffe, Ruhr University, 4630 Boehum, FRG

The strengthening mechanisms of alloys composed of more than one phase are discussed in a systematic manner. A definition is proposed for certain types of microstructures. Plastic strain of fine dispersion structures (precipitation-hardened alloys) can vary between homogeneous and extremely localized in a few slip bands. There are consequences of localization on yield and fracture stress, static and cyclic work-hardening ability, fatigue crack initiation, and propagation.

A distinction is required between fine and coarse two-phase structures. While the strength of the former can be adequately treated with dislocation theory, the latter require a micromechanical approach similar to that of composite materials. As examples will serve some aspects of strength of alloJ~s with dual phase and duplex structures. Finally, it is shown that the microstructure leading to maximum yield stress need not be identical with that of optimum wear resistance.

1. INTRODUCTION

Most engineering materials are heterogeneous, i.e. they are not only polycrystalline but also built of more than one phase (fig. 1). Precipitation-hardened alloys, zirconia modified AlzO3-ceramics or polystyrene with elastomer additions are examples for different heterogeneous materials. They have in common that heterogeneity of the microstructure is hoped to result in an improvement in some mechanical properties.

The term "strength" is hard to define. It implies the resistance of a material against plastic deformation as well as against the propagation of cracks. Not only the response to a uniaxial tensile load may be contained in this term but also, for example, "fatigue strength" and "wear resistance". However, the yield stress o-y is the most important aspect of strength.

By "strengthening mechanisms" one understands [ i ] a microstructural feature that provides an increase in yield stress A a [2]. Out of the different strengthening mechanisms, precipitation- or dispersion-hardening [3, 4] most effectively raises the resistance of an alloy against non-thermally activated plastic deformation as well as against creep [5]. Progress in improving the yield strength of engineering alloys was parallel to and complemented by a physical understanding of precipitation hardening. I t is based on the interactions of dislocations, which must move in a matrix 0~ for distances much larger than the spacing of particles fl which act as obstacles.

The success of dislocation theory to explain the yield strength of single crystals, polyerystals, solid solutions, and fine dispersion structures has overshadowed other aspects of crystal plasticity for almost 50 years: coarse two-phase structures [6],

Czech. J, Phys. B 35 [1985] 193

E. Hornbogen: Strength of alloys with heterogeneous microstruetures

large amounts of plastic strain [7], and plasticity connected with fatigue [8] and friction [9] have attracted considerably less attention of metal physicists.

The purpose of this paper is to put the approach of dislocation theory to strength into context with microstructural features which require modelling at a level beyond that of dislocations, but below that which is taken care of by continuum mechanics. The definition of "fine" and "coarse" heterogeneous microstruotures is of some importance in this connection (table 1).

2. HETEROGENEOUS MICROSTRUCTURES

They are built from more than one phase. This implies the occurrence of interfaces between c~ and ft. Microstructure will be treated here briefly, using the form of a glossary. Some of the features which are explained will be used. for relations between heterogeneous microstructure and strength: a) microstructural-element, b) - component, c) - type, d) - anisotropy, e) fine and coarse microstructures [10].

a) M i c r o s t r u c t u r a l e l e m e n t s are all zero- to three-dimensional structural discontinuities beyond the phase level. Grain boundaries, itlteffaces, stacking faults are two-dimensional elements. Their density 0i is defined as ~AJVwhich is inversely proportional to the spacing D~, for example the grail~ size D~:

E A " ~ 1 [ m2] (la) G = - V D~, m-; '

rd2 (lb) Gr -

, V dp D~

Qi are the densities, Ai the area of grain boundaries ee and interfaces eft, D~ the grain diameter, D~ the spacing d~ diameter of fl particles, c ~ c* ~ 1 are constants which depend on size, shape, and. distribution functions of/3 particles.

