Mechanical Behaviour of Engineering Materials - Metals, Ceramics, Polymers and Composites 2010 Part 5 doc

Plot of the crack-growth resistance KIRand the stress σ versus the cracklength a for the case of constant KIRduring crack propagation 1/2In contrast, figure 5.12c shows a configuration w

de-Fig 5.12 Plot of the crack-growth resistance KIRand the stress σ versus the cracklength a for the case of constant KIRduring crack propagation (1/2)

In contrast, figure 5.12(c) shows a configuration with the same growth resistance curve, but a smaller initial crack length a3 In this case, thestress needed for the crack to propagate decreases from the start, althoughthe crack-growth resistance increases Hence, crack propagation is unstable assoon as KIcis reached The critical stress σc3 is nevertheless larger than thatfor the longer crack (σc2)

crack-As the figures 5.12(b) and 5.12(c) illustrate, an increasing crack-growthresistance KIRdoes not guarantee stable crack propagation because the stresscan still decrease

If the load propagating the crack is displacement-controlled, a decreasingload does not necessarily cause failure of the component As the compliance ofthe component increases with increasing crack length, the stress may becomesmall enough to stabilise the crack, a phenomenon called crack arresting

We now want to estimate the stress intensity factor KI that marksthe onset of unstable crack propagation This happens when the re-quired external stress σ decreases or at least stops to increase on crackpropagation by a distance da During crack propagation, the currentstress intensity factor KI, which depends on the stress σ and the cracklength a, must equal the crack-growth resistance KIR:

5.2 Linear-elastic fracture mechanics 149

σc23σ12σ11σ1

(b) For a crack of length a2, stable crack

propagation begins at a stress σc At a

stress of σ∗, the crack becomes unstable

˛

˛

KI=KI∗

If the crack-growth resistance curve KIR(a) is known, this criteriontogether with the condition KI(σ, a) = KIR(a) allows to calculate thebeginning of unstable crack propagation.

∗ 5.2.6 Subcritical crack propagation

So far, we have assumed that a crack is stationary if the stress intensity factor

is smaller than the fracture toughness KIc This, however, is not always thecase If, for instance, a corrosive medium penetrates the material along thecrack surface, it might weaken the material near the crack tip This decreasesthe fracture toughness locally and the crack can propagate, but only until itreaches undamaged material If this happens, a crack can slowly propagate

at loads below the critical load, a phenomenon called subcritical crack growth.The crack propagates subcritically until the stress intensity factor KI equalsthe fracture toughness KIc and the crack propagation becomes unstable.14

As the subcritical crack growth is time-dependent, it can be described bythe crack-growth rate da/dt, specifying the increment da by which the crackpropagates during an infinitesimal time dt

Generally, subcritical crack growth occurs when the material near the cracktip is weakened by time-dependent processes Different physical phenomenamay be responsible for this

In many metals, corrosive media like electrolytes can cause stress corrosioncracking Two different kinds of stress corrosion cracking can be distinguished

In anodic stress corrosion cracking, loading accelerates corrosion at thecrack tip, resulting in crack propagation This may happen in metals whosesurface is usually protected by a passivating layer The crack tip can be acti-vated, for instance by local plastic deformation, and the crack can propagate

If a new passivating layer forms immediately on the freshly formed surface,corrosion can only proceed at the crack tip, causing the crack to remain sharp-edged Whether anodic stress corrosion cracking can occur depends on thesurrounding media, the material and its state, the temperature, and the me-chanical stress In unfavourable circumstances, the required stress level may

be very small and even residual stresses may be sufficient to cause failure

by a delayed fracture (see section 3.5.3) Anodic stress corrosion crackingcan, for example, occur in the presence of chloride ions (e g., in saline air) inaluminium alloys and austenitic chrome nickel steels.15

14 In section 5.2.5, we saw that even above KIc a crack may still be stable if thecrack-growth resistance increases If subcritical crack growth occurs, this effect

is usually negligible because the crack grows fast at stress intensity values close

to KIc

15

This latter case is especially problematic because these steels are usually called

