Crack Propagation Mechanisms for Creep Fatigue: a Consolidated Explanation of Fundamental Behaviours from Initiation to Failure

Home , Creep (deformation), Diffusion creep

Article Crack Propagation Mechanisms for Creep Fatigue: A Consolidated Explanation of Fundamental Behaviours from Initiation to Failure

Dan Liu and Dirk John Pons * ID Department of Mechanical Engineering, University of Canterbury, Christchurch 8041, New Zealand; [email protected] * Correspondence: [email protected]; Tel.: +64-3369-5826

Received: 10 July 2018; Accepted: 2 August 2018; Published: 8 August 2018

Abstract: Background—Creep-fatigue damage is generally identified as the combined effect of fatigue and creep. This behaviour is macroscopically described by crack growth, wherein fatigue and creep follow different principles. Need—Although the literature contains many studies that explore the crack-growth path, there is a lack of clear models to link these disparate findings and to explain the possible mechanisms at a grain-based level for crack growth from crack initiation, through the steady stage (this is particularly challenging), ending in structural failure. Method—Finite element (FE) methods were used to provide a quantitative validation of the grain-size effect and the failure principles for fatigue and creep. Thereafter, a microstructural conceptual framework for the three stages of crack growth was developed by integrating existing crack-growth microstructural observations for fatigue and creep. Specifically, the crack propagation is based on existing mechanisms of plastic blunting and diffusion creep. Results—Fatigue and creep effects are treated separately due to their different damage principles. The possible grain-boundary behaviours, such as the mismatch behaviour at grain boundary due to creep deformation, are included. The framework illustrates the possible situations for crack propagation at a grain-based level, particularly the situation in which the crack encounters the grain boundary. Originality—The framework is consistent with the various creep and fatigue microstructure observations in the literature, but goes further by integrating these together into a logically consistent framework that describes the overall failure process at the microstructural level.

Keywords: creep fatigue; crack growth; grain boundary

1. Introduction The process of fatigue failure is physically and macroscopically described by crack-growth behaviour. The situation is complicated when creep is also present, because this provides an additional mechanism for plastic deformation and crack growth. In the case of creep-fatigue, an engineering structure fails when the crack length achieves a critical value. The total damage is the accumulated effect of cycle count, temperature, and period (frequency). Some of the ways this has been modelled is to prove a mathematical representation of the various components of facture mechanics [1–3] or to apply curve ﬁtting [4–6]. However, this is not the present topic. Here we are more interested in the progression of crack growth at the microstructural level [7–10], speciﬁcally the way the crack navigates through the grains and along grain boundaries under the combined effects of cyclic loading and creep. Existing works do not provide an integrative treatment of this process. This paper develops a microstructural theory for crack growth. It does so by adapting existing mechanisms of plastic blunting and diffusion creep into an integrative conceptual framework.

Metals 2018, 8, 623; doi:10.3390/met8080623 www.mdpi.com/journal/metals Metals 2018, 8, x FOR PEER REVIEW 2 of 34

MetalsThis2018, paper8, 623 develops a microstructural theory for crack growth. It does so by adapting existing2 of 32 mechanisms of plastic blunting and diffusion creep into an integrative conceptual framework. This provides a logically consistent framework that describes the creep-fatigue failure in terms of This provides a logically consistent framework that describes the creep-fatigue failure in terms of microstructural damage. microstructural damage.

2. Background Literature

2.1. Phases of Crack Growth ItIt isis generallygenerally accepted accepted that that the the process process of crackof crack growth growth under under cyclic cyclic loading loading is divided is divided into three into phasesthree phases (Figure (Figure1)[ 8]: crack1) [8]: initiation,crack initiation, crack propagation, crack propagation, and structural and structural fracture. fracture. Specifically, Specifically, the first stagethe first presents stage presents a threshold, a threshold, below which below there which is no th crackere is growth.no crack The growth. second The stage second shows stage a relatively shows a steadyrelatively state, steady in which state, thein which crack growththe crack rate growth increases rate steadilyincreases with steadily the increasingwith the increasing number of number cycles (thisof cycles is always (this is described always described by Paris’ Lawby Paris’ [11]). Law The [11] final). stageThe final represents stage represents an unstable an situation, unstable wheresituation, the engineeringwhere the engineering structure fails structure within fails a small within number a small of number cycles. of cycles.

Figure 1. Three stages of crack growth.

2.2. Established Principles in the Literature Crack growth caused by fatiguefatigue effect isis attributedattributed toto plasticplastic deformationdeformation at aa crack-tipcrack-tip plasticplastic zone, whereinwherein the the plastic plastic deformation deformation occurs occurs via via dislocations dislocations [7]. Ordinarily[7]. Ordinarily there there is a preferred is a preferred plane, andplane, in and that in plane that thereplane are there specific are specific directions directions along whichalong dislocationwhich dislocation motion motion occurs; occurs; the slip the plane. slip Theplane. axial The (tension axial (tension or compression) or compression) loading is decomposedloading is decomposed into a normal into stress, a normal which isstress, perpendicular which is toperpendicular the slip plane, to andthe slip a shear plane, stress, and whicha shear is stress, parallel which to the is slipparallel plane. to Thisthe slip shear plane. stress This then shear results stress in thethen movements results in the along movements the slip plane.along the This slip implies plane. thatThis shear implies stress that contributes shear stress tocontributes plastic dislocation, to plastic anddislocation, then results and inthen crack results growth. in crack In this growth. case, the In axialthis case, loading the isaxial transformed loading is into transformed the shear stressinto the to produceshear stress fatigue to produce propagation. fatigue propagation. The simplest form of dislocationdislocation is edge dislocation,dislocation, see Figure2 2,, whichwhich representsrepresents aa simplisticsimplistic grain unit. Specifically, Specifically, the dislocation starts from the extra half-plane of atoms (plane I), which is identifiedidentified asas aa defect.defect. When a shear stress is applied, and its magnitude magnitude is sufficient, sufficient, the bond between atom A and atomatom BB isis cut.cut. Then, the bottom of all planes move to the right, and atom B isis bondedbonded withwith itsits neighbouringneighbouring atom (atom C). In this case, a new half-plane of atoms (plane II) is generated. Under thethe shearshear stress, stress, this this irreversible irreversible process process is successivelyis successively and and repeatedly repeatedly presented, presented, with with the final the resultfinal result being being the creation the creation of a step of a at step the at edge the ofedge this of unit. this Theunit. process The process is irreversible is irreversible because because the other the atomsother atoms in the grainin the also grain move also into move new into positions new (notpositions shown (not in theshown figure) in duethe tofigure) other disturbances,due to other hencedisturbances, the atoms hence do not the spontaneously atoms do not revert spontaneously to their original revert positions to their (plastic original deformation). positions (plastic Hence, deformation). Hence, the dislocation is an enduring feature of the grain. It can, however, be sent in a the dislocation is an enduring feature of the grain. It can, however, be sent in a new direction should new direction should the orientation of the shear stress change, as occurs in reversed loading. By this the orientation of the shear stress change, as occurs in reversed loading. By this means, the plastic means, the plastic deformation is produced due to the motion of numbers of dislocations. In addition, deformation is produced due to the motion of numbers of dislocations. In addition, there may be there may be atomic displacement perpendicular to the plane, hence causing a screw dislocation. In atomic displacement perpendicular to the plane, hence causing a screw dislocation. In practical

Metals 2018, 8, 623 3 of 32 Metals 2018, 8, x FOR PEER REVIEW 3 of 34 practicalsituations, situations, the behaviours the behaviours of edge and of screwedge and dislocations screw dislocations are mixed, are and mixed, both contribute and both tocontribute the overall to plasticthe overall deformation plastic deformation of the grain of and the ultimatelygrain and ultimately of the macroscopic of the macroscopic deformation deformation of the part. of the part.

Figure 2. Edge dislocation process. Figure 2. Edge dislocation process. As mentioned above, the half-plane of atoms is regarded as the source of dislocation. The edge of thisAs half mentioned plane is terminated above, the half-planewithin a grain; of atoms hence, is a regarded large number as the of source dislocations of dislocation. are piled The up, edge and thenof this plastic half planedeformation is terminated arises for within the grain a grain; as a whole. hence, This a large implies number that ofthe dislocations cracks caused are by piled fatigue up, areand initiated then plastic within deformation the grains arisesand then for theare grainpropagated as a whole. through This the implies grains, that causing the cracksthe observed caused transgranularby fatigue are fracture initiated phenomenon. within the grains and then are propagated through the grains, causing the observedHowever, transgranular the creep fracture mechanism phenomenon. gives a different crack-growth behaviour; crack growth along the grainHowever, boundary. the creep In general, mechanism crack gives growth a different caused by crack-growth creep is attributed behaviour; to diffusion crack growth behaviour, along whichthe grain is presented boundary. as In the general, movements crack growth of vacancies caused [12,13]. by creep Under is attributed the creep to situation, diffusion behaviour,a constant whichapplied is loading presented leads as theto stress movements concentration of vacancies at the [ 12triple,13]. points Under which the creep are formed situation, by a the constant three adjacentapplied loading grains (Figure leads to 3). stress Normally, concentration diffusion at theis a triple behaviour points of which atomic are shifts formed from by an the area three with adjacent high stressgrains concentration (Figure3). Normally, to an area diffusion with low is stress a behaviour concentration. of atomic In this shifts case, from the an atomic area withmovements high stress due toconcentration the diffusion to mechanism an area with results low stress in the concentration. convergence Inof this vacancies case, the at atomicthe triple movements points, which due to then the segregatediffusion mechanismthe triple grain results boundaries. in the convergence of vacancies at the triple points, which then segregate the tripleBy this grain means, boundaries. the micro-crack is initiated at the grain boundary. After the process of micro- crackBy initiation, this means, the thelocalised micro-crack stress is initiatedhighly concentrated at the grain boundary.at the crack After tip, thewhich process then ofrecruits micro-crack more vacanciesinitiation, thefrom localised the bulk stress of the is grain, highly and concentrated these are diffused at the crack to the tip, crack which tip then along recruits the grain more boundary. vacancies Thisfrom leads the bulk to further of the grain, propagation and these of are the diffused creep crack to the along crack the tip alonggrain theboundary. grain boundary. High temperature This leads providesto further more propagation favourable of the conditions creep crack for alongdiffusio then grainbehaviour, boundary. hence High advances temperature the creep provides crack along more grainfavourable boundaries. conditions Thus, for creep diffusion failure behaviour, is strongly hence temperature advances dependent. the creep crack along grain boundaries. Thus, creep failure is strongly temperature dependent.

Metals 2018, 8, 623 4 of 32 Metals 2018, 8, x FOR PEER REVIEW 4 of 34

FigureFigure 3.3. Crack growthgrowth byby diffusiondiffusion mechanism.mechanism.

Overall, crack-growth behaviours of fatigue and creep follow different principles; specifically, Overall, crack-growth behaviours of fatigue and creep follow different principles; specifically, fatigue effect occurs via cracks through the grains, while the creep effect involves fracture along the fatigue effect occurs via cracks through the grains, while the creep effect involves fracture along the grain boundary. grain boundary. Significant research effort has been exerted to explore the crack-growth path, through Significant research effort has been exerted to explore the crack-growth path, through performing performing fatigue tests and observing the fracture surfaces (fractography). These research efforts fatigue tests and observing the fracture surfaces (fractography). These research efforts mainly focus on mainly focus on the specific features of crack growth for fatigue or creep, such as the creep damage the specific features of crack growth for fatigue or creep, such as the creep damage caused by triple caused by triple points [14–16] and grain-boundary effects for fatigue-crack growth [17,18]. points [14–16] and grain-boundary effects for fatigue-crack growth [17,18]. In summary, the literature proposes multiple mechanisms at the microstructural level to describeIn summary, the crack-growth the literature process. proposes multiple mechanisms at the microstructural level to describe the crack-growth process. Stage I: Crack initiation Stage I: Crack initiation (a) Dislocation behaviour [19] is a key mechanism at crack initiation. This effect is widely (a) Dislocationaccepted behaviour [19]. This effect [19] iscauses a key the mechanism slip planes to at extrude crack initiation. or intrude, Thiswhich effect results is in widely stress acceptedconcentration [19]. This at effect the surface causes and the then slip planesleads to to crack extrude nucleation. or intrude, which results in stress (b)concentration The strain at energy the surface release and rate then [20,21] leads todetermines crack nucleation. the crack-growth rate. This mechanism (b) Theunderpins strain energy the unstable release rate stage [20 after,21] the determines threshold the of crack crack-growth initiation. rate. In this This stage, mechanism increased underpinsenergy the accelerates unstable stagethe increase after the of threshold crack-growth of crack rate initiation. at the beginning In this stage, phase, increased which energyis then acceleratesretarded the due increase to reduced of crack-growth energy and rate finally at the achieves beginning a stable phase, state. which is then retarded due (c)to reducedThe shear energy effects and [22,23] finally provide achieves opportunistic a stable state. directions for crack growth. This mechanism (c) The shearinvolves effects the [direction22,23] provide oriented opportunistic at 45° providing directions a more for crackfavourable growth. condition This mechanism for crack involvesgrowth the directionat the initial oriented phase at of 45 stage◦ providing I. a more favourable condition for crack growth at Stagethe II: initialCrack phasepropagation of stage I. Stage(d) II: CrackThe plastic propagation blunting process [24–27] has asymmetrical effects in the tensile and compressive loading cycles. This mechanism describes crack-growth behaviour in one loading cycle. The (d) The plastictensile loading blunting blunts process the [ 24crack–27 ]tip, has and asymmetrical the new crack effects surface in the is created tensile anddue compressiveto a shearing loadingeffect. cycles. Then, when This mechanismthe loading is describes reversed, crack-growththe surface created behaviour under tensile in one loading loading remains cycle. The tensile(crack extension) loading blunts becaus thee crackof a crushing tip, and effect. the new crack surface is created due to a shearing (e)effect. The Then, crack when tip plastic the loading zone [28–30] is reversed, enhances the crack surface growth. created This under mechanism tensile involves loading remainsa highly (cracklocalised extension) stress because around of athe crushing crack tip. effect. This, on the one hand, results in plastic deformation (e) The crack(cracktip opening) plastic at zone the [crack28–30 tip;] enhances on the other crack hand, growth. it gives This a mechanism more favourable involves situation a highly for localiseddiffusion. stress around the crack tip. This, on the one hand, results in plastic deformation (crack(f) Diffusion opening) creep at the [12,13] crack provides tip; on thea mechanism other hand, that it leads gives to a moregrain favourableelongation and situation then for diffusion.further crack opening. In addition, since atoms diffuse from a high- to a low-concentration (f) Diffusionregion, creep more [12 vacancies,13] provides are generated a mechanism and conver that leadsged at to the grain crack elongation tip, where and highly then localised further crackstress opening. is presented, In addition, which since provides atoms a more diffuse favourable from a high- situation to a for low-concentration creep damage. region,

Metals 2018, 8, 623 5 of 32

more vacancies are generated and converged at the crack tip, where highly localised stress is presented, which provides a more favourable situation for creep damage. (g) Existing precipitates [31] cause fatigue resistance. The mechanism is mainly that precipitates are the obstacle to dislocation, and hence restrain the process of crack propagation. (h) Crack deflection (change of direction) [32] preferentially occurs at the grain boundary. This is determined by the twist angle and the tile angle. (i) Triple-points [14–16] provide a mechanism for significant stress concentration, which provides a more favourable stress field for creep damage. (j) A region with a high density of micro-cracks is particularly weak, and supports crack-branching activities [33]. (k) The slip bands within grains [34] may cause the crack to re-direct within grains.

Stage III: Structural failure

(l) The plastic energy [23] available exceeds the need for producing the new crack surface. The excess energy is applied to form voids, and thus further worsens creep-fatigue resistance.

2.3. Gaps in the Body of Knowledge Although the existing literature [22,23] indicates three stages of crack growth from initiation to failure, the implications of this for the behaviour of the crack at the grain-boundary effect is unclear. The grain boundary is known to play an important role in both fatigue-crack and creep-crack growth in different ways; hence, a grain-boundary effect cannot be ignored. Despite the apparent comprehensiveness of the above list of mechanisms, the literature contains many disparate ﬁndings without an obvious integration into a holistic theory for the crack propagation process. Hence, there is a need to develop a framework to link these details together, to propose a coherent set of mechanisms for the steady stage of crack growth at the grain level, and to involve grain-boundary effects. This ideally needs to consider all the loading effects (fatigue, creep, and creep fatigue) at the microstructural level.