b) Heterogeneous microstructures are made up of different p h a s e s e, /3 with crystal or glass structures f~, f~. ]n the case of coarse microstructures a certain volume fractionfi may even consist of an ultra-fine phase mixture (such as tempered martensite). Such zones will be designated as m i c r o s t r u c t u r a l c o m p o n e n t s , and treated as phases in relations between their volume fraction fl and mechanical properties

(2) f~ + f~ + . . . . E f , = 1.

c) A given volume fraction of phases (or microstructural component) can be arranged in several principal ways. The densities of grain boundaries G~, 0pa, and, phase boundaries Ga are suitable for the definition of these t y p e s of two phase microstructures [11] (fig. 1):

194 Czech. J, Phys. B 35 [1985]

E. Hornbo#en: Strength oJ alloys with heterogeneous microstructures

(3a) d i s p e r s i o n o f fl: Opa = 0 , 0 < fa < 1

(3b) cel l o f fl: ~ = 0 , 0 < fB < 1

(30) d u p l e x o f e and fl: 0,, =0r189 f , = f p ~ 0 . 5 .

They are based on grain structures of e. D u a l - p h a s e , net , o r cel l s t r u c t u r e s are obtained from a grain structure if fi covers the corners, edges, or the surface of the e-grain, respectively. Dual-phase is a special type of coarse dispersion structure.

type of microstructure : ,4 = ~PP/e~zc~

d) All types can be produced as a n i s o t r o p i c microstructures. For their description a system of principal directions has to be introduced in analogy with crystal structures. The microstructural tensor may be isotropic or consists of two or three different axes, either rectangular, or inclined at arbitrary angles [12, 13] (fig. 2).

e) The distinction of a " c o a r s e " a n d " f i n e " m i c r o s t r u c t u r e is useful in connection with the relations to macroscopic properties (table 1). In the subsequent sections, especially in section 5, this question will be resumed.

3. LOCALIZATION OF PLASTIC STRAIN

Plastic strain in metal crystals is not usually homogeneously distributed, as it is assumed by continuum mechanics [7]. Heterogeneous alloys provide additional causes for localization of strain (fig. 4). As an example the behaviour of dispersion structures will be discussed. They are obtained during an aging sequence of precipitation hardened alloys: solid solution -+ dispersion of particles fl with dp < dec --+

Czech. J. Phys. B 35 [1985] 195

E. Hornbogen: Strength of alloys with heterogeneous microstructures

--+ dispersion with da > dec--+ coarse dispersion (for example dual-phase), dpc is the critical particle size at which a transition occurs from shearing to by-passing (Orowan mechanism) by an interacting dislocation line. The calculation of dpc [14] is based on Friedel's equation [15] (see also section 4, footnote 1, eq. (8)):

F is the force exerted by a particle on a moving dislocation (for a cube shaped ordered coherent particle F ~ deTaeu). The requirement for shearing is:

(4b) dp?Ai , a = F < Gb 2

from which critical particle sizes can be obtained (see eq. (8b)). The consequence of shearing of particles at da < dac is a high tendency for localization of strain on one slip alone. This is so, because the critical shear stress Azp (eq. (4a)) is lowered locally if the cross section of particles is reduced by the passage of n dislocation (with atomic spacing b, fig. 3):

(4c) A~p(n) ~ ?APBd# (1- nb) b D a -~ "

non-coherent b~ * b#

coheren t b~ = b#

shearing : coherent and incoherent particles

&T / d Ar <O-~ - I dA'r >0 I T I T i non-coherent

dAt" 1 c o h e r ~ ~ l--d~'n Imo~i ~ possibi[iti :

I I I de' dc d~" " d

perticte diameter

a) b)

Fig. 3. a) Dislocation particle interactions, b) Particle size dependence of local hardening or softening dAr#/dn.

Equation (4c) is applied to coherent dispersions of ordered particle fl in a disordered matrix ct. ?aPs is the specific energy of the anti-phase domain boundary [3, 4]. The quantity I-da /d. I which can be derived from eq. (4c) characterizes the tendency towards strain localization.

