failure diagram and chemical driving forces for subcritical crack growth

Failure Diagram and Chemical Driving Forces for SubcriticalCrack Growth

K. SADANANDA

Kitagawa-Takahashi diagram that is modified for fatigue is now extended to the subcriticalcrack growth behavior under stress-corrosion crack growth. The analogy with the fatigue helpsus to identify several regimes of interest from both the point of understanding of the materialbehavior as well as quantification of the failure process for structural design of components thatare subjected to stress-corrosion and corrosion fatigue crack growths and failure. In particular,the diagram provides a means of defining the mechanical equivalent of chemical stress con-centration factor and the chemical crack-tip driving forces to crack growth or its arrest. Inaddition, threshold stresses, crack arrest, and nonpropagating crack growth conditions can bedefined, which help in developing sound design methodology against stress corrosion andcorrosion fatigue. Chemical crack driving forces under corrosion fatigue can be similarly definedusing the inert behavior as a reference.

DOI: 10.1007/s11661-012-1469-x� The Minerals, Metals & Materials Society and ASM International 2012

I. INTRODUCTION AND BACKGROUND

FOR characterizing the localized material damage interms of crack initiation and its growth leading to failureof a component, it is important to differentiate thecrack-tip driving forces in relation to the materialresistance. Damage in terms of crack initiation andgrowth occurs only if the crack-tip driving forces exceedthe material resistance. It is easier to define bothmaterial resistance and mechanical crack-tip forces fora limiting case of purely elastic material. In the purelyelastic regime, where a simple Griffith-criterion can beapplied, the crack-tip driving force is related to theelastic energy release rate that must exceed the materialresistance in terms of surface energy, c, to create the newsurfaces during separation. The material resistancecomes from cohesive forces that bind the material. Inthe case of the crack-tip plastic relaxation, there isenergy dissipation, and Griffith’s condition is modifiedto include this energy dissipation. Thus, crack-tipdriving force, G, expressed as Irwin’s energy release rateshould be greater than the total material resistance thatincludes the surface energy and the energy dissipated asthe plastic work term, P. This is expressed as

G � 2cþ P ½1�

As P is much greater than c, the latter is generallyneglected. However Rice[1] has shown that the aboveequation can be expressed in an alternate form as

G� P � 2c; ½2�

thereby separating the mechanical forces to the left-hand side and the material resistance to the right-handside, with the requirement that one has to consider thelocalized deformations that contribute to the crack-tipdriving forces. In this formulation, the c term becomesimportant as environment can reduce the surface energyc, thereby requiring reduced mechanical crack-tip driv-ing forces needed to propagate a crack. In the aboveequation, the term P, is reinterpreted not as dissipatingenergy term but as a contributing factor to the netmechanical driving force needed to overcome the mate-rial resistance. In addition, it is understood[2] that (a)plastic deformation at lower temperatures takes placevia generation of dislocations and their motion, (b) dis-locations are conserved (expressed as conservation ofBurgers vector, which implies that they are created asloops with every dislocation segment having its balanc-ing negative part in the loop, 180 deg away from it),and (c) the negative parts of the dislocations emittedfrom the crack tip form ledges that open the crack tip,while the positive parts form the plastic zone ahead ofthe crack tip. From the point of our discussion, the dis-locations contributing to plastic flow are the sources ofinternal stresses which can either augment or retard thecrack-tip driving force because of applied stress. To ex-press Eq. [2] in a more convenient form in terms of dis-location internal stresses, we resort to Irwin’s strainenergy release rate[3] expressed in terms of stress inten-sity factor K, and formally rewrite the above equations,respectively, in terms of K as

K2=E0 � 2cþ P ½3�

and

K2=E0 � P � 2c ½4�

with the restriction that deformation is localized at thecrack tip. The constant term E¢ depends on the plane

K. SADANANDA, Senior Engineer, is with Technical DataAnalysis, Inc., 3190 Fairview Park Drive, Suite 650, Falls Church,VA 22042. Contact e-mail: [email protected]

Manuscript submitted April 26, 2012.

METALLURGICAL AND MATERIALS TRANSACTIONS A

stress or plain strain condition. This can be formallyextended to express in terms of internal stresses arisingfrom plasticity, within the linear elasticity as

Kapl � KInt � fðcÞ or Kth ½5�

where f(c) designates the functional dependence on c, thesurface energy, and should depend on the surface areanewly created because of crack growth. In this gener-alization, we are expressing the plastic work term P interms of opposing force that need to be overcome by theelastic strain energy release rate and the net force thathas to exceed the material resistance for separation.Here, the term Kapl is contribution from the appliedstress, and Kint is contribution from the internal stressesdue to localized plasticity at the crack tip. Whilerecognizing that the internal stresses are self-equilibrat-ing on a larger scale (say integrated over the wholespecimen), we are interested only in the localized crack-tip driving forces that drive the crack growth at anyinstance. In essence, the wave length of the internalstresses may be of the order of plastic zone size near thecrack tip. Hence. the above analysis is strictly valid forsmall-scale yielding conditions. The internal stresses canbe positive (tensile) or negative (compressive) dependingon the nature of plasticity. Thus, for example, under-loading can result in compressive crack-tip plasticitythat can result in tensile internal stresses which canaugment crack trip driving force causing acceleration ofcrack growth. On the other hand, overloading can createextended tensile plastic zone that cause compressivestresses at the crack tip, and hence, can contribute toreduction in crack-tip driving force, and thus theretardation of crack growth rates. The remote disloca-tion stress fields far away from the crack do not affectthe crack-tip driving force as (a) they are remote, and (b)every dislocation has its counterpart that balances itsstress field. The range of their internal stress-contribu-tions is limited to inter dislocation loop spacing orspacing between positive and negative dislocations. Thisglobal dislocation-plasticity, however, contributes toenergy dissipation, as creation of every dislocation loopinvolves energy expenditure. However, they do not enterinto the crack-tip driving force as the force is thenegative gradient in the energy with crack lengthincrement, and their contributions change little withthe increment. (If they contribute to the gradient thenthey have to be considered). The above representation issomewhat similar to Neuber’s approach[4] for plasticnotch-tip stain fields. He considers the local strain-energy density expressed in terms of the theoreticalstress intensity factor Kt consisting of square rootaverage of the Kr and Ke as stress and stain intensityfactors, respectively. For a purely elastic case, Kt reducesto that of elastic crack, Kr, and for a local plastic strainsat the crack or notch tip one has consider Ke also thatdiffers from Kr, as it can be a significant factor.

