Predictive Capability of Fracture Mechanics Concerning Environmentally Assisted Cracking
J. Toribio1 and V. Kharin1,2
1Department of Materials Science, University of La Coruña,
ETSI Caminos, Campus de Elviña, 15192 La Coruña,SPAIN
2On leave from: Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,
290601 Lvov, UKRAINE
Fax: (34-81) 13 28 76 ;E-mail:kharin@udc.es
Abstract.- Analysis of environmentally assisted cracking (EAC)
is performed with emphasis on the validity of the fracture mechanics
approach for evaluation and prediction of crack behaviours in
solids. The keystone of the approach is the idea of the crack
growth kinetics curve "crack growth rate v vs. stress
intensity factor K" as intrinsic characteristic of
a couple {materialóenvironment}. The uniqueness
of v(K)-curves as an attribute of a given material-environment
system forms the backbone for the approach and ensures soundness
of applications. Nevertheless, ample evidences of the v(K)-curves
non-uniqueness do exist. Apart from K, crack growth rate
depends on a family of variables related to the pre-EAC loading
history, geometry and testing/service routine. This produces a
loss of confidence in materials evaluation and life predictions.
The origins of the uncertainty of fracture mechanics characterisation
of EAC are analysed in rather general terms. In parallel, hydrogen
assisted cracking is addressed in appropriate explicit terms as
an important and instructive particular case. Two kinds of responsible
factors of essential weakness of the fracture mechanics approach
to EAC are revealed: (1) crack tip plasticity which breakdowns
the K-dominance of the near tip stress-strain field depending
on crack formation/extension histories even under small scale
yielding within elastic K-controlled round-tip zone; (2)
time dependent material-environment interactions affecting the
tip shape and nearby plasticity. Suggestions to consolidate the
customary fracture mechanics approach are outlined. A stringent
treatment of EAC in terms of local values of governing parameters
just at the crack tip is emphasised. In addition, a safe approach
is recommended for design against EAC basing on the idea of "the
worst crack tip situation" as intrinsic one for each material-environment
system.
CONTENTS
1. Introduction
2. Fracture Mechanics Treatment of EAC (Engineering Approach)
2.1. Bases of the Approach
2.2. Manifestations of the Uncertainty of v(K)-curves
2.2.1. Pre-EAC Loading History
2.2.2. Geometric Variables
2.2.3. Kinematic Variables
2.3. The nature of the vóK-dependence and the Certitude of the Approach
3. Key Events of the EAC Process: The Items of K-dominance
3.1. Environment Supply to Prospective Damage Sites
3.1.1. Environmental transport to towards fracture process zone
3.1.2. Surface interactions
3.1.3.Transport within material to rupture sites
3.2. Damage Enhancement by Environmental Substances
3.2.1. Mechanical background ó the scene for environmental enhancement of damage
3.2.2. Local rupture criteria and the EAC kinetics
3.2.3. The concepts of the EAC threshold
3.3. Stress-Strain State
3.3.1. The two-side bounding for maintenance of the end-region autonomy
3.3.2. The effect of crack advance on K-dominance
3.3.3. Environmental effect on the near tip stress-strain state
4. Consolidation of Fracture Mechanics Assessment of EAC
4.1. The Strictly Local Treatment of EAC
4.2. A Safe Approach to Evaluation of EAC
5. Diffusion Related Preconditions of K-dominance in EAC
5.1. The Effect of Far Field on the Near Tip Diffusion
5.2. The Role of History of the Coupled Diffusion-Cracking Process
6. Closure
1. INTRODUCTION
Assessment of environmental impact on materials and structures involves explication of environmentally assisted cracking (EAC). One of the main tasks in studies of environmentally induced degradation of materials is to provide adequate evaluation of the susceptibility of a given material to cracking caused by particular conditions of harmful surroundings (specified environment composition, temperature and pressure, etc.). This also has to render predictive capability regarding evolution of cracks in structures under coupled influence of loads and aggressive environments. As the minimum desired predictive capability, the transferability of laboratory specimen testing data must be ensured for assessment of crack growths and failures that could occur in structures in service.
Fracture mechanics approach is considered to be effective to manage the tasks of materials evaluation and estimation of residual lives of structures. Its keystone is the crack growth kinetics curve being a plot of crack growth rate (CGR) v vs. stress intensity factor (SIF) K. It is enclosed between the limit crack growth resistance of a material ó fracture toughness Kc ó being the SIF value with which crack can grow with no environmental assistance, and the threshold SIF Kth obviously defined as the maximum one at which crack advancement cannot be detected for reasonably long time or, equivalently, CGR is still zero, v(Kth) = 0, with appropriate accuracy (Fig. 1). The relation between v and K serves to characterise EAC resistance for the purposes of materials evaluation, structural design and maintenance since it is supposed to be unique for the material-environment combination irrespectively of particular solid geometry and mode of loading. The uniqueness of v(K)-curves as intrinsic characteristics of material-environment systems forms the backbone of the approach and ensures the soundness of applications. This uniqueness truly grants matching the similitude of the crack tip events and, consequently, of crack behaviours in test specimens and in structures in service. Thereby it provides the transferability of laboratory testing data and opens the way for reasonable predictions of crack propagation under different circumstances. The extent that the v(K)-curve is indeed unique the fracture mechanics approach is valid, and any discrepancy between predicted and observed behaviours should be attributed to roughness in analysis or experimental scatter but not to the concept [1].
However, noteworthy experimental observations bring doubts regarding intrinsic character of the v(K)-curves in EAC showing numerous manifestations of their non-uniqueness, as reported in details elsewhere [2,3]. These indicate that common fracture mechanics approach is not fully capable of treating EAC processes, and the extent to which the v(K)-curve and the threshold SIF are material properties becomes thus an open issue which requires further research.
This study is confined mainly with linear elastic fracture mechanics
(LEFM) considerations of cracks under small scale yielding with
no relevance to specific effects of the mechanical factor of the
out-of-plane (traverse) strain constraint on crack tip events,
i.e., not emphasising the difference between plane strain, plane
stress or other generalised-plane-strain conditions like addressed
elsewhere [4]. The paper starts with a summary of the customary
fracture mechanics approach to evaluation of EAC accompanied with
a brief survey of experimental manifestations of its uncertainty
as a predictive tool. Afterwards follows the outline of the key
events of EAC with the objective to specify the items of K-dominance
of all the constituents of the entire process. In the subsequent
section some suggestions to supersede the common approach with
more stringent treatments are given. At last, the intrinsic sources
of violation of the SIF dominance over EAC are revealed for the
case when environmental facilitation of fracture involves stress-strain
assisted diffusion of the harmful specie in a material to microstructural
rupture sites.
2. FRACTURE MECHANICS TREATMENT OF EAC
(ENGINEERING APPROACH)
2.1. Bases of the Approach
The keystone of the fracture mechanics approach to analyses of cracks in solids is the notion of the mechanical autonomy of the crack tip zone [1,5]. This suggests that all physical processes related to local rupture and crack advance event rely only on the material itself and on some single variable which brings cumulative characterisation of loading intensity in the near tip fracture process zone irrespectively of particular geometry and mode of loading of a cracked solid. SIF is believed to serve well for this purpose provided small scale yielding limitations on the applicability of LEFM are obeyed and some fixed out-of-plane (through-thickness or traverse to crack front) straining condition of plasticity constraint is maintained, like plane stress, zero (common plane strain) or non-zero (generalised plane strain) traverse strain [4]. Regarding EAC, this basic concept was expanded involving the supposition that crack tip rupture events depend on the environment, too, as a matter of fact the bulk one being obviously taken by default. EAC evidently comprises certain physico-chemical processes being in essence kinetic ones characterised in terms of rates rather than only current values of representative (state) variables. Then it was naturally to incorporate CGR to couple mechanical impetus for fracture with physico-chemical kinetics which in couple render an experienced behaviour of a crack.
Accordingly, it was guessed that in a given system {materialóenvironment}
equal SIF causes equal values of CGR [5-8], so that corresponding
v(K)-curve represents the law of crack propagation
v = v*(K |F1,F2,...,FM)
(1)
where asterisk in the right-hand part stands to emphasise the predetermined nature of material's function, i.e., the relationship determined solely by the material and the environment characterised by its global state variables denoted for generality like Fi (i =1,...,M). In particular, the relevant set of F-s may include environmental pressure, temperature, chemical composition data, electrochemical characteristics like pH and applied potential EV, etc. The right hand part in (1) is considered to be a plain function, i.e., that current magnitudes of listed arguments render certain instantaneous value of CGR.
The v(K)-curve (1) is a powerful tool for solving
problems of EAC evaluation and control since it is believed to
contain comprehensive data about crack behaviour for particular
system {materialóenvironment}. It defines the equations
of crack growth to predict the evolution of crack sizes in solids
with time t for the purposes of structural life assessment
(cf. [8,9]), like the next differential equation with respect
to crack length or depth a (only one geometry parameter
of a crack is taken here for brevity):
= v*(K(F,a) |F1,F2,...,FM)
(2)
where F is applied load. Solving differential equation
(2) crack extension may be calculated depending on load F
and actual or expected initial crack size ai . Particularly,
the durability ó time to failure tf ó can
be estimated for a variety of solid/crack geometries and loadings
having correspondent SIF solution K = K(F,a)
for certain load and geometry of a solid [8]:
tf = ;;; (3)
where critical crack size ac corresponds to achievement of the fracture toughness Kc, i.e., K(F,ac) = Kc. Regarding the problems of crack growth prediction the relation (1) is assigned with a significance of constitutive-like equation dependent only on the material and the environment, and thus, is is supposed to possess uniqueness as intrinsic characteristic of the system {materialóenvironment}. To define it, the v(K)-curve (1) must be determined either experimentally or through a deeper physical theory. The first way is more feasible and usually followed in engineering.
Predictive capability of the described fracture mechanics approach to EAC relies on the uniqueness of the v(K)-curve as the characteristic of a specific couple {materialóenvironment}. The firmness of this uniqueness makes material evaluations trustworthy and comparable, and predictions of crack growth reliable. It is obviously supposed to be granted on the condition that SIF is the adequate parameter to characterise mechanical state in the crack tip zone. This matter is considered to be ensured in obvious manner of LEFM, i.e., provided the small scale yielding is maintained according to standard limitations about LEFM validity in terms of the sizes of the crack, the ligament and the near tip plastic zone (cf. [1,8]) during the whole crack propagation.
Impressive body of experimental data supports the presumption
of the uniqueness of the fracture mechanics characteristics of
EAC [5-10]. However, ample evidences of their non-uniqueness do
exist, too. Spectacular results are available which cannot be
related to limited efficacy of LEFM which fails when too extended
plastic zone appears near a crack tip. Apparently, the conditions
of SIF controlled small scale yielding near crack tip there were
maintained. Observed deviations of crack behaviours in identical
material-environment systems were neither due to the known phenomenon
of the influence of member thickness nor to out-of-plane straining
affecting near tip stress-strain field and crack growth resistance
parameters [4]. The outline of these experimental manifestations
of the weakness of the fracture mechanics treatment of EAC follows
below.
2.2. Manifestations of the Uncertainty of v(K)-curves
In this section a collection of manifestations of non-uniqueness
of the v(K)-curve which produces uncertainty of
EAC evaluation is briefly surveyed (with more details they were
reviewed elsewhere [2,3]). These data are classified here according
to groups of variables ("factors") which ostensibly
influence CGR apart from SIF. Firstly, the effects of pre-EAC
loading factors (fatigue pre-cracking and peak loads, etc.)
on the subsequent EAC behaviour are elucidated. Secondly, geometric
factors such as crack length and bluntness are analyzed. The SIF
gradient dK/da is mentioned in this group assuming that
the term "geometry" includes also the complete design
of test arrangements, i.e., test specimen with loading device
(gripping system) and other peculiarities which yield definite
boundary conditions to determine the near tip stress-strain
state, the SIF solution K = K(a) in particular.
Finally, the role of kinematic factors is discussed, using the
term "kinematics" to comprise test or service variables
which identify particular routines of test execution or structure
operation history, i.e., chronology of loading events (initial
loading or deformation and subsequent load changes with time,
like test interruption, load rate, step load changes, etc.),
as well as variations of environmental conditions.
2.2.1. Pre-EAC Loading History
ï Fatigue pre-cracking regime, i.e., maximum and minimum cyclic SIF values Kmax and Kmin can impressively affect both Kth and the whole v(K)-curve, apparently even after a crack advances beyond the zone of residual cyclic plasticity near initial crack tip. The influence of fatigue pre-cracking is detectable, e.g., in corrosive systems for very different electrochemical processes which promote EAC. Roughly, increasing Kmax mostly produces retarding effect on crack propagation over the entire v(K)-curve (Fig. 2a). However, this is not a strict trend, and in the range of very low Kmax the opposite may occur at least concerning the threshold SIF (Fig. 2b). In addition, the measured threshold SIF is strongly affected by another characteristic of fatigue pre-cracking such as the minimum cyclic SIF Kmin, or in other terms, by the SIF range DK = Kmax ñ Kmin or the stress ratio R = Kmin/Kmax .
ï Pre-loading of a specimen in the form of single overloads or load hold before immersing in harmful environment influences ulterior proceeding of EAC. Prior overload in inert environment can cause apparent increase of Kth (Fig. 2c). Holding of a specimen at some initial value of applied SIF under no environmental action again can cause shifting of the whole v(K)-curve towards slower crack velocities.
Fig. 2. Schemes of trends of variation of measured characteristics of EAC depending on the crack formation history:
(a) ó the influence of Kmax at fatigue pre-cracking on the v(K)-curve appearance;
(b) ó the effect of fatigue pre-cracking SIF on the measured threshold Kth;
(c) ó variation of the measured Kth with pre-EAC
overload SIF value.
2.2.2. Geometric Variables
ï Crack length (depth) a can affect evaluation of Kth, so that thresholds for short cracks were found substantially lower than those for long cracks, e.g., in corrosive environments (Fig. 3a). In addition, there is detectable influence of crack length on the whole v(K)-curve, especially in the plateau-like region (stage II in Fig. 1): CGR for a short crack results higher than for a long one at the same K.
