Cyclic J-integral in relation to fatigue crack initiation and propagation

0013-7944/91 $3.00 + 0.00 Pergamon Press pk.

Engineering Fracture Mechanics Vol. 39, No. I, pp. I-20, 1991 Printed in Great Britain.

CYCLIC J-INTEGRAL IN RELATION TO FATIGUE CRACK INITIATION AND PROPAGATION C. L. CHOW and T. J. LU Department of Mechanical Engineering, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1805 U.S.A. Abstract-Fatigue crack initiation (threshold) and propagation are associated with an appreciable sensitivity to one basic factor, among others (environment, microstructure, frequency, temperature etc.), namely load ratio R. The crack closure concept has frequently been hypothesized as a possible explanation for this. Nevertheless, existing experimental results with respect to crack closure are somewhat conflicting due to significant variabilities of such factors as specimen geometries and stress intensity factor ranges. This paper intends to describe the R-dependent fatigue threshold as well as fatigue crack growth from a different point of view, i.e. from the global energy equivalence concept. The generalized driving force attached to a fatigue crack tip is in the first instance re-examined with the introduction of a new cyclic J-integral. The original definition of a cyclic J-integral is shown to be inadequate as it fails to account for the total energy flux into the crack tip region from external loading mechanisms. Under small scale yielding conditions, the newly defined cyclic J-integral is equivalent to the range of elastic energy release rate, AG, for a linear elastic crack independent of loading processes. It is proposed to use AJ as an appropriate criterion for fatigue crack growth so long as the crack does not extend during the unloading portion of one load reversal. AJ as such is also interpreted as the source supplier for the energy flow rate dissipated on crack tip cyclic plastic deformations. It is then assumed that, under fixed test conditions, the minimum specific work of fracture required for fatigue crack initiation, AJ,,,, is a material property independent of the load ratio R. The so-predicted values of fatigue threshold, AK,,,, are correlated favorably with experimental measurements. For fatigue crack growth (FCG), a unified formulation is capable of being derived from the thermodynamic theory of irreversible processes, if the cracked surface area is taken as an internal variable and the rate of energy dissipated on FCG depends on AJ only. The applicability of the formulation for producing a master FCG diagram is examined for both metals and non-metals, including steel, aluminum alloys, PMMA and PVC, etc.

NOMENCLATURE

>,A’ b

B

D, fi E

E&T G, A;

Gc J A Jcyc,

AJ or AJ* K, AK A& AK,,,, k N N’ m,m+ P, PC’),Pm P msrr P.IN” r, 0 rp 3rpc R T TIZI

u,, Au, WP V X,Y z EFM

39,1-A

half crack length constants present in fatigue crack growth laws half crack length plus plastic zone size constant omitted by conventional Westergaard equations energy dissipation and its rate Young’s modulus E/( 1 - v’) for plane strain and E for plane stress angular distribution of stress and strain components energy release rate and its range critical value of energy release rate J-contour integral original cyclic J-integral newly defined cyclic J-integral stress intensity factor and its range effective range of stress intensity factor fatigue threshold under load ratio R load biaxiality number of load cycles work hardening exponent constants present in fatigue crack growth laws external applied loading maximum and minimum of load P polar coordinates monotonic and cyclic plastic zone sizes P,,/P,,,,, , load ratio temperature force acting on arbitrary crack tip enclosing contour r displacement and its range total hysteretic energy per cycle volume of material within which all dissipative processes occur Cartesian coordinates x + iy

I

C. L. CHOW and T. J. LU dimensionless constant crack tip opening displacement and its range strain and stress components changes in strain and stress stress applied at remote and its range material yield stress and yield strain complex potential arbitrary crack tip enclosing contour entropy Lame’s constants specific fracture energy per unit crack advance Poisson’s ratio strain energy density cyclic strain energy density crack growth resistance critical value of R at which the crack propagates unstably heat generated per cycle due to cyclic plastic deformations fraction of Wp dissipated on fatigue crack propagation