At larger sizes dp > dpr particles have to be by-passed. Consequently strain becomes more homogeneously distributed. We conclude that there are three states for localization of strain during the first stages of aging: (1) the solid solution, (2) da < < dpo high degree of localization, (3) homogenization for da > dac (table 1, fig. 4). (Some values of critical particle sizes are given in table 2.)

196 Czech. J. Phys. B 35 [is

E. Hornbogen: Strength of alloys with heterogeneous mierostruetures

Table 1

Attempt for a definition of "fineness" of dispersion structures (ripe critical particle diameter for transition: shearing--~by-passing). Orowan hardening (eq. (8a)) is negligible in coarse

structures

Fine

hypocritical hypercritical Coarse

d/~ d p < 4 c 4 > 4 c 4 >> 4e particle size d# related to critical particle size ripe

d~/b 2 X 10~ 101--103 >103 ratio of particle size d~ to atomic spacing b

D~/d~ >~ 10 2 > 10 2 < 10 2 ratio of grain diameter D~ to particle diameter d#

Table 2 Critical particle sizes of iron alloys

Phase d#c/nm Coherency

C d i a m o n d 1 "5 - -

TiC, Vd, NbC 3"5 --

Fe3C 8.0 -- Cu 13.0 --

Fe3A1 > 100 4-

60

40

20

0

"t g2o -

~ o , L -

Mk

I I I t i I I

75h 1640~

FT- FFq 4O

Fig. 4. Measurements of transcrystalline localization of strain in Fe 4- 36 at. % 20 Ni + 12 at. % AI by means of replica at deformation ~ 2%, (Mk ~ homogeneous 0

solid solution). 0

75h 17200C

I I I I I i i J

1.2 2.4 3.6 4.8 stip d is tance / p r o

I f the dispersion o f fl becomes coarse, i.e. the particle spacing Dp is about equal to the grain diameter D,, a different situation is found. The particles do not contribute to the yield stress (Avp ~ 0). Plastic deformation takes place in the (as-decomposed)

e-solid solution, only indirectly affected by fl-particles that provide dislocation sources and stress concentrat ions at their interfaces.

Czech. J. Phys. B 35 [1985] 197


Plastic strain of a coarse dispersion can be treated approximately by using a se- quential arrangement of mieromechanical elements, which implies: a~ = cry, ~ ~= gp (fig. 5). The total strain becomes a function of the volume fractions f~, fp, and the partial strains g~, gp:

==

r I i I I i . ~o~M strain C ~'B~"

Fig. 5. Models for deformation of coarse two-phase structures (lamellate, dispersion), partial properties of ~ (soft) and /~ (hard) are indicated. For a more subtle approach of microplasticity on this level sometimes finite elements are used (Rm, Rm~ ~ tensile s t rength of bulk alloy, of

component cr

As long as ] /does not share plastic deformation, i.e. gp = O:

(5b) e = g~f~,

plastic strain is restricted to e. This component shows the strain localization caused by its individual dislocation behaviour. We conclude that two microstructural levels are required for an understanding of strain in coarse two-phase structures: (1) the dislocation level of the mierostructural components and its interfaces, (2) the micromechanieal level determined by the propemes of the components and their morphological arrangement.

4. YIELD S T R E N G T H S A N D F A T I G U E S T R E N G T H OF F INE DISPERSION S T R U C T U R E S

Fine dispersions can be defined by densities of interfaces (eq. (lb)) or particle spacings Da. Moving dislocations require a large number of interactions ( > 10 3) to produce measurable strain or a considerable increase in yield stress Aza (eq. (4a)). This range of a "fine" structure is (eq. (1)):

(6) 0 " 2 n m g b < Dp < D ~ l p m .