We can now formally generalize Eq. [6] as

Kapl � KInt � Kth ½6�

to cover the cases where the internal stresses can eitheraugment the applied forces or retard it depending on the

sign of the internal stress. Thus, we open the possibilitythat the local plastic flow can help in crack growth, as isthe case when we have local tensile residual stresses fromplasticity. In these expressions, the Kth-term representsthe material resistance measured as threshold stressintensity factor below which crack growth does notoccur. In the above representation, Kth, therefore, ismaterial property and does not depend on the crack size,as some have assumed.[5] It includes, besides the surfaceenergy term, any other contribution that opposes de-cohesion of the material along the crack plane. Thesteady-state threshold stress intensity value, Kth, corre-sponds to that of a long crack that samples the averagematerial microstructure (including grain size) that issubject to the applied load and the environment. Theeffect of compressive or tensile internal stresses onincubation time in stress-corrosion crack growth wasdiscussed before[6] along the lines of Eq. [6]. In the caseof fatigue, for example, if the local crack incrementoccurs by the plastic blunting process (formation ofstriations, etc.—see Laird and Smith,[7]) then the thresh-old includes the work required to accomplish that bydislocation nucleation in addition to the surface energyterm that enters into the expression.[8] If there is asteady-state plastic zone associated with the crack tip,then threshold stress intensity includes the stress inten-sity necessary to overcome the resistance due to the saidsteady-state plastic zone.In the analysis of force on a dislocation, it is well

recognized by the materials community that the netforce is expressed as force due to applied stress plus orminus that due to internal stress, with the recognitionthat the internal stresses can be positive or negative.[2]

The same concepts are applicable in terms of the netforce on a crack tip in terms of applied and internalstresses. Internal stresses due to dislocations depend notonly on their number but on their distribution. Hence,they are history dependent. The crack-growth retarda-tion due to overloads and acceleration due to under-loads during fatigue are well known. However,physically, they arise in terms of excess dislocationsover and above the steady-state value that are generatedand that contribute to positive or negative internalstresses which accelerate or retard the crack growth.Crack arrest can occur if the net K in Eq. [6] falls belowthe threshold K. It is difficult to deduce from thecontinuum principles represented in Eqs. [4] through [6]in terms of the magnitude of applied and internalstresses for a given loading conditions. Nevertheless,within in the linear elastic formalism, where the stressescan be additive, we can formally express the crack-tipdriving forces in terms of applied and internal stressesthat are needed to over come the material resistance toseparation. Residual stresses, due to such as welding,shot peening, quenching, cold-rolling, etc., which thefracture-mechanics community routinely considers areonly a subset of the internal stresses that are generatedin a material that can enhance or oppose the appliedstresses affecting the kinetics of crack nucleation and itsgrowth. Crack initiation in the presence of pre-existingstress concentrations such as notches involves pre-existing internal stresses aiding the crack nucleation.


Extending further, and considering that a crack is a highenergy defect, its nucleation even in a smooth specimencan only be facilitated by a built of local in situgenerated internal stresses. In the case of fatigue, thesein-situ generated internal-stress concentrations arisebecause of heterogeneous plasticity of favorably ori-ented large grains near the surface, with the formationof dislocation dipoles, intrusions and extrusions, dislo-cation channels, cell walls, etc. There is a cumulativecyclic strain represented by hysteresis in each cycle, andwith a number cycles, there is a buildup of local strainenergy that helps the applied stress in initiating andpropagating a fatigue crack. The number of cyclesrequired for crack initiation in notched specimen islower than that in a smooth specimen for the samereason that local internal stresses in the form of stressconcentration aids the total crack-tip driving force, inEq. [6]. If the notches have plastic strain fields, thecontribution from the notch tip stress fields also changesas modeled, for example, by Neuber.[4]