3. Approach The approach in the present work was to start with existing principles of fatigue-crack and creep-crack growth (Section 2.2), and then conceptually develop a microstructural model of the creep-fatigue crack-growth process. This was achieved by imagining the process from the perspective of atoms in orderly grains and at the grain boundaries (where the orderly arrangement between atoms breaks down). The bonds between atoms inside a grain are strong, relative to the bonds between atoms at grain boundaries. We sought to identify a coherent set of mechanisms that would affect the bonds between atoms, for all the loading regimes. We integrated multiple existing concepts at the microstructural level, such as the plastic blunting process, the mechanisms of dislocation, and the diffusion creep process. These processes are extant in the literature, and our contribution is to integrate and apply them to the ﬁne scale to give a new understanding of the interaction between the underlying mechanisms. In addition, a number of new mechanisms for crack growth were also proposed to construct a comprehensive conceptual framework. Since the principles of crack growth of fatigue and creep are different (the fatigue effect occurs via cracks through the grains, while the creep effect involves the grain boundary cracking), the fatigue and creep crack-growth behaviours were discussed and illustrated separately. In this microstructural conceptual framework, three stages of crack growth from crack initiation to structural fracture were included. The second stage of steady crack growth is the primary focus. This is where the majority of the lifetime is consumed. It also presents multiple situations regarding the position of the crack relative to the grain boundary. The result is a proposed theory of the failure mechanisms, and this is represented in the form of a schematic representation, see Section4. Metals 2018, 8, 623 6 of 32

4. Results The microstructural conceptual framework mainly focuses on the second stage of crack growth. On the one hand, this is because this stage presents a stable process of crack growth and is numerically described by the well-known Paris’ Law [11]. On the other hand, this is because the majority of the lifetime is consumed at the second stage of crack growth for low cycle fatigue, and this fatigue regime is an important consideration when the creep effect is activated during the design process [8]. In this framework, the crack-growth behaviours under the tensile partition and compressional partition are discussed separately, and crack-growth behaviours caused by the fatigue effect and creep effect are described separately.

4.1. New Propositions The challenge is showing how these multiple premises are integrated together at the microstructural level. There are no models and conceptual frameworks in the literature that provide such integration. To solve this problem, we propose several new effects. Our propositions of mechanisms that operate at the microstructural level during crack growth are:

A That energy dynamics explain the crack initiation and the following unstable phase, and is based on a liberation of energy due to coalescence of dislocations. In general, when the internal energy stored in the structure due to cyclic loading arrives at a critical value, the barrier to initiate the crack is overwhelmed. After this, the released energy is gradually consumed to produce new crack surfaces at an accelerated crack growth rate. B That grain-mismatch occurs due to grain elongation under the tensile loading, resulting in relative movement between two neighbouring grains. Then, the shear stress along the grain boundary is increased and a weaker region along the grain boundary is created. This results in a mismatch band at the grain boundary. This widens the crack body and enhances its growth. C That bonding crushing effects exist at the ﬁner scale. During the process of compression, the atomic bonds, which are distorted and rearranged in the tensile phase, may be further damaged and become potential failure sites for next loading cycle. This process is irreversible, since the atoms cannot return to their original position under the compressional loading. D That a crack net is caused by the aggregation of micro-cracks. This crack net probes a larger volume of material for weaknesses, and then promotes the main crack. E That irregular conﬁguration of the grain boundary causes stress/strain pile up at the grain boundary, and then results in a large driving force for extending the crack tip into the neighbouring grain.

Furthermore, it is proposed that these mechanisms are integrated by the following assumptions:

F That the primary mechanism for failure in the creep-fatigue loading regime is crack growth by mechanisms of crack blunting (including shearing and crushing); hence, fatigue effects. G That the supporting mechanisms that augment the extent of damage are grain elongation and diffusion; hence, creep effects.

We formulated these propositions as part of an iterative process of seeking a consolidated framework. Hence, the above are not so much lemmas that were asserted at the outset, but rather principles that were acquired during the framework development. The resulting conceptual framework, shown below, is a fusion of the principles (Section 2.2) and propositions (Section 4.1) into a more extensive conceptual framework. Metals 2018, 8, 623 7 of 32

4.2. Conceptual Framework of Temporal Development of Crack-Growth for Creep Fatigue

4.2.1. Stage I: Crack Initiation The ﬁrst stage (stage I) is that of crack initiation. We summarise the crack initiation mechanics as follows. Fundamentally, crack initiation is caused by stress concentration, where two situations are presented. A typical situation is that the crack starts at a surface defect such as a machining mark. This provides a highly localised stress concentration, hence a pre-existing micro-crack [8]. By this means,Metals a2018 crack, 8, x isFOR initiated PEER REVIEW at this speciﬁc point, and then the localised plastic strain caused by the7 of 34 ductile micro-tearing events provides opportunity for further propagation. Second,Second, for for a situation a situation with with perfect perfect surfaces surfaces (defect-free), (defect-free), dislocations dislocations play anplay important an important role [19 role] for[19] the initiationfor the initiation of a fatigue of a crackfatigue [35 crack] (Figure [35] 1.9 (Figure therein), 1.9 [therein),36] (Figure [36] 1 therein).(Figure 1 During therein). this During process, this loadingprocess, cycles loading cause cycles dislocations cause dislocations to pile up at to the pile microstructural up at the microstructural level. These dislocationslevel. These aredislocations in the crystalare in lattice the crystal at the nanoscale.lattice at the Under nanoscale. loading, Under they lo alignading, and they coalesce align toand produce coalesce thicker to produce slip planes thicker throughslip planes the crystal through and the eventually crystal and slip bandseventually at the slip macrostructural bands at the levelmacrostructural [37]. Under level cyclic [37]. loading, Under therecyclic is moreloading, relative there movement is more relative between movement the bands, between so the effect the bands, becomes so more the effect pronounced—the becomes more bandspronounced—the are further displaced. bands are Wefurther suggest displaced. that this We is suggest due to that irreversibility this is due causedto irreversibility by the localised caused by hardeningthe localised effects hardening of the dislocations. effects of the dislocations. In theIn the area area of persistentof persistent slip slip bands, bands, the the slip slip planes planes extrude extrude or or intrude intrude to to the the surface surface of of the the object, object, seesee Figure Figure4. This4. This results results in tinyin tiny steps steps in thein the surface, surface, where where the the stress stress concentration concentration then then results results in in crackcrack nucleation. nucleation.

FigureFigure 4. Crack 4. Crack initiation initiation due due to dislocation.to dislocation.

InIn stage stage I (the I (the crack-initiation crack-initiation stage), stage), a crack-growth a crack-growth threshold threshold exists. exists. In In general, general, there there are are two two typestypes of of thresholds thresholds [38 [38]:]: microstructural microstructural and and mechanical. mechanical. The microstructural threshold threshold describes describes the theinitiation initiation of of micro-cracks micro-cracks at at diff differenterent microstructural microstructural features, features, such such as as slip slip bounds bounds and and cavities. cavities. The Themechanical mechanical threshold threshold is is related related to macroscopic geometrygeometry forfor long long crack crack growth, growth, and and is discussedis discussed in in thethe present present work. work. Below Below this this latter latter threshold, threshold, there there is ordinarily is ordinarily no crack no crack growth growth [8]. We [8]. attribute We attribute this to thethis cracks to the being cracks too being short totoo result short in to a meaningfulresult in a stressmeaningful concentration stress concentration effect (which iseffect primarily (which an is effectprimarily of proportional an effect geometry).of proportional Also, ifgeometry). there are manyAlso, superﬁcialif there are defects many at superficial the surface—which defects at isthe usuallysurface—which the case—then is usually the material the case does—then not the have material a geometrically does not have exact a geometrically boundary at the exact nanoscale, boundary andat hencethe nanoscale, the stress and avoids hence this the region: stress the avoids many this small region: defects the effectively many small decrease defects the effectively stiffness decrease of the superﬁcialthe stiffness layer, of andthe hencesuperficial this layerlayer, is and to some hence extent this layer unloaded. is to some extent unloaded. AfterAfter the the threshold threshold stress stress intensityintensity isis exceeded, there there is is a aregime regime of ofhigh high acceleration acceleration of crackof crack-growthgrowth rate. rate. Then, Then, following following this this high-acceleration high-acceleration regime, regime, a atransition transition to to a a slower slower acceleration acceleration of of crack-growth rate is presented.presented. OnOn the the one one hand, hand, thisthis processprocess is is attributed attributed to to the the concept concept of ofstrain strain energy energy release release rate, rate,which whichdescribes describes the energy the energy consumed consumed per unit per of unit newly of newly created created crack cracksurface. surface. This implies This implies that the that more thestrain more energy strain energyis released, is released, the more the new more crack new cracksurfaces surfaces are produced. are produced. Normally, Normally, the strain the strain energy energyrelease release rate ratecould could be numerically be numerically related related to the to the M-integral M-integral [20,21]. [20,21 ].The The relationship relationship between between the thenumber number of of cycles cycles and and M-integral M-integral was was investigated investigated by by Margaritis Margaritis [39] [39 ](Figure (Figure 7 7 therein), therein), where where the the M- integral increases with the increasing number of cycles at the early phase, then it decreases over the peak, and finally it reaches a constant situation. This trend is consistent with observation of the initiation stage shown in Figure 1. Specifically, after the threshold, increased energy (released strain energy) accelerates the increase of crack-growth rate at the beginning phase, then reduced energy retards the acceleration of the crack-growth rate, and finally a steadily increasing crack-growth rate is achieved under a situation with constant energy (this is viewed as the crack-propagation stage). We suggest that the underlying energy dynamics are based on a rapid liberation of energy due to coalescence of strain. Specifically, the internal energy, which is stored in the structure in the form of the irreversible formation of dislocations that increase in number due to cyclic loading. Then, when this energy arrives at a critical value, the vacancies in the dislocations coalesce and form a crack— hence breaking through the barrier to initiate the crack. This qualitative leap from perfection to

Metals 2018, 8, 623 8 of 32

M-integral increases with the increasing number of cycles at the early phase, then it decreases over the peak, and finally it reaches a constant situation. This trend is consistent with observation of the initiation stage shown in Figure1. Specifically, after the threshold, increased energy (released strain energy) accelerates the increase of crack-growth rate at the beginning phase, then reduced energy retards the acceleration of the crack-growth rate, and finally a steadily increasing crack-growth rate is achieved under a situation with constant energy (this is viewed as the crack-propagation stage). We suggest that the underlying energy dynamics are based on a rapid liberation of energy due to coalescence of strain. Specifically, the internal energy, which is stored in the structure in the form of the irreversible formation of dislocations that increase in number due to cyclic loading. Then, when this energy arrives at a critical value, the vacancies in the dislocations coalesce and form a crack—hence breaking through the barrier to initiate the crack. This qualitative leap from perfection to imperfection requires internal energy. This process is accomplished during a short time period by releasing large amounts of energy. After this, the released energy is consumed to produce new crack surfaces in a high acceleration of crack-growth rate, and the whole system is forced into an unsteady condition. Naturally, the need to re-balance this system gradually reduces the amount of released energy (caused by applied loading) until a balanced condition is reached. During this process, the high acceleration of crack-growth rate is gradually retarded, and then this acceleration reaches stabilisation (the whole system is re-balanced). The growth of the crack causes further large-scale deformation of the grain as a whole, which causes more dislocations to occur within the grain, and repeats the cycle. An alternative explanation is that the unstable behaviour of crack growth in stage I is due to shear mechanisms [22,23]. Generally, a crack is propagated through two different methods: the plastic deformation around the crack tip and the shear-stress effect at the planes oriented at 45◦ to the loading direction. During the initial phase of crack growth, a small plastic zone and a small stress field are presented around the crack tip because of the small magnitude of stress intensity. In this case, the mechanism of plastic deformation around the crack tip may not be significant enough to become the driving force for crack growth. Instead, the shear stress at the planes oriented at 45◦ provides more favourable conditions for crack growth. This is because the shear stress at these planes and the relative movements between these planes under cyclic loading provide more vulnerable areas for crack growth. Prior damage and lattice imperfections may exist on these planes from previous loading cycles. In this case, a crack grows along the planes oriented at 45◦ with a minimum of effort, hence resulting in a high acceleration of the crack-growth rate. As the stress intensifies, the shear-stress effect is gradually suppressed by the plastic-deformation mechanism; hence, the acceleration of crack-growth rate is reduced. This is because, during this process, the direction of the crack growth gradually deviates from the surface of the planes oriented at 45◦, and then the behaviour of penetrating the grain boundary results in the deceleration of crack-growth rate. Finally, crack-growth behaviour is stably caused by the crack-tip plastic zone, and a steady situation is achieved (stage II is initiated). We suggest that both mechanisms apply, and complement each other. The key elements (mechanisms) in the crack initial stage are presented by Figure5. We summarise the crack initiation mechanics as follows, see Figure6. Crack nucleation starts at the surface due to highly localised stress concentration. In the case of smooth surfaces, it is attributed to the presence of persistent slip bands (extrusion and intrusion). Stage I (the stage of crack initiation) has a threshold value, below which no crack growth occurs. Once a crack is initiated, it grows quickly at the early phase, then the acceleration of crack-growth rate is reduced, and the crack-growth behaviour finally becomes steady (see next section). This process is proposed to be underpinned by the above mechanisms of strain energy release and shear stress. Metals 2018, 8, 623 9 of 32 Metals 2018, 8, x FOR PEER REVIEW 9 of 34

Metallurgical defect Surface defect Perfect surface Irreversibility caused Stage I by the localised Crack Initiation hardening effects of Dislocation pileup the dislocations

Stress Slip planes Concentration at Stress-intensity Tiny steps in the threshold (an effect of surface or in surface proportional geometry) substrate Energy dynamics: a rapid liberation of strain energy

Irreversible formation Shear & plastic deformation mechanisms Crack nucleation of dislocations due to cyclic loading Shear-stress at the Initial crack growth is planes oriented at on the 45o planes 45o to the load Accumulation of internal strain energy in Increased localised multiple dislocations Possible vulnerability stress intensity at Crack initial growth at due to prior damage crack tip due to high acceleration The vacancies in the lengthening crack Critical value dislocations coalesce of energy and form a crack Plastic-deformation Crack growth Acceleration of crack- mechanism deviates from the growth rate is dominates at higher planes oriented at gradually retarded Released energy is stress intensity 45o consumed to produce new crack surfaces

Crack penetrates and Crack growth in a extends beyond the System rebalance grain boundary steady situcation

Figure 5. Key elements (mechanisms) in the crack initial stage. Metals 2018, 8, x FOR PEERFigure REVIEW 5. Key elements (mechanisms) in the crack initial stage. 10 of 34 We summarise the crack initiation mechanics as follows, see Figure 6. Crack nucleation starts at Loading the surface due to highlyElevated localised Temperature stress concentration. In the case of smooth surfaces, it is attributed to the presence of persistent slip bands (extrusion and intrusion). Stage I (the stage of crack initiation) has a thresholdTensile value, below which no crack growth occurs. Once a crack is initiated, it grows quickly Time at the early phase, then the acceleration of crack-growth rate is reduced, and the crack-growth behaviour finallyCompression becomes steady (see next section). This process is proposed to be underpinned by Fatigue makes more contribution to damage than the above mechanisms of straincreep underenergy the reversedrelease loading and shearwithout stress.hold time.

Grain level Stage I Crack nucleation due to persistent slip bands Log (extrusion and intrusion)

Crack starts at surface defect III III because of highly localized stress concentration

Stage I indicates a threshold value, below which there is no crack growth Threshold

Unstable stage due to mechanisms of energy release and shear stress

FigureFigure 6.6. StageStage I: Crack initiation.