These obstacles enforce a curvature of the dislocation to obtain the increase in critical resolved shear stress (eq. (4a)). The total critical resolved shear stress zr is

198 Czech. J. Phys. B 85 [1985]

E. Hornbogen: Strettgth of alloys with heterogeneous microstructures

additive to the friction stress in ~, z=. It contains the effects of Peierls stress, order (i.e. formation and shearing of APB'@ and solid solution, hardening (i.e. with negligible effect on curvature of dislocation lir~es):

(7a) "ry = z,, + Az'r

(7b) 3zy ,,~ ay.

Minimum radius of curvature, i.e. maximum additional strengthening Avpo is acquired if the particle has to be by-passed, at dpc

G,b _ G~bfA/2 ArBo-

Dp d~ (8)

Then, for dp > dB~

(8a) Gctb f A 12 A'c~ < - - - - < Az~c.

dp

For example, F c = dpeVAPB = Gb 2 for coherent, ordered partiolesl). The critical diameter can be calculated:

Gb 2 (8b) ~o -

~)APB

Smaller particles dg < d~ sustain less curvature and produce less hardening at a given spacing Dp (meaning of c as in eq. (lb)):

~,u 3/2 A

G~/2b2Dg At G/IO or D~ ~ 10b the theoretical upper limit of the yield stress is attained. Usually, particles are sheared far above such small spacings (eqs. (4) and (8)). Only particles of some hard. interstitial compounds or diamond can reach critical diameters, at which the theoretical limit of the yield stress could be approached (fig. 6): dac < < 2nm.

Shearing of particles leads to negative work hardening in a slip plane, while by- passing introduces additional obstacles and. therefore work hardening (eq. (4c)). Besides, to induce localized shear, negative work hardening can become effective close to the surface of the alloy only. Ir~ the interior strong pile-ups of dislocations form, which in turn increase bulk work hardening.

Localization of plastic slip by ultra-fine particles highly affects fatigue behaviour of the alloys. It is, however, difficult to find simple relations between strengthening

1) Besides formation of anti-phase domain boundaries inside the particles a large number of other mechanisms are known, which all lead to dissipation of energy and therefore to an interaction force between dislocation and particle: stress fields, interfaces between particle and matrix, surfaces of pores, differences in elastic constants, stress induced transformation of particles.

Czech. J. Phys. B 35 [1985] 199


under static load Azp, and. fatigue life. Cyclic loading can cause the following pheno- mena:

(1) cyclic work hardening or softening, (2) crack initiation, (3) stable crack growth, (4) critical rupture.

Heterogeneity plays an important role in all 4 stages [17]. In pure metals and sol/d solutions, cyclic work hardening is followed by the formation of soft persistent bands (PSB's) inside a structure which primarily was evenly filled with dislocations [8]. Ultra-fine particles wilt induce soft zones immediately (eq. (4c)). This, in turn, aids crack initiation. The mechanism implies very limited cross slip, and the formation of extrusions and intrusions. There is a direct correlation with the tendency for localization I "dArt~/dnl and the number of cycles for crack initiation (fig. 7, eq. (4c)).

010 [ t )~ - theoreticQ( timit Tth �9 \ / . . . . . . . . .

t /xX..-

O r I

/ / / / / / / / / / ,/

/

1 2 3

/a , . 9 / / / /

( / . ~

1 2 3

; ~ ~ / n - n R ) * | N-N R ) / 2*n-nR / t j - \ . , /

3 (n-nR)* N

dpc d~

Fi.g 6. Fig. 7.

/ / /

/ /

4 7 '

/ . / /

, / / /

4

Fig. 6. Precipitation hardening Arp due to eq. (4) and the upper theoretical limit, rtn, schematic. Fig. 7. Transcrystalline localization (see figs. 3b, 4) and initiation of fatigue cracks by intrusions

(dislocations moving under tension n, under compression N, moving reversibly nR, NR).