ï Initial sharpness of a crack (i.e., initial value of crack tip radius or semi-width r) has strong influence on Kth, so that the threshold rises as a starter crack is more blunt (Fig. 3b). This denotes the role of initial acuity of a crack as stress-strain concentrator in the initiation of EAC. However, the opposite effect may be also expected since crack bluntness in the tip correlates obviously with the crack width (height) along the whole crack area. A decrease of EAC susceptibility with more narrow crack can occur, e.g., because of suppress of the environment access to the crack tip through a less penetrable canal. The limit value r* of sharpness exists below which this toughening effect disappears.
ï SIF gradient c = dK/da affects EAC proceeding.
In particular, crack initiation (where dK/da > 0) and
crack arrest (with dK/da < 0) test techniques (cf. [9,10])
for threshold evaluation can render considerably different values
of Kth. With regard to CGR, noticeable discrepancies were
also found between v(K)-curves obtained from the
tests performed on specimens with different values of the SIF
gradient dK/da (Fig. 3c).
Fig. 3. Schematics of the observed effects of geometry factors on characteristics of EAC:
(a) ó influence of crack length on EAC threshold and crack growth kinetics;
(b) ó dependence of measured Kth on initial crack bluntness;
(c)ó decrease of the "plateau" CGR depending
on SIF gradient as a fraction of the cracking velocity obtained
under sustained SIF K(a) = const, i.e., c
= 0
2.2.3. Kinematic Variables
ï Initial loading conditions (initial applied SIF Ki) effect on Kth was elucidated using crack arrest techniques where dK/da < 0. Apparent threshold SIF was found to cover a too wide scatter band for different values of Ki. In the matter of v(K)-curves, clear systematic deviations from the "single" curve depending on the initial SIF value were observed in some cases (Fig. 4a-b).
ï Interruption of EAC tests started at some value of Ki with recess periods without load and re-start of cracking at about the same Ki rendered significant decrease of CGR in a wide range of SIF variation, apparently beyond the extend of the influence of the previous (residual) plastic zone at the test interruption point with a relatively high value of K (Fig. 4c).
ï Loading/straining rate which can be transferred to the crack tip by means of SIF rate K_ influences spectacularly the threshold SIF and the v(K)-curve, although the sign of the effect may be controversial depending on the environment and the material (Fig. 4d-e). With regard to the v(K)-curve, some data display uplift of CGR on its stage II (cf. Fig. 1) and shifting of the near-threshold part I including Kth to higher SIF magnitudes, while other results showed that faster crack tip straining produced an elevation of the crack growth kinetics curve as a whole with diminishing of Kth (Fig. 4e-f).
ï Load step height (SIF stair dK) is another variable which was reported to affect EAC apart from the value of SIF itself so that CGR varies during an essentially long period tv (of the order of hours) at constant K maintained after a step-wise SIF increment during the time tK of about only few seconds [11], see Fig. 4g. Thus, again, different values of CGR occur at the same SIF. This is the kind of a delayed effect which proceeds in the material-environment system well after SIF alteration have terminated, tK << tv.
ï Environmental conditions change should be expected
to cause alteration of CGR of the similar appearance as the effect
mentioned right above. Indeed, variation of environmental characteristics
in tests [12] was not closely followed by corresponding adjustment
of CGR to its new value according to the hypothesis presented
by relation (1). On the contrary, CGR noticeably continues to
vary at newly established constant parameters of the environment.
Thus, in general, the {materialóenvironment}-couple
in hand may display different velocities of crack growth at constant
SIF, i.e., equal K values do not produce equal v.
Fig. 4. Schemes of the effects of test/service routine on EAC:
(a)-(b) ó experienced influences of the initial loading condition (different initial applied SIF Ki1 < Ki2 < Ki3) on the appearance of v(K)-curves: shift as a whole and transition towards some reference v(K)-curve;
(c) ó the effect of test interruption and re-start: curve 1 corresponds to initial run of EAC interrupted at point A and curve 2 displays crack growth kinetics after re-initiation of EAC at point A';
(d) ó frequently met trend of the effect of loading/straining rate on Kth;
(e)-(f) ó observed modes of variation of v(K)-curves in EAC depending on loading/deformation rate (the arrow indicates the direction of v(K)-curve variation with rising load/deformation rate);
(g) ó delayed effect of a step-wise SIF change of followed
by a variation of CGR.
2.3. The Nature of the vóK-dependence and The Certitude
of the Approach
Despite ample testing and service experience do confirm reasonable
uniqueness of v(K)-curves and efficacy of fracture
mechanics as the keystone of damage control procedures in many
cases, the cited results convince that the same SIF does not always
yield equal crack velocities in otherwise identical couples {materialóenvironment}.
The observed deviations are rather systematic in distinct to obvious
random statistical scatter of test data. Consequently, uncertainty
can arise in determination of the basic characteristics of materials
resistivity against EAC, and what is worst, this can cause invalid
excessively optimistic material evaluation and non-conservative
estimation of structural strength and life. Therefore, this customary
approach is not generally valid. The list of arguments in the
right-hand parts of relations (1) and (2) turns to be incomplete
or inadequate. Not only K and F-s explicitly displayed
in relation (1), but another variables from the mentioned in the
previous subsection must be involved in its right-hand part. Moreover,
the effects of kinematic variables apparently cannot be represented
by a plain function, but rather by a functional
v = v*(K,K_,a,c,...,t
|F1,F2,...,FM) (4)
which renders the value of instantaneous CGR at the time moment t depending not on the current values of the listed variables but on the shapes of the functions staying as its arguments which vary with time t along the interval 0 £ t £ t , i.e., K(t), Fi(t), a(t), etc., for particular realisation of EAC under certain circumstances. The hollow symbol is employed in the right-hand part of the expression (4) just to emphasise this sort of path (history) dependent relationship between process parameters.
Now, expression (4) reflects the empirical conclusion concerning the nature of the vóK-relationship in EAC to which numerous observations of the variability of cracking behaviours definitely drive.
As a matter of fact, a variety of quantities may be monitored during crack growth and used to characterise diverse evolutions of EAC process in the form of inter-relations between selected variables. The aim is to reveal those able to remain the same in spite of variability of particular circumstances that EAC proceeds in a given material-environment system. The efforts to achieve this goal are made basing on the concept of similitude (cf. [1]). This relies on discovering the key variables and relations between them able to match evolutions of all responsible interactions and events, and thus to cover the diversity of particular evolutions of cracks.
EAC obviously results from participation of several physico-chemical
processes. Trying to find the firm way of its quantitative characterisation
the first step is to clarify the constituents of the entire EAC
process. Then, the matter of the uniqueness of v(K)-curves
will be the question of ensuring exclusive SIF dominance over
all the responsible events of EAC so that CGR as their observable
macroscopic consequence could be in every other way the predetermined
function of the material and the environment. To this end, although
the EAC constituent processes have already been discussed in numerous
papers (cf., e.g., [5,13-15]), the matter of their K-dominance
as the key one for the soundness of the fracture mechanics approach
has not got attention. To cover this deficiency the topic of SIF
control is focused in the discussion of EAC contributing events
which follows below. This is expected to bring more insight concerning
the matter and to define the restrictions to be additionally imposed
on materials testing and structural damage evaluations to ensure
reliable (safe) assessments of the tolerance against EAC.
3. KEY EVENTS OF THE EAC PROCESS: THE ITEMS OF K-DOMINANCE
3.1. Environment Supply to Prospective Damage Sites
3.1.1. Environmental transport towards fracture process
zone
This phase of the entire EAC process includes: (i) transport of the relevant agents from a bulk environment to the vicinity of crack tip; (ii) release of the responsible environmental species therein to their active form from bonded states in molecules, e.g., like hydrogen evolution in aqueous corrosive systems. Activity coefficients (thermodynamic activity factors) ziCT (i=1,...,n) of the liable components in near tip environment are important for EAC. They control environmental attack on metal just in the closest neighbourhood of the fracture process zone, like hydrogen evolution and entry into metal (hydrogenation) to cause hydrogen degradation near the crack tip, or local anodic dissolution, or liquid metal embrittlement, or another action peculiar to specific environment.
Activity of a certain chemical specie in the near tip environment
may be presented in terms of its partial pressure PCT (or
better, its fugacity) in gaseous environments or using other appropriate
state variables like electrochemical ones under corrosive conditions,
such as electrode potential EVCT and hydrogen ion exponent
pHCT, etc. The important matter is that they refer to the
crack tip zone where the environmental physico-chemistry
can significantly differ from the bulk characteristics of the
surrounding medium, correspondingly, P or EV and
pH and other relevant parameters [5,13-18]. Relations between
bulk and local environment characteristics are governed by in-crack
environmental currents and kinetic processes of mass-charge transfer
(diffusion and electromigration along the crack) and chemical
reactions within a crack space and on its faces (Fig. 5).
Fig. 5. The sketch of in-crack processes which control the specific
crack tip environment chemistry provided by a certain bulk medium.
In general terms, each parameter fi (i=1,...,m)
from the m-member set of those which characterise the crack
tip environment and determine environmental attack directly on
the fracture process zone in a fixed bulk environment, i.e., at
certain values of Fi (i=1,...,M), depends
more or less strongly on an extended series of the primary variables
[15]: on the geometry of a crack as the transfer canal, the reacting
environment volume and the crack surface area, on strains ei
(i=1,2,3, i.e., principal components are employed) expected
to alter material-environment reaction rates on the strained surface,
and on CGR v and strain rates e_i both interfering
with the kinetics of in-crack reactions (e.g., through control
of the fresh surface creation rate, especially at the crack tip,
or change of the in-crack reaction volume), and on time t.
This may be expressed in the next general form:
fi = fi (F1,...,FM, a, d,
v, e1,2,3, e_1,2,3, t) ;(i
=1,...,m) (5)
where geometry parameters a and d of an opened crack
represent, correspondingly, the characteristic transportation
distance from a bulk environment to the crack tip together with
the crack area (here only crack length or depth is taken for brevity
although more parameters including a member thickness may act
on equal terms) and crack opening displacement which gives canal
width under load (the height of in-crack space) over the whole
crack area. This includes also the special value of the crack
tip opening displacement (CTOD) dt which characterises
the crack tip strains (the strains, although, are explicitly presented
in expression (5) for descriptive purposes). Apparently, just
these extreme surface strains at the crack tip are of importance
in EAC, but not moderate ones out of the tip vicinity which have
less, if any, influence on the in-crack physico-chemistry. The
relationships (5) take their specific shapes depending on a certain
process history, i.e., along a particular trajectory of variation
of all the arguments in the right-hand parts peculiar for a singular
EAC proceeding.
It seems worthwhile to illustrate these general considerations with reference to hydrogen assisted cracking as an important and rather frequently met case of EAC. Here the severity of environment may be characterised by hydrogen activity coefficient in the near tip environment zHCT. It controls hydrogen interaction with metal surface and entry into metal (hydrogenation), both in the closest vicinity of the fracture process zone.
Crack tip hydrogen activity may be presented in terms of its partial
pressure PCT in gaseous environments or defined from electrochemical
characteristics EVCT and pHCT of corrosive ones. Numerous
data prove that all them can differ from their counterparts in
bulk surroundings [13-17]. This may be expressed using relationships
similar to (5):
PCT = PCT (P, a, d, v,
e1,2,3, e_1,2,3, t) (6)
in gaseous environments, and additionally,
pHCT = pHCT (pH, EV, a, d, v, e1,2,3,
e_1,2,3, t) ;and ;EVCT = EVCT
(pH, EV, a, d, v, e1,2,3, e_1,2,3,
t) (7)
in corrosive ones, where again the strain rate terms stand to emphasise the experienced role of strain dynamics on in-crack reactions and on a resulting crack tip chemistry [15].
Given a specified corrosive couple {materialóbulk
environment}, from the data about actual variation of crack
tip electrochemical parameters (7) the intensity of hydrogen evolution
at the crack tip may be evaluated using the shift of the potential
[14]
DHEEVCT = EV*(pHCT) ñ EVCT (8)
which relates current values of the local electrochemical variables
with the thermodynamic stability border for water given by the
equation of Nernst line (see Fig. 6)
EV*(pH) = a + b pH (a = ñ 0.014
V,;b = ñ 0.059 V) (9)
Fig. 6. Typical progress of the crack tip environmental conditions
during EAC for steels in aqueous corrosive media (arrows indicate
the direction of following the displayed trajectory as a crack
grows).
Now, hydrogen activity coefficient in the near tip environment
can be supposed to be proportional [14] to the hydrogen evolution
index or the negativity of the overpotential (8):
zHCT µ ó DHEEVCT(pHCT, EVCT)
= ó DHEEVCT(pH, EV, a, d, v,
e1,2,3, e_1,2,3, t) (10)
where the last expression in the right-hand part results from the superposition of relations (7) and (8). The value of zHCT may be converted into units of equivalent hydrogen gas pressure (fugacity) or equilibrium volume concentration Ce of hydrogen in metal.
Dealing with environmental in-crack processes, other related factors
able to influence metal-hydrogen interactions do worth of note.
Namely, chemical factor needs more careful consideration since
some accompanying environmental species can either aggrandise
or inhibit the action of hydrogen, like sulphur or oxygen, correspondingly
[16,19]. Its accounting again requires to consider in-crack environmental
transportation processes coupled with chemical reaction kinetics
which control evolutions of the crack tip concentrations of whichever
reacting specie zCT other than hydrogen. At any rate, relation
similar to (6) or (10) describes evolution of each of the relevant
concentration (activity) at the crack tip apart from the one for
hydrogen:
zCT = zCT(P, pH, EV , z, a,
d, v, e1,2,3, e_1,2,3, t) (11)
where z stands to represent chemical composition of a bulk environment. Thus, for the case of hydrogen assisted cracking the set of state variables Fi of a bulk environment is given by P, pH, EV , and z, whereas local crack tip environment characteristics PCT, pHCT, EVCT and zCT act as f-s.