4.2.2. Stage II: Crack Propagation

The second stage (stage II) shown in Figure 1 presents the behaviour of crack propagation, wherein the crack growth undergoes a relatively steady process. This stage is numerically presented by Paris’ Law (Equation (1)) [11], which shows a power-law relationship between the crack growth rate and the range of the stress intensity factor during the fatigue cycle: =∆ (1) where is the crack growth rate; ∆ is the effective stress intensify factor, which is identified as the difference between maximum and minimum stress intensify factors for one cycle; is the stress intensify factor; is the crack growth; is the number of cycles; and C and m are constants. The crack-growth process of creep fatigue is mainly discussed under the second stage of crack growth, and this discussion is based on the idea of the plastic blunting process [24–27]. At the beginning of the loading cycle, the crack tip is sharp (Figure 7a). Then, the applied tensile stress leads to crack opening, and the stress concentration at crack tip causes the slip along planes at 45° (Figure 7b). This leads to the gradual increase of crack-tip area, and finally the crack tip widens to its maximum plastic deformation (Figure 7c) due to the plastic shear effect. At the same time, the crack tip becomes blunt and grows longer because a new crack surface is created. When the loading is changed to a compressional situation, the crack is closed, and the slip at the crack tip is reversed (Figure 7d). Finally, the crack surfaces are crushed together, and the new crack surface created under tensile loading remains (Figure 7e) (crack extension). Consequently, the sub- cracks promoted by each loading cycle are accumulated, and the structure fails as the total damage achieves a critical value.

Metals 2018, 8, 623 10 of 32

4.2.2. Stage II: Crack Propagation The second stage (stage II) shown in Figure1 presents the behaviour of crack propagation, wherein the crack growth undergoes a relatively steady process. This stage is numerically presented by Paris’ Law (Equation (1)) [11], which shows a power-law relationship between the crack growth rate and the range of the stress intensity factor during the fatigue cycle:

da = C(∆K)m (1) dn

da where dn is the crack growth rate; ∆K is the effective stress intensify factor, which is identiﬁed as the difference between maximum and minimum stress intensify factors for one cycle; K is the stress intensify factor; a is the crack growth; n is the number of cycles; and C and m are constants. The crack-growth process of creep fatigue is mainly discussed under the second stage of crack growth, and this discussion is based on the idea of the plastic blunting process [24–27]. At the beginning of the loading cycle, the crack tip is sharp (Figure7a). Then, the applied tensile stress leads to crack opening, and the stress concentration at crack tip causes the slip along planes at 45◦ (Figure7b). This leads to the gradual increase of crack-tip area, and ﬁnally the crack tip widens to its maximum plastic deformation (Figure7c) due to the plastic shear effect. At the same time, the crack tipMetals becomes 2018, 8, bluntx FOR andPEER grows REVIEW longer because a new crack surface is created. 11 of 34

FigureFigure 7. 7.Plastic Plastic blunting blunting process. process.

WhenThe idea the loadingof plastic is changedblunting toprocess a compressional was integrated situation, with thea general crack is understanding closed, and the of slip creep at themechanism crack tip isto reversedshow crack-growth (Figure7d). behaviour Finally, the of crack creep surfaces fatigue. are The crushed process together, of crack and propagation the new crack(Figures surface 8–19) created is described under (at tensile a grain-based loading remainslevel) by (Figure three typical7e) (crack situations: extension). (1) crack Consequently, tip within a thegrain sub-cracks (Section promoted4.2.2.1); (2) by crack each tip loading at the cycle grain are boundary accumulated, (Section and 4.2.2.2 the structure (1)) or at fails the astriple the totalpoint damage(Section achieves 4.2.2.2 (2)); a critical and (3) value. crack tip through grain boundary (Section 4.2.2.3). TheIn the idea present of plastic work, blunting fatigue processis identified was as integrated crack growth with by a general overloaded understanding local atomic bonds of creep, and mechanismcreep is regarded to show as crack-growththe flow of atoms behaviour relative to ofeach creep other fatigue. with transfer Theprocess of bonding of crack to new propagation partners. (Figures8–19) is described (at a grain-based level) by three typical situations: (1) crack tip within a grain4.2.2.1. (Section Crack 4.2.2.1Tip within); (2) a crack Grain tip at the grain boundary (Section 4.2.2.2 (1)) or at the triple point (Section. Conceptual 4.2.2.2 (2)); framework and (3) crack for tipcrack through growth grain in the boundary tensile regime (Section 4.2.2.3). In the present work, fatigue is identiﬁed as crack growth by overloaded local atomic bonds, and creep Under tensile loading (Figure 8), the fatigue component follows the idea of the plastic blunting is regarded as the ﬂow of atoms relative to each other with transfer of bonding to new partners. process. In this process (1A in Figure 8), tensile stress leads to crack opening, and shear behaviour is presented at the crack tip. Then, by the plastic shear effect, the new surface is created; meanwhile, the crack tip widens and becomes blunt (the blunting behaviour was illustrated in Figure 12 in Ref. [40] and Figure 32 in Ref. [41], where a stretch region is presented). Diffusion creep is identified as the main creep mechanism for creep-fatigue damage in the present work. It is generally believed that diffusion creep leads to grain elongation, which then results in a further opportunity for crack opening (1B in Figure 8).

Metals 2018, 8, 623 11 of 32 Metals 2018, 8, x FOR PEER REVIEW 12 of 34

Stage II

Situation 1: Tip of crack within grain-Tensile loading Stage II N cycles Loading Log Elevated Loading Temperature (tensile) Per each load cycle III III Time 1A Tensile stress Crack Crack opening Crack Threshold Propagation opening Plastic shearing Plastic shearing New surface Loading (tensile) Tip widens and Tip widens and Loading Fatigue bluntens bluntens (tensile) Plastic blunting process A

Crack extends in length ❹ Creep-fatigue Fatigue capacity reduces Intrinsic defects in grain due to creep damage, then may also open in tension takes the form of (1-X) Interaction Tip becomes blunter Grain elongation 1B Diffusion Creep Grain elongation Creep Crack opening Crack Diffusion Temperature opening ❶ Linear relation with plastic strain Loading Crack tip plastic zone Natural exponential relation with stress (tensile) Localised stress around Higher temperature leads to more creep tip causes increased creep Cyclic time Creep ❷ by increased diffusion Logarithmic relation with temperature Long cyclic time causes more creep

Grain size ❸ Power-law relation with applied loading Grain elongation Small grain size is good for fatigue Big grain size is beneficial for creep

Figure 8. Crack tip within a grain—Tensile loading. Metals 2018, 8, x FOR PEER REVIEWFigure 8. Crack tip within a grain—Tensile loading. 13 of 34 In addition, diffusion creep has a dependency on applied loading, and larger loading produces more creep damage. Specifically, the idea of a crack tip plastic zone [28–30] indicates that the localised stress is highly significant around the crack tip, and gradually decreases in the direction of crack growth (Figure 9). In particular, under the theory of linear elasticity, stress distribution around the crack tip can be numerically presented as (Equation (2)): = = (2) 2 √2

where is the stress distribution, is the crack length, K is the stress intensity factor, and and are polar coordinates. This equation encapsulates a stress-concentration phenomenon around the crack tip. This relation is commonly accepted as being applicable to the creep-fatigue condition on the assumption that the crack-tip plastic zone is small. Higher localised stress around the crack tip causes more creep damage due to increased diffusion, and thus contributes to crack opening. Specifically, atoms normally diffuse from a region of high concentration to a region of low Figure 9. Stress around crack tip. concentration. In this case, more vacanciesFigure 9. areStress generated around crackand converged tip. at the crack tip, and then a more favourable situation for creep damage is presented. Consequently, the crack tip surface caused by the fatigue effect is further extended and becomes blunter due to the creep effect (1C in Figure 8), which implies the combined effects of fatigue and creep. A defected-related state, such as intrinsic defects, around the crack tip may also promote crack growth (negative to fatigue resistance), where the weakest zone of atomic bonds is represented. An example of presenting a defected-related effect is shown in Figure 32 from [41], where the crack is promoted due to void coalescence and the bridges between voids are sheared off. In addition, precipitates may also be considered to be defected-related effects, but ones which are normally beneficial to fatigue resistance. On the one hand, this benefit results from the obstacle to further dislocation [31], whereby the precipitates restrain the process of crack propagation. On the other hand, this benefit could also be explained by the fatigue-softening behaviour [42]. Generally, softening behaviour is caused by disordering the stacking sequence of precipitates due to reversed loading, whereby the crack encounters a tougher situation to penetrate this area. In addition, the fatigue-softening behaviour can also be attributed to the reduced area intercepted by the dislocation of the slip planes due to the gradual consumption of precipitates under the reversed loading (the precipitates are cut into smaller sizes under reversed loading, and thus are dissolved). Creep effect has strong dependencies on temperature, cyclic time and grain size [43]. Generally, a higher temperature leads to more creep damage, and a linear relation between temperature and applied loading is presented (❶ in Figure 8). In addition, a longer cycle time gives more creep damage, and a logarithmic relation between cyclic time and temperature is presented (❷ in Figure 8). Furthermore, a small grain size is positive for fatigue while big grain size is beneficial for creep resistance, and a power-law relationship between grain size and applied loading is presented (❸ in Figure 8). Consequently, the negative creep effect caused by elevated temperature, or prolonged cyclic time, or smaller grain size gives weaker atomic bonds; then, this results in adversely affected material. The concept of ‘fatigue capacity’ indicates that the total fatigue capacity is gradually consumed by creep effect. This process is numerically presented by taking the form of “1-X” (❹ in Figure 8) [43,44]. . Conceptual framework for crack growth in the compressional regime Under compressional loading, crack closure is presented based on the idea of the plastic blunting process, see Figure 10. In this process, the crack surface, especially the tip zone, is crushed together. Since the crushing process is inelastic and irreversible, the new crack surface created under tensile loading remains (1E in Figure 10). This implies that compressional loading also contributes to crack growth.

Metals 2018, 8, 623 12 of 32 Metals 2018, 8, x FOR PEER REVIEW 14 of 34

Stage II

Situation 1: Tip of crack within grain-Compressional loading Stage II N cycles Loading Log Elevated Temperature

Time III III

Crack Threshold Propagation

Loading Loading (tensile) (compression) Plastic blunting process A Plastic blunting process B 1D 1C

Crack extends in length Crack Creep-fatigue closing Tip crush (Unreversed) Intrinsic defects in grain A un-stressed state may also open in tension New crack surface (defects) Crack extension through grain Tip becomes blunter Potential for further damage

1E Crack closing Tip crush Un-stressed state

Loading New crack surface (compression) Crack extension (through grain)

Figure 10. Crack tip within a grain—Compressional loading. Figure 10. Crack tip within a grain—Compressional loading. This phenomenon may be explained by the observation of the crack tip opening displacements Metals4.2.2.1. 2018, 8, x Crack FOR PEER Tip REVIEW within a Grain 16 of 34 [45,46] under compressional loading, where the crack is not entirely closed in this loading regime due .to theConceptualConceptual plastic effect framework framework (this is forillustrated for crack crack growth growthin Figure in in the 3the of tensile compressionalRef. [45]). regime regime In addition, the contribution of compressional loading may also be explained by the behaviour In theUnder situation tensile with loading compressional (Figure8 ),loading the fatigue (Figure component 12), the same follows creep-fatigue the idea of theprocess plastic shown blunting of atomic bonds. Specifically, creep fatigue does its damage in the tensile phase by changing the in ‘situationprocess. 1-1E’ In this is presented, process (1A where in Figure reversed8), tensile loading stress crushes leads the to material crack opening, ahead of and the shear crack, behaviour even landscape of atom arrangements and hence the bond vulnerability (which implies the bonds are acrossis the presented grain boundary. at the crack As tip. a result, Then, the by thenew plastic crack surface shear effect, caused the by new the surface tensile isloading created; remains, meanwhile, extended and become weaker). During the process of compression, because of the crushing effect, due tothe the crack tip-crush tip widens effect, and and becomes the crack blunt is further (the blunting extended behaviour across the was grain illustrated boundary. in Figure 12 in Ref. [40] some of these partly broken bonds may be further damaged and become potential failure sites for the and Figure 32 in Ref. [41], where a stretch region is presented). Diffusion creep is identified as the main next loading cycle, while others may be totally damaged and then lead to crack extension. This is creepStage mechanism II for creep-fatigue damage in the present work. It is generally believed that diffusion significantly not a reversible process, since the atoms cannot return to their original positions after creep leads to grain elongation, which then results in a further opportunity for crack opening (1B in one tension-compression cycle. Furthermore, the defect-related state around the crack tip may further FigureSituation8). 2-1: Tip of crack at grain boundary-Compressional loading propagate cracks due to the crushing effect. Specifically, the crushing effectLog causes the crack tip to be In addition, diffusion creep has a dependency on applied loading, and larger loading produces squeezed, and then the tipLoading is possibly extruded towards the zone withStage the II weak atomic bonds. In this more creep damage. Specifically, the idea of a crack tip plastic zone [28–30] indicatesIII that theIII localised case, the crack is further extended (1D Elevatedin Figure 10) by cutting these vulnerable bonds with the stress is highly significant around theTemperature crack tip, and gradually decreases in the directionCrack of crack assistance of the crushing effect. Consequently, the sub-crack under oneThreshold loadingPropagation cycle is formed, growth (Figure9). In particular, under the theory of linear elasticity, stress distribution around the which contributes to the final fracture. Time crack tip can be numerically presented as (Equation (2)): After N cycles, the accumulated crack arrives at thePlastic grain blunting boundary process orB it encounters the triple r point. This results in the following situation. a K Crack deflection σ = σ f (θ) = √ f (θ) (2) ij r Loading Loading 2 2πr (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Same mechanism as 1E Zone of material where bonds are partly broken Crack closing Creep-fatigue Tip crush A un-stressed state Un-stressed state Tip becomes blunter (defects) OR New crack surface Crack extension (through grain)

Tip crush (Unreversed) Crack closing New crack surface Loading Crack extension along (compression) grain boundary 2F-1 Crack closing Tip crush Weak boundary Potential crack growth direction New crack surface Arriving at triple point Crack extension (along grain boundary) Penetrating into neighbouring grain

Figure 12. Crack tip at grain boundary—Compressional loading.

In particular, the crack path may be deflected (this behaviour is illustrated in Figure 12 of Ref. [49]) at the boundary. Specifically, crack deflection at the grain boundary is generally determined by the twist angle (a) between two slip planes on the grain boundary and the tilt angle (b) between the intersection lines of two slip planes on the cross section [32]. This behaviour is illustrated in Figure 13, where the crack path is redirected at the grain boundary of the sample surface.

Metals 2018, 8, x FOR PEER REVIEW 15 of 34

4.2.2.2. Crack Tip at Grain Boundary or Triple Point (1) Crack tip at grain boundary . Conceptual framework for crack growth in the tensile regime Under tensile stress (Figure 11), the fatigue component experiences the same process shown in ‘situation 1-1A’, wherein the new crack surface is created due to the plastic shearing effect. Then, this Metals 2018, 8, 623 13 of 32 causes the crack tip to widen and become blunt. For the creep component (2B-1 in Figure 11), diffusion creep results in grain elongation along the stress direction. On the one hand, this results in wherecrack opening,σij is the which stress was distribution, discusseda inis Section the crack 4.2.2. length,1; onK theis other the stress hand, intensity shear stress, factor, which and rresultsand θ fromare polar grain coordinates. elongation, leads This equationto elongation encapsulates of the grain a stress-concentration boundary, which then phenomenon causes the aroundcrack tip the to widen;crack tip. meanwhile, This relation a mismatch is commonly band accepted appears as[47,48]. being Specifically, applicable to the the grain creep-fatigue boundary condition presents a on zone the ofassumption irregular atomic that the bonds, crack-tip and plastic thus the zone bonds is small. at the Higher grain localised boundary stress presen aroundt anisotropic the crack behaviour. tip causes Therefore,more creep the damage relative due movement to increased of points diffusion, caused and by thus grain contributes elongation to at crack the opening.grain boundary Speciﬁcally, may presentatoms normally different diffuseresults, from the extension a region ofor highbreakage concentration of atomic tobonds. a region This of results low concentration. in a zone of material In this wherecase, more bonds vacancies are partly are generatedweakened, and and converged provides atpo thetential crack for tip, further and then damage. a more Consequently, favourable situation creep intensifiesfor creep damage the plastic is presented. blunting process caused by the fatigue effect, and a blunter crack tip arises.

Stage II

Situation 2-1: Tip of crack at grain boundary-Tensile loading Stage II Loading Log Elevated Temperature Loading Same mechanism as (tensile) III Time 1A III Tensile stress Crack Crack opening Threshold Propagation Crack Plastic shearing opening New surface Loading Tip widens and Loading (tensile) (tensile) Tip widens and blunts bluntens Plastic blunting process A Fatigue

Zone of material where bonds are partly broken Creep-fatigue

Grain elongation Tip becomes blunter Interaction

Crack Tip becomes blunter opening Wider crack tip due to shear at grain boundary Temperature Mismatch band Loading 2B-1 Cyclic time (tensile) Creep Diffusion creep Creep Diffusion Grain elongation Grain size Crack opening Weaker atomic bonds Shear at boundary Grain elongation Tip widens and blunts Adversely affected material

Figure 11. Crack tip at grain boundary—Tensile loading. Figure 11. Crack tip at grain boundary—Tensile loading.

