number of cleavage steps

8/9/2019 Number of Cleavage Steps

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2 Mechanisms of fracture and crackgrowth

number of cleavage steps, which may join and form a river pattern. The

convergence of r iver patterns is a lways downstream; this gives a

possibility to determine the direction of local crack propagation in a

micrograph. Examples of cleavage steps and river patterns, as observed by

electron fractography. arc presented in figure 2.10. They arc places where

small scale plastic deformation is likely to occur. Plastic deformation

requires energy, and therefore river patterns and steps are observed more

abundantly on cleavage fractures produced at temperatures close to the

transition temperature.

A cleavage tongue is another typical feature of a cleavage fracture. It is

called a tongue because of its apparent shape. Cleavage longues of various

sizes are shown in figure 2.11. They are believed [23, 27] to be formed by

local fracture along a twin-matrix interface, (Twins are formed as a result

of the high rates of deformation in front of the advancing crack). Tongues

in iron are thought to be generated when a cleavage crack, growing along a

(100) plane, intersects a (112) twin interface and propagates along the

interface for some distance, while (100) cleavage continues around (he

twin Final separation occurs when the twin fractures in an unidentified

manner.

Lvidence for this formation process can be obtained from slcreograpliie

measurements It turns out that different ways of interaction with twins are

possible [51. One of these possibilities will now be illustrated. A section

along a (110) matrix plane through a twin in a bee lattice is shown in figure

2,12. The composition plane of a twin in this lattice is the (112) plane.

This plane is perpendicular to (llO), i.e. perpendicular to the plain: of the paper, and it intersects (M0) along the [111] direction. The crack is

assumed to propagate along (001)[ 110] from A to B. where it impinges on

the twin. Then it continues along (he matrix-twin interface (112)| 111 ]

from B to C. Simultaneously, (he main crack may circumvent the (win

1

K

36

2.2 Cleava e fracture

Figure 2.11. ("lea \ agelongues (arrows)

* [no]Figure 2.12. Formation of cleavage tongue (BCD) due to

passage of twin. Cul along (110) plane through a coherent twinin bec lattice


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^Mechanisms of fracture and crack growth

a,t ^rie bones seem lo support this point of view.

1Jnder normal circumstances face-centcred-cubic (fee) crystal structures cJo not

exhibit cleavage fractures: extensive plastic deformation will always cid^ v i r in

these materials before the cleavage stress is reached. Cleavage does occur in body-

centered-cubic (bec) and many hexagonal-close-packed hc=p>) structures. It occurs

particularly in iron and low carbon steel (bec). I i i ngstcn, molybdenum, chromium

(all bec) and zinc, beryllium and magnesium (all hep) are materials capable of

cleaving.

2.3 Ductile fracture

Fracture occurring under the single application of a continuously in-creasing load

can either be brit t le c leavage fracture or fracture associated wi t-H plastic

deformation, wich is essentially ductile. For the latter the amount of plastic

deformation required to produce fracture may be so lirrited in certain cases, that

relatively l i tt le energy is consumed. Then the fructure is s t il l brit t le in an

engineering sense and can be initiated at a shiirp notch or crack at a comparatively

low nominal s tress, particularly w lien a state of plane strain reduces the

possibilities for plastic deformation.

The most familiar type of ductile fracture is by overload in tension, w l- ich produces

the classic cup and cone fracture. After the maximum load hiis been reached the

plastic elongation of a prismatic-tensile coupon becomes inhomogeneousand

concentrates in a small portion of the specimen such that necking occurs. In extremely

pure metals, which are virtually fre e of second phase particles, it is possible for plastic

deformation on c "jugate slip planes to continue until the specimen has necked down

to an arrow point by 100 per cent reduction of area (figure 2.14). Such a fiiilure is a

geometric consequence of the slip deformations. As an example, figure 2.15 shows a

single crystal that has almost failed by shear on a single slip plane.