In addition, with respect to the majority of EAC realisations in hydrogenating corrosive environments distinct mechanisms of crack extension often can act simultaneously [14,20-22]: hydrogen degradation (embrittlement), local anodic dissolution, and specific crack tip material-environment chemistry (creation/rupture of surface brittle films of oxides or salts, etc.). The two latter, apart from possible direct influence on hydrogen entry into metal and from crack extension by dissolution, can affect the near tip plasticity of a material and the crack tip morphology ó crack tip blunting magnitude, shape, etc. In due course these modify crack tip stress-strain fields and damage accumulation.
In particular, the value of crack tip anodic overpotential
DADEVCT;= EV ñ EVCT (12)
or with account of the relation (6)
DADEVCT;= EV ñ EVCT(pH, EV,
a, d, v, e1,2,3, e_1,2,3, t)
(13)
determines the anodic current density, and hence the rate of crack
tip anodic dissolution according to Faraday law (cf. [14]). Consequently,
removing of a material in the crack tip by dissolution can make
it either blunter or sharper depending on the character of dissolution,
i.e., would it be more homogeneous or selective along specific
narrow paths (active path dissolution). As a result, through modification
of the crack tip morphology, the specific crack tip chemistry
may affect the near tip mechanics, and hence, the proceeding of
EAC, in particular, by hydrogen degradation mechanisms.
As a matter of fact, the right-hand parts of the above expressions characterising crack tip environments in general are not plain functions of instantaneous values of displayed bulk variables but rather functionals over their time histories, in particular, over the trajectories a(t) and d(t) in the time interval 0 £ t £ t. This implicitly involves also the role of rates, i.e., of the CGR v = da/dt and of the loading rate represented by the strain rate term. Nevertheless, they are displayed above explicitly for better descriptiveness in discussion. Beside, the roles of rates worth reminding since CGR not only competes with the velocity that environmental agents are able to follow the moving crack tip and maintain there a certain activity level, but also characterises creation of fresh fracture surface and affects variation of the crack width. The strain rates worth of special notion here because they are known to interfere with the kinetics of in-crack reactions and this way influence the in-crack chemistry.
Therefore, the reasons exist for more or less sensible influence of the not exclusively K-controlled both loading/cracking kinematics and crack geometry outside the crack tip zone on the near tip environment, and thus, on the instantaneous CGR at a given SIF under fixed bulk environment.
The falsehood of the supposition that fixed bulk environment provides in the vicinity of a tip of arbitrary crack the identical or at least somehow constant local environment conditions depending solely on the material and SIF seems to be well established for most environments [13-18]. However, in gases due to their high mobility (like hydrogen gas) the access of harmful specie to crack tip by Knudsen type diffusion along a crack canal may be sufficiently rapid yielding minute difference between bulk and local environment parameters at a bulk gas pressure above a certain limit [16,23]. In this case environmental transportation phase of the entire EAC process has negligible effect on K-dominance and the uniqueness of v(K)-curve. Under special circumstances another kind of definite correspondence between bulk and local environments might be expected to exist in less mobile environments ó the autonomous near tip environmental cell. This occurrence could be favoured in thin-wall members with rather long through-thickness cracks so that the near tip environment would be determined by a relatively short-distance (half-a-thickness) mass/charge exchange with lateral surrounding, and a crack-size dependence of local environment would be eliminated.
To conclude this item, in general quite limited possibility exists
that local near tip environment could be one-to-one determined
by the properties of the material and instantaneous SIF and parameters
of the bulk environment. Participation in EAC of the in-crack
processes which proceed in time and in space, the latter being
determined by a crack size and a width under load, argues that
specimen/member geometry and loading mode affect environment-related
events of EAC just at the crack tip so that K-dominance
with respect to them becomes dubious.
3.1.2. Surface interactions
On the way to assist crack extension environmental agents interact with material surface at the crack tip. This may comprise the next sequential phases: (i) physical adsorption of relevant components (molecules) on the surface; (ii) dissociative chemisorption of certain harmful agents; (iii) entry of the species into surface layer (dissolution or absorption) to start their way to rupture sites within the process zone or chemical reaction on the surface to create brittle films, or activate by any other way the triggers of damage in material. Not all these stages must always act in a particular EAC process depending on the operative mechanism of environmentally affected fracture peculiar to the material-environment system in hand [5,8,14,20]. Adsorption effect on strength is apparently the most relevant for glasses and ceramics in humid environments, anodic dissolution of metals can dominate in corrosive ones, absorption-related degradation is ascribed to hydrogen assisted fracture of metals. Any of the stages may be affected by catalysed surface reactions, surface contamination with accompanying environmental species (like oxygen in hydrogen assisted cracking [19]) or products of chemical reactions, e.g., corrosive, etc. Mechanical factor also can participate here due to strain-dependence of the integrity and the penetrability of surface films, or through stimulation of surface catalytic efficiency by lattice strains.
The governing parameters of these three kinetic processes with
respect to specific harmful agent ó the sorption-desorption
rate coefficients kr (r=±1,±2,±3)
ó must depend then on strains ei or stresses si
(i=1,2,3, i.e., principal components are still used) as
well as on surface/environment chemistry determined by the activities
zCT of catalysing or contaminating species or, equivalently,
by environment state variables fj (j=1,...,m)
in the vicinity of crack tip. Beside, to account for interference
of transient physico-chemical kinetics and stress-strain dynamics
the rates of strains and stresses e_i and s_i (i=1,2,3)
must be involved, too. Therefore, in general case the responsible
characteristics of surface interactions will be the functions
of the sort
kr = kr (f1, f2,..., fm, s1,2,3,
e1,2,3, v, s_1,2,3, e_1,2,3) (r
=±1,±2,±3) (14)
where stress-strain state components are taken on the solid's
surface at the crack tip and environment state variables f1,
f2,..., fm depend on certain processes within a
crack which result in their evolutions expressed by relations
(5).
Concerning hydrogen assisted cracking as a very exemplary case
of EAC, explicit description of the mentioned stages of the hydrogen
sorption processes for molecular hydrogen environment may be given
by the system of kinetic equations like next (cf. [24]):
(15)
where q and h are the fractional coverages of surface adsorption sites by hydrogen molecules and chemisorption ones by adatoms, correspondingly, and c is the fractional occupation by hydrogen atoms of interstitial lattice sites in the surface layer, whereas the values of qa, ha and ca represent correspondent densities of the available sites (at saturation). The saturation values are dependent on strain and surface contamination (e.g., oxidation), whereas for the equations coefficients the general presentations (14) are in force. All that affects the sorption evolution, i.e., alters characteristic process relaxation times (time scales of approaching the equilibrium in whichever of the three process phases).
The surface sorption phase may be the rate determining step for
crack growth when surface processes have relaxation times longer
or comparable to those for the diffusion transport of hydrogen
within metal to microscopic rupture sites. As one can deduce from
the sorption kinetics studies [24], with penetration distances
of about the process zone size in metals which is typically of
the order of 10-100 mm this may take place under relatively weak
hydrogenation conditions at the crack tip, apparently at environmental
activity of hydrogen equivalent to 0.1 MPa pressure of hydrogen
gas at utmost. This level of hydrogen activity is unusual for
most practical cases of gaseous hydrogenation or electrochemical
one under corrosion when at entry sites rather high hydrogen activity
(fugacity) is obviously achieved [7,25]. Then hydrogenation of
metal at the crack tip can be defined in terms of the volume concentration
C of absorbed hydrogen using whichever of the appropriate
sets of parameters of the crack tip environment:
C(entry surface) = Ce(zHCT, zCT,...)
= Ce(f1, f2,..., fm) (16)
where Ce is the equilibrium value of concentration (i.e., hydrogen solubility) in metal correspondent to local environmental conditions. Anticipating further detailed consideration, note that hydrogen solubility in metals depends on stresses and strains, too.
Taking into account the above discussion of the inter-relation
between the global and the crack tip environments, by superposition
with (16) of the relevant expressions from (6)-(11) the crack
tip hydrogenation condition in a given bulk corrosive environment
my be presented like next:
C(entry surface) = Ce(P, pH,
EV, z , a, d, v, s1,2,3,
e1,2,3, e_1,2,3, t) (17)
where the hollow symbol outline is used with the same meaning
as above, i.e., to emphasise that the right hand part in (17)
in general is a functional of the displayed variables.
Therefore, regarding this constituent of the EAC process, verifying
of K-dominance and the uniqueness of v(K)-curve
in EAC continues to be the matter of the exclusive SIF control
over both the in-crack transportation processes which render certain
crack tip environment and the stress-strain field near the crack
tip surface as it follows from the above relations (14) for general
case and from (16)-(17) for hydrogen induced cracking in particular.
Equations presented in this section show that stress-strain rate
values are involved as the controlling variables of crack-tip
environmental processes so that ensuring of K-dominance
for EAC becomes doubtful, or at least needs some rate limitations
in EAC testing procedures to obtain reliable and conservative
assessments of materials resistivity against EAC in terms of the
v(K)-curve.
3.1.3. Transport within material to rupture sites
Certain cases of EAC involve transportation of the damage facilitating
agent to internal fracture nuclei in the near tip process zone.
This relates not only to hydrogen assisted fracture of metals
where it is a common place [5,14,26], but also to other material-environment
systems, e.g., concrete-like materials degraded by aqueous substances
[27]. Consideration of hydrogen related effects performed within
general framework of the thermodynamics of mass transfer brings
rather instructive general insight on this item.
Two transport modes of hydrogen to microfracture sites in loaded
solids must be distinguished: diffusion and transport by
moving dislocations. Driving force XD for diffusion is
determined by chemical potential of hydrogen mH associated
with its concentration C and solubility coefficient Ks
[28,29]:
XD = ñ ómH = ñ ó
(18)
where R is universal gas constant and T absolute
temperature. Solubility factor Ks is known to depend on
hydrostatic stress s, alloy chemical and phase composition
and density of hydrogen traps in metal lattice [28-30]. The overall
density of traps, and in certain alloys their phase composition,
e.g., in metastable austenitic steels [31,32], both depend on
plastic deformation which may be represented in terms of the effective
plastic strain ep [29]. Finally, hydrogen solubility coefficient
is:
Ks = Ks0 (ep,T) exp (19)
where Ks0 is the plastic strain dependent component of solubility, VH is partial molal volume of hydrogen in metal. Gradient of any of the mentioned solubility affecting factors can induce diffusion flux. Thus, plastic strain magnitude ep should be placed among governing factors of hydrogen diffusion.
In addition, since traps density affects average mobility of a
diffusible specie, diffusion coefficient depends on plastic strain,
too [32,33]: D=D(ep). In general, diffusion coefficient
of hydrogen depends also on stress state in a solid, although
this effect seems to be rather weak. At any rate, D should
be considered spatially inconstant as stress-strain state does.
Finally, assuming uniform temperature distribution in a solid
and using equations (18)-(19) hydrogen diffusion flux in metal
may be found following the common way of thermodynamics:
JH = ñ D (20)
Obvious condition of mass balance leads to the equation of stress-and-strain
assisted diffusion with respect to hydrogen concentration:
= D + óD _ (21)
where vector and scalar coefficients correspondingly are the next M = ólnKs and N = ó2lnKs. If only stress gradient exists, equations (20)-(21) reduce to well known ones for stress-only assisted diffusion [14,26,34].
Another representative macroscopic variable should be implemented if one considers hydrogen transport by dislocations. This process is evidently associated with a mean velocity of dislocations, which can be expressed in macroscopic terms through plastic strain rate [35]. The two transport modes ódiffusional and dislocationaló are essentially different. The first one is operative under both sustained and transient (time-dependent) stress-strain states, has instantaneous plastic strain ep as one of the responsible variables, and evolves towards equilibrium hydrogen distributions [29]. In contrast, the other one proceeds exclusively during continuing (dynamic) straining, and has plastic strain rate e_p as the governing variable. Here for simplicity only the rate of effective plastic strain ep is mentioned as the measure of the averaged plastic strain in a point, although precise definition of dislocation-dragged hydrogen flux would explicitly involve all the components of the plastic strain rate tensor. Dislocational transportation results in non-equilibrium hydrogen distributions ó localised temporal over-saturations. They are fed by newly arriving dislocations which bring hydrogen into specific microstructural sites, from which hydrogen escapes by diffusion to restore thermodynamic equilibrium with local surroundings. Quickly after termination of progressive straining, i.e., at zero plastic strain rate, those hydrogen over-saturations completely relax to restore local equilibrium by short-range diffusion. The efficiency of the dislocational mode of hydrogen accumulation in prospective damage sites results from competition between these two effects of in- and out-flow of hydrogen. Under sustained or slow quasi-static loading local over-saturations created in certain microstructural sites due to hydrogen supply by dislocations have time to relax by short-range diffusion, and thus the significance of dislocational transport seems to be negligible in this case. Then the long-range (in relative values) diffusion driven by near tip stress-strain field is the main operative mode of hydrogen transport to fracture nuclei in metals.
Diffusion is often the slowest step among all the phases of hydrogen
supply to fracture sites, and then the kinetics of hydrogen assisted
cracking is diffusion controlled. This is the case when hydrogen
entry into metal obeys the dynamic equilibrium on the crack tip
surface according to relationship (16). Taking into account the
stress-strain dependence of hydrogen solubility in metal given
by expression (19), and adjusting the solubility factor for reference
temperature in such a manner that Ks0 = 1 at ep =
0, boundary condition for hydrogen diffusion to fracture sites
in the process zone is the next:
C(crack tip surface) = C0(f1, f2,...,
fm)Ks0(ep,T) exp (22)
where C0 is hydrogen solubility (equilibrium concentration)
in stress-strain free metal, and strain and stress values in the
right-hand part are taken on the surface, too.
Although the above consideration addresses hydrogen assisted cracking,
this springs out from general thermodynamics and reveals rather
common features of transport processes of the harmful substances
to damage sites in materials which may be the intermediate step
of the entire process of environmentally assisted degradation.