The fracture mechanical approach to fatigue crack growth predicts an inverse relation between the yield stress O-y (eq. (7b)) and the crack growth A a pro cycle N, at a given amplitude of stress intensity A K (Young's modulus E):

(9a) da _ A . . . . . . A K 2 = Ad~ = A n b .

d N frye

We must remember that~ the model assumes large amounts of irreversible, micro- scopically homogeneous plastic displacement 5 at the crack tip and. negligible work hardening [18]. Localization of strain by ultra-fine particles will, in turn, provide

200 Czech. J. Phys. B 35 [1o85]

E. Hornbogen: Strength of alloys with heterogeneous microstruetures

for reversible motion of n R dislocations and therefore for a fraction of reversible strain, nRb. If nb dislocations produce the crack opening displacement 6, crack growth will be retarded if n R dislocations move reversibly inside small plastic zones at the crack tip:

da (9b) - - = A(n - nR)b = A(6 - fiR).

dN

Consequently, if particles are sheared, cracks move much more slowly than expected from the yield stress of the material.

It may be noted that particle-induced localization of strain can have different effects on initiation and propagation of cracks. A surface treatment which affects crack formation will improve fatigue life of an. underaged alloy: rapid initiation, but slow growth of cracks [19].

Finally, it has been noted recently that particle strengthening can take place in fine dispersion structures of a glass matrix with crystalline particles, Inherent localization of strain in a glass will be refined and the tensile strength raised during aging of an originally homogeneous embrittled glass [20] (fig. 8).

, ~~veiRs brittte \

b)

ceins fine

trans-, inter-

crystaltine

c) ~ d)

Fig. 8. Localization of plastic strain and fracture of aged metallic glasses: a) localized slip b) brittle fracture of glass, c) homogenization of slip by primary crystallization, d) brittle fracture

of crystalline microstructure.

5. STRENGTH OF COARSE TWO-PHASE MICROSTRUCTURES

It has already been discussed that plastic strain of coarse dispersion structures can be restricted to one phase only (fig. 5, eq. (5b)). Micromechanioal models derived for composite materials [21] are also useful for an understanding of strength of different types of "natural" coarse two-phase structures [6, 22]. The lamellated (a = const) and the fibre structures (e = const) will serve as the base for an approxi, mate understanding of other two-phase structures (figs. 5 and 9). The case ay e,~ fly#

(fig. 5), a = constant, resembles the dispersion of (hard) fl, while net-, or cell-structure of fl, as well as a duplex structure, is better described by e = const. The bulk yield

Czech. J. Phys. B 35 [1985] 201


stress of a coarse dispersion such as a dual-phase structure is therefore determined by that of the matrix e:

(10a) O'y ~ ~ y ~ t "

For the other types of structure, the higher yield stress of fl becomes effective proportional to its fraction in the cross-section perpendicular to the direction of loading:

(10b) O'y g 5y~f, + 5y~f~. a a

b . . . . , , , , . . . . . , , I

a)

(7

/ �9 o e . o 6 - ~ F �9 * t �9 %'/ t

L.. . . . ;+ie" .: �9 �9 e �9 �9

e � 9 �9 � 9 1 4 9

2 . " ; " �9 �9 O � 9 e �9

b}

Fig. 9. a) Modes of crack propagation in coarse two-phase microstructures (left:

= const, right: e = const), b) Structure of a particle-free zone (PFZ) and its role in the formation of a quasi-cell structure

(PFZ ~ e).

The ultimate tensile strength follows corresponding rules of mixture. In all cases, perfect adhesion of e-f l- interfaces is a prerequisite, i.e. higher strength of the interfaces as compared to those of the phases.

A frequent phenomenon in precipitation hardened alloys and tempered steels are p a r t i c l e - f r e e z o n e s (PFZ) [23]. In principle, a soft microstructural component can be defined, which forms a cell structure along the grain boundaries of e. The interior of the grain c~ contains the ultra fine dispersion and is consequently very hard (eq. (8)). Primary yielding will take place exclusively in the cell walls fl (eq. (5b)):

( l l a ) =

Even if the fracture strain gltfp is large, the bulk ductility can be very small, because of a minute volume fraction fa in which plastic deformation takes place. Additional geometrical models lead. to a grain size dependence with D~- ~ of the crack extension Gic, for this type of pseudo-intercrystalline deformation and fracture inside the cells of constant wall thickness [17] :

(rib) C,c = e,,cdp.