‘stainless’, implying a high level of corrosion resistance However, in saline sphere, the corrosion resistance of these steels is reduced

atmo-5.2 Linear-elastic fracture mechanics 151

A different case is hydrogen-induced stress corrosion cracking It occurs inmaterials showing hydrogen embrittlement (see section 3.5.3, page 117) Thestress concentration near the crack tip dilates the crystal lattice Hydrogen,generated during the corrosion process, is therefore preferentially stored inthis region near the crack tip, reducing the fracture toughness in the moststressed region The crack propagates through the weakened material, and,subsequently, hydrogen diffuses to the new crack tip Hydrogen-induced stresscorrosion cracking is observed particularly in high-strength steels because hereelastic strains near the crack tip can be large and large amounts of hydrogencan thus be stored

Polymers show a similar effect in the presence of solvents Solvents ferredly enter the material near the crack tip because the distance betweenthe molecules is increased there by the large tensile stresses If, for instance, arod made of polymethylmethacrylate (Plexiglas) is bent and the tensile side

pre-is wetted with acetone or alcohol, brittle fracture can occur after a short posure time In this case, the cleavage strength is reduced because the dipolebonds between the molecules are replaced by bonds formed with the solvent(see also section 8.8)

ex-Ceramics can also fail by subcritical crack growth due to stresses andlocalised chemical reactions In glasses, for instance, water can enter surfacedefects and can attack the bonds between the silicon and the oxygen atoms ifthese are strained by an external stress [9]:

Si−O−Si + H2O → Si−O−H + H−O−Si

This reaction can cause a seemingly sudden failure of glasses Subcritical crackgrowth also occurs in crystalline ceramics, especially if they have a glassyphase (see section 7.1) on the grain boundaries Among the most sensitiveceramics are silicate ceramics like porcelain or mullite, for they usually contain

a large amount of more than 20% glassy phases [19, 142] Subcritical crackgrowth can also be present in engineering ceramics, for instance in aluminiumoxide (Al2O3) in humid atmosphere or saline solution [104]

At elevated temperatures, metals and ceramics exhibit time-dependentplastic deformation, called creep, a phenomenon to be discussed in detail inchapter 11 If a pre-cracked material is loaded at high temperatures, the crackcan grow In metals and ceramics, pores are frequently responsible for thisbecause they form and coalesce in the highly-stressed region in front of thecrack tip, often on grain boundaries [119] (see section 11.3) The crack thusfrequently propagates between the grains (intercrystalline fracture) This pro-cess is called creep crack growth (ccg) In polymers, time-dependent plasticdeformation occurs already at ambient temperature (see section 8), and theyare thus also susceptible to creep crack growth

Subcritical crack growth is determined by the temperature, the material,and, for the low-temperature processes, the environment In a given system,the crack-growth rate frequently depends on the stress intensity factor KI

only Below a certain, temperature-dependent limiting value K , crack growth

Fig 5.13 Creating an initial crack by cyclic loading

vanishes completely or almost completely [104] If KI exceeds KI0, the growth rate rapidly increases with increasing stress intensity factor In manycases (for example, in stress corrosion cracking), a plateau region follows i e.,the crack-growth rate is nearly constant for a certain range of KI-values Whenthe stress intensity factor approaches KIc, the crack-growth rate increasesrapidly again Examples for crack-growth rate curves and a mathematicaldescription of the crack-growth rate are discussed in section 7.2.6

crack-∗ 5.2.7 Measuring fracture parameters

In the previous sections, we introduced several important material ters: The fracture toughness KIc, the critical energy release rate GIc, and thecrack-growth resistance curve We now want to see how these quantities aremeasured

parame-A common feature of all experiments is that the test specimens are cracked To achieve this, a notched sample is used and a crack is propagatedfrom this notch by cyclic loading (see chapter 10) as shown in figure 5.13.Creating the initial crack in this way is necessary because the notch tip isusually not sharp enough to behave like a true crack Cyclic loading allows toproduce the initial crack at a load that is much smaller than that needed forstatic experiments (see chapter 10)

pre-Several specimen geometries are standardised, and the corresponding ometry factors are given in tabular form or by approximation functions [21,133,138] Among the most common specimen geometries are the three-point bend-ing specimen (figure 5.14(a)) and the compact tension specimen, or ct speci-men for short (figure 5.14(b))

ge-For the compact tension specimen, the stress intensity factor can be lated from the external load F by

calcu-KI= F

B√

Wf (

5.2 Linear-elastic fracture mechanics 153

(a) Three-point bending specimen

B

GW

4

is the geometry factor In calculating KI, it is important to note that the cracklength a is measured from the point of loading, not from the beginning of theinitial crack This is easily understood because it is completely irrelevant forthe stress state near the crack tip how ‘wide’ the crack is some distance away