Whichever operative in particular material-environment system,
mechanisms of this internal transportation phase of the EAC process
involve either diffusion-like permeation trying to restore thermodynamic
equilibrium (diffusion itself, diffusion-convection mode of transport
through porous media like concrete, etc.) or transfer compelled
by progressive straining. All them produce an excessive concentration
of the specie in the fracture process zone. At any rate, mass
transfer to rupture sites may be assisted or induced by stress-strain
state and deformation rate. Thus, again, the strength of K-dominance
in EAC depends on the extent to which the processes of the internal
mass transfer of responsible agents to fracture nuclei are controlled
exclusively by SIF. From the above discussion follows that this
is the question of SIF dominance over crack tip environment and
stress-strain field first, and in addition, of verification that
no more effects/variables interfere sensibly with mass transfer
into and within the fracture process zone. This latter item will
be addressed below more scrupulously.
3.2. Damage Enhancement by Environmental Substances
The aim of this section is to reflect the damaging effect of the
environmental agent by a set of proper variables on the way to
explore afterwards the matter of K-dominance in EAC.
3.2.1. Mechanical background ó the scene for environmental
enhancement of damage
Mechanical stress-strain state creates the background for damage facilitated by environment and drives it. Elucidation of macroscopic mechanical variables which govern operative fracture mechanism is complicated with respect to whichever cracking process. Moreover, during EAC several micromechanisms of fracture may take turns in dominating the process. Particularly, taking hydrogen assisted cracking as an example, different fracture micromechanisms and degree of degradation (embrittlement) may take place depending on combination of load intensity and the amount of hydrogen, as is consistent with available observations [36,37]. Correspondingly, operative fracture event may convert from strain- to stress-controlled one. Thus, to quantify mechanical conditions responsible for EAC both stress and strain components should be controlled. According to phenomenological as well as to micromechanical studies of fracture, there is no reason to neglect one in favour of another whichever fracture micromechanism appeared to be dominative.
The set of mechanical variables to characterise pre-fracture state
according to dislocational study [38] in general contains the
next three: effective plastic strain ep, effective stress
se, and maximum tensile stress s1
from the three principal ones. At certain circumstances one of
these parameters of stress-strain state may dominate fracture
event, thus yielding common criteria of rupture in terms of the
maximum stress or the critical strain as particular cases. Following
the macroscopic phenomenological studies and the mesomechanical
void growth analyses of fracture [8,39] hydrostatic stress s
participates in the family of fracture controlling variables instead
of the maximum stress. However, in practical cases this may really
make no difference since these two alternative partakers may be
algebraically recalculated one into another, e.g., in plane strain/stress
state, or uniaxial one, or others. The change of any of the three
key mechanical variables affects the pre-fracture situation, i.e.,
alters critical values of the others for fracture.
3.2.2. Local rupture criteria and the EAC kinetics
Environmental facilitation of damage proceeds by very different
physical mechanisms depending on the material and the environment
[5,8,14,20,22,30]. At any rate, damaging depends on the amount
(concentration) x of the relevant substance accumulated
in damage nuclei and on the stress-strain state in a material
cell (material point). The degradation action itself at a certain
value of x in a prospective rupture site may be instantaneous
or kinetic, e.g., like in metals hydrogen-induced decohesion (weakening
of interatomic bonds) from one hand or hydride forming reaction
from the other, correspondingly. With account of the above conclusion
about the responsible mechanical variables, this may be expressed
in the next general form:
w = w(x ,ep, s1, se, t)
(23)
where w is appropriate damage variable which at rupture event
reaches certain limit value w = wc, e.g., a unity in dimensionless
terms. The right hand part of expression (23) becomes a functional
that depends on function x(t), and possibly, on
the remainder ones placed as the arguments in (23), along the
time interval 0 £ t £ t in general case
of kinetic-type damaging of the sort of chemical reaction. Otherwise
it well can be presented by a plain function of the current values
of the arguments. At any rate, time evolution of the harmful agent
concentration x = x(t) plays the key role
in the kinetics of EAC as it defines a succession of rupture events
which proceed whilst satisfying the rupture criterion
w(x(t),..., t) = wc ;(0
£ t £ t ) (24)
Taking again hydrogen assisted fracture as an explanatory example
of EAC, at relatively rapid damaging reaction elementary fracture
events according to general damage evolution equation (23) and
rupture criterion (24) correspond to critical combination of instantaneous
values of hydrogen concentration C (it will substitute
there x) and components of stress-strain state in material
point. Damage evolution equation like (23) becomes then a plain
function, and thus local rupture criterion like (24) can de resolved
with respect to concentration to define its critical value Ccr
depending on the stress-strain state. Hence, with account
of the micromechanical considerations [14,40] and phenomenological
studies [41,42], the criterion of fracture in a responsible material's
cell (physical "grain", or idealised material point,
or other) may be presented in general like next:
Ccr = Ccr*(s1, se, ep) (25)
where, as usual in this paper, the asterisk emphasises the predetermined character of the material's function defined solely by intrinsic attributes of the material like its composition, microstructure, etc. One explicit version of the criterion (25) built up on the dislocational-decohesive concept of the micromechanism of hydrogen assisted fracture allows to elucidate, in particular, microstructural preconditions for stress- or strain-controlled fracture. Apart from the nature of an alloy, this strongly depends also on both hydrogen concentration and stress triaxiality (on the ratio s1/se or s/se).
Criterion of rupture in a material point like (25) apparently has no direct applicability in engineering evaluations and predictions because local concentration is poorly evaluable (measurable or calculable). However, it can serve to explicate and justify the links between measurable macroscopic quantities, and thus, to develop models for predictions of crack behaviours and extrapolations beyond the available data (cf. [14,40,43]).
Concerning crack growth induced by hydrogen, the criterion of
local rupture correspondent to (25) may be formulated supposing
crack to extend when hydrogen concentration C(t)
reaches a critical value Ccr in a relevant material's cell
at some specified location near crack tip. Assuming, for definiteness,
that crack propagates along x-axis of a rectangular coordinate
system attached to the crack tip, crack growth criterion takes
the form:
C(xc,t) = Ccr*(s1, se,
ep);
= Ccr*(xc) (26)
where the second representation of the right hand part is a superposition
of function (25) and spatial distributions of stresses and strains
in a solid, and a definite value of x = xc must
be specified. The latter is usually associated with the adopted
concept of a responsible cell, e.g., the worst material unit in
the fracture process zone, or the location of extreme stress or
strain in crack tip vicinity, or the fixed microstructural size,
or other.
The left-hand parts in local rupture conditions (24) for general case and (26) for hydrogen "embrittlement" are defined by transport processes of the relevant agent to fracture sites which have been discussed in the previous section. However, these criteria relate only to the critical event but do not define flaw growth yet. Strictly speaking, fine physical modelling of the evolution of damages and their consolidation to form main crack increment is required to interpret CGR. Nevertheless, for the purposes of the present study rather general model assumptions will be sufficient. Substantially, the two approaches can be distinguished that have received consideration. According to one of them crack growth proceeds discontinuously by a series of jump-like steps [5,14,26,34,40,43]. The time intervals Dt between discrete crack increments Da are just the periods to attain in a certain responsible cell at x = xc the level of damage or to accumulate the amount of hydrogen which satisfies a criterion like (24) or (26), correspondingly. Here again, alike regarding the location xc, the value Da must be definitely specified, too, e.g., from microstructural considerations. Then CGR is defined as the averaged value v = Da/Dt . Following the other way, crack growth is assumed to go on continuously [5,22] in such a manner that criterion (24) or (26) is constantly fulfilled at x = xc. The instantaneous CGR is then the derivative of crack size v = da/dt .
These two approaches usually are considered to be different with
respect to the background physics. Despite numerous speculations
trying to support either of them, no one has been generally accepted.
The two hardly are firm physical concepts, but they rather represent
analytical tools having certain advantages or shortcomings for
particular studies. Meanwhile, for the purposes of the examination
of the special matter of the K-dominance in EAC these distinctions
are of minor, if any, importance. As a matter of fact, concerning
the item of SIF control over local rupture event as an element
of EAC proceeding, whichever of the two fashions would be chosen,
it draws the conclusion that the firmness of this asset totally
relies on the certitude of K-dominance regarding both the
transport phase of EAC process and the mechanical state in the
fracture process zone. Thus, again, the problem of K-dominance
in EAC is still the question about exclusive SIF control over
the crack tip environment and the near tip stress-strain field.
However, this apparently does not exhausts the subject of the
concern. This matter will be addressed in Section 5 below.
3.2.3. The concepts of the EAC threshold
Environmentally assisted mechanical behaviour obviously exhibits the existence of a certain threshold representing the upper load limit or stress/strain level at which environmental facilitation of fracture is not detectable. The threshold SIF is the topic of primary engineering importance since it defines the safe load in service below which no crack propagation is expected or, at least, it is negligible from the engineering point of view, thus maintaining structural integrity.
As a matter of fact, two rather different ideas are often confused as the same one. The first of them, the conventional threshold [9,10,44], means Kth as the limit SIF value below which crack does not extend for "infinite" time despite the environmental action (or, in practice, for a reasonably long time base tB obviously determined from service requirements or experience). According to the other , the threshold SIF is thought to reflect the maximum environmental degradation attainable in a material under specific set of external (global) conditions due to the most severe of possible mechanisms of environmental degradation, cf. [45]. The second concept of the EAC threshold is more general and embodies the former in cases where this one is workable at all.
Really, in certain cases the maximum of degradation effect is
achieved in a manner that presumes establishing of the equilibrium
between the crack tip environment and the amount in rupture sites
of the concentration of a harmful specie x, or of the correspondent
to (23) damage w if damaging is markedly a kinetic process. Under
stationary external conditions the equilibrium obviously corresponds
to attainment of the maxima of both them in the steady state manner.
This is the maximum harmful action that the near tip zone endures
for a long time, so that the time to prepare crack advance is
thought infinite, Dt = _. Correspondent CGR in environmentally
assisted fracture is still zero at a constantly maintained SIF
level (provided at this time scale no other time-dependent fracture
mechanisms irrelevant to environmental degradation could operate,
e.g., creep-like processes). To this end, the time base tB
for valid Kth testing must be sufficient to approach reasonably
this equilibrium of the responsible kinetic processes of transportation
or of the damaging reaction, i.e., it must be as long as the characteristic
time to attain the equilibrium with a certain margin (the relaxation
time of the rate determining kinetic process). The value of Kth
determined with this precaution will be the true physical threshold
with respect to EAC. This corresponds to zero CGR if the relaxation
time for environment-related processes in EAC is much shorter
than the time scale necessary for other possible time-dependent
creep-like mechanical effects to become noticeable. Otherwise,
the notion of the EAC threshold should be modified, e.g., it may
be associated with environmental acceleration of
cracking that can somehow proceed without environmental assistance.
This first meaning of the EAC threshold is quite relevant to cracking
by hydrogen embrittlement mechanisms enabled by hydrogen transportation
to rupture sites by diffusion. The criterion of hydrogen assisted
local rupture (26) draws the threshold condition for cracking
taking in its left-hand part the steady state hydrogen concentration
distribution C_(x) which is the maximum approached
by the transient one C(x,t) at tÆ_
whilst diffusion proceeds to equilibrium (cf., e.g., [46]).
The second of the cited concepts of the EAC threshold is the ampler one since it relates to the the maximum environmental degradation achievable in the system which can be attained either on the way of establishing a steady state equilibrium, or temporarily (instantaneously) as the transient extreme severity of environment attack. This is more relevant when different mechanisms of environmental effect on fracture operate simultaneously, like hydrogen embrittlement, oxide film creation-rupture and active-path dissolution in stress-corrosion cracking [20,22]. Certain components of such EAC processes may be dictated by dynamical factors like crack tip deformation rate and fresh surface creation, where both the strain rate and the crack growth velocity may be involved as governing variables [5,15,20,22]. Clearly, there it would be impossible to maintain a certain SIF rate which renders the weakest resistance against EAC together with fixed SIF level for a prolonged time to evaluate Kth. Here common long-term quasi-static tests to define Kth [44] as a safety margin turn to be inadequate for assessment of EAC tolerance because the background idea of equilibrium as the worst situation is no more relevant.
In terms of general consideration of local damaging (23) and rupture criterion (24) presented in the previous subsection, this second treatment comprises both the anticipated worst cases concerning environmental impact on material behaviour:
ó the attainable maximum of the damaging specie concentration z and commensurate development of damage w itself which may be achieved in a system whilst approaching the steady state equilibrium, so that to terminate local fracture requires only to overcome definite minimal SIF level;
ó the fastest accumulation of the damaging specie and correspondent evolution of damage w itself promoted by the worst instantaneous in-crack physico-chemistry and straining dynamics, so that local rupture may be accomplished whilst passing certain minimal SIF in dynamical (transient) manner.
The second case really worths of special attention in view of
reported data on the effect of strain dynamics on the severity
of crack tip environment and about the decrease of the lower-shelf
SIF for noticeable environmental enhancement of cracking [15,20].
Focusing again on hydrogenous mechanism of cracking facilitation,
as particular situations where the strain dynamics can come in
play provoking the most severe environmental attack may be referred
the next. In various corrosive systems obviously both competitive
mechanisms of environmental attack operate simultaneously [14,15,20,22]:
the rate of crack tip hydrogenation competes with the rate of
passivation since hydrogen evolution follows after film rupture
rate dominated by strain dynamics. Beside, hydrogen entry into
metal and subsequent embrittlement at a given hydrogen evolution
intensity (hydrogen evolution index (7) or (9)) is controlled
by the hydrogen coverage on entry surface which depends on degree
of passivation [20] as well as by the barrier properties (permeability)
of surface films [22]. The surface straining rate participates
in both as it was commented above with respect to equations (14)
and (15) describing surface interactions. Moreover, dislocational
transportation of hydrogen also can come in play under dynamical
straining not only causing accelerated hydrogen accumulation in
prospective rupture sites, but possibly making operative another
microstructural fracture nuclei being more favourable for pick-up
of hydrogen brought by dislocations. This way, apart from its
influence on hydrogen transporation rate, strain dynamics can
modify the microstructural features of fracture micromechanism
itself and the local response on hydrogen degradation.
With this in mind, the comprehensive concept of the true (physical) threshold for EAC may look as follows: Kth is the maximum SIF at which environmental facilitation of cracking (i.e., either crack advance of growth acceleration due to environment) cannot occur in given combination {materialóenvironment}, or for practical purposes, cannot be detected within reasonable {timeócrack-size} resolution margins.