We can further modify Eq. [6] to include the envi-ronmental contributions. Fatigue is driven by plasticflow contribution to crack growth, in terms of cyclicstrains. The internal stress term in Eq. [6] depends on thenature and distribution of dislocations at the crack tip.Similarly, stress corrosion is driven by localized chem-ical driving force which depends on the specific kineticsof the governing mechanism for a given material/environment system. Chemical potential gradient thatprovides the chemical driving force is difficult todetermine unless we know the specifics of the crack-tipreactions. There is a general consensus that these aredifficult to determine locally at the crack tip. In addition,as the crack growth rates increase, the reaction kineticsalso can change making their determination and quan-tification extremely difficult. Here, environment caninclude aqueous, gaseous, or liquid metal environment;internal or external source; and thus all embrittlingprocesses that provide an additional local crack-tipdriving force. For the purpose of life prediction, weresort to generality separating the crack-tip drivingforces from the material response, both of which can bequantified without knowing the details of the mecha-nisms or chemical reaction kinetics involved. For thecase of environment, Eq. [6] remains the same except forthe fact that the Kint term includes the contribution fromthe internal stresses because of dislocations as well as themechanical equivalent of chemical driving forces thatcontribute to crack growth. In addition, as surfaceenergy term c is also affected by the environment, thethreshold for crack growth Kth in Eq. [6] is also affected.That is, in addition to the crack-tip driving forces, thematerial resistance also changes by environment. Thethreshold Kth, being a measure of material resistance,has to be determined experimentally. The questionremains in terms of how to separate chemical andmechanical effects on crack growth as both the mechan-ical crack-tip driving forces and the material resistanceare affected by the environment. This can be done only ifwe have a proper reference state as a basis forquantification of the mechanical equivalent of chemicaldriving forces vs purely mechanical driving force.

Following methodology provides a suggested tool forquantification of the chemical equivalent crack-tipdriving force both under static loads (stress-corrosioncrack growth) and cyclic loads (corrosion fatigue crackgrowth).

II. CHEMICAL EQUIVALENT OF MECHANICALFORCE

It is well known that in fatigue the stress concentra-tion factor Kf differs from the elastic stress concentra-tion factor Kt. While Kt is defined as ratio of maximumstress at the notch tip to the nominal stress, Kf is definedusing a failure criterion.[9] It is the ratio of nominalamplitude stress to fail a smooth specimen to that of anotched specimen. Hence, it has been found that Kf £ Kt

depending on the notch severity; for blunt notches theyare equal, while for sharp notches, Kf is less than Kt.This difference arises because a nucleated crack at sharpnotches can get arrested as Eq. [6] is not fulfilled unlessapplied K is increased. The reason is the internal stressgradients are too high to the extent that the net force atthe incipient crack that nucleated at the notch root canfall below the material threshold because of sharp dropin the net crack-tip driving force. Generalizing theseconcepts we can formally define the chemical drivingforce using either failure or crack growth criterion.These definitions become relevant in quantifying thechemical crack-tip driving forces in par with themechanical driving forces, as in the ultimate analysis,we are interested in the reduction in the applied force tocause damage due to the embrittlement process.

III. CHEMICAL STRESS CONCENTRATIONFACTOR

Based on the above reasoning we can define themechanical equivalent of the chemical stress concentra-tion factor Kch (not to be confused with KIC) as thestress for initiation and/or propagation as the ratio ofthe stress in an inert environment to that in theembrittling environment. Thus, we define the chemicalstress concentration factor for initiation as,

Ki-ch ¼ ri-inert=ri-ch � 1 ½7�

Similarly for propagation or failure we define thechemical stress concentration factor as

Kf-ch ¼ rf-inert=rf-ch � 1 ½8�

where ri-inert and ri-ch are the nominal stresses for crackinitiation, and rf-inert and rf-ch are for crack propagationand failure in inert and chemical environments, respec-tively. These definitions can be applied to all cases withKt ‡ 1 including the limiting case of Kt = 1 when thespecimen is smooth uniaxial specimen. In the case ofsmooth specimen, the above equation reduces to theratios of crack initiation and failure stresses in inert andaggressive environmental conditions. Essentially, the


chemical driving force is defined indirectly in terms ofthe reduction in the applied stresses to cause crackinitiation and failure by the specific environment, fromtheir corresponding stresses in an inert environment,which forms a reference state. It is recognized that theinertness of the environment again depends on thematerial; ultra high vacuums some times are notsufficiently inert for some reactive materials, while theambient air is inert for noble metals like gold. By using areference state, we are now defining the incrementalforces due to environment as excess (or deficit) quan-tifies over and above the reference state. The aboveprocedure is not new as many thermodynamic quantitiesare generally defined by excess (or deficit) quantitiesusing ideal or inert system as reference. Hence, theseconcepts are important in quantification of the chemicaldriving forces with less ambiguity. With this backgroundwe can now discuss the failure diagram that can help usin quantifying and providing a physical perspective inunderstanding the above crack definitions.

IV. FAILURE DIAGRAM

We now analyze the behaviors of smooth, notch, andfracture mechanics specimens and show how we canapply the above definitions in quantifying the mechan-ical, chemical, and materials behavior in terms of crackinitiation and propagation.[10] The analysis is againgeneric and is applicable for all the localized damagesaccentuated by aggressive environments: aqueous, gas-eous, liquid metal, and internal or external environ-ments. From the point of crack growth, the analysisrelays in the representation of growth rate da/dt or da/dN in terms of the linear elastic stress intensity factor, K.In essence, the presumption is that the plastic deforma-tions around the crack tip are localized and are limitedclose to the crack tip, thus justifying the application of Kfor all cases of subcritical crack growth.