Stage II

Situation 2-1: Tip of crack at grain boundary-Compressional loading Log

Loading Metals 2018, 8, x FOR PEER REVIEW Stage II III13III of 34 MetalsMetals 20182018,, 88,, xx FORFOR PEERPEER REVIEWREVIEW Elevated 1313 ofof 3434 Temperature Crack Threshold Propagation Time

Plastic blunting process B

Crack deflection

Loading Loading (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Same mechanism as 1E Zone of material where bonds are partly broken Crack closing Creep-fatigue Tip crush Figure 9. Stress around crack tip. FigureFigureA un-stressed 9.9. StressStress state aroundaround crackcrack tip.tip. Un-stressed state Tip becomes blunter (defects) OR New crack surface Consequently, the crack tip surface caused by the fatigue effect is further extended and becomes Consequently,Consequently, thethe crackcrack tiptip surfacesurface causedcaused byby thethe fatiguefatigue effecteffect isis furtherfurther extendedextendedCrack extension andand becomesbecomes blunter due to the creep effect (1C in Figure 8), which implies the combined effects(through of fatigue grain) and blunterblunter duedue toto thethe creepcreep effecteffect (1C(1C inin FigureFigure 8)8),, whichwhich impliesimplies thethe combinedcombined effectseffects ofof fatiguefatigue andand creep. A defected-related state, such as intrinsic defects, around the crack tip may also promote crack creep.creep. AA defected-relateddefected-related state,state, suchsuch asas intrinsicintrinsic dedefects,fects, aroundaround thetheTip crackcrushcrack (Unreversed) tiptip maymay alsoalso promotepromote crackcrack growthgrowth (negative(negative toto fatiguefatigue resistance),resistance), wherewhere thethe weakestweakestCrack zonezone ofof atomicatomic bondsbonds isis represented.represented. AnAn growth (negative to fatigue resistance), where the weakestclosing zone of atomicNew crackbonds surface is represented. An exampleexample ofof presentingpresenting aa defected-relateddefected-relatedLoading effecteffect isis shownshown inin FigureFigure 3232Crack fromfrom extension [41],[41], along wherewhere thethe crackcrack isis example of presenting a defected-related(compression) effect is shown in Figure 32 from [41], where the crack is promoted due to void coalescence and the bridges between voids aregrain sheared boundary off.2F-1 In addition, promotedpromoted duedue toto voidvoid coalescencecoalescence andand thethe bridgebridgess betweenbetween voidsvoids areare shearedsheared off.off. InIn addition,addition, precipitates may also be considered to be defected-related effects, but ones Crackwhich closing are normally precipitatesprecipitates maymay alsoalso bebe consideredconsidered toto bebe defectdefected-relateded-related effects,effects, butbut onesones whichwhich areare normallynormally beneficial to fatigue resistance. On the one hand, this benefit results from the Tipobstacle crush to further beneficialbeneficial toto fatiguefatigue resistance.resistance. OnOn thethe oneone hand,hand, thisthis benefitbenefit resultsresults fromfrom thethe obstacleobstacle toto furtherfurther dislocation [31], whereby the precipitates restrain the process of crack propagation.Weak boundary On the other dislocationdislocation [31],[31], wherebywhereby thethe precipprecipitatesitates restrainrestrain thethe processprocess ofof crackcrack propagation.propagation. OnOn thethe otherother hand, this benefit could also be explained Potentialby the crack fatigue-softening growth direction behaviourNew crack [42]. surface Generally, hand,hand, thisthis benefitbenefit couldcould alsoalso bebe explainedexplained byby thethe fatigue-softeningfatigue-softening behaviourbehaviour [42].[42]. Generally,Generally, softening behaviour is caused by disordering the stacArrivingking sequence at triple point of precipitatesCrack extension due to reversed softeningsoftening behaviourbehaviour isis causedcaused byby disorderingdisordering thethe stacstackingking sequencesequence ofof precipitatesprecipitates(along grain duedue boundary) toto reversedreversed loading, whereby the crack encounters a tougherPenetrating situation into toneighbouring penetrate grain this area. In addition, the loading,loading, wherebywhereby thethe crackcrack encountersencounters aa toughertougher sisituationtuation toto penetratepenetrate thisthis area.area. InIn addition,addition, thethe fatigue-softening behaviour can also be attributed to the reduced area intercepted by the dislocation fatigue-softeningfatigue-softening behaviourbehaviour cancan alsoalso bebe attributedattributed toto thethe reducedreduced areaarea interceptedintercepted byby thethe dislocationdislocation of the slip planes dueFigure to the 12. gradual Crack tip consumptio at grain boundary—Compressionaln of precipitates under loading. the reversed loading (the ofof thethe slipslip planesplanes duedueFigure toto thethe 12. gradualgradualCrack tip consumptioconsumptio at grain boundary—Compressionalnn ofof precipitatesprecipitates underunder loading. thethe reversedreversed loadingloading (the(the precipitates are cut into smaller sizes under reversed loading, and thus are dissolved). precipitatesprecipitates areare cutcut intointo smallersmaller sizessizes underunder reversedreversed loading,loading, andand thusthus areare dissolved).dissolved). CreepIn particular, effect has the strong crack dependenciespath may be deflectedon temperat (thiure,s behaviour cyclic time is illustratedand grain sizein Figure [43]. Generally, 12 of Ref. CreepCreep effecteffect hashas strongstrong dependenciesdependencies onon temperattemperature,ure, cycliccyclic timetime andand graingrain sizesize [43].[43]. Generally,Generally, a[49]) higherCreep at the temperature effectboundary. has strong leadsSpecifically, dependenciesto more crack creep deflection ondamage temperature, ,at and the agrain linear cyclic boundary timerelation and isbetween grain generally size temperature [determined43]. Generally, and by aa higherhigher temperaturetemperature leadsleads toto moremore creepcreep damagedamage,, andand aa linearlinear relationrelation betweenbetween temperaturetemperature andand athe higher twist temperatureangle (a) between leads two to more slip planes creep damage, on the grain and boundary a linear relation and the between tilt angle temperature (b) between and the appliedapplied loadingloading isis presentedpresented ((❶❶ inin FigureFigure 8).8). InIn addition,addition, aa longerlonger cyclecycle timetime givesgives moremore creepcreep appliedintersection loading lines is of presented two slip planes (❶ in on FigureFigure the 8 cross8).). InIn se addition,addition,ction [32]. aa This longerlonger behaviour cyclecycle timetime is illustrated givesgives moremore in creepFigurecreep damage, and a logarithmic relation between cyclic time and temperature is presented (❷ in Figure 8). damage,damage,13, where andand the a a crack logarithmic logarithmic path is relation relationredirected between between at the cygrain cycliccyclicclic boundarytime time and and temperature temperatureof the sample is is presentedsurface.presented (( ❷❷ inin FigureFigure 88).8).). Furthermore, a small grain size is positive for fatigue while big grain size is beneﬁcialbeneficial for creep Furthermore,Furthermore, aa smallsmall graingrain sizesize isis positivepositive forfor fatiguefatigue whilewhile bigbig graingrain sizesize isis beneficialbeneficial forfor creepcreep resistance, and a power-law relationship relationship between between grain grain size size and and applied applied loading loading is is presented presented (❸ in resistance,resistance, andand aa power-lawpower-law relationshiprelationship betweenbetween graingrain sizesize andand appliedapplied loadingloading isis presentedpresented ((❸❸ inin Figure8 8).). Consequently, Consequently, the the negative negative creep creep effect effe causedct caused by elevated by elevated temperature, temperature, or prolonged or prolonged cyclic FigureFigure 8).8). Consequently,Consequently, thethe negativenegative creepcreep effeeffectct causedcaused byby elevatedelevated temperature,temperature, oror prolongedprolonged time,cyclic ortime, smaller or smaller grain size grain gives size weaker gives weaker atomic bonds;atomic then,bonds; this then, results this in results adversely in adversely affected material.affected cycliccyclic time,time, oror smallersmaller graingrain sizesize givesgives weakerweaker atomicatomic bonds;bonds; then,then, thisthis resultsresults inin adverselyadversely affectedaffected material. The concept of ‘fatigue capacity’ indicates that the total fatigue capacity is gradually material.material. TheThe conceptconcept ofof ‘fatigue‘fatigue capacity’capacity’ indicaindicatestes thatthat thethe totaltotal fatiguefatigue capacitycapacity isis graduallygradually consumed by creep effect. This process is numerically presented by taking the form of “1-X” (❹ in consumedconsumed byby creepcreep effect.effect. ThisThis processprocess isis numerinumericallycally presentedpresented byby takingtaking thethe formform ofof “1-X”“1-X” ((❹❹ inin Figure 8) [43,44]. FigureFigure 8)8) [43,44].[43,44]. . Conceptual framework for crack growth in the compressional regime .. ConceptualConceptual frameworkframework forfor crackcrack growthgrowth inin thethe compressionalcompressional regimeregime Under compressional loading, crack closure is presented based on the idea of the plastic blunting UnderUnder compressionalcompressional loading,loading, crackcrack closureclosure isis prespresentedented basedbased onon thethe ideaidea ofof thethe plasticplastic bluntingblunting process, see Figure 10. In this process, the crack surface, especially the tip zone, is crushed together. process,process, seesee FigureFigure 10.10. InIn thisthis process,process, thethe crackcrack susurface,rface, especiallyespecially thethe tiptip zone,zone, isis crushedcrushed together.together. Since the crushing process is inelastic and irreversible, the new crack surface created under tensile SinceSince thethe crushingcrushing processprocess isis inelinelasticastic andand irreversible,irreversible, thethe newnew crackcrack surfacesurface createdcreated underunder tensiletensile loading remains (1E in Figure 10). This implies that compressional loading also contributes to crack loadingloading remainsremains (1E(1E inin FigureFigure 10).10). ThisThis impliesimplies ththatat compressionalcompressional loadingloading alsoalso contributescontributes toto crackcrack growth. growth.growth.

Metals 2018, 8, x FOR PEER REVIEW 13 of 34

Figure 9. Stress around crack tip.

Consequently, the crack tip surface caused by the fatigue effect is further extended and becomes blunter due to the creep effect (1C in Figure 8), which implies the combined effects of fatigue and creep. A defected-related state, such as intrinsic defects, around the crack tip may also promote crack growth (negative to fatigue resistance), where the weakest zone of atomic bonds is represented. An example of presenting a defected-related effect is shown in Figure 32 from [41], where the crack is promoted due to void coalescence and the bridges between voids are sheared off. In addition, precipitates may also be considered to be defected-related effects, but ones which are normally beneficial to fatigue resistance. On the one hand, this benefit results from the obstacle to further dislocation [31], whereby the precipitates restrain the process of crack propagation. On the other hand, this benefit could also be explained by the fatigue-softening behaviour [42]. Generally, softening behaviour is caused by disordering the stacking sequence of precipitates due to reversed loading, whereby the crack encounters a tougher situation to penetrate this area. In addition, the fatigue-softening behaviour can also be attributed to the reduced area intercepted by the dislocation of the slip planes due to the gradual consumption of precipitates under the reversed loading (the precipitates are cut into smaller sizes under reversed loading, and thus are dissolved). Creep effect has strong dependencies on temperature, cyclic time and grain size [43]. Generally, a higher temperature leads to more creep damage, and a linear relation between temperature and applied loading is presented (❶ in Figure 8). In addition, a longer cycle time gives more creep damage, and a logarithmic relation between cyclic time and temperature is presented (❷ in Figure 8). Furthermore, a small grain size is positive for fatigue while big grain size is beneficial for creep resistance, and a power-law relationship between grain size and applied loading is presented (❸ in Metals 2018 8 Figure 8). Consequently, ,the, 623 negative creep effect caused by elevated temperature, or prolonged 15 of 32 cyclic time, or smaller grain size gives weaker atomic bonds; then, this results in adversely affected material. The conceptThe concept of ‘fatigue of ‘fatigue capacity’ capacity’ indica indicatestes that thatthe thetotal total fatigue fatigue capacity capacity is isgradually gradually consumed by consumed by creepcreepMetals effect. 2018 effect., 8This, x This FOR process processPEER REVIEW is isnumeri numerically cally presented presented by by taking taking the the form form of of “1-X” ( ❹ inin Figure8)[ 4317 ,of44 34]. Figure 8) [43,44]. . Conceptual framework for crack growth in the compressional regime Under compressional loading, crack closure is presented based on the idea of the plastic blunting process, see Figure 10. In this process, the crack surface, especially the tip zone, is crushed together. Since the crushing process is inelastic and irreversible, the new crack surface created under tensile loading remains (1E in Figure 10). This implies that compressional loading also contributes to crack growth.

Metals 2018, 8, x FOR PEER REVIEW FigureFigure 13.13. CrackCrack deﬂection.deflection. 18 of 34

Alternatively,Stage II the tip-crush effect during the process of crack closure may lead to crack growth along the grain boundary. Specifically, grain band sliding, which occurs during crack opening, may result in a weakerSituation region 2-2: along Tip of the crack grain at triple boundary. point-Tensile This loading then provides a more likely direction for crack propagation than the direction passing through the grain boundary. After this, the crack may Loading arrive at a triple point or may penetrate into the neighbouring grain (deviated progressionStage II along the Elevated grain boundary). TheseTemperature two situationsLoading will be discussed in the next section.Log (tensile) Same mechanism as 1A (2) Crack tip at triple pointTime Tensile stress III III . Conceptual framework for crack growth in theCrack tensile opening regime Plastic shearing The triple point, whichCrack is defined as the connection point among three adjacentCrack grains, provides opening New surface Threshold Propagation a different crack-growth mechanism for fatigue and creep. Specifically, the fatigue effect under tensile Loading Tip widens and (tensile) stress (Figure 14) presents the sameTip widensprocess and shownblunts in ‘situationbluntens 1-1A’, where a blunt tip is presented at the triple point accordingFatigue to the idea of plastic blunting process. LoadingCompared (tensile) with the situation shown in Section 4.2.2.2 (1), creep presents a different behaviour atPlastic the triple blunting point. process On A the one hand, diffusion creep results in grain elongation along the direction of tensile stress, and then this leads to crack opening (2B-2 in Figure 14); on the other hand, the triple point presents a significant stress concentration [14–16], which provides an effective stress field for crack growth along the grain Creep-fatigue Tip becomes blunter boundary (2B-2 in Figure 14). In this situation, multiple opportunities arise for crack propagation along the grain boundary. Generally, the weakest microstructural zone willDirection finally of crack determine growth the direction of crack growth. Specifically, the weakest zoneInteraction shows an intensified state, which is normally Grain elongation Crack elongates in an caused by defects. These include vacancies, Multiplewhich opportunitiesare beneficial for crack to diffusionopportunistic creep [10,50], way where and themicro- cracks, which aggravate the concentration growthof stress along andgrain boundaryhence provide favourablebonds are most conditions overloaded for (might change direction) diffusion. These defect-related effects may alsoWeaker change grain boundarythe direction of crack growth. In addition, Crack due to stress concentration opening accordingLoading to the idea of the crackTriple tip pointplastic zone (shown in Figure 9), the stress concentration is (tensile) Potential for further presented at the crack tip, where a strongA un-stressed high-str stateess (defects) field is providedcreep damage hence more creep damage is produced. This implies that this area/directionCrack along may grain haveboundary more potential for crack growth.Temperature As a result, Creep Creep with the combined effect of fatigue and creep, 2B-2a blunter crack tip is presented. CyclicThe timecrack grows Diffusion Diffusion creep opportunistically in the direction where the grain boundaries are most vulnerable (for propagation Grain elongation Stress concentration into the grain see below). Grain size Crack opening Multiple opportunities Grain elongation Crack along boundary Weaker atomic bonds

Adversely affected material

Figure 14. Crack tip at triple point—Tensile loading.loading.