This definition implies consideration of the worst crack tip
conditions with respect to crack tip chemistry, harmful agent
sorption process, etc., which produce the most severe environmental
attack on material. These are in general variable during typical
EAC course, so this worst-state ought not to be a stationary
or equilibrium one, but it can be achieved temporarily at intermediate
times. Such the worst-state renders the weakest resistance
of to EAC in a given combination {materialóenvironment}.
And thus, correspondent threshold SIF as the lowest bound of detectable
sensitivity of the cracking resistance to environment must be
the intrinsic characteristic of the system.
3.3. Stress-Strain State
Evaluation or control of the listed above responsible mechanical
variables in the close vicinity of the crack tip with nonlinear
material behaviour would be a too complicated problem. Fortunately,
within a certain near tip region the stress-strain state, being
poorly controllable in explicit terms, can possess the self-similitude
which enables to quantify it in implicit manner using a single
parameter of local field intensity, like SIF within the domain
of applicability of LEFM. Here the key idea of fracture mechanics
[1,5,8,47] about the crack tip autonomy comes in play. For the
purposes of this study confined mainly with the field of LEFM
it may be interpreted as follows.
Fig. 7. The scheme of the crack tip vicinity with relevant zones
of specific stress-strain fields and material behaviours (lightly
shadowed is the domain of dominance of the r ó1/2-singularity
elastic field, medium density shadowed is the plastic zone, densely
shadowed is the fracture process zone, and a wavy pattern covers
the far field area).
Around the crack tip exists a region of characteristic size RSIF (Fig. 7) so that within a distance from the tip r £ RSIF stress-strain field defined by elastic constitutive equations is K-dominated, i.e., accurately enough represented solely by the universal r ó1/2 singular term of the complete series solution. From the other side, in the closest near tip vicinity really exists the fracture process zone of the size RFPZ where microscopic damaging proceeds and macroscopic constitutive equations obviously fail to describe highly nonlinear behaviour of material. Beside, this domain is usually surrounded by the plastic region of the size RY where elastic modeling is no longer accurate, too. However, the whole inelastic zone (fracture process zone and plastic region) may be so small that it does not disturb noticeably the linear elastic solution somewhat outside this special area (small scale yielding condition). When this happens, the K-dominated annular elastic region may still exist at distances from crack tip RY < r £ RSIF. Outside this region at r > RSIF the remote stress-strain field (the far field) is not governed by K and may be arbitrary.
As long as this ring of the K-dominated elastic field completely
shields the inelastic near tip region from any other external
influence except that provided by SIF, the state of the whole
near-tip domain including the fracture process zone is likely
dependent solely on SIF and the material itself (cf. [47]). Hence,
K is here the only variable which defines the crack tip
mechanical pre-damaged state, despite the lack of explicit exact
consideration of nonlinear behaviour and microfracture events
therein.
3.3.1. The two-side bounding for maintenance of the end-region
autonomy
From the above outlook it follows that SIF dominance over the
near tip mechanics is the asset which may be acquired with certain
accuracy the better the smaller is that exceptional end-region,
the plastic zone when speaking about LEFM, if compared with the
crack (or ligament) length [47]. Correspondent tolerance limit
for the accomplishment of the K-driven crack tip autonomy
is established obviously by the upper bound on the admissible
end-region size [1,5,8,47] at the crack tip critical event, i.e.,
at the moment of local rupture. Practically, the empirical standard
regulations of the LEFM validity [1,8] serve well for this purpose.
Namely, the next upper-bound limitation is used to provide the
soundness of co-relation of the local rupture event with SIF:
2.5 £ a (27)
where sY it the yield stress in tension. Using known estimate
of the plastic zone size in plane-strain state [1]
RY = (28)
it is easy to check that the above condition (27) is approximately
equivalent to the next one in terms of characteristic sizes:
a ³ 25RY (29)
However, this is likely the constraint imposed "from the outside" on K-dominance in the nonlinear end-region "enclosed" between the far-off domain and the crack tip itself. The other confinement "from the inside" may be supposed to be desired, too, owing to the fact that fracture mechanics works well with flaws ó cuts, slots, or cracks ó sharper than certain limit r* (see, e.g., [8,48]). To explore the matter more thoroughly the refined studies of the near tip zone should be addressed.
Namely, extensive scrupulous studies of the near tip situation
in elastoplastic solids including large-strain and damaging simulations
[49-55] convince that CTOD dt scales the intensity of the
crack tip fields throughout all specific near tip domains, one
embraced by the subsequent, i.e., the damaged and the large-strain
plastic zones controlled by CTOD, the small strain plastic one
described by the Hutchinson-Rice-Rosengren solution controlled
by J-integral [1], and the elastic domain with SIF dominated
asymptotic field, if any of the latter exists in some particular
case like the small scale yielding. In purely mechanical problems,
i.e., with no account for possible environmental influences, these
fields have universal self-similar appearance of the fixed material
dependent functions of CTOD and spatial coordinates attached to
the crack tip, the polar coordinates of distance r and
angle j for definitness, as follows:
ei = ei*(r/dt,j) ;and ;si
= si*(r/dt,j) at r £ RSSF
(i = 1,2,3) (30)
where asterisks in the right-hand parts, as usual herein, emphasise
the predetermined nature of the material's functions dependent
solely on the mechanical constants of a material like Young modulus
E, tensile yield stress sY and strain hardening
exponent, and RSSF denotes the distance within which stress-strain
field is self-similar with a reasonable accuracy. It is worth
mentioning that the near tip stress-strain field, i.e., the functions
in the right-hand parts of (30), depend on the shape of the blunted
crack tip as far as it defines kinematically admissible plastic
slips (slip line fields) [56], see Fig. 8. However, in the absence
of any other specific impact on the tip geometry (e.g., environmental)
blunting evolution under load also depends solely on the material,
and thus, and the problem remains self-consistent, so that the
dependence of stress-strain distributions on the tip shape does
not arise explicitly in (30). Under small scale yielding SIF can
serve as a nominator of the intensity of the near tip nonlinear
fields (30) by substituting there CTOD with K according
to well known relation of the sort [1,49,51]:
dt = l (31)
where l is known numerical factor. Then, expressions (30) and (31) represent the SIF controlled autonomous mechanical state near the crack tip in solids.
Fig. 8. The schemes of smooth (a) and cornered (b) blunted crack
tips with their kinematically admissible slip line fields which
cause different stress-strain states on the way to completion
of the local rupture (cf. [56]).
Meanwhile, analyses [49,51,53] of the nonlinear large-strain zone
within SIF governed elastic one show that the self-similitude
of elastoplastic stress-strain field does not arise from the very
beginning of straining, but establishes since some value
of CTOD has been achieved. This CTOD magnitude dtSSF, and
correspondingly, the minimal SIF KSSF µ according
to relation (31), necessary to accomplish the formation of the
K-dominated self-similar field in the fracture process
zone is proportional to initial bluntness of the crack tip,
dtSSF = lSSFr (32)
where lSSF is a certain numerical factor, according to available estimates [49,51,53] roughly lSSF 1.5 or more.
In order to grant K-dominance of crack propagation, local
rupture must occur whilst near-tip situation is really controlled
by SIF. To this end, apart from preservation of the small scale
yielding according to limitation (27), the SIF must exceed certain
level to attain self-similar state (30) before a critical event.
Therefore, during the whole crack propagation the two-side limitation
must be obeyed:
KSSF < K < KSSY (33)
where KSSY is the maximum SIF at which small scale yielding in a certain cracked solid is still maintained according to the condition (27). The lower bound for SIF in (33) is just what allows to discriminate between a crack as an object of fracture mechanics methodology and an arbitrary flaw ó a cut, a notch, or of any other kind ó with which fracture mechanics, the LEFM in particular, does not work because local rupture there occurs at nominal applied K < KSSF(r), i.e., when the very near tip stress-strain field is not controlled solely by K.
Cyclic pre-cracking regime impacts evidently crack bluntness [1]. However, above discussion of the lower-bound limitation is not the whole story since the mentioned results refer to the crack in a virgin material, but not take into account residual plastic stresses and strains caused by preceding history of the crack (fatigue pre-cracking regime, peak loads, etc.) which affect the resultant very near tip field. Residual plasticity can destroy the self-similitude therein, i.e., violate the unique co-relation (30)-(31) between SIF and field components inherent for a stationary crack under monotonous loading.
Residual stresses from the previous history of K variation (i.e., the loading history) may affect the very near tip state at SIF levels below the historical maximum (the highest one achieved before). Moreover, despite the stress state is apparently recovered at applied K exceeding this historical maximum [51], but the cumulative plastic strain does not which will affect SIF dominance over local fracture if strain-controlled fracture mechanisms were operative. Therefore, the lower bound for K-dominance in (33) established for virgin material basing on the relationship (32) represents only the necessary condition. Available data about the fine peculiarities of the very near tip fields with account for residual plasticity after unloadings are too scarce to derive more stringent lower limit to ensure establishing of the self-similar near-tip fields (30).
Nevertheless, for the case of environmentally unaffected fracture ó for standard fracture toughness testing ó both these matters of ensuring the K-dominance, i.e., the items of crack tip bluntness and residual stress-strain field, are apparently resolved at once by the standard empirical ruling guide for fatigue pre-cracking (cf., e.g., [1]) which dictates, in particular, the admissible level of Kmax < 0.6Kc.
In EAC when environmental degradation makes local rupture possible
at applied nominal SIF values substantially below the limit toughness
Kc this standard bounding is no longer reliable. Here to
assure K-dominance over the purely mechanical crack tip
situation (i.e., not referring for the moment to any environmental
aspects) clearly calls for more severe limitation on both crack
sharpness and the level of residual strains from the pre-EAC loading-cracking
history. In other words, the soundness of fracture mechanics with
respect to EAC requires confirmation of the exclusive SIF control
over the very tip stress-strain state in the whole range of SIF
from the very low limit when environmental degradation can be
detected. This is the way to improve fracture mechanics evaluation
of EAC eliminating the mentioned in the Section 2 uncertainties
associated with pre-EAC loading history and crack tip geometry.
3.3.2. The effect of crack advance on K-dominance
Considerations of the mechanical autonomy of the near tip process zone in the previous subsection refer to stationary crack. Correspondingly, the derived conclusions are valid for non-propagating cracks. Firmness of the key conclusion about the self-similitude of elastoplastic near tip stress-strain state (30) when crack starts to grow has been addressed by numerous analyses [1,47,57-61]. Although in many of them large scale yielding was studied, the result to be emphasised below is of general validity for elastoplastic deformations near the crack tip since it is the consequence of the drastic deviation from the predominantly proportional straining which takes place near the tip of non-extending crack.
In brief, when macroscopic crack growth starts, unloading occurs because of stress release on newly created fracture surface along substantial neighbourhood of the end-region. Consequently, crack advancement destroys the definiteness of the antecedent near tip field of a stationary crack, i.e., changes the shapes of material dependent functions in the right-hand parts of expressions (30). Roughly, weakening of strain- and rising of stress-concentration together with alteration of stress-strain state triaxiality accompany crack propagation. This affects near tip damaging (23) and local rupture criterion (24) for EAC (or critical hydrogen concentration (25) and critical event condition (26) for hydrogen assisted cracking, in particular). Moreover, this can even cause qualitative change of fracture micromechanism from stress- to strain-controlled rupture or vice versa. Beside, alteration of stresses and strains influences stress-strain assisted transportation of a harmful environmental specie to prospective fracture nuclei as discussed in subsection 3.1. Consequently, after a change of the local rupture criterion and of the rate of the environmental agents supply to damage sites, e.g., of hydrogen accumulation in the process zone, crack growth kinetics must exhibit alteration, too. Note, that all that is true (although, may be less spectacular quantitatively) under small scale yielding in spite of the fact that SIF continues to be the only parameter governing the elastic field in a ring RY < r £ RSIF around the crack tip (the SIF dominated elastic zone, see Fig. 7). In effect, it further does not with respect to inelastic domain in the close vicinity of the crack tip.
By analogy with incremental theory of plasticity capable of accounting
for non-proportional loadings, variation of the near tip fields
(30) when crack grows may be represented through the infinitesimal
increments
dei = dei*(si,ei,dK,da),
dsi = dsi*(si,ei,dK,da)
;at;r < RY (i = 1,2,3) (34)
where, again, asterisks mark certain material dependent forms. The resulting near tip stress-strain field after a finite increment of SIF or crack size is to be determined by integration of expressions (34) over a particular Kóa-variation path. Since the increments (34) proceed from incremental plasticity theory, these integrals must be path-dependent. Every such trajectory is characterised by a certain course of the derivative dK/da, i.e., it is associated with a specific SIF gradient c. The latter turns to be one more variable which governs crack propagation. Hence, its appearance in the equation of EAC kinetics (4) is predetermined at least by the nonlinear mechanical effects accompanying crack growth.
Quantitatively the role of the factor under consideration may be rather moderate, and even nearly insignificant, for stress-strain field at small scale yielding [61]. However, this does not yields the same for the kinetics of EAC.
Indeed, confining now for the purpose of discussion with hydrogen
assisted cracking, let us suppose that crack propagates at constant
SIF so that the criterion (26) is satisfied at diffusion time
t = Dt when hydrogen concentration attains the critical
value Ccr for instantaneous values of stress and strain
components. This renders definite value of CGR v = Da/Dt.
Due to crack extension, despite constant K, stress-strain
state in the end-region changes causing according to the relationship
(25) also some alteration dCcr of the critical concentration
necessary for local rupture. This evidently yields the deviation
dCt of the previous value Dt of diffusion time to
reach this new critical concentration level now being equal to
(Ccr + dCcr). Approximately this yields
dCt = (35)
where C't = dC/dt in a certain critical location. At relatively slow hydrogen accumulation in the process zone, dC/dt << 1 in relative units, this alteration of time to the next crack increment in spite of small dCcr will be significant and able to yield noticeable variation of CGR, roughly (1 + dCt/Dt) times, despite all other related quantities remain essentially the same.