V. SMOOTH SPECIMEN BEHAVIOR

Figure 1(a) shows the schematic behavior of a smoothspecimen in an aggressive environment with concentra-tion C1. Figure 1(b) shows the applied stresses for crackinitiation and failure. It decreases with increase in timeand reaches asymptotic limits defined as threshold stressfor initiation, rth, and for propagation and failure rPR

or rf, respectively. The stress for crack initiation, rth,can differ from that for its propagation, rPR and/or forfailure rf, depending on the material work hardeningcharacteristics. Time factor is not represented in theabove equations as they are defined for a static orsteady-state condition. Hence, at any given time, theabove factors can be defined. Figures 1(b) and (c)indicate that for any given concentration there are someminimum stresses required for crack initiation andgrowth. Furthermore, there is a saturation effect ofenvironment, i.e., after some concentration, calledsaturation limit, CS, there is no further change in theminimum stress value. These imply that environment

alone cannot do complete damage in terms of nucleationand propagation of a crack; it only assists the appliedstresses. In contrast, applied stresses alone can contrib-ute to damage in terms of crack initiation and growth asis the case in an inert environment. Hence, to quantifythe effect of environment we have to use inert environ-ment as a reference state, as is done in the aboveequations. Here the time required for crack initiationforms a variable just as in the case of fatigue the numberof cycles required to contribute to damage is a variable.The time-dependency can be related to the details of themicromechanics involved such as reaction kinetics at thecrack tip, hydrogen generation rate, hydrogen diffusionrate or accumulation rate ahead of the main crack forembrittlement to occur, etc. Saturation effect comes asthe number of lattice sites available for occupancy byaggressive atoms is limited depending on the material,crystallographic planes (cleavage vs slip), availability ofother dispersing traps for environmental species, etc.Stress corrosion literature[11–15] is overwhelmed in termsof controversies and complexities governing the detailsof the mechanisms of crack initiation and growth.From the point of mechanics, however, there is a net

reduction in the material resistance, irrespective of thedetails of the specific mechanisms involved. The mech-anisms essentially reflect in terms of material resistanceto separation, measured in terms of limiting threshold,rth for smooth specimen, KISCC for fracture mechanicsspecimen, etc. In a smooth specimen, crack initiationoccurs at localized stress concentration facilitated byheterogeneous slip in large grains that are favorablyoriented. Thus, stress concentration either pre-existingor in situ created by heterogeneous deformation isessential for embrittlement, as shown by Kamdar[16]

earlier. Our discussion here is limited to localized

C1

(a)

σ

Time

σth

σPR ≈ σf

ConcentrationC1

(b)

Initiation

Failure

σth

Concentration

σth*

σPR∗ ≈ σf*

Saturation

Effect

(c)

Initiation

Failure

Fig. 1—Behavior of a smooth specimen under tensile load in a cor-rosive environment. (a) Specimen’s configuration and environment,(b) stress vs time for crack initiation and propagation for a givenconcentration, and (c) variation of thresholds for crack initiationand propagation as a function of concentration.


corrosion where crack nucleation occurs rather than togeneral corrosion, such as exfoliation, general galvaniccorrosion, etc.

Figure 2 shows the behavior of a fracture mechanicsspecimen in an environment with concentration C1. Theresults are somewhat parallel to that of smooth speci-men in Figure 1, except that the applied stress contri-bution is expressed in terms of K, the stress intensityfactor range. There is threshold stress intensity forinitiation and propagation for a given concentration,and these thresholds also reach asymptotic limits as afunction of concentration indicating a saturation effectof the environment. The stress intensity factor takes care

of stress-crack length dependency. Finally Figure 3shows the behavior of an intermediate case, a notchwith a finite notch tip radius. In the limiting case ofinfinite notch tip radius, the behavior should reduce tothat of a smooth specimen, and in the limit of zeroradius, the behavior reduces to that of a crack. Hence,notch of finite radius falls in between those of a crackand a smooth specimen. Important to note here is, as infatigue, there is a regime at high Kt a range where crackcan initiate under stress corrosion, but not propagate.Understanding and quantifying the regime where non-propagation to propagation occurs obviously becomeimportant from design considerations.Figure 4 shows a failure diagram along the lines

described as modified Kitagawa-Takahasi diagram[17]

for fatigue. The applied stress vs crack length is plottedin a log-log scale. This diagram does not consider thetime or duration that the specimen is in the environmentunder stress. Incubation times and propagation kinetics,and role of applied stress on these are important forprediction of the total life. For concentration C1, thelimiting threshold stress for a smooth specimen, rth,forms a limiting value on the ordinate indicated in thefigure. We will consider that the Y-axis represents thesmooth specimen behavior, as zero crack length cannotbe represented in the log-scale. This limiting stress rthdecreases with the increase in concentration (Figure 1)and reaches a limiting value r�th at the saturation limitindicated by the concentration, CS. Likewise, the prop-agation stress, r�PR or failure stress, r�f , which may alsobe concentration dependent, are indicated for thesaturation case in the Figure 4. For a fracture mechanicsspecimen, the limiting thresholds K�ISCC and K�IC areindicated by the corresponding lines, considering therelation K = Yr�(pa) where Y is the geometric factorand is taken �1, r is the nominal or applied stress, and ais the crack length. The slopes of the KISCC and KIC linesare, therefore, �0.5 each. The KISCC and KIC lines areextended meeting the propagation a�PR and a�f , of thesmooth specimen on the Y-axis. For the sake ofconvenience, we drop (*) from our discussion, withoutlosing the generality. KISCC corresponds to the thresholdbelow which stress-corrosion crack growth does not

K

Time

KISCC

KPR ≈ ΚIC

ConcentrationC1

(b)

(a)

Initiation

Failure

K

Concentration

KISCC*

KPR∗ ≈ Κ IC*

Saturation

Effect

(c)

Initiation

Failure

Fig. 2—Behavior of a fracture mechanics specimen in a corrosiveenvironment where the stresses are expressed as stress intensity fac-tors. Time and concentration dependences are similar to those ofsmooth specimen in Fig. 1.