. Conceptual framework for crack growth in the compressional regime Under compressional loading (Figure 15), according to the idea of the plastic blunting process, the crushing effect maintains the new crack surface created under the tensile situation, and thus the crack is extended along the grain boundary (2E-2 in Figure 15). Crack growth in the next cycle or the next few cycles may have two possible paths (2F-2 in Figure

15). Specifically, on the one hand, a crack may traverse (or grow along) the grain boundary, where the weakest atomic bonds are present. This may be attributed to stress concentration at the grain boundary, since the existence of the stress gradient results in the convergence of vacancies [12,13], and in particular, the vacancies at the grain boundary provide favourable conditions for diffusion creep [10,50]. In this case, the crack along the grain boundary gradually accumulates, and then reaches the next triple point, where the opportunity for re-direction is provided. On the other hand, the crack may be deflected away from the grain boundary. This behaviour is illustrated in Figure 3 of Ref. [47] and Figure 7 of Ref. [48]. This phenomenon may result from defects, such as vacancies in the grain or on its boundary and micro-cracks in the vicinity of the crack. These provide locations for re-direction of crack growth. Specifically, vacancies provide better conditions for diffusion creep, as discussed in the previous sections. In addition, the aggregation of micro-cracks, which are attached to the main crack, also contributes to crack deflection at the grain boundary. This

Metals 2018, 8, x FOR PEER REVIEW 19 of 34

phenomenon is identified as branching activities, and is illustrated in Figure 10 of Ref. [33] and Figure 13 of Ref. [51]. In this situation, a crack-based tree is constructed at the microstructural level, where the main crack is the trunk of this tree and micro-cracks form the branches. The weakest area normally appears at the region with the highest density of micro-cracks, and it plays an important role in crack propagation. This is because the aggregation of micro-cracks weaves a crack net which offers multiple sites for shear strain under tensile loading. The shear stress at the inter-atomic level for any one crack is not decreased by having more micro-cracks. Consequently, the crack net probes a larger volume of material for weaknesses than a single crack could do on its own. This promotes the main crack to automatically grow in a direction most favourable to increasing the total strain (hence most injurious to the integrity of the part). In addition, the deviating progression may also be caused by the stress concentration at the grain boundary due to irregular configuration thereof. Specifically, this grain-boundary condition results in stress/strain pileup at the boundary (weak boundary condition is presented) and then leads to larger driving force for extending the crack tip into the neighbouring grain, whereby the barrier of the grain boundary is overcome. This is consistent with the general understanding that cracks always Metals 2018, 8, 623 16 of 32 propagate towards the direction which requires the minimum energy (stress).

Stage II

Situation 2-2: Tip of crack at triple point-Compressional loading Stage II

Log Loading Elevated III Temperature III

Time Crack Threshold Propagation

2E-2 Crack closing Loading Loading (tensile) (compression) Tip crush Plastic blunting process A Plastic blunting process B Un-stressed state New crack surface Crack extension (along boundary) Crack closing Tip becomes blunter Tip crush (Unreversed) Creep-fatigue New crack surface Direction of crack growth A un-stressed state Crack extension along boundary (defects) Crack elongates in an opportunistic way where the bonds are most overloaded (might change direction)

2F-2 Loading (compression) Opportunity 1 transverse grain boundary Opportunity 2 Arrive next triple point Go through grain

Opportunity 1 Opportunity 2 Along grain boundary Encounter defects Change direction Potentialgrowth crack direction Go through grain

Figure 15. Crack tip at triple point—Compressional loading. Figure 15. Crack tip at triple point—Compressional loading. Metals 2018, 8, x FOR PEER REVIEW 16 of 34

. ConceptualConceptual framework framework for for crack crack growth growth in in the the compressional compressional regime regime In theUnder situation compressional with compressional loading, crack loading closure (Figure is presented 12), the same based creep-fatigue on the idea of process the plastic shown blunting in ‘situationprocess, 1-1E’ seeFigure is presented, 10. In thiswhere process, reversed the lo crackading surface, crushes especially the material the ahead tip zone, of the is crushed crack, even together. acrossSince the grain the crushing boundary. process As a isresult, inelastic the new and crack irreversible, surface thecaused new by crack the surfacetensile loading created remains, under tensile due toloading the tip-crush remains effect, (1E and in Figure the crack 10). is Thisfurther implies extended that across compressional the grain boundary. loading also contributes to crack growth. StageThis II phenomenon may be explained by the observation of the crack tip opening displacements [45,46] under compressional loading, where the crack is not entirely closed in this Situation 2-1: Tip of crack at grain boundary-Compressional loading loading regime due to the plastic effect (this is illustrated in Figure 3 of Ref.Log [45]).

Loading Stage II III III Elevated Temperature Crack Threshold Propagation Time

Plastic blunting process B

Crack deflection

Loading Loading (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Figure 12. Crack tip at grain boundary—Compressional loading.

Metals 2018, 8, 623 17 of 32 Metals 2018, 8, x FOR PEER REVIEW 21 of 34

Stage II

Situation 3: Tip of crack through grain-Tensile loading

Loading Stage II Elevated Temperature Loading Same mechanism as Log (tensile) 1A Time Tensile stress III Crack opening III

Plastic shearing Crack Crack New surface Propagation opening Threshold Loading Tip widens and (tensile) blunts Tip widens and blunts Loading (tensile) Fatigue Plastic blunting process A

Creep-fatigue

Grain elongation Interaction Tip becomes blunter Potential for further creep damage A un-stressed state (defects) Crack Mismatch band due to shear opening at grain boundary

Loading 3B Temperature (tensile) Creep Diffusion creep Diffusion Grain elongation Creep Cyclic time Crack opening Shear at boundary Grain size Grain elongation Mismatch band

Figure 16. Crack tip through grain boundary—Tensile loading. Figure 16. Crack tip through grain boundary—Tensile loading. . Conceptual framework for crack growth in the compressional regime In addition, the contribution of compressional loading may also be explained by the behaviour of atomicUnder bonds. compressional Specifically, loading creep (Figure fatigue 17), does the its crack-closure damage in the process tensile maintains phase by the changing new crack the landscapesurface created of atom under arrangements tensile loading. and Meanwhile, hence the bondthe fatigue vulnerability crack is extended (which implies during the the bondsprocess are of extendedcrushing, and becomethis is enhanced weaker). by During any nearby the process defects of in compression,the material. because of the crushing effect, someIt of is thesenotable partly that brokenthe slip bonds band may may result be further in a re-direction damaged and of the become crack potential within grains, failure following sites for thethe nextslip bands loading inside cycle, the while grain. others We maypropose be totally that th damagedis is caused and by then the lead relative to crack movements extension. between This is significantlyslip planes under not a cyclic reversible loading, process, which since provides the atoms a better cannot condition return for to theircrack original growth positions along the after slip oneplanes. tension-compression Microstructurally, cycle. at the Furthermore, surfaces of the the defect-related slip bands, statethe atomic around bonds the crack are tip distorted may further and propagateelongated under cracks the due reversed to the crushing loading. effect. Thus, Specifically,these bonds thebecome crushing vulnerable, effect causes and then the the crack weakest tip to beregion squeezed, (the surfaces and then of thethe tipslip is bands) possibly is extrudedformed. In towards this case, the at zone these with surfaces, the weak the atomicatomic bonds.bonds Inrequire this case, the least the crackeffort is to further cut, which extended subsequently (1D in Figure results 10 in) bycrack cutting propagation these vulnerable along the bonds slip bands. with theThe assistance slip band ofactivities the crushing are illustrated effect. Consequently, in Figure 11 the of sub-crackRef. [34], wherein under one the loading crack behaviour cycle is formed, of re- whichdirection contributes within a tograin the is final presented. fracture.

Metals 2018, 8, 623 18 of 32 Metals 2018, 8, x FOR PEER REVIEW 22 of 34

Stage II

Situation 3: Tip of crack through grain-Compressional loading Stage II Loading Elevated Temperature Log

Time III III

Crack Threshold Propagation

Loading Loading (tensile) (compression) Plastic blunting process A Plastic blunting process B

Crack closing Tip crush (Unreversed) New crack surface

Tip becomes blunter Creep-fatigue Crack extension through grain

Same mechanism as 1E Crack closing Possibility: Loading Crack growth along slip band Tip crush (compression) Un-stressed state New crack surface Crack extension Slip bands (through grain)

Figure 17. Crack tip through grain boundary—Compressional loading. Figure 17. Crack tip through grain boundary—Compressional loading. The above mechanisms for one loading cycle in the stage of crack propagation for tension and compressionAfter N cycles, are summarised the accumulated in Figure crack 18, and arrives for multiple at the grain load boundarycycles in Figure or it encounters19. the triple point. This results in the following situation.

4.2.2.2. Crack Tip at Grain Boundary or Triple Point

Metals(1) Crack2018, 8, tipx FOR at grainPEER REVIEW boundary 16 of 34

. ConceptualConceptual framework framework for for crack crack growth growth in in the the tensile compressional regime regime InUnder the situation tensile stress with (Figurecompressional 11), the loading fatigue (Figure component 12), the experiences same creep-fatigue the same process shown inin ‘situation ‘situation 1-1E’ 1-1A’, is presented, wherein the where new reversed crack surface loading is crushes created the due material to the ahead plastic of shearingthe crack, effect.even acrossThen, thisthe causesgrain boundary. the crack tip As to a widenresult, andthe becomenew crack blunt. surface For thecaused creep by component the tensile (2B-1 loading in Figure remains, 11), duediffusion to the creep tip-crush results effect, in grain and the elongation crack is alongfurther the extended stress direction. across the On grain the oneboundary. hand, this results in crack opening, which was discussed in Section 4.2.2.1; on the other hand, shear stress, which results from grainStage elongation, II leads to elongation of the grain boundary, which then causes the crack tip to widen; meanwhile, a mismatch band appears [47,48]. Speciﬁcally, the grain boundary presents Situation 2-1: Tip of crack at grain boundary-Compressional loading a zone of irregular atomic bonds, and thus the bonds at the grain boundaryLog present anisotropic behaviour. Therefore, the relative movement of points caused by grain elongation at the grain Loading Stage II boundary may present different results, the extension or breakage of atomic bonds.III This resultsIII Elevated in a zone of material where bonds are partlyTemperature weakened, and provides potential for furtherCrack damage. Threshold Propagation Time Plastic blunting process B

Crack deflection

Loading Loading (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Figure 12. Crack tip at grain boundary—Compressional loading.

Metals 2018, 8, 623 19 of 32

Consequently, creep intensiﬁes the plastic blunting process caused by the fatigue effect, and a blunter crack tip arises. Metals 2018, 8, x FOR PEER REVIEW 23 of 34

Figure 18. Key elements for one loading cycle in the stage of crack for tension and compression. Figure 18. Key elements for one loading cycle in the stage of crack for tension and compression.

Metals 2018, 8, 623 20 of 32 Metals 2018, 8, x FOR PEER REVIEW 24 of 34

- Multiple loading Cycles -

The twist angle between two slip planes on the grain Crack deflection boundary and the tilt angle between the intersection lines of two slip planes on the cross section Situation 1 Cracks continue to go through GB Crack redirection Slip planes effect within grains Relative movements between slip planes under cyclic loading

Triple-point effect Atomic bonds are distorted and High stress concentration elongated under reversed loading

Atomic bonds between slip planes Situation 2 Beneficial to diffusion creep become vulnerable Crack arrives triple point

Tearing stress effect: tearing stress Transverse GB to peel off the surface at GB

Atomic bonds at GB are further Diffusion effect overloaded Stress gradient along GB

Atomic bonds at GB broken due to the previous damage caused by creep Convergence of vacancies

Penetrate into the Irregular configuration of GB Crack net effect (branching activities) neighbouring grain Geometric anisotropy or defects Aggregation of micro-cracks at GB

Multiple sites for shear strain Plastic strain pile up at GB under tensile loading

Large driving force for extending A larger volume of material for crack tip weaknesses

Figure 19. Key elements for multiple load cycles in the stage of crack propagation. Metals 2018, 8, x FOR PEER REVIEW 16 of 34 4.2.3. Stage III: Structural Failure . ConceptualConceptual framework framework for for crack crack growth growth in in the the compressional compressional regime regime The final stage shown in Figure 20 illustrates the fracture of an engineering structure: In the situation with compressional loading (Figure 12),12), the same creep-fatigue process shown inin ‘situation ‘situation 1-1E’ 1-1E’ is is presented, presented, where where reversed reversed loading loading crushes crushes the material the material ahead ahead of the of crack, the crack, even acrosseven across the grain the boundary. grain boundary. As a result, As a the result, new the crack new surface crack caused surface by caused the tensile by the loading tensile remains, loading dueremains, to the due tip-crush to the tip-crusheffect, and effect, the crack and theis further crack is extended further extended across the across grain theboundary. grain boundary. In particular, the crack path may be deflected (this behaviour is illustrated in Figure 12 of Ref. [49]) at the boundary.Stage II Specifically, crack deflection at the grain boundary is generally determined by the twist angle (a) between two slip planes on the grain boundary and the tilt angle (b) between the Situation 2-1: Tip of crack at grain boundary-Compressional loading Log

Loading Stage II III III Elevated Temperature Crack Threshold Propagation Time

Plastic blunting process B

Crack deflection

Loading Loading (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Figure 12. Crack tip at grain boundary—Compressional loading.

Metals 2018, 8, 623 21 of 32 intersection lines of two slip planes on the cross section [32]. This behaviour is illustrated in Figure 13, where the crack path is redirected at the grain boundary of the sample surface. Alternatively, the tip-crush effect during the process of crack closure may lead to crack growth along the grain boundary. Speciﬁcally, grain band sliding, which occurs during crack opening, may result in a weaker region along the grain boundary. This then provides a more likely direction for crack propagation than the direction passing through the grain boundary. After this, the crack may arrive at a triple point or may penetrate into the neighbouring grain (deviated progression along the grain boundary). These two situations will be discussed in the next section. Metals(2) Crack2018, 8, tipx FOR at triplePEER REVIEW point 16 of 34

. ConceptualConceptual framework framework for for crack crack growth growth in in the the tensile compressional regime regime InThe the triple situation point, with which compressional is defined as theloading connection (Figure point 12), amongthe same three creep-fatigue adjacent grains, process provides shown a indifferent ‘situation crack-growth 1-1E’ is presented, mechanism where for reversed fatigue and loading creep. crushes Specifically, the material the fatigue ahead effect of the under crack, tensile even acrossstress (Figurethe grain 14 )boundary. presents theAs samea result, process the new shown crack in surface ‘situation caused 1-1A’, by where the tensile a blunt loading tip is presented remains, dueat the to triplethe tip-crush point according effect, and to the the crack idea is of further plastic extended blunting across process. the Comparedgrain boundary. with the situation shown in Section 4.2.2.2 (1), creep presents a different behaviour at the triple point. On the one hand, diffusionStage II creep results in grain elongation along the direction of tensile stress, and then this leads to crack opening (2B-2 in Figure 14); on the other hand, the triple point presents a significant Situation 2-1: Tip of crack at grain boundary-Compressional loading stress concentration [14–16], which provides an effective stress field for crack growthLog along the grain boundary (2B-2 in Figure 14). In this situation, multiple opportunities arise for crack propagation along Loading Stage II the grain boundary. Generally, the weakest microstructural zone will finally determineIII the directionIII Elevated of crack growth. Specifically, the weakest zoneTemperature shows an intensified state, which is normallyCrack caused Threshold Propagation by defects. These include vacancies, which are beneficialTime to diffusion creep [10,50], and micro-cracks, which aggravate the concentration of stress and hence provide favourable conditions for diffusion. Plastic blunting process B These defect-related effects may also change the direction of crack growth. In addition, according to the idea of the crack tip plastic zone (shown in Figure9), the stress concentration is presentedCrack deflection at the crack tip, where aLoading strong high-stress fieldLoading is provided hence more creep damage is produced. This implies (tensile) (compression) that this area/direction may have more potential for crack growth.Tip crush As(Unreversed) a result, with the combined effectPlastic of fatigue blunting andprocess creep, A a blunter crack tip is presented.Crack The crack grows opportunistically in the closing New crack surface direction where the grain boundaries are most vulnerable (for propagationCrack extension into through the grain see below).