Engineering materials a lways contain large amounts of second phase

particles. Three types of particles can be distinguished:

a. Large particles, visible under the optical microscope. Their size may vary

from 120 um. They usually consist of complicated compounds of trie various

alloying elements. The alloying elements may be added to mprove castability or

some other property. The particles are not essential to the strength of the

material and generally serve no purpose at all

2

7


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Somelimcs. however, particles of this size may be produced on purpose, as

in the case of carbides in certain steels.

b. Intermediate particles, only visible by means of the electron micro-

scope. Their size is in the order of 500-5000 Angstrom units. These

particles may also consist of complex compounds of the various alloying

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2.3 Ductile /nature

figure 2.14. Failure by pure sliciir deformation

(slip) in pure mclal

Figure 2.16. Cracking of large particles in AI-Cu-Mg-alloy. A. 3per cent strain: B. (> pel I oil slrain; C. 14 per cent strain: I). 25

per cent strain. Note development of crack between NOP.Straining direction vertical

Figure 2 I 5. Sheared single crystals of pure

copper (courtesy Weiner)


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The large particles are often very brittle and therefore they cannot

accommodate the plastic deformation of the surrounding matrix. As a result

they fail early on. when the matrix has undergone only a small amount of

plastic deformation. This means that voids are formed. The initiation of

voids at large particles can be observed with an optical microscope [16.

29], Various stages in this process are shown in f igure 2 .16. When

comparing the distance between two particular inclusions in the various

stages, one obtains an impression of the increasing strain.

The figure indicates that voids are already initiated by large particles at

small strains in the order of a few per cent, whereas final fracture look

place al s trains in the order of 25 per cent. This being so. the large

inclusions visible in the optical microscope cannot be essential to the

fracture process, although the) will decrease the ductility of the material.When fractured they cause a stress concentration giving a local increase of

the strain, which could ill the absence of the particles be obtained as an

overall strain throughout the material. This implies that the large inclusions

can determine the instant and location of ductile fracture, but they do not

play a role m the process of ductile fracture itself

fracture is finally induced by the much smaller intermediate particles of

the sub-micron size [4. 16. 29]. Since these particles cannot deform as

easily as the matrix, they lose coherence with the matrix when extensive

plastic How takes place in their vicinity. In this way tiny voids arc formed,

which grow by slip: the material between the voids necks down to the lull

100 per cent in the same way as in figure 2.14. This necking takes place al

a micro-scale and the resulting tolal elongation remains small.

The mechanism of initiation, growth and coalescence of micro-voids

gives rise to characteristic fraclographic features. When observed in the

electron microscope the fracture surface consists of small dimples which

represent the coalesced voids (fig. 2.17). In most of the dimples the smallparticle that initiated the void can easily be recognized.

Dimples always have an irregular shape, due to the random occurrence of

voids. However, dimples can be roughly divided into two categories

according to their apparent shape, namely equiaxed and parabolic. The

shape m which they appear in the microscope depends upon the stress

systems that were active during their formation [9. 10], and upon the angle

of observation in the microscope [4, 30]. Equiaxed dimples may be formed

if the stresses arc predominantly tensile (figure 2.18a). and elongated

dimples occur in the shear or tear mode (figure 2.18b, c). Both dimj types

arc present in figure 2.19.

Putlick [31], Roberts [32] and Crussard et al. [33] were among the ft to

consider inclusions or intermelallic particles as the initiation sites the

voids. Various models have been established for void growth a coalescence,

but so far none is fully compatible with the observations tl can be made

from ductile fracture surfaces, although some models m apply in certain

cases [4, 16, 29, 34].

An impression of the process of void growth and coalescence can

obtained from the study of dimple profiles. Cross sections of replicas g a

good idea of the topography of the fracture surfaces [4, 16, 29]. This

il lustrated in f igure 2 .20 which gives an example obtained al hi

magnification. The cut will not always be through the center of a dimj but

the apparent average depth-to-width ratio in the cross sections v be

approximately the same as the actual depth-to-width ratio of the dimp

441


Figure 2.17. Dimples initialed at intermediate size

particles (arrows). Al-alloy


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2 Mechanisms oj fracture and crack growth!I M1111........................................................................................ F11! i n i ' 11 ip 11'i F11111 ir 11 Mr 11 r 11' 1! 11 n fpi r 11 p > i ' i r' i r i i r I r' r l r r r ' r r i r iT ' i r i r i r ' i r r i r 11 r i r 11 n I r ir: i nr r l r! I r 11 Mr 11 pi r Mr i; Mi [; r 11 [ 111 i r 11 r 11 M111 r i' 11 [ 11........IIIMIIIIIIIIIIIMIMIIIMIIIIIIMIIIIIIII...............................................................

2.J Ductile fracturespecimen and stresscondition

t t t t

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fracture surfaces

eguiaxed dimples

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shear dimples

l same direction

tear dimples

1 I *

igurc 2. IK. Occurrence of different types of dimples

[16]. Thus it can be asserted from figure 2.20 that the depth-to-width ratio

is low and that dimples are relatively shallow holes. The latter is confirmed

by stereoscopic measurements of dimple topography. Apparently, voids

grow mainly in lateral directions, causing them to remain shallow. In oxide-

dispersion strengthened materials the initiation and growth of voids at the

dispersed particles can be made visible when a thin foil is subjected to a

tensile deformation while under observation in the electron microscope [35.