σ

Time

ConcentrationC1

(b)

Kt1

Kt2

Changing ρand/or Kt

σ

Kt

σPR

σIN

Non-PropagatingCracks

No-cracks

(c)

(a)

Fig. 3—Behavior of notched specimen similar to those of smoothand fracture mechanics specimens for a given stress concentrationfactor. Nonpropagating conditions can prevail for sharp notcheseven after crack initiation conditions are met.

σth*

Log (Crack Length)

Log

(Str

ess)

σPR*

σf*

Internal stress

Non-Propagating Cracks

c1

KISCC

KIC

Failure Diagram

A

B

C D

cS

σthac

Fig. 4— Failure diagram connecting behaviors of smooth and frac-ture mechanics specimens as two limiting conditions with nonpropa-gating and internal stress regimes defined. Mechanical equivalent ofchemical internal stress is defined.


occur. As the crack size decreases, the stress to accom-plish the crack growth increases until the applied stressmeets the propagation threshold, rPR, in a smoothspecimen. Similarly is the KIC behavior, indicating therequired fracture stress as a function of crack length.Here, as in the case of fatigue, we assume that the longcrack growth behavior represents intrinsic or funda-mental material response or resistance, and the thresh-olds and limiting values are independent of crack size.Implication is that any deviations from the long crackgrowth behavior arise because of fluctuations in thecrack-tip driving forces. Thus, the diagram combines thebehaviors of a smooth and the fracture mechanicsspecimens. The limiting lines in Figure 4 define a regimeresembles a triangle, and is being referred henceforth asinternal stress triangle. We may note here that Gangl-off[18] has considered electro-chemical short cracks,essentially defining contributions from electrochemicalcrack-tip driving forces that become dominant in somematerial-electrolyte conditions, particularly when thecracks are small. These are additional forces that arise insome systems under consideration as localized forces;but the generic behavior and representation in thefailure diagram remains the same as in Figure 4.

Of interest now is to examine the regions bounded bythese limiting curves and their physical significance.Looking at the smooth specimen data, while going fromthe concentration C1 to CS, the applied stress to initiatea crack decreases with increase in concentration. Onecan define the efficacy of the environment in contribut-ing damage in terms of crack initiation by the gradientdr/dC, see Figure 1. It is the mechanical equivalent ofchemical damage parameter for the specific material/environmental system, with the assumption that damagemechanism remains the same during this change. Takingthe inert behavior as reference, we can define themaximum efficacy of the specific chemical agent as½fri-inert � r�thg=CS�: The dependence may not be asimple linear function, as shown in Figure 1(c). How-ever, the limiting or steady-state conditions can be notedin Figure 4, while for dynamic conditions for any givenstress and time, the actual curves for each concentrationas shown in Figures 1 to 3 have to be generated.

Examining Figure 4, it shows that the crack initiationcan occur at r�th for a smooth specimen when theconcentration is CS or above. However, based on thefracture mechanics analysis the initiated crack does notgrow at that stress unless the minimum size of the crackis given by ac. For the incipient crack to propagate, tothe size ac, two scenarios are possible: (a) The nominalstress on the smooth specimen can be increased to r�PRfor it to meet the propagating condition and subse-quently to failure stress r�f . (b) The local internal stresscan be raised to the rPR level, and further, as the internalstresses decrease with distance, the gradient of theinternal stresses should be such that KISCC condition iscontinuously met as the incipient crack grows to the sizeof ac. Internal stresses can only be met by dislocation-stress fields by their appropriate distribution (e.g., as ina pile-up). In many theories[11,16,19] of stress corrosionand even liquid metal embrittlement (LEM), plasticity isinvoked as the necessary ingredient for crack growth.

These include[12] hydrogen-enhanced localized plasticity(HELP), hydrogen-enhanced decohesion (HED), inter-nal hydrogen-assisted cracking (IHAC), or sustainedload crack growth (SLC), hydrogen environmentassisted cracking (HEAC), LEM, solid metal embrittle-ment (SME), temper embrittlement (TE), etc. In allthese mechanisms, implication is that the environment isaffecting either P or c or both in Eq. [2]. Most ofcomplexities that are stated in the literature in under-standing stress-corrosion crack growth for a givenmaterial-environment-loading system arise because ofthe inability of knowing or in defining exactly how theseterms are affected and/or how to compute or quantifytheir contributions to either to the net crack-tip drivingforce or to the material resistance or both. Figure 4circumvents the problem to some extent and examinespurely on the basis of the net resultant crack-tip drivingforces needed for crack growth in relation to thematerial resistance which is also affected by environ-ment. Here HELP comes by building the necessaryinternal stresses that can help the remote applied stressto overcome crack growth resistance represented byKISCC. It is recognized that KISCC is a function ofconcentration as shown in Figure 2(c) and has to beexperimentally determined. Figure 4 points out thatessential aspect involved in crack propagation at lowremote stresses. Specifically not only the magnitude ofthe internal stress has to be sufficiently high to meet thecrack propagating condition, the gradient of the internalstresses also should be sufficient to meet by KISCC-line atthe imposed nominal stress. If there is a pre-existingcrack of Size A, Figure 4, with remote stress r�th, thencrack can only propagate along the path A fi B andB fi C by generating sufficient internal stresses andtheir gradients via localized plasticity to reach the size ac.For a> ac, crack can grow under the remote-appliedstress, r�th, along the path C fi D to meet the failurecriterion or overload fracture criterion. The decohesionaspect is involved in the reduction of KISCC while thecrack-tip plasticity is involved in building up thenecessary internal stresses and their gradients to meetthe total force required for crack growth. Environmentcan also reduce the flow stress of the material andthereby contribute to the local plasticity that helps in theaccumulation of dislocations that are needed for internalstress buildup. Environment can also affect the decohe-sion process in lowering the KISCC required for crackgrowth. Thus, we have a case here where environment,while decreasing flow stress contributing to embrittle-ment also reduces the cohesive forces thereby reducingKISCC. Plastic flow can build up the internal stresses toaid applied stresses in overcoming the reduced materialresistance to failure. Figure 4 mimics the mechanics ofcrack growth considering the material resistance intoconsideration.There is always plasticity associated with a crack,