It is only important that the initial crack is sharp-edged with a small radius

of curvature, and this is ensured by creating it through cyclic loading Thequantities G, H, B, W , and s from figure 5.14(b) must, according to thestandard astm e 399, be related in a certain way: As an example, B = W/2,

s = 0.55W , H = 1.2W , and G = 1.25W are used for a standard ct specimen.The initial crack length should be limited by 0.45W ≤ a ≤ 0.55W

95% line

Fmax

F5

(c) F5left of Fmax, but right

of the first maximumFig 5.16 Determination of the critical force FQ The 95% lines are drawn with aslope of 90% only

∗ Measuring the fracture toughness

The procedure to measure KIcor GIcis independent of the specimen geometry.The loading points are displaced with constant speed and the required force ismeasured If the force is plotted against the displacement of the loading points,

a load-displacement curve results as shown in figure 5.15 The onset of unstablecrack propagation can be seen from this curve because the force reaches amaximum and drops, resulting in a larger compliance of the specimen As theexperiment is displacement-controlled, the crack usually stabilises again due

to the unloading and propagates only when the crack is opened further If thesize of the plastic zone near the crack tip is small compared to the volume

of the specimen, crack propagation starts without any noticeable deviation ofthe loading curve from linear-elastic behaviour (see figure 5.16(a))

5.2 Linear-elastic fracture mechanics 155

In contrast, the material behaviour shown in figure 5.16(b) indicates icant plastic deformation The crack propagation is stable at first and becomesunstable at a force Fmax This shape of the curve is typical for ductile materi-als In this case, it has to be ensured that linear-elastic fracture mechanics isstill valid Furthermore, it is not possible to determine from the curve alone atwhich force crack propagation has started because stable crack propagationand plastic deformation both reduce the slope of the curve To determine thefacture toughness KIc, a pragmatic approach is taken, similar to the definition

signif-of the yield strength Rp0.2 This will be described below

A special case is shown in figure 5.16(c) On reaching a load FQ, the crackpropagates unstably for a certain distance and then becomes arrested This iscalled pop-in

To determine the fracture toughness KIc, the following procedure has beenagreed upon (see e g., standards astm e 399 and iso 12737):

We start by drawing a line with a slope of 95% of that of the elastic linefrom the experiment (figure 5.16) The intersection of this line with the load-displacement curve determines the force F5.16Two cases can be distinguished:

• If F5lies to the right of the force value at which the first reduction in theload occurs, the force FQ is determined by this maximum (FQ= Fmax infigure 5.16(a), FQ in figure 5.16(c))

• If F5 is left to this maximum, FQ = F5 is used as a critical value ure 5.16(b)) In this case, it has to be ensured that plastic deformationwas small enough to allow using linear-elastic fracture mechanics A nec-essary condition for this is

(fig-Fmax

If this condition does not hold, the experiment has to be evaluated cording to the rules of elastic-plastic fracture mechanics, discussed in sec-tion 5.3

ac-For a ct specimen, the critical stress intensity factor KQis calculated from

Fig 5.17 Measuring the initial crack length

plain stress

plain stress

plain strain

plastic zone

Fig 5.18 Size and shape of the plastic zone near the crack tip (dog bone, after [58])

the specimen geometry, may influence this value This can be illustrated byinspecting the plastic zone near the crack tip (see figure 5.18)

At the surface, the specimen is in a state of plane stress because no mal forces can be transmitted here The smallest principal stress is thus zero.Within the specimen, the stress state is a state of nearly plane strain becausethe transversal contraction near the crack tip is constrained by the surround-ing material The stress state is thus a state of triaxial tension Therefore, theequivalent stresses and thus the plastic deformations are larger near the sur-face than within the specimen As plastic deformation dissipates energy, thecrack-growth resistance increases with decreasing thickness of the specimen(see figure 5.19) For sufficiently thick specimens, the influence of the surfacezone with its state of plane stress can be neglected and the crack-growthresistance KQ approaches a constant value, the fracture toughness KIc

nor-To ensure independence of the geometry and to determine the fracturetoughness as lower (and therefore safe) limiting value for the crack-growthresistance of a material, a state of plane strain is required Only if this can beguaranteed, the measured value KQ is called fracture toughness KIc Accord-ing to the standards astm e 399 and iso 12737, this requires