36, 37]. In normal structural materials voids at the intermediate particles

can seldom be observed. In a study of 13 different aluminium alloys Broek

5 44

figure 2.19. Equia\cd dimples at A. parabolic dimples al 1! 1urge cleaned panicle (ol l\po sliov.il in figure 2.16) al I). 2024-

T.1 Al-alloy
http://sliov.il/http://sliov.il/


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2 Mechanisms oj fracture and crack growth

[4, 16. 29] showed the occurrence of voids at the intermediate particles:

slender particles have a tendency to fracture (figure 2.21) while particles of

other shapes lose coherence with the matrix (figure 2.22). However, in

general only very few voids were found.

Apparently, the cohesive forces between the matrix and particles are

extremely large in conventional materials. They are so large that void

6 44


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I-mure 2.21). Cross section throughreplica of dimples

2 Mechanisms of fracture and crack growth

|

|

7

5


Figure 2.21. Crackedparticles in AI-allo\


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Initiatio

n does

not take

place

until

very late

in the

fracture

process.

The luck

of many observable voids leads to the assumption that immediate

coalescence must take place at the moment that voids are formed in some

quantity. This implies that voids can only be initiated at such high stresses HI

strains that the conditions for coalescence are already fulfilled. This requires

a model for void initiation which predicts immediate and spontaneous

void.growth. (Where void growth in oxide-dispersion strength-i tud

materials was reported, it occurred in the direction of the tensile stress.

This mode ofvoid growth is a normal consequence of longitudinal Straining,

which would also occur under elastic conditions). From a physical point of

view, one would expect a cavity to spread preferentially in a direction

perpendicular to the tensile stress, as in the ease of a crack This lateral void

growth is confirmed by the shallowness of the dimples A model that is

reasonably compatible with the observed facts is the following[16, 29].

During plastic deformation dislocation pile-ups will form at the particles.

These piled-up loops are depicted in figure 2.23a, TllC loops arc repelled by

the particle through the action of their image forces On the other hand, the

leading loop will be pushed towards the parlicl< by stresses set up by the

pile-up and the applied shear stress. When one or a couple of loops are

pushed to the interface a dccohcsion ol ilu

4S

8

5



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interface will ultimately lake place. If this occurs, a void is formed. The

consequence is that the repelling forces on subsequent loops are drastically

reduced "and the greater part of the pile-up can empty itself into the newly

formed void. The dislocation sources behind the loops, which became

inactive because of the constraint of the pile-up ahead, can resume action

and hence the process may lead to unstable lateral void growth and

coalescence as soon as the voids have been initiated (figure 2.23c, d). In

figure 2.24 the model is shown in terms of displacements.

Contrary to cleavage, where the action of a tensile stress is sufficient for

the separation, ductile fracture cannot occur without plastic deformation.

The mechanism of final separation is a direct consequence of dislocation

movements and slip displacements necessary for the growth and

coalescence of voids. Apart from a stress to induce dislocation movement a

certain plastic strain is required for ductile separation to occur. This plastic

deformation may be confined to a small volume of material through which

the fracture passes. Then failure occurs with relatively little plastic-

deformation on a macroscale. requiring only little energy. The fracture is

brittle in an engineering sense. Fractures induced by cracks in high

strength materials are usually of this type.

2.4 Fatigue cracking

Under the action of cyclic loads cracks can be initiated as a result of cyclic

plastic deformation [38, 39]. Even if the nominal stresses are well below the

elastic limit, locally the stresses may be above yield due to stress

concentrations at inclusions or mechanical notches. Consequently, plastic

deformation occurs locally on a microscale, but it is insufficient to show in

engineering terms.

9

l l l i l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l i .. .