This nonlinear mechanics effect, however, can be of minor quantitative importance at small scale yielding. On the contrary, it undoubtedly worths of much more attention in studies of EAC under large scale yielding using the concepts of nonlinear fracture mechanics since the discrepancy between near tip fields of stationary and extending cracks becomes there pronounced. In particular, persistent attempts to use J-integral as the measure of the intensity of the near tip stress-strain field in studies of EAC seem to be quite dubious, and even miserable if one takes into account the weakness of this basic concept of non-linear fracture mechanics to characterise cracking resistance on the propagation stage even with no relevance to environmental aggravation [62,63].
Nonlinear mechanics effects substantially complicate the matter
of well-grounded evaluation of EAC causing the dependence of cracking
process on loading and cracking history (on a particular path
in Kóa-space). Somewhat helps the matter
the fact that after a certain amount of crack extension under
constant intensity of the embracing K-dominated elastic
field (or J-dominated elastoplastic one as well) the very
near tip inelastic situation stabilises (although it does not
restore its initial shape) and returns anew to be governed by
the single parameter of stress-strain field intensity (cf. [1,61]).
Thus, maintaining constant SIF during EAC one may attain the regime
of crack growth under well K-dominated near tip stress-strain
field. This eliminates the deal of uncertainty regarding characterisation
of EAC using fracture mechanics since at least mechanical situation
in the crack tip end-region becomes dependent exclusively on SIF
and on material itself. However, a definite conclusion can hardly
be reached yet about how conservative (i.e., safe) might be assessments
and predictions of EAC basing on data about CGR at that steady
state K-controlled stress-strain field, if they really
could be conservative. Apparently, they rather cannot as far as
the studies of nonlinear mechanics effects in the inert environment
cracking imply that the crack extension reduces the severity of
the near tip mechanical situation so that to continue crack growth
an increase of the mechanical impetus is needed (cf. [1,61]).
This is clearly displayed by the JR-curves which indicate
the rise of crack growth resistance in terms of bearable value
of J-integral vs. subcritical crack growth Da. However,
environmental degradation can change this trend. Nevertheless,
correspondent v(K)-curve at steady state elastoplastic
near tip field have a reasonable premise to be the intrinsic characteristic
of EAC.
3.3.3. Environmental effect on the near tip stress-strain
state
One more history related factor of EAC worths of comment. In contrast to the discussed above role of the loading routine, here the time history is referred associated with kinetic physico-chemical processes participating in EAC. This is the environmental effect on the crack tip morphology, i.e., on the shape and magnitude of crack tip blunting. It has already been mentioned above that the preferable (or kinematically admissible) slip line field depends on the shape of the blunted crack tip (see Fig. 8). Consequently, the near tip elastoplastic stress-strain field and local rupture do the same. Purely mechanical problem about crack tip situation is self-consistent: the tip shape (deformation displacements) and stress-strain field are dependent solely on material and load (cf., e.g., [50,52,53]). But if any other cause is impinging on the width and the shape of a tip, it alters stress-strain field and crack behaviour, EAC in particular. Crack tip environments are capable of doing that by various ways.
Different modes of hydrogen interference in metals plasticity have been reported which give rise to this ability: dislocation multiplication [64,65], especially near the surface, enhancement of slip localisation [66], etc. This may trigger near surface plastic instabilities or even cause certain local alteration of the yield surface of a metal in some vicinity of the crack tip which is known to have consequences for elastoplastic state therein (cf., e.g., elastoplastic solutions in [50,52,53]).
Regarding corrosive environments, local anodic dissolution may affect both crack tip shape and width . It chemically removes metal from the surface in the vicinity of the crack tip less selectively (i.e., homogeneously) or more, e.g., by slip dissolution mechanism, affecting the resulting crack tip shapes, forcing them to be more smooth, or cornered, or even (micro)branched. Surface films of corrosion product at the crack tip, e.g., oxide, having mechanical properties different from the base material, also affect evolution of the crack blunting and the near tip elastoplastic situation. Whichever in origin, observations of the alteration of the crack tip shape during EAC have been reported in numerous studies since long ago [11, 67,68]
The mentioned stimuli to environment induced variability of the
near tip plasticity are time-dependent kinetic processes. They
proceed in accordance with their own rules from the very beginning
of the contact of material and environment and along with crack
growth. The superposition of physico-chemical kinetics and loading-cracking
kinematics (i.e., the time history effect) takes place here producing
the sort of sinergistic action dependent on a particular trajectory
of a process in the Kóaót-space. It is expected
to violate the K-dominance of EAC, too.
4. CONSOLIDATION OF FRACTURE MECHANICS ASSESSMENT OF EAC
4.1. The Strictly Local Treatment of EAC
Considerations of the previous sections draw that v(K)-curves in EAC not only appear to be variable that could be related to imperfect experimentation or interference of other processes escaped accounting. Moreover, CGR intrinsically must depend apart from SIF on ample set of variables. Correspondingly, the idea of the (bulk-)environment dependent material's function (1) capable of matching a variety of EAC proceedings in a given system {materialóenvironment} over-simplifies the matter. After the exploration in Section 3 of the matters of K-dominance concerning constituents of EAC it follows that within the customary approach there is nearly no hope to avoid the ambiguity of EAC evaluation caused in part by poor characterisation of the environmental physico-chemistry in the tip neighbourhood. This deal of uncertainty in characterisation of EAC by means of the v(K)-curve can be eliminated by adopting exclusively local characterisation of both external impetuses for cracking [5,14], namely, the mechanical and the physico-chemical influences just at the crack tip. This consists of matching the crack tip environment beside of SIF assumed to be responsible (although, apparently not exclusively) for local mechanical events. Monitoring or control of the environmental physico-chemistry in the crack tip vicinity during EAC will surely improve the certainty and comparability of the evaluations of materials resistivity to EAC as well as the predictive capability for engineering purposes.
The key point of the consolidated fracture mechanics approach
to EAC [5,14] is the concept of the crack growth kinetics curve
as the characteristic of the material-environment combination
where the local near tip environment is meant, but not the global
one. Thus, this purified treatment relies on the modification
of the engineering v(K)-curve (1) which in distinction
to that customary one is associated with a certain couple {materialólocal
environment}:
v = v*(K | f1, f2, ..., fm)
(36)
where the variables f1, f2,..., fm must represent the complete set of characteristics of the crack tip chemistry to define environmental attack on the fracture process zone, e.g., its hydrogenation or other effects. This is a quite natural deed as far as the fracture mechanics approach is in essence a local one which strengths arise mostly from the autonomy of the end-region of a crack.
In most systems {materialóbulk environment} being the subject matter in engineering analyses and safety concerns the near tip environment is essentially variable depending on environmental in-crack processes, crack geometry, loading, etc., as discussed above in section 3.1. Then, to utilise v(K)-curve (36) for crack propagation predictions for a variety of solid/crack geometries and loadings on the way analogous to the described in section 2.1, the evolution of crack tip environment parameters f1, f2,..., fm must be predicted, too, dependent on global physico-chemical characteristics F1, F2,..., FM of the environment and on parameters of a crack as a canal of environment access to the process zone. This requires to simulate in-crack processes with proper transport and reactions equations to determine the environmental variables, the f-s, which directly govern crack growth apart from SIF.
Therefore, in general, predictive capability of fracture mechanics concerning EAC is provided by the set of coupled equations which comprises the next [14]:
(i) equations of the mechanics of solids to determine not only local stress-strain state near the crack tip, i.e., the SIF solution which was sufficient within customary engineering approach, but also the whole shape of the opened crack as the space where proceed environmental physico-chemical processes which regulate the near tip chemistry;
(ii) relevant equations of mass-charge transfer and physico-chemical kinetics within a crack to evaluate variation of the crack tip environment during a particular course of EAC in a certain loaded solid, i.e., to find the f-solutions of the kind given by expressions (5) for specified loading, environmental and crack growth histories;
(iii) governing equation of crack growth kinetics to couple physico-chemical
and mechanical factors of EAC, this now being the evident adjustment
of the equation (2) with account of the modified crack propagation
law (36), i.e.,
= v*(K(F,a) |f1, f2,
..., fm) (37)
To follow this way of EAC prediction, the v(K)-curves (36) must by somehow established to determine the right-hand part of the crack growth equation (37). This may be performed either by EAC testing with direct monitoring and/or regulation of the crack tip environment, or by mathematical simulation of the in-crack transport and reaction processes, combining either of them, if helpful, with simulations utilising working (physical) models. All these ways have got considerable exploration, so that correspondent techniques do exist now (relevant insight and numerous references on this matter one can find in papers [5,13-18]). Beside, utilisation of this approach in engineering predictions of crack growth involves also prediction of the crack tip environment parameters fi (i=1,2,...) in service, i.e., again, calls for correspondent mathematical or working models.
This consolidated fracture mechanics approach, though substantially more sophisticated and costly, may be beneficial in particular applications where more accurate evaluations of the susceptibility to EAC are required. However, for common engineering to follow this way may be too complicated and expensive. Moreover, to monitor the crack tip environments a difficult but relatively feasible job in laboratory testing in contrast to the assessment of the local in-crack environment in service which could hardly be performed with certitude. Thus, although more legitimate, this approach seems to be impractical for ordinary engineering practice, but rather beneficial in materials research to conduct purer experiments and derive more trustworthy comparative data.
Beside, dealing with the couples {materialólocal environment}
yet does not ensure the uniqueness of correspondent v(K)-curves
(36) as intrinsic characteristics of material, and therefore,
does not renders the certitude and safety of EAC evaluations.
Apart from the role of the process history, a series of values
from the discussed above in Section 3 still have not been turned
to good account as they deserve in building up safe predictive
procedures for EAC. This list of process variables intrinsically
affecting EAC includes those representing the near tip stress-strain
dynamics (velocities e_i and s_i, i=1,2,3)
governed by loading dynamics presented by SIF rate K_ together
with crack growth kinetics given by CGR v. In addition,
the representative of the effects of non-proportional plasticity
in the end-region of a moving crack, i.e., SIF gradient c,
also needs account. Recall, that the parameters of the near tip
stress-strain state dynamics and crack growth kinetics not only
influence near tip environment chemistry (see subsection 3.1.1),
the effect already taken into account implicitly in the v(K)-curve
(36) through using of the local environment characteristics, but
they also affect surface interactions, namely, the entry phase
and subsequent transport of a harmful environmental specie to
microfracture nuclei (see subsections 3.1.2-3.1.3) which still
need assessment. Therefore, the v(K)-curve (36)
can be not more than a somehow fair approximation of evolutions
of EAC. This "local" interpretation of the crack growth
kinetics curve still remains incomplete and does not meet the
requirements for the intrinsic material's curve. Ignoring of the
role of these variables still may cause insecure evaluation of
EAC and yield excessively optimistic reliability estimates.
4.2. A Safe Approach to Evaluation of EAC
Following the "local" treatment of EAC more strictly,
the key relationship (4) which represents the crack growth kinetics
is reduced to the next form
v = v*(K, K_, c,...,t
|f1, f2,..., fm) (38)
This still remains a history-dependent functional which renders different instantaneous values of CGR for the same SIF when EAC proceeds along various Kóaót-paths, but not the unique value of v as the function (36) does by definition. The intrinsic prerequisites for this kind of inter-relation between v and K in EAC will be addressed in the next section, as well as some efforts to estimate their quantitative significance will be presented there with respect to hydrogen assisted cracking. Meanwhile, now we focus on another possibility to gain safe evaluation and prediction of EAC which, although sometimes less precise, may be rather practical.
Common engineering prefers to characterise EAC using solely SIF as a fitting variable and a bulk environment parameters F1,F2,..., FM which are the best suitable with available practical means of monitoring and control. Owing to the fact that CGR indeed is a function, and even a functional, of several variables displayed in forms (4) and (38), the promising strategy to grant the safe evaluation of EAC is to find in each case the worst combination of all the other (i.e., apart from K) directly influencing local variables in (38) which may whenever occur in a system {materialóbulk environment} that one deals with. Naturally, here as the worst is meant as the assemblage of mechanical and local environmental variables which produce the most drastic environmental impact on the crack tip process zone (damaging and material degradation therein) at a certain K, i.e., provides the maximum CGR attainable at given SIF in a particular couple {materialóbulk environment} under consideration.
Given an environment with global characteristics F1,...,FM,
whichever EAC proceeding under arbitrary circumstances (member's
geometry, service/test loading routine, etc.) is tightly
associated with its particular v(t)-evolution along
a certain trajectory in the space of all variables which represent
crack tip events and participate in expression (38). A projection
on vóK-plane of the tangle of all these trajectories
conceivable in the {materialóenvironment} couple
is surely bounded with some envelope which for every SIF gives
the definite value of maximum attainable CGR vm = vm(K),
see Fig. 9. This envelope ó the reverence crack growth
kinetics curve ó may be considered as the really intrinsic
curve dependent solely on the material and the environment. Using
it as the master curve
v = vm*(K |f1, f2,..., fm)
(39)
within otherwise the same framework of engineering fracture mechanics as described in subsection 2.1 will provide safe evaluation of EAC.
From the other hand, the projection of that tangle of v(t)-trajectories on the sub-space of all the responsible variables placed in the right hand part of the expression (38) behind of SIF, i.e., K_, c and f-s, also occupies a certain bounded region. For these parameters which are not convenient or suitable for monitoring and control in practice this domain defines the extremes of their possible variation. These bounds apparently may be fixed for a given system as its intrinsic attributes, too, providing thus a suitable guide to narrow the amount of necessary experimentation to establish the reference vm(K)-curve for safe assessment of EAC for a {materialóbulk environment} combination.
Fig. 9. The worst crack growth kinetics curve (the bold curve)
for a combination {materialó (bulk) environment}
as the envelope of all possible v(K)-curves (fine
lines) ó the reference or master curve.
To exemplify this matter can serve a brief outline of the studies
of the stress-corrosion cracking of steels in aqueous electrolytes
(for substantial test data see paper [14] and references therein).