INTRODUCTION ALTHOUGH

FATIGUE fracture is not a rigorous science and is based essentially on semi-empirical formulations, numerous efforts devoted by designers, engineers and scientists since the 1850s have made “fatigue” of metal and non-metal materials less mysterious. It is now well recognized, following Forsyth[l], that fatigue crack growth (FCG) can be classified into three distinct stages (Fig. 1). Early studies of fatigue crack propagation were mostly concerned with the intermediate stage (Region HI in Fig. 1) of growth rates, typically 10-6-10-3 mm/cycle, where the well-known Paris-Erdogan power law[2]

$- =A (AK)m is applicable. For high cycle fatigue fracture, however, stage II may not occur until three-quarters of the total life of the component have elapsed and the FCG law of (1) can then be applied only to the final quarter of the life. Consequently, more recent attention has been focused on how to properly describe FCG behavior in the threshold region and also to identify the various governing mechanisms there[3,4]. From the form of the da/dN vs AK curve in Region I (Fig. 1), the crack growth rate decreases rapidly for small decrease of AK and the curve will approach asymptotically a AK threshold, AK,,,. AK,, is defined as the threshold value of stress intensity factor range below which crack growth stays dormant. So far, the exact value of AK,,, cannot yet be measured accurately and the values quoted for AKrh for most materials are usually those AK values corresponding to low FCG rates in the range 10-6-10-9mm/cycle. Practically, crack lengths in most cases are continuously monitored by the electrical potential-drop technique, and AK,,, is established from the highest stress intensity factor range at which no crack growth is detectable within a prescribed life range, say 10’ cycles[5]. t REGION It \ ‘CYCLIC’ PROCESS DOMINATES

REGION III ‘STATIC’ FRACTURE MODES DOMINATE (UNSTABLE)

\

R INCREASING

NEAR

“Kth(o) \ AKth(R1)

A’%(&)

THRESHOLD

Log

AK

Fig. I. Schematic illustration of the effect of load ratio, R, on sigmoidal variations of fatigue crack growth -..._

Cyclic J-integral

3

Fatigue crack initiation and propagation are associated with an appreciable sensitivity to one basic factor, among others (environment, microstructure, frequency, temperature etc.), namely load ratio R. It has been identified that, for the initiation stage or Region I of Fig. 1, the closer AK is reduced towards AKrh, the more significant is the effect of increasing R in promoting crack growth such that AKrhbecomes very sensitive to its variations. The crack closure concept due to Elber[6] has been hypothesized as a possible explanation for this as it assumes that, during the loading part of a cycle, the crack faces immediately in front of the crack tip do not open instantly owing to the residual compressive stresses in this region. The “effective AK”, AKcff, is defined as AK,, = AK - AKOP where AK, corresponds to the stress intensity factor range over which the crack remains dormant. In this sense, it might be postulated that AK,h could be directly related to AK,. Nevertheless, existing experimental results with respect to crack closure are somewhat conflicting due to significant variabilities of such factors as specimen geometries, stress intensity factor ranges, etc. Furthermore, crack closure fails to explain the environmental effects on FCG behaviors[3]. Clearly, closer examinations need to be done to ascertain whether or not it is indeed an experimentally observed intrinsic natural property, before the concept of crack closure can be applied to fatigue cracking with sufficient consistency and confidence. This paper intends to explain the R-dependent fatigue threshold as well as fatigue crack growth from a different point of view, by considering the energy balance relation in the crack tip region where profound cyclic plastic deformations occur. The generalized driving force attached to a crack tip under fatigue loading is in the first instance re-examined, with the introduction of a new cyclic J-integral. The original version of the cyclic J-integral is shown to be associated with its inherent inability in accounting for the total energy flux into the crack tip region from external loading mechanisms. Under the small scale yielding (SSY) condition, the newly defined cyclic J-integral is equivalent to the range of elastic energy release rate, AG. It is then assumed that, under fixed test conditions (namely, environment, temperature, frequency of cycling and so on remain unchanged), the minimum specific work of fracture required for fatigue crack initiation, AJlh, is a material property independent of the load ratio R. The cracked surface area of a separating body is taken as an internal state variable and assuming that the rate of energy dissipation on crack propagation depends on AJ only, then a unified FCG law is derived based on the well-established thermodynamic theory for irreversible processes. Favorable correlations between experimental measurements and theoretical predictions are obtained.