I igiire 2 2: IHxvIksion of particles 111 MK>I-\l-llo>. Tophull and lioltom half show live same localion. taken al ilillcrenl

angles of llic election beam [2')\(courtesy I'ergamim)

Figure 2.23. Dislocation model for void initiation andgrowth a. Piled up loops; b. Cross section; c. Detail;

d. Cavity (courtesy Pergamon)

2 Mechanisms of fracture anil crackgrowth


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Several equivalent m ode ls have been p roposed [38, 40, 41] to expla in

the initiation of fatigue cracks by local plastic deformation. The model of

_ Wood [38] is depic ted in f igure 2 .25 . D ur ing the r ising- load par t of the

cycle, slip occurs on a favourably oriented slip plane. In the falling-load

I part, slip takes place in the reverse direction on a parallel slip plane,

since slip on the first plane is inhibited by strain hardening and by

* ox ida tion of the newly c rea ted f ree su r face. This f ir s t cyc lic s lip can g ive

rise to an extrusion or an intrusion in the metal surface. An intrusion can

grow into a crack by continuing plastic flow during subsequent cycles

(figure 2.25). If the fatigue loading is cyclic tension-tension this

mechanism can still work since the plastic deformation occurring at

increasing load

'will give rise to residual compressive stresses during load release An

example of fatigue cracking initiated by cyclic slip [42] is presented in

figure 2.26.

49

10.4 Fati ue

crackin

Figure 2.24. Voidcoalescence by slip

Figure 2.25. Wood's model forfatigue crack initiation


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A fatigue crack, once started, can also grow by a mechanism of reversed

49

11.4 Fati ue

crackin


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2 Mechanisms of fracture and crack growthI........IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM

52 51

2.4 Fati ue crackin

Figure 2.27. Possible model for

Figure 2.28. Striations on fatigue cracksurface of Al-C'u-Mg alloy

opening

opening

closing


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2 Mechanisms of fracture and crack growthslip [43-52], Spveral stages of fatigue crack growth are shown in figure

2.27. A sharp crack in a tension field causes a large stress concentration at

its tip where slip can occur fairly easily. The material above the crack

(stages I and 2 in figure 2.27) may slip along a favourable slip plane in the

direction of maximum shear stress. Due to that slip the crack opens, but it

also extends in length. Slip can now occur on another plane (stage 3).

Work hardening and increasing stress will finally activate other parallel

slip planes, which leads to a blunt crack tip (stage 4). During the rising

loadpart of the cycle the crack has propagated by an amount Aa.

Plastic deformation has occurred in a small region embedded in elastic

surroundings. During load release the elastic surroundings will contract

and the plastically deformed region, which has become loo l arge, does noi

fit any more in its surroundings. In order to make it fit the elastic material

will exert compressive stresses on the plastic region during the decreasing

load part of the cycle. These compressive stresses will be above yield

again, at least at the crack tip. This means that reversed plastic

deformation occurs, which will close and resharpen the crack lip, as is

shown in stage 5 of figure 2.27.

The cyclic opening and closing of the crack (stages 1-5 and 6-7) will

develop a typical pattern of ripples, every new cycle adding a new ripple

52 51

2.4 Fati ue crackin


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These ripples show up on the fracture surface in the electron microscope;

they arc called fatigue slrialions. Figure 2.28 shows fatigue striations hra

commercial Al -Cu-Mg alloy. The model of striation formation given in

figure 2.27 is a general representation of crack blunting and resharpening. It

is a synthesis of the various models proposed in the literature [46 52]. kind

it may give an appreciation of the mechanism of fatigue crack growth,

lufficient as a background for the study of fracture mechanics principles. A

more detailed model allowing a limited quantitative analysis has been put

forward recently by Neumann [52]. A mechanism of cleavage ma\

KOllKlimes be involved in fatigue crack propagation. This gives way to the

lo'i-iiiation of brittle striations [44. 45] as opposed to the ductile striations

dim ussed above.

Striations represent the successive positions of the crack front during

i i .uk propagation. This can be deduced from figure 2.29, showing an clcFurther, although phenomenologically sound, the

energy approach has an intangible. Quality that makes it hard to visualize. For these and other

reasons, the focus of attention in the theory of fracture mechanics in the 1960s was on the stress

field approach to LEFM. From this viewpoint, the theory of fracture is treated as a subset of the

engineering discipline of solid mechanics. In particular, fracture mechanics is a study of the

mechanics of deformable bodies containing crack-like singularities. In order to develop the concept

of fracture mechanics from this perspective, it is first necessary to master a working knowledge of

linear elasticity. Accordingly, in the next few chapters, we will first treat those aspects of solid

mechanics necessary to develop a theory of fracture based on the stress analysis approach and then

go on to formulate a suitable theory and study its consequences.

REFERENCES

ASTM, 1960, "Fracture Testing of High-Strength Sheet Materials (A Report of a Special ASTM

Committee),"ASTM Bulletin, No. 243, January 1960, pp. 29^0, and No. 244, February 1960,

pp. 18-28.

number of cleavage steps

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