5.2 Linear-elastic fracture mechanics 157

∗ Measuring the crack-growth resistance curve

As explained in section 5.2.5, the crack-growth resistance curve is a plot ofthe stress intensity factor versus the crack length a Experiments are usu-ally displacement-controlled to enable measurement of the load-displacementcurve after the maximum force has been exceeded

To measure the crack-growth resistance curve according to astm e 561, theload is applied step-wise, and the crack length is measured for each load valueafter the crack has stabilised For a complete curve, 10 to 15 measurementvalues are needed To avoid using several specimens for a single curve, thecrack length is measured during the experiment This can be done in severalways [21, 43]

Optical methods measure the crack length directly on the polishedsurface of the specimen This method is rather simple, but its maindisadvantage is that the crack is only measured on the surface; thecrack length within the specimen is unknown

The compliance method measures the compliance of the specimen

by unloading it during the experiment Comparing the measured valuewith a calibration curve determined on specimens with known cracklength, the crack length can be determined

The electrical potential drop method uses the electrical resistance ofthe specimen to measure the crack length A constant electrical current

is applied between two points of the specimen far away from the crackand the potential drop in the vicinity of the crack is measured Com-parison with a calibration curve allows calculation of the crack length.Obviously, the specimen has to be electrically isolated from the testingmachine and the displacement transducer

As before, it is necessary that the deformation of the specimen is mainlyelastic to allow use of linear-elastic fracture mechanics For the crack-growthresistance curve measurement, this can be ensured according to astm e 561

by the following condition:

From the measured values for the force Fiand the crack length increment

∆ai, the crack-growth resistance curve is calculated using equation (5.30)

∗ 5.3 Elastic-plastic fracture mechanics

In the previous sections, it was frequently stressed that linear-elastic fracturemechanics can only be used if the plastic zone near the crack tip is sufficientlysmall If this is not the case, we enter the domain of elastic-plastic fracturemechanics (epfm) which can deal with a large plastic zone The method,however, cannot be used for arbitrarily large plastic zones – plastic behaviourmust still be restricted to the region around the crack tip and must be mainlydetermined by the surrounding elastic stress field

Two alternative methods are commonly used to describe the state nearthe crack tip: The crack tip opening displacement and the J integral Bothmethods can be shown to be mathematically equivalent

∗ 5.3.1 Crack tip opening displacement (ctod)

In the crack tip opening displacement method (or ctod-method for short), it

is assumed that crack propagation is not determined by the stress intensityfactor, but by the amount of plastic deformation near the crack tip This can

be measured by the opening δtof the crack tip If this reaches a critical value

δc, the crack propagates

The crack tip opening displacement can be defined in different ways Theyall have in common that it is assumed that the crack tip is blunted by the

5.3 Elastic-plastic fracture mechanics 159

Fig 5.20 One possible definition of the crack tip opening displacement δt

Fig 5.21 Coordinate system and integration contour for the J integral

plastic deformation and that both crack surfaces are almost parallel, see ure 5.20 One possibility to determine δtis to draw two lines at an angle of 45°

fig-to the crack line and measure the distance between their intersections withthe crack surface

ex-(but not necessarily linear-elastic) material behaviour As long as the ing is monotonous and no unloading occurs, it can also be used for plasticdeformations.17

load-The choice of the path C is arbitrary as long as it encloses the crack tip

In practice – for instance when doing finite element simulations – it is usuallywise to put it not too close to the crack tip so that it runs only throughelastically deformed regions

Equations (5.10) and (5.35) are equivalent Therefore, for linear-elasticmaterials and a state of plane stress, the equation

GI= J = K

2 I

is elaborated further in appendix D.6

We start by looking at the second term of equation (5.10), dU(el)/da.Using the elastic energy density w(el), we get

dU(el)

da =

Z Z ZV

dw(el)

dx1

dx1dx2.Gauss’ theorem [24] allows to convert an area integral over an area Ainto a line integral along its boundary C.18For the integral in question,

w(el)dx2,which corresponds to the first term in equation (5.35)