CYCLIC PLASTIC

ZONE AND CYCLIC J-INTEGRAL

(a) Limitation of AK in characterizing cyclic plastic deformations A crack tip under external loading (monotonic or cyclic) is now well known to be enclosed usually by a zone comprised of such microstructural rearrangements or energy dissipation mechanisms as microcracking and/or -voiding, crazing, slipping, homogeneous yielding, change in crystal lattice structure and multiple species of damaging. Denoting the total energy dissipated or the unavailable energy by D, then due to stress concentrations in the vicinity of the crack tip, the rate of energy dissipation, d, is positive in the zone adjacent to the crack tip. This zone is identified as the active zone (an example is shown in Fig. 2 for a propagating crack tip). In the wake of the active zone, the rate of energy dissipation is negligible due to unloading. Since the major difference between a monotonically loaded and a cyclically loaded crack tip zone is that a cyclic plastic zone well contained by the monotonic one (Fig. 2) occurs in the latter case, one is tempted to suggest that it is the microstructural occurrences in this cyclic plastic zone which establish the necessary condition of atomic bond rupture at stresses much lower than that needed to break bonds in monotonic tension, provided that the cycling numbers at these low stresses are sufficiently large. Attention must then be paid to the importance of the cyclic crack tip plastic zone in the description of crack propagation under fatigue loading. Originating from the now classical work of Rice[7], under SSY conditions, most investigators

C. L. CHOW

and T. J. LU

NEUTRAL

LINE

6WA(7

Fig. 2. Schematic

illustration

of various

damage

zones in the crack tip region during

fatigue crack growth.

in the field of fatigue study refer to the following relation for the determination tip plastic zone size r,,:

of the cyclic crack

where crOis the yield stress of the material and u is the dimensionless constant which may depend upon Poisson’s ratio, strain hardening exponent, etc., but is independent of specimen geometry as well as the applied load level. In the case of the rigid strip yield model employed by Rice, a = 7r/8. Consequently, the cyclic plastic zone size (and its subsequent evolution) is often claimed to be dependent only upon the varying amplitude of the applied load but independent of its initial value, which may serve as the theoretical basis for the applicability of the Paris-Erdogan power law of (1) in describing FCG. It should be noted, however, that the derivation of relation (2) was based essentially on the simplified Dugdale model (or the rigid strip yield model) which is most appropriate for plane stress problems and perfectly plastic materials obeying the Tresca yield criterion, although plane strain conditions and strain- or work-hardening material models more generally prevail in fatigue crack growth of common fracture specimens. Experimental verification of eq. (2) has also not been carried out exhaustively, partly due to the difficulty of identifying accurately the precise size and location of the cyclic plastic zone as opposed to the monotonic one. Furthermore, the simplified model also implies that the load applied at a direction parallel to the crack line exerts no influence on the rate of fatigue crack propagation, which is in general not in accord with experimental observations[&lO]. Miller[9] suggested that the difference in FCG rates under varying biaxial stress states could be related to the difference in the crack tip plastic zone size. He further recommended that a two-parameter description of fatigue fracture be adopted rather than a one-parameter description such as AK. On the other hand, vast quantities of experimental data collected during the last 30 years or so confirm that the Paris-Erdogan power-law of (1) cannot fully account for the mean stress or load ratio R effect, even in the mid-range of da/dN data. For example, several-orders-of-magnitude change in A and approximately one-order-of-magnitude change in m in eq. (1) have been observed in the case of 300 M steel as R is varied at 0.50,0.02 and - 1.OO[l 11. In addition, the two asymptotes existing on the sigmoidal form of the da/dN vs AK curve (Fig. 1) at crack initiation (threshold) and final failure respectively, are not able to account for the power law of (1). It appears that the development of some physically more appealing quantity (if this quantity does exist) which is superior to AK in characterizing fatigue crack propagation is desirable. In passing, we notice that AK has the dimension ML-3’2 expressed in units like ksi& or hbar times square root of centimeters, which are awkward and difficult to envisage in engineering terms[l2, 131.