particularly for metals as sharp crack cannot be sus-tained in a metal. Hence, KISCC-line in Figure 4 includesthis intrinsic plasticity associated with it, as it isexperimentally determined material property that con-siders the steady-state conditions existing at the cracktip. Internal stress triangle, Figure 4, therefore, defines


the additional internal stresses required to reach thissteady-state condition. Hence, the internal stress trianglecorresponds to the minimum stress-distance profile atthe stress concentration to insure continuous crackgrowth. For the case of a smooth specimen, major partof life may involve initiation of a crack, and thereforedamage process involves in building the necessaryinternal stresses and their gradients for an incipientcrack that is formed to grow to the size until the remotestresses will become sufficient to enable its continuedgrowth—until KISCC line is met. If there are overloads orunderloads then local perturbations in the steady stateare accounted by the local perturbation in terms of thedislocation-generated internal stresses whose effect maylast until steady-state conditions are again established.The fact that the overloads and underloads do affect theincubation time and that their effects can be accountedfor by the excess (or deficit) internal stresses has beenshown before.[6]

In the case of notches, we have an intermediate casewhere crack can initiate when the maximum stress at thenotch tip meets the rth condition for the concentrationused. The crack, however, does not necessarily propa-gate. The notch-tip stresses normally decrease withdistance from the notch. If there is local plasticitybecause of notch tip stresses, then the internal stressesaround the notch tip also get altered. Incipient cracknucleated at the notch tip grows in the stress field of thatplastic notch. From Figure 4 these internal stresses fromthe notch have to meet the minimum criteria both interms of magnitude and the gradient for the incipientcrack formed to grow. For a sharp notch at low stresses,the initial magnitude of the internal stresses can be highbut, as the gradients are also high, the stresses can fallbelow the required stress for propagation, therebycontributing to crack arrest. Thus, as in fatigue, wecan have nonpropagating cracks whenever the localstress falls below that required for propagation. This isschematically shown as in Figure 3. Figure 4 defines theregime of nonpropagating cracks wherein the appliedstresses are not sufficient to generate appropriate localplasticity that can provide the sufficient internal stressesfor crack growth. Hence, from design point, the stressesshould be within the nonpropagating conditions for thematerial to prevent failure of a component under SCC.The concepts are similar to the endurance limits andnonpropagating cracks in fatigue. As discussed above,the Kf differs from Kt for the same reason that crackpropagation conditions need to be met for fatiguefailure to occur. By the above analysis, the sameconcepts are applicable for all subcritical crack growthprocesses. In essence, the differences between Kf and Kt

in fatigue and similarly in SCC, and other subcriticalcrack growth phenomenon arise because of the require-ment that not only the minimum in the magnitude of theinternal stress but also the minimum in the gradient ofthe internal stress have to be met for crack propagation,as envisioned in the failure diagram.

The biggest problem in all these is how to predict thenature of the internal stresses and their gradients ornature and distribution of dislocations required; or inessence, how material generates the required internal

stresses. Analytic solutions are difficult, and even thediscrete dislocation calculations are also difficult becauseof number of dislocations involved in each step. Inaddition, the strain history becomes important as thereis slow buildup of these internal stresses as function oftime or function of number of cycles in fatigue. We haveanalyzed the fatigue problem separately, but here wepresent some limited data under static load to show thevalidity of the above concepts. Major problem in theanalysis of stress-corrosion crack growth as well asothers is the lack of systematic crack growth rate datafor a given environment in relation to an inert environ-ment. In many cases, the threshold data that provide themeasure of material resistance is lacking for bothsmooth specimens as well as fracture mechanics speci-mens. Local chemical forces that augment local mechan-ical stresses should also be included in contributing tolocal crack-tip driving forces. These are difficult todetermine other than by conducting experiments in thecorresponding chemical and inert environments. Futureefforts should be directed to generate the requiredexperimental data that is necessary for any life-predic-tion model for components in service.

VI. ANALYSIS OF HIROSE AND MURA DATA

In the following, we present analysis of systematic butlimited data by Hirose and Mura[20] on 4340 Steel testedin 0.1 N H2SO4 acidic solution using deep notches butwith different notch tip radii. Figure 5 shows their datain terms of the time for initiation of a micro crack aheadof the notch tip for different apparent K values, whereapparent K was calculated considering the deep notch ascrack of the same length, as the radius is very smallcompared to the depth of the notch. They have shownthat the micro crack initiates within the plastic zone ofthe notch (Figure 6). Figure 5 shows the variation ofthreshold K as a function of root radii. They havecalculated the maximum stress at the notch tip, as wellas the nominal stress applied for each case. These areplotted as function of root radius q in Figure 7. Themaximum stress and the nominal stress follow a powerlaw relation with root radius. The extrapolated lines inFigure 7 meet at a point when the radius is large. Atsuch large root radius, the nominal stress and themaximum stress at the notch tip are essentially equal,which corresponds to case of a smooth specimen. Thisprocedure is adopted as there are no data available for acrack nucleation in a smooth specimen for that envi-ronment. Using all the available information includingthe extracted smooth specimen data, reported KISCC