17 This is the reason why the energy density in equation (5.35) is called w, not w(el).18

Here we use a simplified version of Gauss’ theorem for two dimensions (see pendix D.1)

ap-5.3 Elastic-plastic fracture mechanics 161

We now look at the first term in equation (5.10), dW/da To late the work dW done during an infinitesimal propagation of the crack,

calcu-a closed surfcalcu-ace ccalcu-an be put calcu-around the crcalcu-ack tip The work d(dW ) doneduring this infinitesimal crack growth by da on an area element dS isgiven by the product of the force dF acting on the area element andthe change in the displacement field du:

The force dF acting on the surface element dS can be calculated bymultiplying ∆S with the normal stress on the surface, σ n, where n isthe normal vector of the surface:

dW

da =

Z ZS

∗ 5.3.3 Material behaviour during crack propagation

We saw in section 5.2.5 that the crack-growth resistance KIR in linear-elasticfracture mechanics depends on the crack length increment ∆a Similarly, the

value of the J integral also changes during crack propagation Analogous

to KIR, the value of the J integral during crack propagation is called JRgrowth resistance As for KIR, a crack-growth resistance curve can be drawn

crack-by plotting JRagainst the crack length increment ∆a (figure 5.22)

Crack propagation in ductile materials is different from that in brittle ones.With increasing load, the material deforms plastically near the crack tip andblunts it (subfigure○ in figure 5.22) This blunted region is called the stretch2

zone

Subsequently, due to the high stresses and large plastic deformations, ities form in front of the crack tip (cf section 3.5.1), depicted in subfigures○3

cav-and ○ During formation of the stretch zone and of cavities (subfigures4 ○1

to ○), the relation between ∆a and J4 R is almost linear

With increasing load, the cavities coalesce with each other and the crack,causing a ‘true’ growth of the crack (subfigure ○) This is called crack ini-5

tiation and occurs when J reaches Jc [58].19 The slope of the crack-growthresistance curve now decreases (see figure 5.22) The formation and coales-cence of cavities or pores during crack propagation is characteristic for shearfracture (section 3.5.1), producing the typical dimple fracture surface Thestate○ is different from the initial configuration5 ○ for two reasons: The ma-1

terial near the crack tip is now plastically deformed and has thus hardened,and the crack surface is dimpled, resulting in a blunted crack tip

19

Sometimes, this value of the J integral is denoted as Ji (where ‘i’ stands for tiation’) [21] In this case the value at the transition between stable and unstablebehaviour (called J∗here) is called J

‘ini-5.3 Elastic-plastic fracture mechanics 163

Contrary to brittle materials, in ductile materials even small loads cancause (very small) crack growth If the load is not raised further, this will notcause any trouble, but if the load is cyclic, each repetition of the load will cause

a small crack propagation and finally cause a sufficiently large crack to destroythe component This so-called fatigue fracture is the topic of chapter 10.The smaller the plastic region near the crack tip is, the steeper is theline characterising the formation of the stretch zone In the limiting case oflinear-elastic material behaviour, the line is vertical and the curve is similar

to that in figure 5.12 The smaller the unloading of the crack tip due toplastic deformation is, the higher are the stresses near the crack tip As thestress state is triaxial, the danger of crack propagation by cleavage fracture(see section 3.5.2) grows The transition between dimple surface fracture forductile materials and cleavage fracture in brittle ones is not clear-cut, andmixtures of both cases can occur

The value of the J integral that marks the beginning of unstable crackpropagation can be calculated analogously to the stress intensity factor KI

in section 5.2.5 The critical J value J∗ is not the crack initiation value Jc

At Jc, stable crack growth by coalescence of cavities begins; at J∗, the crackstarts to become unstable

As unstable crack propagation causes an unloading of the material, the

J integral must not be used during this stage because, according to tion 5.3.2, the equations are not valid in this case, even when the experiment

sec-is stabilsec-ised by dsec-isplacement control

∗ 5.3.4 Measuring elastic-plastic fracture mechanics parameters

As in linear-elastic fracture mechanics, specimens in elastic-plastic fracturemechanics are also standardised An initial crack is produced in the same way

by cyclic loading The values of the J integral for crack propagation, Jc and

J∗, are read off the JR crack-growth resistance curve, so there is no need toperform additional experiments

We will now discuss the procedure using the example of a ct specimen(figure 5.14(b)) The specimen is loaded using displacement-control, and aload-displacement curve is measured The resulting graph will look like theone in figure 5.23 The area beneath the curve corresponds to the work done