For a typical path of evolution of electrochemical characteristics
pHCT and EVCT of the environment near the crack tip during
EAC sketched in Fig. 6 correspondent variation of the shift of
electrode potential DHEEVCT defined by expression (8) appears
as given in Fig. 10. According to relation (9) the same way behaves
hydrogen evolution or hydrogen activity in the vicinity of the
tip. It follows, that the hydrogenation index DHEEVCT during
EAC tests of steels approaches asymptotically the most negative
level (DHEEVCT)min (Fig. 10) which provides the
most severe hydrogenation. For the case when EAC under corrosion
goes by hydrogen assisted fracture mechanism, this brings an example
of the worst state attained as a stationary one whilst process
continues. Having confirmed both this trend of the variation of
the intensity of hydrogenation and the nature of fracture mechanism
as the hydrogen assisted one, the master vm(K)-curve
can be obtained conducting crack growth kinetics tests maintaining
steadily these worst conditions of hydrogenation at the crack
tip.
Fig. 10. Typical progress of the hydrogen evolution index (8)
for the alteration of crack tip environment during EAC of steels
in aqueous corrosive media displayed in Fig. 6.
However, additional precautions should be taken also with respect to the remainder of the potentially influencing variables, the SIF rate K_ and gradient c, which also must be the most unfavourable for EAC resistance. Apparently, in the range of rather slow loadings when dislocational mode of hydrogen supply to rupture sites is not operative and hydrogen diffusion in metal is the controlling step of transport, the longer the time for hydrogen to diffuse the more facilitation of fracture occurs [69] and some low bound for the rate K_ exists below which the effect of hydrogen on fracture is maximum. Regarding the SIF gradient, available test data [70] allow to suggest that negative values of c cause the retardation effect on cracking, cf. Fig. 3c, and so, positive are advisable for safe assessment of EAC (although, more confirmations of this trend are desirable).
On the other hand, the transient or instantaneous worst situations may occur in cases of stress-corrosion cracking of metals with significant contribution of anodic factors of fracture facilitation. For example, in certain systems [22] competition of the rates of the anodic reaction of protective film formation over the crack tip metal surface and of the deformation-controlled film damaging affects hydrogen entry into metal and subsequent hydrogen degradation (embrittlement). At each instantaneous value of SIF this may produce maximum instantaneous CGR at certain crack tip deformation rate represented by K_. The other examples of transient (dynamical) worst situation may be expected in EAC processes where similar forms of corrosion-deformation interaction ó reaction-straining competition ó take place, like mechanisms of crack growth via creation-rupture of brittle films, slip-dissolution, etc. [20,71].
Nevertheless, the worst state seems to be indeed the intrinsic attribute of the system {materialóenvironment} considered in a global sense, i.e., referring to bulk surroundings. Accordingly, the same may be expected with regard to the vm(K)-curve as fracture mechanics characteristic of EAC. The idea of this master curve is compatible with the discussed in subsection 3.2.2. concept of the EAC threshold as the border when the maximum efficiency of local environmental impact on material is attained so that crack extension becomes the easiest from the part of mechanics. Then, the vm(K)-curve starts from zero CGR at this true physical threshold Kth (provided other thinkable time-dependent effects like those of creep-like fracture are out of play).
Correspondent materials testing techniques for safe assessment
of EAC should involve artificial maintenance of this the most
severe level of crack tip interactions which provide the strongest
environmental impact on the crack tip zone. This calls for bounding
procedures to establish the proper limits for characteristics
of crack tip mechanics and physico-chemistry to evaluate the weakest
EAC resistivity. Obviously, more extensive testing is required
there to find the worst among all possible behaviours. This could
be reduced by appropriate modelling and development of simulation
techniques regarding involved interactions.
5. DIFFUSION RELATED PRECONDITIONS OF K-DOMINANCE IN EAC
In the previous section the ways to improve the certainty (and safety) of the fracture mechanics approach were explored to arrive at some concept of crack growth kinetics curve which could better fit the duties of the intrinsic material-environment dependent function able to characterise EAC. The check-point concerning this capability consists in that one must be convinced about exclusive K-dominance over all components of EAC process. The ways to shield from or to diminish the ambiguity of EAC testing caused by factors intrinsically not-controllable by sole SIF were in essence control them artificially: to manipulate local environment at the crack tip and to maintain fixed (or better, the worst) other manageable influencing factors like SIF rate and gradient. In some cases the problems of this kind fortunately may not arise or their role can be negligible, e.g., with respect to hydrogen assisted cracking in gaseous environments when crack tip and global environment parameters often are really the same, and when loading may be essentially quasi-static. Nevertheless, having eliminated the matters of SIF non-controlability of such phases of the EAC process as the environmental transport towards the end-region of a crack, the entry of a harmful agent into material through crack tip surface, and assured the reasonable K-dominance over inelastic stress-strain state and damaging in the crack tip process zone, the question about the overall SIF control in EAC still remains unresolved in case when diffusional transportation of a damaging specie to microstructural fracture sites is involved.
Really, to ensure the firmness of the fracture mechanics approach to EAC and the uniqueness of the crack growth kinetics curve the two preconditions must be simultaneously fulfilled during whichever proceeding of EAC:
ó the family of crack tip environment parameters f1, f2,..., fm closes the set of variables representing environmental attack at the end-region;
ó SIF is the only external variable (i.e., not the intrinsic material-environment parameter) which governs the stress-strain field throughout the physical fracture process zone and beyond to the extent sufficient to dominate in the rupture sites booth relevant aspects, the one of damage and the other of accumulation of harmful environmental species.
Having this, the listed variables K and f-s jointly do match the similitude of all EAC constituents, and then, correspondent v(K)-curve (36) can acquire the uniqueness as the intrinsic material-dependent function.
Revision of the relevant aspects of the near tip diffusion in
solids is presented below for hydrogen assisted cracking of metals
as the most pertinent case because there is expected more pronounced
quantitative significance of the effects under consideration due
to the extreme mobility of hydrogen within metals. Nevertheless,
this study deals with the general problem of the stress and strain
assisted diffusion, and therefore, derived analytical results
are applicable to whichever case of near tip diffusion in EAC.
Presenting here the brief outline of these studies, for more details
we refer the reader to another papers [72-75].
5.1. The Effect of Far Field on the Near Tip Diffusion
The matter of K-dominance with respect to hydrogen diffusion in the near tip zone, alike the same item concerning the near tip stress-strain field (cf. section 3.3), is a question of the accuracy of the approximate hydrogen concentration distribution Ca governed by the K-dominated component of the near tip stress-strain field (i.e., by the asymptotic controlled term effective at distance r £ RSIF) to represent the actual one Cf being driven by the complete mechanical field containing the whole series (i.e., influenced by the non-autonomous far field) which arbitrary diverges from the K-controlled stress distribution at r > RSIF. To this end, the discrepancy E = Ca - Cf between the two concentration distributions is the subject of interest to estimate the effect of the far field on hydrogen accumulation in the fracture process zone, i.e., at r £ RFPZ (see Fig. 7).
Hydrogen activity in the near tip environment is supposed to be constant which provides a certain level of equilibrium hydrogen concentration C0 in virgin material or correspondent value of concentration Ce in deformed one, so that boundary conditions for diffusion may be equivalently presented in whichever of the two forms (16) or (22).
Focusing first on the steady state solution of the equation (21)
of stress-strain assisted hydrogen diffusion which is asymptotically
attained at tÆ_, it is easy to get it exactly for
arbitrary stress-strain field throughout a solid. Really, this
hydrogen distribution corresponds to the equilibrium state when
diffusion flux (20) is zero, or equivalently, diffusion driving
force (18) is nullified which is provided at C/Ks = const.
Taking into account relation (19) the steady state solution is
then as follows:
C_ = C0 Ks (r) or ;C_
= C0Ks0 (ep(r)) exp (40)
where r is radius vector of a spatial point. With plastic strain ep = 0 the last expression coincides with known solution for stress-only driven concentration [34].
As far as this steady state concentration is a plain function of the stress-strain components, the C_-distribution immediately acquires the same sort of K-dominance as the mechanical state has. In other words, steady state near tip concentration is insensitive to remote not SIF controlled stress-strain field. Correspondingly, because this solution represents the maximum attainable level of hydrogenation, and of hydrogen induced degradation, too, this means that the threshold condition for hydrogen assisted cracking must be exclusively SIF controlled, i.e., defined by the threshold SIF Kth as the intrinsic characteristic of a material-environment system (provided the stress-strain field does so).
What about transient situation, considering equation (21) of stress-strain
assisted diffusion in solids for two particular cases, the first
with only the asymptotic K-controlled term of stress-strain
field throughout the entire solid and the second for whichever
actual whole term solution dependent on a specific solid geometry
and loading, the following diffusion-like equation can be derived
with regard to the discrepancy [72,73]:
= D + óD _ + Q (41)
with the source-type term
Q = -D (DM_óCa + DNCa)
- óD _DMCa (42)
where the subindices f and a denote that the marked variables are determined by different stress-strain fields, the far- and the asymptotic-field correspondingly, vector and scalar coefficients here are DM = Ma - Mf and DN = Na - Nf. Since initial and boundary conditions for concentrations Ca(r,t) and Cf(r,t) are identical, the ones for E (r,t) are zero.
With regard to the source term (42) some simplifications can be
made taking into account that at r £ RSIF both
stress-strain fields under consideration practically coincide
with each other. This brings the next:
Q = ; (43)
where sf and sa represent hydrostatic stress components of the far- and asymptotic-fields, correspondingly.
The last expression represents the source term for diffusion of the discrepancy E which emanates at r > RSIF. With zero initial and boundary conditions for E its absolute value increases at r £ RSIF with time as the source works. This "error" generated at r > RSIF by the source Q diffuses from there towards the fracture process zone. The effect of the far field on K-dominance over hydrogen accumulation therein becomes more noticeable when a greater amount of "error" E can reach the region r < RFPZ by diffusion from the distant source situated at r > RSIF.
Roughly, according to expression (43) the source intensity Q becomes significant after at r > RSIF had noticeably arisen the concentration Ca of hydrogen diffusing under the influence of the asymptotic-controlled elastic stress field (plasticity is supposed located solely near the crack tip, and thus, stress-strain state at r > RSIF is elastic, i.e., ep = 0) . Evidently, the time to achieve this noticeable source productivity depends on diffusion distance for Ca from the tip surface which is as large as the value of RSIF. Beside, after the source started to work intensively, the increase of E in the process zone also needs time for "error" diffusion towards crack tip from the source over the distance about RSIF - RFPZ. Thus, the evolution of the discrepancy between the two hydrogen concentration distributions essentially depends on the characteristic sizes of the near tip zones RSIF and RFPZ (Fig. 7).
From validity requirements of K-based LEFM [1,8], like
limitation (27), and available data about characteristic near
tip scales, like given by expressions (28), (31), etc.
[1,8,49,54,55,61], it was derived the next estimate [72]
³ 0.1 > 10 (44)
To provide a tangible sensation, with illustrative purpose the last numerical bound is here presented which is quite proper for steels as far as they usually have sY < E/100. The estimate (44) seems to be rather conservative.
Approximate upper bound solution [72] of the problem of stress-strain
assisted diffusion may be given by analogy with the stress-only
affected one (cf. [14,43]) like follows:
Ca(x,t) = C0Ks(x) erfc
;= C_(x) erfc (45)
where erfc(_) is the complementary error function. Having
it, from expression (43) the next conservative estimate is obtained
for the time tQ when the "error" source can start
to produce a noticeably disturbance of the near tip K-controlled
hydrogenation process [72]:
tQ >; (46)
From the other hand, analysing concentration evolution (45) one
can conclude that Ca in the fracture process zone (i.e.,
at x £ RFPZ) exceeds 95% of the steady state
level C_ at diffusion times longer than a certain value
tss for which conservative estimate is as follows [72]:
tss;=;130 ; (47)
Increase of the K-driven concentration Ca at t > tss falls into narrow 5% strip near the steady state concentration C_ which defines the threshold for hydrogen assisted cracking.
Combining the estimates (44), (46) and (47) we conclude the next
regarding the time when the source of discrepancy turns on:
tQ > tss;;;³;tss;;
(48)
and therefore
tQ;>;tss ;if ;;>;161
(49)
The derived inequality springs from the empirically proved standard fracture mechanics condition which ensures mechanical K-dominance in the crack tip zone. Relation (49) means that not SIF controlled stress-strain field (the far field) can affect the asymptotic-driven hydrogenation near the crack tip well after the transient concentration therein falls within 95% scatter band near the steady state hydrogen distribution. Before this there is no reason to anticipate noticeable discrepancy between concentrations Ca and Cf in the fracture process zone. But within the narrow 5%-width strip in the vicinity of the steady state limit C_ the supposition about SIF control over hydrogen diffusion becomes erroneous. This fact is conditioned with some rather conservative bounding regarding elastic-plastic properties of materials which is displayed in (49). But the last limitation does not restrain too much the validity of fracture mechanics to characterise stress-strain dependent hydrogenation of the near tip process zone as far as it apparently covers the majority of engineering alloys.
Stationary hydrogen concentration C_ determines threshold conditions for crack propagation when local rupture criterion (26) is fulfilled at tÆ_ yielding v = 0. With critical hydrogen concentration somewhat below this steady state level but within 5%-width band, 0.95C_ < Ccr < C_, to achieve local rupture event needs long diffusion times t > tss. Correspondingly, CGR results slow. Thus, the mentioned above 5% band near the steady state concentration is just what can be associated with the near-threshold portion of a typical v(K)-curve (Fig. 1).
The performed analysis implies that variable not K-dominated mechanical far fields are able to destroy SIF control over hydrogenation of the fracture process zone at long times, when concentration alteration looses its K-dominated uniqueness. Consequently, the same relates to the near threshold part of the v(K)-curve which becomes dependent not solely on SIF, but on geometry and loading of a peculiar test specimen or structure component as far as mechanical far field does.