data, and crack length, we reconstruct the flow diagram,and it is represented in Figure 8. The analysis shows thatall the experimental data fall outside the KISCC butslightly below KIC line except for a datum. After thecrack initiation the authors have stopped the test. Crackinitiation is defined in terms of time for the formation ofdetectable size of a crack. From the figure it is clear thatif the test continued the crack that formed will growunder the same applied stress until overload fractureoccurs, meeting the KIC line. Although the experiments


are limited, the results are consistent with the failurediagram. In order to evaluate and quantify completely,we need the required data for notches of different Kt

with different root radii, as a function of concentration.For notches, one can use the elastic-plastic notch tip

stress field analysis to arrive at the local K value for aninitiated and growing crack, as is done for fatigue.[21]

The emphasis here is to show how to integrate thecorrosion data of smooth specimen to fracture mechan-ics specimen in one unified frame work using the failurediagram. Considering the divergent corrosion mecha-nisms and complexities associated with materials andenvironments involved, a simple approach is suggestedfor quantification of the mechanical equivalent ofchemical driving force for component life prediction.We extend these concepts to corrosion fatigue to showhow using the behavior of an inert environment it ispossible to follow even the dynamic changes in theenvironmental contribution to the fatigue crack growth.

VII. APPLICATION TO CORROSION FATIGUE

In contrast to subcritical crack growth under sus-tained load where Kmax can adequately describe thecrack-tip driving force because of applied mechanicalload, fatigue involves both peak load and amplitude,requiring two load parameters for accurate description.In terms of stress intensities, we have for fatigue,amplitude DK, and Kmax; and corresponding thresholdsare DKth and Kmax,th. Both thresholds need to besatisfied for crack growth to occur under fatigue. Ingeneral, Kmax,th is greater than DKth, and hence it is thecontrolling parameters at low R-ratios, while DKth iscontrolling parameter at large R ratios, as Kmax condi-tion is met. Environmental contributions during fatiguecome predominantly via Kmax parameter. In pure inertenvironment, the R-ratio effects are minimum, whileDKth and Kmax,th in this case are essentially equal. Thefatigue behaviors in inert and aggressive environments issummarized in Figure 9. In Figure 9(a), the basic crackgrowth rate behavior is expressed in terms of da/dN vs

102

Hirose & Mura, 1984

4340 Steel – 0.1N H2SO4

2.5

ρ

101

0.5

0.110.25

1 01.0

10

101 102 103 104

Time, sec

Kap

plie

d

Fig. 5—Experimental data showing applied stress intensity factor Kvs time for crack initiation for notches with different root radii, q. Kis calculated assuming that notch as a sharp crack.

0.35

0.3

4340 SteelHirose and Mura, 1984

0.2

0.25

Crack length

PZS

Nucleated ahead of the notch-tip

0.15

Cra

ck le

ngth

or

PZ

S (

mm

)

b

0.05

0.1

00 0.5 1 1.5 2 2.5 3 3.5

ρ (mm)

Fig. 6—Crack initiation occurs within the plastic zone of a notch. bdefines the distance at which the microcrack nucleates ahead of themain crack.

10

100

1000

0.1 1 10 100

175 MPa

Slope=0.26

Slope=0.24

4340 Steel

ρ,mm

σnominal

σmax

σ ,M

pa

Fig. 7—Calculated maximum stress at the notch tip and nominalstress for crack initiation as a function of notch tip radius. Theextrapolated values meet at large notch root radius indicative of asmooth specimen conditions.

1880

1000

4340 Steelσf = 1880 MPa

B

KIC-LineChemicalStress C

100Exptal data

σth = 175 MPa

Stress

A D

KISCC-Line

Non-Propagating Cracks

Crack length, m

Nom

inal

Str

ess,

Mpa

1010-5 10-4 10-3 10-2 10-1 100

Fig. 8—Failure diagram connecting limiting stress values for crackinitiation and failure in a smooth and fracture mechanics specimens.


DK for different R-ratios (for simplicity only high andlow R ratio data are illustrated). At any given crackgrowth rate one can plot DK-Kmax plot showing the L-shaped behavior. With increasing crack growth rate theL-shaped curves shift to higher DK-Kmax values, and acrack growth trajectory path can be described in termsof the variation of these two parameters as a function ofcrack growth rate. For an inert case, the two parametersare equal with the trajectory path represented in termsof DK-Kmax following a 45 deg-line. This line representspure fatigue phenomenon, where crack growth is essen-tially determined by cyclic stains with no monotoniccomponents that involves only Kmax component. Forexample, these monotonic components include processsuch as void or crack nucleation ahead of the maincrack, cleavage faceting or cracking of ceramic inclu-sions ahead of the main crack, etc., which are moreprevalent during overload fracture. If environmentaffects via Kmax, then the limiting DK and Kmax valuesdeviate from each other, and correspondingly thetrajectory path deviates from the 45 deg-line. Note thatthe limiting values are positive in the L-shaped curve.The degree of deviation depends on the environmentalcontribution, which, in turn, depends on the relativekinetics fatigue damage vs environmental damage. Thereare various types of mechanisms involved, and they varydepending on the relative kinetics of the fatigue andenvironmental process, which again depend on specificsof material and environment. These have been discussedextensively before providing experimental support foreach case. The emphasis here is that chemical equivalent