5

Cyclic J-integral

(b) A modified Dugdale model for cyclic crack tip plasticity For the first instance, it will be demonstrated that definite influences of load ratio R, as well as loads applied in a direction parallel to the crack line, on the rate of fatigue crack propagation are capable of being expounded by modifying the Dugdale model or the rigid strip yield model. As this has been done in detail in a separate study[l4], only a short illustration of the proposed approach relevant to the present investigation is given below. The prototype problem considered is a horizontal crack with length 2a which is embodied in a very thin infinite plane sheet and symmetrically loaded by remote uniform in-plane stresses ka, and grn in the x- and y-directions, respectively (Fig. 3). Yielding in the sheet is confined to a narrow strip-like band, extending from the crack tip and lying along the crack line, with band height being approximately equal to the thickness of the sheet. The basic assumption made by Dugdale[lS] was that opening of prospective fracture surfaces ahead of the crack tip is opposed by a cohesive stress, its magnitude being equal to the yield stress crO.The length over which the cohesive stress acts is then determined by the postulation that the stress singularity arising from it should be such that it cancels that due to the applied loads. This last condition, together with the method of complex variable theory of elasticity due to Muskhelishivili[l6], led Dugdale to the relation

‘p-

(3)

a-

which, under the SSY assumption, is reduced to K2

7c

rp=i

(4)

0 a,

where K = a,,,&. Along the crack line, aXX(x,0) and ayy(x, 0) are two principal stresses due to the symmetry of external loading, whereas the third principal stress is zero for plane stress. For the Dugdale model where the Tresca yield criterion was employed, the yield locus for the material strip lying along the crack line is depicted in Fig. 4 by the dashed line, implying that whenever a,,(x, 0) reaches a0 the material strip becomes yielded regardless of whether aXX(x,0) = a,,(x, 0) or not. In other words, the so-called rigid strip yield model assumes that the material strip does not extend or contract in the y-direction when \a,,(x, O)l< a0 but is capable of unlimited deformation whenever la,,(x, O)l

The virgin sharp crack The imaginary sharp crack \

The physical blunted crack \

I

-

1 1 1 1 1

urn

1 1 1 1 1

Fig. 3. Dugdale strip yield model under biaxial tension.

1’

C. L. CHOW and T. J. LU

*

0

00

u~*(x,o)

Fig. 4. Yield loci for the material strip band lying along the crack line based on von Mises yield criterion and Tresca yield criterion, respectively.

reaches c,,, while it offers no resistance to extension or contraction in the x-direction. On the other hand, while the proposed modified model keeps to the elastic-perfectly plastic material model used by the original Dugdale model, it employs the von Mises yield criterion instead of the Tresca yield criterion such that the influence of biaxial stress states on the material yield condition may be appropriately taken into account. It has been shown by Chen[l8] that the von Mises yield criterion can be proved in a rational way, based on a novel theory of material strength, and no semi-empirical assumptions are needed a priori. The locus of the von Mises yield condition for the material strip in the bX,(x, O)dYY(x, 0) plane is also included in Fig. 4. To make use of the von Mises yield condition, which is written explicitly for the case considered by &(x, 0) + &(x, 0) - ~JX, O)Q&, 0) = a;, (5) the relation between gXX(x,0) and (TJx, 0) under biaxial loading is required. This may be achieved by using the following relations u xx

= Re[24’(z)] - y Im[$“(z)] - B

avv = Re[2$‘(z)] + y Im[+“(z)] + B 0 XY =

-y

(6)

Re[24”(z)]

where z = x + iy and d(z) is a complex potential. These relations were first derived by Sih[l9] in 1966 in an effort to correct the traditional Westergaard equations for which the constant B was either overlooked or omitted arbitrarily, arguing that it had no influence on the stress intensity factor. It can be readily verified that the solution to the problem shown in Fig. 3, which satisfies all the requirements, is

Consequently, it is seen from eq. (7) that the solution given by the Dugdale model or the rigid-plastic strip model applies rigorously only to the equal biaxial load case with k = 1. Under such biaxial loading conditions, u,,(x, 0) = o,,.(x, 0) at all points ahead of the crack tip so that the material is yielded whenever a,,(~, 0) reaches the uniaxial yield stress uO. When k # 1, on the other hand, the relation between ~~.~(x,0) and a,,(~, 0) is derived from eqs (6) and (7) as cr,,(x, 0) + (1 - k)a, = gyY(x,0)

(8)

Cyclic

J-integral

>

Fig. 5. Plastic superposition for unloading in the modified Dugdale strip yield model. (a) Stress distribution for two load levels, u: and ui (~6, > u:). (b) Stress distribution for compressive load -Au, with doubled yield stresses 2~: and 2~:. (c) Stress distribution after unloading from uI, to 06, - Au, and from u’, to u&-Au,.