Conceptual framework for crack growth in the compressional regimeSame mechanism as 1E Zone of material where bonds are partly broken Crack closing Under compressional loading (Figure 15), accordingCreep-fatigue to the idea of the plastic blunting process, Tip crush the crushing effect maintains the new crackA un-stressed surface state created under the tensile situation,Un-stressed and state thus the crack isTip extended becomes blunter along the grain boundary(defects) (2E-2 in Figure 15). OR New crack surface Crack growth in the next cycle or the next few cycles may have two possibleCrack pathsextension (2F-2 in Figure 15). Specifically, on the one hand, a crack may traverse (or grow along) the(through grain grain) boundary, where the weakest atomic bonds are present. This may be attributedTip tocrush stress (Unreversed) concentration at the grain boundary, since the existence of the stress gradient resultsCrack in the convergence of vacancies [12,13], closing New crack surface and in particular, the vacancies atLoading the grain boundary provide favourableCrack extension conditions along for diffusion (compression) creep [10,50]. In this case, the crack along the grain boundary gradually accumulates,grain boundary and2F-1 then reaches the next triple point, where the opportunity for re-direction is provided. Crack closing On the other hand, the crack may be deflected away from the grain boundary.Tip crush This behaviour is illustrated in Figure 3 of Ref. [47] and Figure 7 of Ref. [48]. This phenomenonWeak boundary may result from defects, such as vacancies in the grain or onPotential its boundary crack growth and direction micro-cracksNew in crack the surface vicinity of the Arriving at triple point Crack extension crack. These provide locations for re-direction of crack growth. Specifically, vacancies(along grain boundary) provide better conditions for diffusion creep, as discussed in thePenetrating previous into neighbouring sections. grain In addition, the aggregation of micro-cracks, which are attached to the main crack, also contributes to crack deflection at the grain boundary. ThisFigure phenomenon 12. Crack is tip identified at grain boundary—Compressional as branching activities,and loading. is illustrated in Figure 10 of Ref. [33] and Figure 13 of Ref. [51]. In this situation, a crack-based tree is constructed at the In particular, the crack path may be deflected (this behaviour is illustrated in Figure 12 of Ref. [49]) at the boundary. Specifically, crack deflection at the grain boundary is generally determined by the twist angle (a) between two slip planes on the grain boundary and the tilt angle (b) between the intersection lines of two slip planes on the cross section [32]. This behaviour is illustrated in Figure 13, where the crack path is redirected at the grain boundary of the sample surface.

Metals 2018, 8, 623 22 of 32

microstructural level, where the main crack is the trunk of this tree and micro-cracks form the branches. The weakest area normally appears at the region with the highest density of micro-cracks, and it plays an important role in crack propagation. This is because the aggregation of micro-cracks weaves a crack net which offers multiple sites for shear strain under tensile loading. The shear stress at the inter-atomic level for any one crack is not decreased by having more micro-cracks. Consequently, the crack net probes a larger volume of material for weaknesses than a single crack could do on its own. This promotes the main crack to automatically grow in a direction most favourable to increasing the total strain (hence most injurious to the integrity of the part). In addition, the deviating progression may also be caused by the stress concentration at the grain boundary due to irregular configuration thereof. Specifically, this grain-boundary condition results in stress/strain pileup at the boundary (weak boundary condition is presented) and then leads to larger driving force for extending the crack tip into the neighbouring grain, whereby the barrier of the grain boundary is overcome. This is consistent with the general understanding that cracks always propagate towards the direction which requires the minimum energy (stress). These two situations reflect the fact that fatigue makes a greater contribution than creep in the present case, where the cyclic loading is without hold time. Specifically, the first option implies that the tensile stress, which is perpendicular to the grain boundary, provides a tearing stress to peel off the surface at the grain boundary. This is a significant fatigue behaviour, where the bonds at the grain boundary are further overloaded and then broken due to the previous damage caused by creep. In addition, the surface of the grain boundary is normally not ideally smooth; the geometric anisotropy or defects at the grain boundary may cause that plastic strain to pile up at one specific point, which means that less stress is needed at this point to break the barrier and penetrate the grain boundary, thereby providing for re-direction for crack growth [52]. This transgranular behaviour (which is consistent with the principle of fatigue-crack growth) is illustrated in Figure 4 of Ref. [53]. This combined intergranular and transgranular process represents the mixture of fatigue and creep effects [54]. The intergranular behaviour may become more significant when the dwell time is applied and prolonged [54,55]. After this process, the crack goes through the grain boundary, and the crack tip is extended to the next grain, which is the third situation and is examined below.

Metals4.2.2.3. 2018, 8, x Crack FOR PEER Tip REVIEW through Grain Boundary 16 of 34

. ConceptualConceptual framework framework for for crack crack growth growth in in the the tensile compressional regime regime In theUnder situation tensile with loading compressional (Figure 16 loading), the fatigue (Figure component 12), the same experiences creep-fatigue the same process process shown shown in ‘situationin situation 1-1E’ 1-1A is presented, (Figure8 ),where where reversed the new lo crackading surface crushes is the produced material due ahead to plasticof the crack, shearing even effect, acrossand the then grain the boundary. crack tip As widens a result, and the becomes new crack blunt. surface For thecaused creep by partition the tensile (3B loading in Figure remains, 16), on the due toone the hand, tip-crush diffusion effect, creepand the results crack inis further grain elongation, extended across which the leads grain to boundary. crack opening due to the conﬁgurational effect of the grain. On the other hand, the extension of the grain boundary caused by grainStage elongation II gives a relative movement between two neighbouring grains, which generates shear stress along the grain boundary. In this case, a mismatch band at the grain boundary is generated due Situation 2-1: Tip of crack at grain boundary-Compressional loading to the distortion of the existing crack under shearing conditions. This bandLog may be further intensiﬁed due to pre-existing damage caused by the grain-boundary effect (shown in Section 4.2.2.2 (1), situation Loading Stage II 2-1). As a result, this mismatch band widens the crack body and further promotesIII crackIII opening. Elevated Finally, the combination of fatigue and creepTemperature results in a blunter crack tip. Crack Threshold Propagation Conceptual framework for crack growth in theTime compressional regime Plastic blunting process B Under compressional loading (Figure 17), the crack-closure process maintains the new crack surface created under tensile loading. Meanwhile, the fatigue crack is extendedCrack deflection during the process of

crushing,Loading and this is enhancedLoading by any nearby defects in the material. (tensile) (compression) Tip crush (Unreversed) Plastic blunting process A Crack closing New crack surface Crack extension through grain

Figure 12. Crack tip at grain boundary—Compressional loading.

Metals 2018, 8, 623 23 of 32

It is notable that the slip band may result in a re-direction of the crack within grains, following the slip bands inside the grain. We propose that this is caused by the relative movements between slip planes under cyclic loading, which provides a better condition for crack growth along the slip planes. Microstructurally, at the surfaces of the slip bands, the atomic bonds are distorted and elongated under the reversed loading. Thus, these bonds become vulnerable, and then the weakest region (the surfaces of the slip bands) is formed. In this case, at these surfaces, the atomic bonds require the least effort to cut, which subsequently results in crack propagation along the slip bands. The slip band activities are illustrated in Figure 11 of Ref. [34], wherein the crack behaviour of re-direction within a grain is presented. The above mechanisms for one loading cycle in the stage of crack propagation for tension and compression are summarised in Figure 18, and for multiple load cycles in Figure 19.

4.2.3. Stage III: Structural Failure

MetalsThe 2018 ﬁnal, 8, x stageFOR PEER shown REVIEW inFigure 20 illustrates the fracture of an engineering structure: 25 of 34

❺ N cycles A power-law relation between life and applied loading (The Coffin-Manson-type and Basquin-type relation) More cycles accumulate more creep-fatigue damage

Loading Tensile Compression Elevated Temperature

Tensile Time Compression

Break Stage III presents very high crack growth rates due to the approach of instability, and thus little fatigue life is involved to the final fracture

Log

III III Fracture

Threshold Tensile Compression

Figure 20. 20. StructuralStructural failure. failure.

In this stage, crack growth rate rapidly increases, and part-separation commences when the total In this stage, crack growth rate rapidly increases, and part-separation commences when the total crack length reaches a critical value (final facture). This stage represents an unstable state, where the crack length reaches a critical value (final facture). This stage represents an unstable state, where the equilibrium shown in the second stage of crack growth is broken due to intensified accumulation of equilibriumdamage and shown reduction in the of remaining second stage cross-sectional of crack growth area. This is broken is attributed due toto a intensified high stress accumulation field at the ofcrack damage tip due and to reduction high stress of intensity remaining [23]; cross-sectionalhence, the plastic area. zone This becomes is attributed large compared to a high to the stress crack field atsize. the crackUnder tip this due stress to highstate, stress the plastic intensity energy [23 av]; hence,ailable exceeds the plastic that zone needed becomes for producing large compared the new to thecrack crack surface. size. Under In this thiscase, stress on the state, one hand, the plastic a part energy of the plastic available energy exceeds is applied that neededto create for new producing crack thesurface; new crackone the surface. other hand, In this the case,rest of on plastic the oneener hand,gy is applied a part to of form the plasticvoids [23] energy (see Figure is applied 4.11 to createtherein). new Then, crack under surface; the one applied the other stress, hand, these the vo restids are of plasticfurther energy expanded is applied and then to formcoalesced voids by [ 23] (seeinternal Figure necking 4.11 therein). [56]. This Then, phenomenon under the worsens applied the stress, creep-fatigue these voids resistance, are further and then expanded leads to and rapid then coalescedgrowth of by the internal crack. Therefore, necking [56 in]. this This stage, phenomenon a small contribution worsens theof loading creep-fatigue cycles leads resistance, to big creep- and then leadsfatigue to rapid damage, growth which of then the crack. results Therefore, in failure. in this stage, a small contribution of loading cycles leads to big creep-fatigueThe key elements damage, (mechanisms) which then in the results stage in of failure. structural failure are presented in Figure 21.

Plastic energy effect Stage III Structural failure Plastic energy

High stress intensity at crack tip Create new crack

Large plastic zone Rapid growth of crack Form voids

Coalesce by internal necking Structural failure

Figure 21. Key elements in the stage of structural failure.

Metals 2018, 8, x FOR PEER REVIEW 25 of 34

❺ N cycles A power-law relation between life and applied loading (The Coffin-Manson-type and Basquin-type relation) More cycles accumulate more creep-fatigue damage

Loading Tensile Compression Elevated Temperature

Tensile Time Compression

Break Stage III presents very high crack growth rates due to the approach of instability, and thus little fatigue life is involved to the final fracture

Log

III III Fracture

Threshold Tensile Compression

Figure 20. Structural failure.

In this stage, crack growth rate rapidly increases, and part-separation commences when the total crack length reaches a critical value (final facture). This stage represents an unstable state, where the equilibrium shown in the second stage of crack growth is broken due to intensified accumulation of damage and reduction of remaining cross-sectional area. This is attributed to a high stress field at the crack tip due to high stress intensity [23]; hence, the plastic zone becomes large compared to the crack size. Under this stress state, the plastic energy available exceeds that needed for producing the new crack surface. In this case, on the one hand, a part of the plastic energy is applied to create new crack surface; one the other hand, the rest of plastic energy is applied to form voids [23] (see Figure 4.11 therein). Then, under the applied stress, these voids are further expanded and then coalesced by internalMetals 2018 necking, 8, 623 [56]. This phenomenon worsens the creep-fatigue resistance, and then leads to24 rapid of 32 growth of the crack. Therefore, in this stage, a small contribution of loading cycles leads to big creep- fatigue damage, which then results in failure. The key elements (mechanisms) in the stage of structural failure areare presentedpresented inin FigureFigure 2121..

Plastic energy effect Stage III Structural failure Plastic energy

High stress intensity at crack tip Create new crack

Large plastic zone Rapid growth of crack Form voids

Coalesce by internal necking Structural failure

MetalsMetals 20182018,, 88,, xx FORFOR PEERPEER REVIEWREVIEWFigure 21. Key elements in the stag stagee of structural failure. 2626 ofof 3434

DuringDuringDuring thethethe process processprocess of ofof crack crackcrack growth, growth,growth, the thethe creep-fatigue creep-fcreep-fatigueatigue damage damagedamage gradually graduallygradually accumulates accumulatesaccumulates withwithwith the thethe increasingincreasingincreasing numbernumbernumber ofofof cycles,cycles,cycles, andandand thisthisthis processprocessprocess isis presentedpresentedpresented inin aa power-lawpower-law relationrelation ( (❺❺ inin FigureFigure 2020).20).).

AsAsAs discussed discusseddiscussed in in Sectionin SectionSection 4.2.2.1 4.2.2.1,4.2.2.1,, the idea thethe ofideaidea the crackofof thethe tip crackcrack plastic tiptip zone plasticplastic implies zonezone that impliesimplies the stress thatthat concentration thethe stressstress aroundconcentrationconcentration the crack aroundaround tip the playsthe crackcrack an tiptip important playsplays anan role importanimportan in cracktt rolerole growth. inin crackcrack growth. Speciﬁcally,growth. Specifically,Specifically, the more thethe the moremore stress thethe isstressstress concentrated, isis concentrated,concentrated, the larger thethe largerlarger the plastic thethe plasticplastic zone, zone,zone, and anan thendd thenthen the thethe more moremore the thethe crack crackcrack is isis promoted. promoted.promoted. Therefore,Therefore, wewewe assume assumeassume that thatthat crack crackcrack growth growthgrowth is a isis geometry-related aa geometry-relatedgeometry-related behaviour, behaviour,behaviour, and the andand crack thethe growthcrackcrack growthgrowth rate (crack raterate growth (crack(crack ingrowthgrowth one cycle) inin oneone may cycle)cycle) be related maymay bebe to related therelated size toto of the the plasticsizesize ofof zone.thethe plasticplastic Normally, zone.zone. the Normally,Normally, area of a the zonethe areaarea can ofof be aa presented zonezone cancan asbebe a presentedpresented second-order asas aa powersecond-ordersecond-order relation powerpower with a relationrelation certain dimension,withwith aa certaincertain such dimension,dimension, as radius forsuchsuch circle asas radiusradius and the forfor length circlecircle ofandand a sidethethe lengthlength for square. ofof aa sideside In this forfor square. case,square. since InIn thisthis the case,case, stress sincesince amplitude ththee stressstress is directly amplitudeamplitude related isis directlydirectly to size relatedrelated of the plastictoto sizesize of zone,of thethe weplasticplastic believe zone,zone, that wewe the believebelieve applied thatthat loading thethe appliedapplied could loadingloading be related couldcould to bebe the relatedrelated crack toto growth thethe crackcrack rate growthgrowth in a power-law raterate inin aa power-power- form. Inlawlaw addition, form.form. InIn this addition,addition, relation thisthis is alsorelationrelation consistent isis alsoalso consiste withconsiste thentnt idea withwith of thethe damage ideaidea ofof accumulation damagedamage accumulationaccumulation shown in Figure shownshown 22 inin. InFigureFigure the ﬁrst 22.22. stage,InIn thethe the firstfirst original stage,stage, the steadythe originaloriginal state (asteadysteady perfect ststateate body) (a(a perfectperfect is suddenly body)body) broken isis suddenlysuddenly due to brokenbroken the large duedue amount toto thethe oflargelarge accumulated amountamount ofof energy, accumulatedaccumulated which implies energy,energy, at thiswhichwhich stage impliesimplies a small atat amount thisthis stagestage of damage aa smallsmall is amountamount produced ofof with damagedamage a large isis numberproducedproduced of withwith loading aa largelarge cycles numbernumber (stage ofof I in loadingloading Figure cyclescycles 22). Then, (stage(stage after II inin theFigureFigure process 22).22). Then, ofThen, re-balancing, afterafter thethe processprocess the damage ofof re-re- accumulationbalancing,balancing, thethe gives damagedamage a relatively accumulationaccumulation steady givesgives state, aa where relativerelative thelyly rate steadysteady of accumulation state,state, wherewhere the stablythe raterate increases ofof accumulationaccumulation (stage II instablystably Figure increasesincreases 22). Finally, (stage(stage when IIII inin FigureFigure the total 22).22). damage Finally,Finally, reaches whenwhen thethe or istotaltotal close damagedamage to a critical reachesreaches value, oror isis the closeclose load-bearing toto aa criticalcritical capacityvalue,value, thethe collapses load-bearingload-bearing in a short capacitycapacity time. collapsescollapses This stage inin implies aa shortshort that time.time. much ThisThis damage stagestage impliesimplies is produced thatthat muchmuch within damagedamage a small isis numberproducedproduced of withinwithin loading aa cycles smallsmall number (stagenumber III ofof in loadingloading Figure 22cyclescycles). (stage(stage IIIIII inin FigureFigure 22).22).