Fig 5.23 Load-displacement curve to determine the JRcrack-growth resistance

η = 2 + 0.5221 − a

W



As can be expected because of the definition of the J integral as an energyrelease rate, the value of the J integral is directly connected to the externalwork WF The derivation of equation (5.41) can be found in Gross / Seelig [58]

To measure the JRcrack-growth resistance curve, a direct or indirect surement of the crack length during the experiment is needed This is doneusing one of the methods described in section 5.2.7 It is not possible to mea-sure the development of the crack length during the test after the experiment

mea-on the fractured specimen

Mechanical behaviour of metals

As already mentioned in section 1.2, metals are characterised by their lent plastic deformability that is of great technical importance On the onehand, it allows to produce complex metallic components by forming processeslike forging or drawing On the other hand, it causes metals to deform plas-tically when the yield strength is reached, instead of failing catastrophically

excel-by fracturing This improves safety because overloading can often be detectedbefore disaster happens

In this chapter, we will explain the mechanisms behind the plastic tion of metals Afterwards, we will discuss how the stress required for plasticdeformation can be increased, thus strengthening the material

crys-If we measure the strength of single crystals of pure metals, the valuesfound are several orders of magnitudes below this theoretical value and evenlie below that of engineering alloys Typical values are in the range of a few

1

For simplicity, we usually use a simple cubic lattice in the sketches of crystalsshown in this chapter, although this is not a Bravais lattice found in any techni-cally important metal

Fig 6.1 Sliding of atomic planes in a perfect crystal

megapascal As single crystals always contain lattice defects, one possibleexplanation could be that these are responsible for the reduced strength If,however, the number of defects is reduced further, for instance by a heattreatment, the yield strength becomes even smaller Only an absolutely perfectsingle crystal without any defects would possess a yield strength agreeingwith the theoretical prediction This can only be nearly realised in so-calledwhiskers (see section 6.2.8) which, however, are extremely small

The reason for this spectacular failure of the theoretical prediction is thatplastic deformation does not occur by sliding of complete layers of atoms.Instead, it proceeds by a mechanism that is based on a special type of latticedefect, the dislocations To understand plastic deformation of metals thusrequires an understanding of dislocations

6.2 Dislocations

6.2.1 Types of dislocations

Dislocations are one-dimensional (line-shaped) lattice defects Figure 6.2(a)shows an edge dislocation, one of the two basic types Its spatial structure canmost easily be visualised by imagining that an additional half-plane of atoms

is put into the crystal In the vicinity of the line where this half plane ends,the crystal is distorted, further away from it, it still is perfect

An edge dislocation can be described by two vectors The first is the linevector t, the vector pointing in the direction of the dislocation line The secondvector is the Burgers vector b that can be determined in the following way:

We draw a so-called Burgers circuit around the dislocation line that takes thesame amount of steps from one atom to the next in each direction as visualised

in figure 6.2 If the crystal were perfect, the circuit would be closed, but, due

to the lattice defect, it is not An additional step is required to get back to thestarting point The vector describing this step is the Burgers vector b As long

as the Burgers circuit encloses the dislocation line, the Burgers vector defined

in this way is independent of the size and the shape of the circuit As shown

in figure 6.2(a), the Burgers vector and the line vector of an edge dislocationare perpendicular

In our definition, we did not specify the direction of the circuit, but it has

to be chosen consistently One simple way of doing this is to use a ‘right-handrule’ oriented on the line vector t of the dislocation line as shown in figure 6.3

6.2 Dislocations 167

bBurgerscircuit

dislocation line(b) Screw dislocation

Fig 6.2 Types of dislocations In the lattice models, a possible Burgers circuit ismarked

The choice for the direction of the line vector is arbitrary as well If it isreversed, the orientation of the Burgers vector reverses as well

The second basic type of a dislocation, the screw dislocation is shown infigure 6.2(b) It can be visualised by imagining that the crystal has slipped byone atomic distance on a half plane ending at the dislocation line The screwdislocation can also be characterised by its line vector and Burgers vector Thefigure shows that both are parallel If we move along a crystal plane aroundthe dislocation, the resulting path is helical and thus looks like a screw, whichexplains the name of this dislocation type

Dislocation lines are always either closed or end at the surface of thecrystal, but they can never end within the crystal Why this is so can be seenfrom figure 6.4 Imagine that a dislocation would end somewhere within thecrystal The crystal is distorted in the vicinity of the dislocation line, but isperfect at a sufficient distance away from the dislocation We now walk on a