mechanical force contributing to crack growth can bedefined using the inert crack growth as a reference. InFigure 9(d), A and B represent the values of DK-Kmax

values in inert and aggressive environments, respec-tively, for the same crack growth rate, da/dN. Here, weare using the same crack growth rate in the referenceand the aggressive environments as the basis forquantification of the environmental contribution to thedriving force. The environment can affect components,DK and Kmax, thus affecting the kinetics of corrosionfatigue crack growth. The relative contributions dependnot only on material, environment, loads, and testfrequency. Increasing frequency can enhance fatiguecontribution, while decreasing frequency can enhancethe corrosion contribution. There is a saturation effectof environment at prolonged times as discussed before.In Figure 9(d), the vector AB is decomposed into AC

and CB, indicating specific changes in the DK and Kmax

components because of respective environments. Thesetwo contributions can change independently with crackgrowth rate, as well as vary with the material andrelative kinetics of fatigue vs superimposed environmen-tal effects. For example, when crack growth rates due tofatigue increase rapidly, the time for environmentalreactions decreases resulting in decreasing contribution.On the other hand, when applied stresses increases, theaggressiveness of the environment can also increase as instress-corrosion crack growth. Thus, competing factorsenter here determining the net crack growth rate. Insome cases, the CB component in Figure 9(d) can bezero indicating that for those cases, the environmentaffects the cyclic component only. For most of the cases,however, the CB component increases rapidly withapplied stress or crack growth rate. In essence, thesetrajectory paths indicate the environmental contribu-tions do not remain constant for a given material andenvironment, but varies with the applied load, testfrequency, temperature, or more correctly with background crack growth rates because of cyclic loads.[22]

However, without going into details of the particularmechanism involved, one can quantify the materialresponse and the mechanical forces involved. Basicexperimental data for the material in inert and the givenenvironments are needed to quantify chemical crack-tipdriving forces operating in a given system.Figure 10 combines both fatigue and stress-corrosion

crack growth into one failure diagram representingfatigue endurance in terms of peak stress. It is assumedthat fatigue thresholds are lower than the thresholdsunder stress corrosion. The analysis is similar to that forFigure 4. We have internal stresses that are generated bycyclic plasticity on one side and the internal stressesgenerated by the chemical process on the other. Syner-gistic effects occur as plastic flow properties near cracktip are affected by chemisorption, etc. which in turnaffect monotonic and cyclic strains. The thresholdsKmax.th, and DKth along with KISCC can all be affected.Figure 9(c) depicted continuous changes in the govern-ing mechanisms of environmentally affected crackgrowth via trajectory map. Recognizing that thesecontinuous changes that can occur during crack growth,the best strategy for life prediction is to use the inert

da/d

N

K

Inert

Corrosive

High R

Low R

A

B

C

Kmax*

(a)

K*

Kmax*

da/dNIncreasing

Crack growthTrajectory

Non-Propagating

(b)

(d)(c)

K*

∇

Kmax*

I

II

IIIIV

45o line

∇

∇∇

Fig. 9—Schematic diagram for corrosion fatigue data under constantamplitude as a function of load ratio, R. (a) da/dN curves in inertand aggressive environments, (b) L-shaped curves for any givencrack growth rate defining limiting values for DK and Kmax. (c) Tra-jectory map for pure fatigue vs corrosion fatigue indicating differentmechanisms can be operating during crack growth. (d) Determina-tion of environmental contribution to DK and Kmax during corrosionfatigue using same crack growth rate as reference for inert and givenenvironments.


environment as the reference state to define the super-imposed environmental effects under fatigue as isillustrated in Figure 9(d). A working compromise is touse the specific material/environment experimental cor-rosion fatigue data under constant amplitudes as intrin-sic to the system, and use the data as a basis to predictthe life for a structural component under variableamplitudes, assuming that the similar conditions prevailfor the component in use. This is currently beingimplemented in the development of the life predictionmethodology for fatigue under corrosive environments.

VIII. SUMMARY AND CONCLUSIONS

A failure diagram is presented by extending themodified Kitagawa-Takahashi diagram for fatigue.The behaviors of smooth specimen and that of fracturemechanics specimen are integrated through their limit-ing values in terms of thresholds, KISCC, KIC, andfracture stresses. Crack initiation in both smoothspecimen as well as notch specimen, incubation in afracture mechanics specimen are all accounted by therequirement of internal stresses to aid the applied stressto meet the threshold stress intensity for crack growth.The internal stresses are achieved by the dislocationplasticity. Using these concepts, the chemical drivingforce, chemical stress concentration factor, and mechan-ical equivalent of chemical driving forces are defined andjustified. Analysis of limited experimental data onnotches shows that the results follow closely the failurediagram presented. The procedures that were used in thepast to define and quantify the changing environmentalcontributions during corrosion fatigue follows. The

analysis also points out the need for systematic exper-imental data to characterize the material response togiven applied loads and environments for both smoothand fracture mechanics specimens.

ACKNOWLEDGMENTS

The author acknowledges the helpful discussionswith Dr. A. K. Vasudevan of ONR. The research issupported by ONR under the contract N00014-10c-0359 with Dr. Vasudevan as Technical Monitor.

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Fig. 10—Failure diagram connection fatigue and stress-corrosioncrack growth defining limiting conditions, minimum crack growthconditions, and nonpropagating conditions. Internal stress triangledefines both chemical forces and mechanical forces involved duringcorrosion fatigue.


failure diagram and chemical driving forces for subcritical crack growth

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