FigureFigureFigure 22. 2222.. Damage DamageDamage accumulation accumulationaccumulation

5.5. DiscussionDiscussion

5.1.5.1. SummarySummary ofof thethe CrackCrack GrowthGrowth ProcessProcess andand MechanismsMechanisms InIn thethe presentpresent work,work, aa microstrucmicrostructuraltural graphical-basedgraphical-based modelmodel forfor creepcreep fatiguefatigue isis proposed.proposed. InIn thisthis microstructuralmicrostructural conceptualconceptual frframework,amework, wewe proposepropose thatthat differentdifferent crackcrack growthgrowth mechanismsmechanisms areare activeactive forfor differentdifferent stagesstages inin thethe temporaltemporal evolutionevolution ofof thethe crackcrack (stages(stages II toto III),III), differentdifferent partsparts ofof thethe tension-tension- compressioncompression cycle,cycle, andand forfor differentdifferent regiregionsons withinwithin thethe microstructuremicrostructure (grain(grain boundary,boundary, tripletriple point,point, insideinside thethe grain).grain). InIn general,general, thethe temporaltemporal evolutionevolution ofof thethe crackcrack isis mainlymainly basedbased onon thethe mechanismsmechanisms ofof thethe plasticplastic bluntingblunting process,process, stress-concentrationstress-concentration statestate andand didiffusion-creepffusion-creep behaviour.behaviour. TheThe illustrationillustration ofof thethe crack-growthcrack-growth processprocess isis presentedpresented inin threethree differentdifferent stages:stages: StageStage I:I: CrackCrack initiationinitiation InIn thethe firstfirst stagestage (the(the crackcrack initialinitial stage),stage), aa thresholdthreshold isis presented,presented, belowbelow whichwhich nono significantsignificant crackcrack isis detected.detected. TheThe existenceexistence ofof thisthis thresholdthreshold cancan bebe attributedattributed toto thethe effecteffect ofof stressstress concentration.concentration. ForFor aa surfacesurface withwith defectsdefects (such(such asas aa machiningmachining mamark),rk), thethe stressstress concentrationconcentration isis causedcaused byby thethe pre-pre-

Metals 2018, 8, 623 25 of 32

5. Discussion

5.1. Summary of the Crack Growth Process and Mechanisms In the present work, a microstructural graphical-based model for creep fatigue is proposed. In this microstructural conceptual framework, we propose that different crack growth mechanisms are active for different stages in the temporal evolution of the crack (stages I to III), different parts of the tension- compression cycle, and for different regions within the microstructure (grain boundary, triple point, inside the grain). In general, the temporal evolution of the crack is mainly based on the mechanisms of the plastic blunting process, stress-concentration state and diffusion-creep behaviour. The illustration of the crack-growth process is presented in three different stages: Stage I: Crack initiation In the first stage (the crack initial stage), a threshold is presented, below which no significant crack is detected. The existence of this threshold can be attributed to the effect of stress concentration. For a surface with defects (such as a machining mark), the stress concentration is caused by the pre-existing micro-cracks. For a defect-free surface, the behaviours of extrusion and intrusion between slip plans give some tiny steps in the surface, which then results in the stress concentration. Stage II: Crack propagation In the second stage (the steady stage of crack growth), the crack-growth behaviours under tensile loading and compressional loading were discussed separately. The primary differentiating factor between the tension-compression cycles can be explained by the plastic blunting process. Specifically, tensile loading creates a new crack surface due to the shearing effect around the crack-tip area, and then the crushing effect results in crack extension (by maintaining the new surface created by the tensile loading) under the compressional loading. Regarding the microstructure, the plastic blunting process implies that the damage produced within one tension-compression cycle is not a reversed process since the atoms cannot return to their original positions after one cycle. In addition, the crack-growth behaviours of fatigue and creep were also illustrated separately. Specifically, the fatigue component was generally presented by the plastic blunting process shown above, and the creep component was explained by the mechanism of diffusion creep. On the one hand, stress concentration at the crack tip provides a favourable condition for diffusion to form voids due to the stress-gradient effect, and then the void coalescence by shearing of the bridges between voids results in crack growth. On the other hand, diffusion creep gives grain elongation under tensile loading, which then further opens the crack created by fatigue effect. In addition, grain elongation also gives the shear stress at the grain boundaries between two adjacent grains, which probably results in the mismatch behaviour of the crack due to weak atomic bonds at the boundaries; then, the crack widens. Furthermore, the crack-growth behaviour with respect to the grain-boundary effect was discussed, and this effect plays a more important role in creep-crack growth than fatigue behaviour. In this case, due to the shear stress between two adjacent grains caused by grain elongation, the grain-boundary effect results in a blunter crack tip or wider crack body (mismatch). In particular, this effect can be presented by a special situation, the triple-point effect. In the triple points, the high stress concentration provides better conditions for diffusion. Stage III: Structural failure In the final stage (structural failure), the fracture occurs within small loading cycles. This can be attributed to the high stress field at the crack tip. Specifically, under this stress state, the plastic energy exceeds the needs for producing the new crack surface. In this case, a part of energy is applied to create the new surface, and the rest of the energy is applied to form voids, which further promote crack growth through internal necking. Metals 2018, 8, 623 26 of 32

5.2. Original Contributions The original contribution of the present work is the proposal of a microstructural conceptual framework for the multiple crack propagation mechanisms at a grain-based level. In general, the plastic blunting process was introduced to describe fatigue-crack growth, the mechanism of diffusion was included to present creep-crack growth, and the grain-boundary effect is proposed to describe the phenomena taking place when a crack encounters a grain boundary. While the ideas of crack growth are well known, the literature does not provide a comprehensive explanation of the process from initiation to failure. The novel contribution of the above conceptual framework is therefore the integration of multiple mechanisms at the microstructural level, inclusion of the grain boundary effect, and coverage of the stages from initiation to failure. To achieve this, some new propositions (such as the grain mismatch and the bonding crush effect) were proposed, and some existing concepts (such as the energy dynamics and the micro-cracks aggregation) were extended to explain the crack-growth behaviours in the present work. These principles connect the existing but separate crack-growth principles into a consolidated explanation of the whole failure process. We have proposed a number of new mechanisms (propositions A-G in Section 4.1) and phenomena for crack growth: (1) We proposed the grain-mismatch mechanism which occurs at the grain boundary under the tensile loading. At the microstructural level, tensile loading results in relative movement between two neighbouring grains, which then generates shear stress along the grain boundary. This shear stress may be caused by the stretch of atomic bonds. In this case, a weak region arises at the grain boundary. On the one hand, when a crack arrives at the grain boundary, the grain-mismatch behaviour results in a wider crack tip, which creates more new surface under tensile loading, and then intensifies the crack-extension process due to the crushing effect under the compressional loading. On the other hand, when a crack goes through the grain boundary, the grain-mismatch behaviour widens the crack body, and then a mismatch band is formed. This further promotes the crack-opening behaviour under the tensile loading, then a blunter crack tip is created. The new surface generated in this stage is then retained under compressional loading. The grain-mismatch mechanism gives a wide-ranging explanation of crack-growth behaviour at the grain boundary. This effect is transient and is not expected to be easy to detect. (2) We proposed the idea of the bonding crushing effect. Specifically, during the compressional phase, the distorted and rearranged atomic bonds caused by tensile loading are further damaged. Then, the atomic bonds may break during the compressional process or become potential failure sites for the next loading cycle. The compressional phase is also presented by the existing principle of plastic blunting process, which indicates that the crack is extended because the new surface created in tensile phase is crushed together at the compressional phase. This existing principle gives a general understanding of crack extension, but not to the atomic level. Our crushing effect offers a deeper explanation of the process of crack closure at the atomic level. This key element of the bonding crushing effect is that the damage (distorted and rearranged atomic bonds) under the tensile loading cannot be recovered when the loading is reversed. Then, the damage is retained by this irreversible process, since the atoms cannot return to their original positions under compressional loading. (3) We proposed that the crack growth is a geometry-related behaviour. In general, crack-growth rate is determined by the size of the plastic zone, and a larger plastic zone gives longer crack growth. This is combined with our other propositions, namely that fatigue-crack growth is caused by overloaded local atomic bonds, and creep-crack growth is defined as the flow of atoms relative to each other with transfer of bonding to new partners. These two damage behaviours are stress-motivated processes. On the one hand (for the fatigue process), larger plastic zone results in more overloaded atomic bonds. Thus, more bonds become vulnerable. They may totally break under the compressional process, or may become potential failure sites during the next loading cycles. This implies that more completed or potential crack surfaces are formed. On the other hand (for the creep process), a larger plastic zone leads to more space and enhanced stress conditions for diffusion. Thus, at a Metals 2018, 8, 623 27 of 32 given volume, more atoms are transferred to new positions and then form new bonds with other atoms. This phenomenon is more significant in the crack-tip area. During the process of crack growth, the increased crack length leads to the expansion of the plastic zone, and higher localised stress around the crack tip. Hence, more vacancies are generated and converged in this area, and a more favourable situation arises for creep. Hence, the plastic-zone-based explanation for crack growth presents an explanation of the loading contribution for fatigue and creep behaviours at the atomic level. (4) We proposed a mechanism for crack behaviour at the triple point. This situation may not happen during the whole process from crack initiation to component failure. However, it is a possible situation; thus, it is valuable to discuss it. At a triple point, the stress concentration (which arises from the geometry of the grains) results in weaker atomic bonding between the grains, and thus a more favourable situation for both fatigue and creep damage. In particular, we proposed two possible paths of crack growth over one or more cycles. On the one hand, the crack may be deflected away from the grain boundary. This phenomenon may be explained by two existing mechanisms—the crack-net behaviour and the irregular configuration of the grain boundary—which have been extended in the present work. Specifically, the crack net around the grain boundary gives a larger volume of material for weaknesses, and then promotes and deflects the main crack. In this case, a crack penetrates into its neighbouring grain, where the crack net is highly concentrated. The irregular configuration of a grain boundary causes stress/strain pile up at the grain boundary, and then leads to a larger driving force for extending the crack tip into the neighbouring grain. On the other hand, a crack may grow along the grain boundary. This phenomenon may result from the larger contribution of creep than fatigue. In this case, convergence of vacancies caused by diffusion along the grain boundary gives a favourable condition for crack growth in this area. The situation of the crack tip at a triple point introduces further opportunities for changes in the direction and extent of crack growth. (5) We applied the mechanism of energy dynamics to qualitatively describe the whole process of crack growth from initiation to failure. Although this mechanism is extant, it was then further extended in the present work. Generally, in the initial stage of crack growth, internal energy is continually stored in the structure under the cyclic loading in the form of irreversible dislocations. Dislocations are vacancies in the atomic lattice; hence, they comprise localised vacancies and strain between atoms. When this energy arrives at a critical energy level, the vacancies in the dislocations coalesce and form a crack. This results in a cascade effect of a rapid liberation of energy, which is consumed to form new crack surfaces. During this process, the high crack-growth rate is gradually reduced, and then achieves a steady situation due to the need of the system to rebuild new dislocations. We propose that this cyclic energisation process underpins the steady second stage of crack growth. When the accumulated energy/damage is large enough, the crack is re-energised even as it progresses; hence, the growth rate accelerates. This process is consistent with the representation of the final stage of crack growth. In this case, the completed process from crack initiation to component failure is described in the perspective of energy accumulation and release. Using these, it has been possible to construct a comprehensive conceptual framework of the crack growth process at the microstructural level. This conceptual framework is also consistent with the existing concepts in the literature (premises a-l in Section 2.3). In itself, this does not prove that these propositions are true, but it does show that these concepts are useful in better understanding the creep-fatigue mechanisms.

5.3. Implications for Practitioners The conceptual framework presents a theory for the multiscale mechanisms of crack development and growth. This theory has implications for various practitioners. For those who are studying microstructure, the implications are that an explanatory framework has been provided. This integrates multiple existing microstructural phenomena, and hence it is expected that existing microstructural observations would be able to be reconciled to the theory. For those who are designing engineering components, e.g., design engineers, the implications are more abstract, in that it emphasizes the need Metals 2018, 8, 623 28 of 32 to avoid the reverse loading condition, where possible. Practical materials used by engineers are invariably full of crack initiators; hence, the default state of the material is Stage I, with many of the cracks propagating into Stage II. The aim of engineering design is to avoid Stage III failure, as this terminates the life of the part/machine and can have catastrophic consequences for human safety. Hence, from the engineering perspective, there is a need to keep the part in the Stage II regime for as long as possible. Designers affect this by controlling the geometry of the part, the external loading, the material selection and treatment. The theory shows, per Figures 19 and 20, that the combination of tension and compression is key to crack growth. This is already well known, but the theory makes the relationships more explicit. In particular, this theory makes explicit the mechanisms whereby the compressional process contributes crack growth. This is because the distorted and rearranged atomic bonds caused by the tensile loading are further damaged, such that some atomic bonds break during the compressional process, and others become weak areas for the next loading cycle.

5.4. Limitations The analysis presented here does not completely include all possible loading situations, though it is believed to be broadly representative. This schematic representation of creep-fatigue crack growth is based on the assumption that fatigue damage makes a greater contribution to failure than creep damage. In this case, the crack mainly grows through the grains. The creep behaviour is regarded as an assistant effect. This may not be true of all situations, especially not at higher temperatures, longer cyclic times, or cyclic loading with hold time, where creep begins to dominate. In these situations, the creep effect may contribute to more damage, and crack growth may take a different direction. This is because the creep damage gradually increases with the crack growing due to increasing number of creep voids, but the fatigue damage gradually decreases during this process [57,58]. For a creep-dominated structure, we believe that the crack growth along the grain boundary is more significant. In particular, at the situation of the cyclic loading with hold time, a component of pure creep effect is introduced. This period may result in a more complex situation, where the stress relaxation occurs under strain-controlled loading. In this case, the stress-related mechanisms, such as the stress condition around the crack tip, may need further examination. In addition, at the situation of the cyclic loading with hold time, the effect of oxidation may become more significant. This effect normally results in brittle crack surfaces, and then reduces the failure life. The research [55] shows that this reduction in life is more severe at the creep-fatigue-oxidation condition than the creep-oxidation condition. In this case, the combination of creep, fatigue and oxidation affects crack growth in a more complex manner. Therefore, other possibilities for crack growth, such as the influences of stress relaxation and oxidation on the fatigue component, may need to be considered. These may be an opportunity for future research.

5.5. Implications for Further Research The graphical-based crack-growth model proposed in this paper is a microstructural conceptual framework. Some of the fatigue and creep behaviours have been well demonstrated and explained in previous research, such as stress distribution at the crack tip for fatigue and the stress concentration at the triple point for creep. However, there are still some behaviours which are difﬁcult to validate through experiments, such as the situation with the crack tip at the grain boundary. Our microstructural conceptual framework has offered an explanation of these behaviours, but this work is conjectural. It would be valuable to test the validity of this at the microstructural level. To achieve this, it would be necessary to combine fractography and fatigue tests. Overall, the present work presents some unanswered questions that require experiments. This opens multiple opportunities for future research: (1) The crack-growth behaviours at the grain boundary when crack tip at/through the grain boundary. The question is that it is difﬁcult and complex to empirically examine and investigate Metals 2018, 8, 623 29 of 32

the phenomena at the grain boundary since it will be difficult to dynamically capture those moments when the crack tip arrives at the grain boundary. To catch this microstructural behaviour, a new experimental method may also need to be developed. (2) The grain-mismatch behaviour due to grain elongation under the tensile loading. It is valuable to explore the shape change of grains under tensile loading, and observe the relative movements and investigate the interfacial phenomena between two neighbouring grains. This research may also be extended to the situation of compressional loading. (3) The bonding crushing effects during the process of compression ahead of the crack. This is a call for an atomic-level investigation, e.g., using atomic force microscopy. Existing approaches have been focussed on statically describing the force-distance relation and force field between two neighbouring bonds. This provides valuable information on the local material properties [59–61]. However, the dynamical process (such as the time-depended force/distance variances and the trace of atomic movements) under a cyclic or constant loading have not been addressed. In addition, it may also be valuable to explore both the situations of tensile and compressional loadings. (4) The stress/strain pile-up at the grain boundary because of the irregular configuration of the grain boundary. This is the question of investigating the dislocation pile-up [62] and measuring the stress field at the grain boundary [47]. The future research may start from the existing theories and methods, and then go further to evaluate the configuration-based effects at the grain boundary. In addition, finite element analysis may also be involved to investigate the stress/strain distributions and the shape distortion at the simulative grain boundaries. (5) Numerical modelling and its experimental validation. This could be done at several levels. At a high level of abstraction, there are already formulations for describing the failure process. For example, Paris’s law [11] gives the crack growth in one cycle based on the stress intensity factor, the Arrhenius equation [63,64] presents the creep behaviour at the steady stage based on diffusion creep, and the conventional loading-life equations (the Basquin [65,66] equation and Coffin-Manson [67,68] equation) model life based on empirical data. At the level of detail providing in this paper, as represented in Figures 19 and 20, the crack-growth process is graphically described in the microstructural level.