2 0 2 Czech. J. Phys. B 35 [1985]


In a duplex structure both phases have to be passed by crack and therefore a rule of mixtures is valid for bulk crack extension energy:

( l lc) Gxc = G,c~f~ + G~c~f~.

Similar considerations can be applied to fatigue crack growth in a dual-phase structure. Plasticity and therefore the crack path is limited to ~. Therefore, the bulk growth rate is determined by that of c~, and a geometrical factor g g 0.5 because of a non-planar crack surface [17-19]

01d) da Ida[ dN = IdNl~ g"

6. C O M M E N T S ON W E A R RESISTANCE [9]

The results of the former sections indicate that high strength is obtained by ultra fine microstructures. In coarse mierostructures tensile strength can be determined by the weakest phase alone (eq. (10a)). Wear resistance w -1 is used as an example which shows that fine structures are not always desirable.

The wear rate w is defined as the removal of matter da per sliding path dx:

d a o- (12a w - - k \ /

dx H

1 - a[ ~ a is the normal compressive stress, H the bulk hardness of a homogeneous alloy (about proportional to the yield stress in the work hardened surface), k is the wear coefficient as a measure of the probability of decohesion in the effective asperity area a/H.

A comparison is made of (1) a fine dispersion (tempered martensite) and (2) a coarse dual-phase structure (ferrite + 30~ martensite) of equal bulk hardness H1 = H2. The wear system is characterized by an abrasive with H,b >> H. The second coml~onents fl, of the two microstructures are harder than the abrasive Ha~ > Hab ,

Hp2 > H,b (fl~ >> Fe3C, fla - Fe (0.8 wt .~ C) - martensite). Fig. 10 shows that the wear resistance of the coarse microstructure surpasses by far that of the fine dispersion. While for the latter equation (12a) is valid a relation to the partial hardness Ha2 is in accord with the behaviour of the coarse structure:

(12b) w2 = k _ p . H#2

In figure 10 a model is proposed to explain the differences. For the fine dispersion abrasion takes place by plastic shear (ploughing and chip formation). The hard particles fl are "floating" away together with the deforming matrix ~. If the hard particles are of a size comparable with the groove width and (because of their size)

Czech. J. Phys. B 35 [1985] 2 0 3

E. Hornbogen: Strength of alloys with heterogeneous mierostruetures

are solidly anchored in the matrix, a shielding effect occurs and the abrasive is kept

f rom penetrating ir~to the soft matrix. Thus, coarse heterogeneous microstructures which are undesirable for high yield strength and toughness can be just suitable

to provide a high wear resistance [9, 13].

b)

2.0 • -4

1.8

T

1.6

1.4

I I / DP DP o I I

I I

I I M S / / ~-I

el/MA /i , , /

200 400 600 800 H, Holm

Fig. 10. a) Effect of a frictional shear stress on a fine (A) and coarse (B) two-phase microstructure. Prerequisite: yield stress (hardness) of A and B are equal. Micro- hardness of second phase fl is higher, bulk hardness lower than that of abrasive agent, Sg groove width larger or smaller than spacing Sp. b) Comparison of wear resistance of a steel (Fe 4- 0"1 wt. % C, 1.5 wt. % Mn, 0.1 wt. % Mo) with different microstructures. DP dual phase, MS martensite, MF marten-

site + ferrite, MA temperated martensite.

7. SUMMARY AND CONCLUSIONS

Two structural levels have to be considered for relations between heterogeneous microstructures and mechanical properties.