On the other hand, the value of critical concentration Ccr(K)
at fixed SIF depends on local fickleness of alloy microstructure,
etc., as it is familiar for real materials on rather fine
scale level. That is, some natural random scatter dCcr
is peculiar for critical concentration which induces the scatter
dCt of the time to satisfy the fracture criterion like
(26) even with unique K-dominated concentration augmentation
curve, say Ca(t). Because of the small slope of
concentration alteration curves C(t) in a long times
domain, i.e., C't << 1, this slight "microstructural"
scatter dCcr produces manifold widening of the correspondent
scatter dCt of diffusion times to reach critical event
as this shows relation (35) engaged before concerning analogous
matter. This statistical effect should be rather substantial at
long diffusion times, i.e., at slow CGR values in near threshold
part of the v(K)-curve. On this background of a
wide random scatter the systematic one caused by a lack of autonomy
of the near tip hydrogenation process may be considered to be
of minor, if any, importance. Therefore, standard preconditions
of LEFM for adequate characterisation of the mechanical state
near the crack tip using SIF seem to be capable of ensuring negligible
effect of diverse far field stress distributions on the kinetics
of the near tip hydrogenation, and thus on the diffusion controlled
v(K)-curve in EAC (at least, outside some minor
near-threshold portion not including Kth itself).
5.2. The Role of History of the Coupled Diffusion-Cracking
Process
As a matter of fact, crack propagation and hydrogen diffusion are coupled in hydrogen assisted cracking in that they have inseparable influence one onto another which cannot be separated. This must be because they both ó diffusion and crack growth ó simultaneously proceed through the same solid. In terms of boundary value problems this situation is qualified as diffusion with moving boundary. Thus, in particular, the crack growth history turns to be a factor capable of affecting the near tip hydrogenation, and thus, the kinetics of hydrogen assisted cracking, as discussed elsewhere [75].
To analyse this issue, crack size is assumed to be a smooth function
of time a = a(t). To study near tip hydrogenation,
it is suitable to consider a movable rectangular coordinate system
(x,y) attached to the crack tip. This will cause
transformation of the equation of stress-strain affected hydrogen
diffusion (21) which is associated with arbitrary stationary coordinates
(x1,x2) pinned down to a solid. The latter system
may be attached so that x = x1 ñ a(t),
y=x2. Then the total time derivative of hydrogen concentration
C is:
= ó = ó v (50)
which after substitution into the left hand part of equation (21)
yields the equation of stress-strain assisted diffusion in moving
coordinates. It may be presented in nearly the same form as for
a stationary crack before:
= D (51)
where coefficients with asterisks differ from their previous analogs
M and N only in that they are determined by the
modified solubility-like term
Ks* = Ks*(s,ep,v) = Ks(s,ep)
exp (52)
Diffusion equation (51) by analogy with the previous diffusion
formalism (18)-(21) may be associated with the next fictitious
diffusion driving force:
XD* = ó RT ó= ó RT ó
(53)
Assuming K-dominance over the near tip stress-strain field
under small scale yielding, coefficients of the equation (51)
in the vicinity of the crack tip, apart from spatial coordinates,
depend parametrically on SIF and CGR:
M* = M*(K,v) ;and ;N*
= N*(K,v) (54)
Therefore, concentration of hydrogen near the moving crack tip
according to the equation (51) of stress-strain assisted diffusion
with coefficients (54) should have not only SIF, but also CGR
among its governing parameters. Thus, C = C(x,t;K,v)
beyond the crack tip. Considering hydrogen assisted cracking,
v becomes one more unknown variable in the coupled problem
of hydrogen diffusion and crack growth. To close the system of
equations of this problem a criterion of crack growth (26) serves.
It may be rewritten to emphasise participation of some variables
being now the subject of special interest:
C(xc,t; K,v) = Ccr(K,xc)
(55)
Having established the right-hand part in relation (55) this formulation of the diffusion-cracking problem is completely closed if SIF is somehow known, e.g., is maintained constant as in some of fracture mechanics test specimens, cf., e.g., the paper [76]. For this particular case, from equation (55) having in the left-hand part a solution of the diffusion problem (51) parametrically dependent on v, the CGR does not come a constant, but a time dependent function v = v (K,t). That is, even in this specific situation at fixed SIF the value of CGR must not be constant, but varying with the process time t.
In general case SIF depends on both applied load and current crack length, so that the left-hand part of equation (55) to determine CGR becomes not a function parametrically dependent on K and v, but a functional over the whole history of the process. Correspondingly, at the time moment t instantaneous CGR v(t) as a solution of this coupled diffusion-cracking problem also becomes a functional dependent on the process history from its very beginning in a particular solid with its individual variations of K(t) and v(t) during time t £ t. It looks then quite doubtful that resulting CGR could take the same values at equal SIF values if they were approached along different process histories. On the contrary, variability of CGR at the same SIF turns to be unavoidable. The question is would it be significant quantitatively or a variety of diffusion controlled crack growth kinetics curves v = v(K,t) can be enclosed within a reasonably narrow scatter band around some definite single curve v = v(K).
Parallelism between driving forces (18) and (53) for diffusion
near stationary and moving cracks, correspondingly, allows to
make some estimates for the latter problem. In particular, by
analogy with solution (45) the rough approximation for the concentration
near the tip of a growing crack may be presented in the form:
C(x,t; K,v) = C0Ks*(x;
K,v)erfc (56)
or taking into account expression (52)
C(x,t; K,v) = C0Ks(x;
K)experfc (57)
Using this solution in the left-hand part of the crack growth
criterion (55), CGR may be calculated for given SIF:
v(K,t) = ó ln (58)
This solution renders non-negative values of CGR, and thus has
physical sense, at t ³ tin, i.e., after the
incubation time have elapsed
tin = (59)
where erfcñ1 is the inverse function to erfc.
With constant crack tip environment which provides certain equilibrium
hydrogen concentration C0 at entry surface, CGR at fixed
SIF according to (58) is a rising function of time which starts
from v = 0 at t = tin and asymptotically
approaches the steady state value vss as t Æ
_. Approximate expression (58) at t Æ _ small scales
the next:
vss = ó ln = ln (60)
which turns to be the exact steady state solution of the coupled diffusion-cracking problem [74,75].
Obviously, the lower is the load intensity K, the higher is the ratio of Ccr/C_ (otherwise CGR could be a descending function of SIF). According to expression (59) the same does the incubation time tin: for SIF values K1 < K2 corresponding times tin1 > tin2. At the same time, according to expression (60) the steady state CGR is likely a rising function of SIF. This manner of variation with K of the incubation time for initiation of hydrogen assisted cracking as well as the appearance of the whole v(K,t)-dependence (58) which both yield from the performed theoretical studies agree with available experimental observations [77,78], see Fig. 11.
In spite of the roughness of the presented solutions for the coupled diffusion-cracking problem, the results agree well with mentioned experiments and clearly prove the intrinsic variability of v values at fixed K. Hence, in general CGR is not a single-value function of SIF, and v(K)-curve does not possess the intrinsic uniqueness as a characteristic curve of a material-environment system must do.
Meanwhile, a way of hydrogen assisted cracking testing does exist
when it is possible to maintain K-dominance over all the
constituents of hydrogen assisted cracking process, and thus to
provide the uniqueness of the crack growth kinetics curve as a
material's property. Namely, time dependent fractures of different
natures often reach the regime of steady state crack growth [1]
when controlling parameters of the process, like SIF and environmental
hydrogen activity at the crack tip, are kept constant for sufficiently
long time. During such steady state crack growth all near tip
processes are time independent when viewed by an observer fixed
to moving crack tip. This obviously happens in hydrogen assisted
cracking as follows from the existence of the steady state solution
(i.e., with _C/_t = 0 and _v/_t = 0) of the considered
above coupled diffusion-cracking problem at constant SIF and crack
tip environment. This CGR is a single-value function of SIF. Correspondingly,
the crack growth kinetics curve as a plot of the steady state
CGR values vs. SIF, the vss(K)-curve, possesses
the uniqueness of a material's characteristic curve.
Fig. 11. Schematics of the variability of CGR during EAC tests
[77,78] at constant K conditions with different SIF levels
K1 < K2.
Determination of these "reference" curves under steady
state regimes requires special specimens or loading devices to
provide negligible (slight enough) variability of SIF over a sufficiently
long distance (cf. [76]) to ensure approaching the steady state
growth. This is costly and time consuming procedure. Nevertheless,
despite steady state growth mode can hardly be expected in structures,
the vss(K)-curves may be useful as far as they apparently
provide conservative estimates of structure lives. This is because
steady state CGR values are the maximum ones attainable at each
SIF as it follows from expression (58), cf. Fig. 11.
6. CLOSURE
Extensive experimental and modelling efforts during the recent decades have contributed immensely the knowledge and understanding of EAC. Based on this, evaluation and prediction models have been developed for adequate materials testing and reliable estimations of structure lives. The approach of fracture mechanics provides the foundation for comparable assessments of EAC tolerance and management of related problems in engineering. Its backbone forms the concept of the crack growth kinetics curve, the v(K)-curve, which is designated an intrinsic characteristic of a combination of the material and the surrounding environment. However, ample evidences reveal that in general the v(K)-curve could hardly possess the uniqueness as the really intrinsic curve must do. Due to uncertainty of the crack growth kinetics curve errors can appear in determination of the EAC resistance of materials, and what is worst, these errors can lead to excessively optimistic material evaluation and, consequently, non-conservative estimates of structural strength.
In spite of that phenomenology and background interactions related to overall EAC process have been addressed in depth in numerous papers, no attempt has been made to comprehensive mechanistic analysis of environment-deformation interactions with the aim to elucidate the meaning and significance of the fracture mechanics approach as a predictive tool. To this end, the presented study focuses on the matter of K-dominance over the whole family of EAC process constituents as the check-point of the soundness of the fracture mechanics treatment of EAC.
Performed thorough examination shows the conceptual lack of the uniqueness of the v(K)-curve. It is rare that environment-deformation interactions in EAC can be comprehensively presented by a single-value function of SIF defined solely by the material and the environment. The vóK-relation must be dependent on a wider set of process variables, and this dependence must be not of the kind of their plain function but rather a functional so that instantaneous value of CGR for certain SIF is determined by the whole preceding history of a particular course of EAC process.
Some deal of uncertainty of EAC characterisation caused by complicated inter-relations of local (in the crack tip neighbourhood) and bulk environment parameters is eliminated in updated fracture mechanics approach where both mechanical and environmental physico-chemical factors of EAC are treated in terms of local variables related to the crack tip.
However, using the "local" v(K)-curves of improved certitude does not resolve completely the matter of characterisation of materials susceptibility to EAC. For certain key items of cracking process the K-dominance can hardly be ensured for several reasons.
First, even under small scale yielding when the inelastic near tip domain is well shielded from the surrounding body with a ring of K-controlled elastic stress-strain state, employing solely SIF the very near tip stress-strain field turns to be ill-defined at rather low load levels, or for extending crack, or under some kinds of environment attack in crack tip. The most important matters of concern here are path-dependence of crack tip plasticity so that evolution of the very near tip inelastic fields becomes dependent on both crack extension and variation of SIF. Beside, environmental effect on crack tip morphology (on blunting mode) and on near tip plasticity also affect mechanical state in the process zone. Thus, LEFM as a mechanical tool does not work fairly well when crack grows and endures environment attack. For large scale yielding the capability of J-integral to monitor the crack tip mechanics should be much worse, in particular on the propagation stage.
The other thing is that, having granted local control of the crack tip environment and SIF dominance over stress-strain state in the crack tip domain, this does not ensures exclusive K-dominance over the accumulation of harmful environmental species, e.g., hydrogen, in microstructural rupture sites. Two effects are capable of violation of the SIF driven similitude of the stress and strain assisted diffusion of the damaging agent near crack tip: the role of remote SIF not-controlled stress field (the far field) and the coupling of diffusion and crack propagation in a solid which makes them mutually dependent. The estimations performed in the paper for the case of hydrogen assisted fracture reveal that the first factor causes the variability of v(K)-curves mostly in its limited near threshold portion, whereas the other one affects crack growth kinetics in the whole SIF range from the threshold Kth to the limit crack growth resistance Kc.
In general, customary approach to evaluate crack growth kinetics curve and threshold SIF for EAC cannot pretend to produce valuable intrinsic characteristics of material-environment systems. In the paper some approaches are identified that may be followed to lessen the uncertainty of EAC evaluations and to improve conservatism of safety assessments. The reasonable way to achieve this seems to find out for a specific couple {materialóenvironment} the worst possible crack tip situation which can happen at each SIF value causing the highest CGR vm attainable in given system. Corresponding master curve v = vm(K) represent the envelope for all possible v(K)-curves that could occur in a particular {materialóenvironment} combination. This reference curve must be really intrinsic characteristic representing the weakest resistance against EAC. It worths to note that rather frequently EAC is strain-rate driven process and the worst state may be a transient instantaneous one. Very often it is not taken into account in definition of the threshold SIF in EAC which must represent the maximum degree of environmental degradation. This latter is attainable not always on the way to approaching the equilibrium at long interaction times, but may occur temporarily whilst crack tip strain dynamics promotes worsening of the local crack tip environment or environmental attack on material.
Utilisation of outlined fracture mechanics approaches of improved certitude calls for more detailed analysis and modelling of the phenomenon, more careful planning of experimentation and identifying potential extents of variability of influencing variables under nominally equivalent bulk conditions (bounding procedures).
The aim of this paper was not to exemplify comprehensively and
to explain the variety of influences and involved environment-deformation
interactions neither to give exhaustive recipes or remedies against
EAC. These would be quite specific dependent on the material,
the environment, and other peculiarities of particular cases.
The above analysis itself is believed to be instructive to guide
development of specific fitness-for-purpose testing and evaluation
programmes adjusting certain application-related objectives in
the optimum manner with respect to the safety requirements, the
extends and acceptable costs of experimentation, etc. Further
contributions to a better understanding of the meaning of conventional
approach, clarifying of its limits of validity together with discovering
the ways of improvement of evaluation and prediction procedures
should be welcome, both from the scientific point of view and
for practical and economical reasons.
Acknowledgements
This work was funded by the Spanish DGICYT (Grant UE94-001) and
Xunta de Galicia (Grants XUGA 11801A93 and XUGA 11801B95).
One of the authors (VKh) is also indebted to the Spanish Office
of NATO Scientific Affairs Division and DGICYT (Grant SAB95-0122)
for supporting his stay as a visiting scientist at the University
of La Coruña.
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