and hence the stress level of a,,,(~, 0) at which the material within the strip band becomes yielded should generally be altered accordingly. Instead of eq. (3), the modified Dugdale model predicts that[l4] (9) which, under the SSY assumption, reduces to

In the special case of uniaxial vertical loading with k = 0, we have

(11) where the term enclosed by the last bracket gives the additional influence of applied loading, besides the usual K-dominated term, on the crack tip yield zone size, even within the usual range of small scale yielding. The procedure of the plastic superposition method developed by Rice[7] for the present modified Dugdale model is illustrated in Fig. 5 where, to obtain expressions for changes in crack tip stress and deformation fields due to load reduction, it has been assumed that the yield stress crOcan be approximately replaced by

8

C. L. CHOW and T. J. LU

The resulting cyclic plastic zone size rPc is

(12) where AK = Ao,& is the stress intensity factor range. In the presence of loads that are applied in the direction parallel to the crack line, eq. (12) is modified as (13) It is thus seen from eqs (12) and (13) that, contrary to eq. (2) which is based on the rigid-plastic strip model, the present model predicts that, besides AK, both vertical stress TV, and horizontal stress kg, will exert additional influence on the cyclic crack tip plastic zone size. Furthe~ore, negative k increases rPc while positive k reduces it, if comparisons are made with respect to the uniaxial vertical loading case with k = 0 at the same nominal applied stress level. These results are in accord with the general trends observed in experiments[&lO] if the rate of fatigue crack growth is related to the size of the cyclic plastic zone ahead of the crack tip. They are also in general agreement with the findings of ref. [20], in which a fairly similar D.B.C.S. model is used for comparison, and also to earlier results of monotonic elastic-plastic finite element analyses[21]. (c)

CycZicJ-integral

The discussion given above has demonstrated that AK alone is insufficient in the complete description of fatigue cracking behaviors. In the search for a physically more reasonable quantity other than AK, the so-called “cyclic J-integral” proposed by Dowling[22,23] was considered as a possible candidate. Unfortunately, the definition of cyclic J-integral given by Dowling[22,23] and elaborated later by others[24-261 has been elucidated recently by us[27] as questionable, due primarily to the fact that the proposed cyclic J-integral was derived as an equivalent parameter to the range of stress intensity factor, AK, under SSY conditions. Straightforward modification can be realized, however, resulting in a new and physically more appealing definition of the cyclic J-integral which, under the assumption of SSY, will be shown to be equivalent to the range of elastic energy release rate, AG. Based on the definition of the J-integral by Tanaka[26] for monotonic loading, for Cartesian coordinates with the x-axis parallel to the crack plane and any crack tip encircling contour r beginning from the bottom surface of the crack and ending at the top surface (Fig. 6), the original definition of the cyclic J-integral[24-26] is A&,, =

$,,,l(As,,) dy - ATmy

ds

1,

(14)

where $,,,,(A&,,,) represents a cyclic strain potential given by ILi,r,=~~Ao,,dn,=~[.,-o~~lde,,,

(15)

whereas A&,,, Aemn, Al;, and As,,, denote respectively relative changes between values corresponding to the load change from one state to another. It has been proved[24,26] that AJc+, like its monotonic correspondence J, is path-independent for any material obeying the following relation

In the case of small scale yielding, AJcyc,is related to AK through A&,=%,

(17)

Cyclic J-integral

9

where E* = E/( 1 - vz) for plane strain and E* = E for plane stress. For a fixed R, it has been shown that the FCG law in the form