F o r ultra-fine and fine microstruetures the dislocation theory is adequate to deal with strength and plasticity. Precipitation induced transcrystalline localizatiol~ o f plastic strain is an impor tant phenomenon on this level.

2 0 4 Czech. J. Phys. B 35 [1985]

E. Hornbogen: Strength o f alloys with heterogeneous mierostruetures

Components of coarse micros~ructures with dimensions (>1 gm) by which no considerable dispersion hardening can occur may be arranged in different ways. The types of coarse heterogeneous microstructures are: dispersion, net, cell, duplex. Micromeohanical models, beyond the dislocations level, are required for relations to mechanical properties.

Components of coarse heterogeneous microstructures can be phases or fine microstructures. Their properties are designated as partial properties. A second type of localization of plastic strain is found in alloys in which plastic deformation takes place exclusively or preferably in one microstructural component.

The nature of the microstructure of most alloys requires a two level treatment to interpret the mechanical properties. Dislocation theory provides an understanding of the partial properties, the "rules of mixture" for the second micromechanical level.

Examples of the necessity of the combined "dislocation" and "composite" approach are precipitation hardened alloys with particle-free zones.

Received 11. 9. 1984.

References

[1] Hornbogen E.: in Reinstoffe in Wissenschaft und Technik. Akademie-Verlag, Berlin, 1972, p. 431 -- 442.

[21 Hornbogen E.: in Proc. ICSMA 5, Aachen, Vol. 2, 1979, p. 1337--1342. [3] Kelly A., Nicholson R. B.: Prog. Mater. Sci. lO (1963) 161--383. [4] Hornbogen E.: in Proc. ICSMA 3, Cambridge, Vol. 2, 1973, p. 108--136. [5] Hornbogen E.: in Proc. 3, Riso. Syrup. Deformation of Polycrystals, 1981, p. 23--34. [6] Fischmeister H., Karlson B.: Z. Metallk. 68 (1977) 311. [7] Haetherly M.: in Proc. 1CSMA 6, Melbourne, 1982, p. 1181--1197. [8] Mughrabi H.: in Proc. 4, Riso Syrup. Deformation of Multi-Phase and Particle Containing

Materials, 1983, p. 65--82. [9] Kuhlmann-Wilsdorf D.: in Fundamentals of Friction and Wear of Materials, (ed. D. A.

Rigney). ASM, Metals Park, Ohio, 1980, p. 1--13, 119--186. [10l Hornbogen E.: Acta Metall. 32 (1984) 625--627. [11] Lticke K.: Z. Metallk. 75 (1984) 948--956. [12] Harrington T. P., Mann R. W.: J. Mater. Sci. 19 (1984) 761--767. [13] Hornbogen E.: in Friction and Wear of Composite Materials, (ed. K. Friedrieh). Elsevier,

Amsterdam, 1985, to be published. [14] Hornbogen E., Zum Gahr K. H.: Metallography 8 (1975) 181--202. [15] Friedel J.: Dislocations. Pergamon Press, London, 1964, p. 371. [161 Foreman A. J. E., Makin M. J.: Philos. Mag. 14 (1966) 911. [17] Hornbogen E.: in Proc. ICSMA 6, Melbourne, 1981, p. 1059--1073. [18] Paris P. C., Erdogan F.: Trans. ASME Ser. D. -- J. Basic Eng. 85 (1963) 528. [19] Graf M., Hornbogen E.: Scr. Metall. 12 (1978) 147--150. [20] Hillenbrand G., Hornbogen E.: Metall 36 (1982) 1059--1064. [21] Hull D.: An Introduction to Composite Materials. Cambridge University Press, t98t. [22] Friedrich K., Hornbogen E.: J. Mater. Sci. 15 (1980) 2175--2182. [23] Hornbogen E., Kreye H.: J. Mater. Sci. 17 (1982) 979--988.

Czech. J. Phys. B 85 [1985] 205

strength of alloys with heterogeneous microstructures

Documents