An ideal model would be one that represented the physics of the interactions between atoms within the finer level of the grain. Of interest are the interactions between the imperfections in the crystal lattice (e.g., dislocations), the grain boundaries, and the way the cracks propagate. However, this is likely to be a challenging proposition for some time. It would be ideal to observe the stages of crack growth using methods such as atomic force microscopy (AFM). Some work in this area includes the investigation of the force-distance relation and force field between two neighbouring bonds [59–61]. However, the difficulties are the extremely small area of examination afforded by AFM, and the practical difficulties of arranging a crack to grow in the sample. Furthermore, AFM measures free surfaces, whereas cracks have a three-dimensional interaction with the grains. Possibly a more immediately achievable model could be to use finite element methods to represent the grains and grain boundary regions. It would be necessary to incorporate the anisotropy of the crystal lattice. This may be able to explore the interaction of the crack with the microstructure. Potentially this could be validated by microscopy. Neither observed fracture surfaces nor cross-sections show the full development of the crack architecture in the 3D volume. Also, fracture surfaces only show the Stage III development and termination of the cracks. A FEA approach might help better understand the evolution of the crack architecture across the Stages. Our framework has proposed a number of variables, and the general direction of causality. If it were possible—which does not currently appear to be the case—to measure all these variables, then a statistical method such as factor analysis or structural path modelling might be used to determine the relative importance of the variables and the strength of the relationships. Metals 2018, 8, 623 30 of 32

6. Conclusions Based on the general understanding of crack-growth behaviour, a microstructural conceptual framework has been proposed, and represented graphically, wherein fatigue and creep effects are treated separately due to the different damage principles. This microstructural model illustrates the process of damage accumulation from crack initiation to crack propagation then to structural failure. In particular, the analysis of crack propagation is mainly based on existing mechanisms of plastic blunting and diffusion creep. The possible grain-boundary behaviours, such as the mismatch behaviour at grain boundary due to creep deformation, are included. This conceptual framework is consistent with the various creep and fatigue microstructure observations in the literature, but goes further by integrating these premises and our proposed propositions together into a logically consistent framework that describes the overall failure process.

Author Contributions: D.L. and D.J.P. conceptualised the overall framework, D.L. worked out the details and produced the graphical model, D.L. and D.J.P. contributed to writing the paper. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. The research was conducted without personal financial benefit from any funding body, and no such body influenced the execution of the work.

References

1. June, W. A continuum damage mechanics model for low-cycle fatigue failure of metals. Eng. Fract. Mech. 1992, 41, 437–441. [CrossRef] 2. Metzger, M.; Nieweg, B.; Schweizer, C.; Seifert, T. Lifetime prediction of cast iron materials under combined thermomechanical fatigue and high cycle fatigue loading using a mechanism-based model. Int. J. Fatigue 2013, 53, 58–66. [CrossRef] 3. Seifert, T.; Riedel, H. Mechanism-based thermomechanical fatigue life prediction of cast iron. Part I: Models. Int. J. Fatigue 2010, 32, 1358–1367. [CrossRef] 4. Jing, H.; Zhang, Y.; Xu, L.; Zhang, G.; Han, Y.; Wei, J. Low cycle fatigue behavior of a eutectic 80Au/20Sn solder alloy. Int. J. Fatigue 2015, 75, 100–107. [CrossRef] 5. Shi, X.; Pang, H.; Zhou, W.; Wang, Z. Low cycle fatigue analysis of temperature and frequency effects in eutectic solder alloy. Int. J. Fatigue 2000, 22, 217–228. [CrossRef] 6. Solomon, H. Fatigue of 60/40 solder. IEEE Trans. Compon. Hybrids Manuf. Technol. 1986, 9, 423–432. [CrossRef] 7. Callister, W.D.; Rethwisch, D.G. Materials Science and Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 5. 8. Dowling, N.E. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue; Prentice: London, UK, 2012; p. 954. 9. Evans, R.W.; Wilshire, B. Introduction to Creep; The Institute of Materials, University of Michiganb: Ann Arbor, MI, USA, 1993; p. 115. 10. Kassner, M.E. Fundamentals of Creep in Metals and Alloys; Butterworth-Heinemann: Oxford, UK, 2015; p. 337. 11. Paris, P.; Erdogan, F. A critical analysis of crack propagation laws. J. Basic Eng. 1963, 85, 528–533. [CrossRef] 12. Liu, C.; Han, Y.; Yan, M.; Chaturvedi, M. Creep crack growth behaviour of alloy 718. Superalloys 1991, 178, 537–548. 13. Sadananda, K. A theoretical model for creep crack growth. Metall. Mater. Trans. A 1978, 9, 635–641. [CrossRef] 14. Cocks, A.; Pontern, A. Mechanics of Creep Brittle Materials 1; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1989; p. 310. 15. Ejaz, N.; Qureshi, I.; Rizvi, S. Creep failure of low pressure turbine blade of an aircraft engine. Eng. Fail. Anal. 2011, 18, 1407–1414. [CrossRef] 16. Král, P.; Dvoˇrák, J.; Kvapilová, M.; Svoboda, M.; Skleniˇcka,V. Creep damage of Al and Al-Sc alloy processed by ecap. Acta Metall. Slov. Conf. 2013, 3, 136–144. [CrossRef] 17. Dai, C.; Zhang, B.; Xu, J.; Zhang, G. On size effects on fatigue properties of metal foils at micrometer scales. Mater. Sci. Eng. A 2013, 575, 217–222. [CrossRef] Metals 2018, 8, 623 31 of 32

18. Hanlon, T.; Kwon, Y.-N.; Suresh, S. Grain size effects on the fatigue response of nanocrystalline metals. Scr. Mater. 2003, 49, 675–680. [CrossRef] 19. Laird, C. Mechanisms and theories of fatigue. Fatigue Microstruct. 1979, 149–203. 20. Banks-Sills, L.; Motola, Y.; Shemesh, L. The m-integral for calculating intensity factors of an impermeable crack in a piezoelectric material. Eng. Fract. Mech. 2008, 75, 901–925. [CrossRef] 21. Warzynek, P.; Carter, B.; Banks-Sills, L. The M-Integral for Computing Stress Intensity Factors in Generally Anisotropic Materials; National Aeronautics and Space Administration: Washington, DC, USA, 2005. 22. Miller, K. Materials science perspective of metal fatigue resistance. Mater. Sci. Technol. 1993, 9, 453–462. [CrossRef] 23. Rodopoulos, C.A. Fatigue damage map as a virtual tool for fatigue damage tolerance. In Virtual Testing and Predictive Modeling; Springer: Berlin/Heidelberg, Germany, 2009; pp. 73–104. 24. Chowdhury, P.; Sehitoglu, H. Mechanisms of fatigue crack growth—A critical digest of theoretical developments. Fatigue Fract. Eng. Mater. Struct. 2016, 39, 652–674. [CrossRef] 25. Laird, C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In Fatigue Crack Propagation; ASTM International: West Conshohocken, PA, USA, 1967. 26. Laird, C.; de La Veaux, R. Additional evidence for the plastic blunting process of fatigue crack propagation. Metall. Mater. Trans. A 1977, 8, 657–664. [CrossRef] 27. Peralta, P.; Choi, S.-H.; Gee, J. Experimental quantification of the plastic blunting process for stage ii fatigue crack growth in one-phase metallic materials. Int. J. Plast. 2007, 23, 1763–1795. [CrossRef] 28. Shi, K.; Cai, L.; Qi, S.; Bao, C. A prediction model for fatigue crack growth using effective cyclic plastic zone and low cycle fatigue properties. Eng. Fract. Mech. 2016, 158, 209–219. [CrossRef] 29. Wang, G. The plasticity aspect of fatigue crack growth. Eng. Fract. Mech. 1993, 46, 909–930. [CrossRef] 30. Weertman, J. Fatigue crack propagation theories. In Fatigue and Microstructure; American Society for Metals: Geauga County, OH, USA, 1979; Volume 279. 31. Ham, R.; Broom, T. The mechanism of fatigue softening. Philos. Mag. 1962, 7, 95–103. [CrossRef] 32. Zhai, T.; Jiang, X.; Li, J.; Garratt, M.; Bray, G. The grain boundary geometry for optimum resistance to growth of short fatigue cracks in high strength al-alloys. Int. J. Fatigue 2005, 27, 1202–1209. [CrossRef] 33. Kinloch, A.; Wang, C.; Wu, S.; Ladani, R.; Zhang, J.; Bafekrpour, E.; Ghorbani, K.; Mouritz, A. Aligning Graphene Nanoplatelets with an External Electric Field to Improve Multifunctional Properties of Epoxy Nanocomposites. Carbon 2015, 94, 607–618. [CrossRef] 34. Villechaise, P.; Cormier, J.; Billot, T.; Mendez, J. Mechanical behaviour and damage processes of udimet 720li: Influence of localized plasticity at grain boundaries. In Proceedings of the 12th International Symposium on Superalloys, Pennsylvania, PA, USA, 9–13 September 2012. 35. Claude Bathias, A.P. Fatigue of Materials and Structures: Fundamentals; John Wiley & Sons: Hoboken, NJ, USA, 2013; p. 512. 36. Sangid, M.D. The physics of fatigue crack initiation. Int. J. Fatigue 2013, 57, 58–72. [CrossRef] 37. Wang, Z.; Beyerlein, I.; LeSar, R. Slip band formation and mobile dislocation density generation in high rate deformation of single fcc crystals. Philos. Mag. 2008, 88, 1321–1343. [CrossRef] 38. Zerbst, U.; Vormwald, M.; Pippan, R.; Gänser, H.-P.; Sarrazin-Baudoux, C.; Madia, M. About the fatigue crack propagation threshold of metals as a design criterion—A review. Eng. Fract. Mech. 2016, 153, 190–243. [CrossRef] 39. Margaritis, G.; Botsis, J. Energy evaluations during crack initiation. Eng. Fract. Mech. 1991, 40, 1123–1134. [CrossRef] 40. Lach, R.; Adhikari, R.; Weidisch, R.; Huy, T.; Michler, G.; Grellmann, W.; Knoll, K. Crack toughness behavior of binary poly (styrene-butadiene) block copolymer blends. J. Mater. Sci. 2004, 39, 1283–1295. [CrossRef] 41. Zerbst, U.; Klinger, C.; Clegg, R. Fracture mechanics as a tool in failure analysis—Prospects and limitations. Eng. Fail. Anal. 2015, 55, 376–410. [CrossRef] 42. Fournier, D.; Pineau, A. Low cycle fatigue behavior of inconel 718 at 298 k and 823 k. Metall. Mater. Trans. A 1977, 8, 1095–1105. [CrossRef] 43. Liu, D.; Pons, D.J. Physical-mechanism exploration of the low-cycle unified creep-fatigue formulation. Metals 2017, 7, 379. 44. Liu, D.; Pons, D.; Wong, E.-h. The unified creep-fatigue equation for stainless steel 316. Metals 2016, 6, 219. [CrossRef] Metals 2018, 8, 623 32 of 32

45. Pippan, R.; Grosinger, W. Fatigue crack closure: From lcf to small scale yielding. Int. J. Fatigue 2013, 46, 41–48. [CrossRef] 46. Prasad, K.; Kumar, V.; Rao, K.B.S.; Sundararaman, M. A comparative assessment of crack closure mechanisms in timetal 834 near α titanium alloy under isothermal and thermomechanical fatigue loading. J. Alloys Compd. 2016, 688, 8–11. [CrossRef] 47. Guo, Y.; Collins, D.; Tarleton, E.; Hofmann, F.; Tischler, J.; Liu, W.; Xu, R.; Wilkinson, A.; Britton, T. Measurements of stress fields near a grain boundary: Exploring blocked arrays of dislocations in 3d. Acta Mater. 2015, 96, 229–236. [CrossRef] 48. McMurtrey, M.; Was, G.; Cui, B.; Robertson, I.; Smith, L.; Farkas, D. Strain localization at dislocation channel–grain boundary intersections in irradiated stainless steel. Int. J. Plast. 2014, 56, 219–231. [CrossRef] 49. Pineau, A. Crossing grain boundaries in metals by slip bands, cleavage and fatigue cracks. Philos. Trans. R. Soc. A 2015, 373, 20140131. [CrossRef][PubMed] 50. Kumar, Y.; Venugopal, S.; Sasikala, G.; Albert, S.K.; Bhaduri, A. Study of creep crack growth in a modified 9cr–1mo steel weld metal and heat affected zone. Mater. Sci. Eng. A 2016, 655, 300–309. [CrossRef] 51. Pretty, C.J.; Whitaker, M.T.; Williams, S.J. Thermo-mechanical fatigue crack growth of rr1000. Materials 2017, 10, 34. [CrossRef][PubMed] 52. Kacher, J.; Eftink, B.; Cui, B.; Robertson, I. Dislocation interactions with grain boundaries. Curr. Opin. Solid State Mater. Sci. 2014, 18, 227–243. [CrossRef] 53. Benz, J.K.; Wright, R.N. Fatigue and Creep Crack Propagation Behaviour of Alloy 617 in the Annealed and Aged Conditions; Idaho National Laboratory (INL): Idaho Falls, ID, USA, 2013. 54. Tang, Z.; Jing, H.; Xu, L.; Zhao, L.; Han, Y.; Xiao, B.; Zhang, Y.; Li, H. Creep-fatigue crack growth behavior of g115 steel under different hold time conditions. Int. J. Fatigue 2018, 116, 572–583. [CrossRef] 55. Zhao, L.; Nikbin, K. Characterizing high temperature crack growth behaviour under mixed environmental, creep and fatigue conditions. Mater. Sci. Eng. A 2018, 728, 102–114. [CrossRef] 56. Scheyvaerts, F.; Pardoen, T.; Onck, P. A new model for void coalescence by internal necking. Int. J. Damage Mech. 2010, 19, 95–126. [CrossRef] 57. Xu, L.; Rong, J.; Zhao, L.; Jing, H.; Han, Y. Creep-fatigue crack growth behavior of g115 steel at 650 ◦C. Mater. Sci. Eng. A 2018, 726, 179–186. [CrossRef] 58. Xu, L.; Zhao, L.; Han, Y.; Jing, H.; Gao, Z. Characterizing crack growth behavior and damage evolution in p92 steel under creep-fatigue conditions. Int. J. Mech. Sci. 2017, 134, 63–74. [CrossRef] 59. Butt, H.-J.; Cappella, B.; Kappl, M. Force measurements with the atomic force microscope: Technique, interpretation and applications. Surf. Sci. Rep. 2005, 59, 1–152. [CrossRef] 60. Neuman, K.C.; Nagy, A. Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. Nat. Methods 2008, 5, 491–505. [CrossRef][PubMed] 61. Williams, J.M.; Han, T.; Beebe, T.P. Determination of single-bond forces from contact force variances in atomic force microscopy. Langmuir 1996, 12, 1291–1295. [CrossRef] 62. Kato, M. Hall–petch relationship and dislocation model for deformation of ultrafine-grained and nanocrystalline metals. Mater. Trans. 2014, 55, 19–24. [CrossRef] 63. Arrhenius, S. Über die dissociationswärme und den einfluss der temperatur auf den dissociationsgrad der elektrolyte. Zeitschrift Physikalische Chemie 1889, 4, 96–116. [CrossRef] 64. Arrhenius, S. Über die reaktionsgeschwindigkeit bei der inversion von rohrzucker durch säuren. Zeitschrift Physikalische Chemie 1889, 4, 226–248. [CrossRef] 65. Basquin, O. The exponential law of endurance tests. Am. Soc. Test. Mater. 1910, 10, 625–630. 66. Wöhler, A. Theorie rechteckiger eiserner brückenbalken mit gitterwänden und mit blechwänden. Zeitschrift Bauwesen 1855, 5, 121–166. 67. Coffin, L.F., Jr. A Study of the Effects of Cyclic Thermal Stresses on a Ductile Metal; Knolls Atomic Power Lab.: New York, NY, USA, 1953. 68. Manson, S. Behavior of Materials Under Conditions of Thermal Stress. Available online: https://ntrs.nasa. gov/archive/nasa/casi.ntrs.nasa.gov/19930092197.pdf (accessed on 8 March 2018).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).