is capable of correlating with experimental results for the mid-range of da/dN data[28,29]. It has also been observed that AJ,,,, E* and AK data nearly coincide, indicating that they are equivalent for high cycle fatigue[28]. A plausible explanation of the validity of the AJ,,,, approach (or, equivalently, the AK approach under SSY) to FCG studies may come from the observation that, conforming to the usual definition of high cycle fatigue fracture where the stress amplitude is lower than the yield strength of the material, and the strain cycles are largely confined to the elastic range such that a high number of cycles (usually exceeding lo5 cycles) is necessitated to produce fatigue failure, there exists a characteristic plastic zone ahead of the crack tip which is unique for a given crack length and loading condition[7]. This may imply that global parameters like AJ,,,, and AK are sufficient (under fixed R) for the description of crack tip cyclic deformation behaviors avoiding anything related to underlying “mechanisms” or “mechanical properties” which are usually very poorly known. The relative success of AJ,,,, in predicting a limited amount of experimental data does not, however, guarantee its validity in a wider range. AJ,,,, is basically a semi-empirical parameter with ambiguous physical reasoning. A peculiar observation is that AJcyc,occupies only a part of the total potential energy how into the crack tip region and as such it cannot be regarded as the general crack driving force attached to a crack tip. The nomenclature “AJ” for the cyclic J-integral defined through eq. (14), as used by most investigators[22-26,28,29], is misleading simply because AJ+ p= p(i)* ap

1

P t

p(j) P

+

P(i)

-

t II!Z!E

/

t

a+na

/

Fig. 6. Crack tip encircling contour r used for cyclic J-integral.

IO

C. L. CHOW and T. J. LU

is less than the range of the J-integral which is denoted here and below by AJ. For a linear elastic crack or a plane strain crack under SSY, one has

in which (AK)* = (K,,,, - Kmin)2< AK2 = (Kk,, - Kiin) and it is AJ, instead of AJ,,,, which drives the crack forward during the loading part of the cycle based on Griffith’s global energy balance considerations[30]. Instead of the cyclic J-integral of (14), a new integral is defined through AJ* = AJcycl+ A,,

(19)

where

(20) which is valid for load-controlled cyclic loading, with pz and 0% corresponding value of the load cycle (Fig. 6). Combining (14), (19) and (20) together yields AJ* =

T, 2 (du,) ds],

to the minimum

(21)

and the new strain potential is given by

(22) The major difference between the present definition of (21) and the original definition of (14) for the cyclic J-integral lies in the fact that, instead of Aa, and AT, in AJ,,, of (14), am and T, for the values relevant to the applied external load P (Fig. 6) are taken as the thermodynamic work conjugate forces, corresponding respectively to the strain increment de,, and the displacement increment du, in AJ* of (21). Following the procedures introduced by Rice[31] to prove the path-independence of his J-integral, and assuming that the strain energy is a single-valued function of the strain generated during the change of loading, and further the stress and defo~ation fields outside the “end-region” ahead of the crack tip are specified for the original loading state PCn[32],then AJ* of (21) can be proven with straightforward algebra to be path-independent. Limited to the range of the non-linear elastic or deformation plastic material model, it can be justified that the stress and strain components at every material point are approximately proportional. Another restraint on the validity of AJ* is similar to that given by Broberg[32] in the sense that, everywhere in the region outside the “end-region”, the magnitude of effective stress will not decrease for the loading portion of a cycle (i.e. from state i to j) and it will not increase for the unloading stage (i.e. from state j to i). Upon substitution of elastic crack tip fields into (21) and under the SSY assumption, it can be shown that AJ*=A

$ (

=AJ=AG, >

(23)

and hence AJ* is indeed the range of the J-integral during fatigue loading. For displacementcontrolled fatiguing, procedures for evaluation of AJ* or AJ are essentially the same as described above for load-controlled cycling. Equation (23) illustrates that, if SSY prevails ahead of a crack tip, AJ does not change its value in the whole loading cycle (provided that the crack is nonextending during the process), i.e. AJlij = AJi,j = AG. This may imply that a macroscopic parameter AG, instead of AK, can be used to characterize high cycle fatigue cracking, even though the various microprocesses occurring in the immediate vicinity of the crack tip are still poorly defined and specified.

11

Cyclic J-integral

(d) Cyclic J-integral in relation to the Dugdale model At first sight, it appears that the monotonic J=

J-integral

of Rice,

$dy-T,,,%ds

S(I-

, >

for which II/ is the strain energy density, and the cyclic J-integral of (21), cannot be applied to a Dugdale-type crack where extensive plastic deformation occurs. Surprisingly, this is untrue, and following Rice[33], it can be proven that the Dugdale model provides a very special case for which strict path independence of J as well as AJ does actually result, even though the non-linear elastic assumption is invalidated. Indeed, for an integration path f’, which extends from the bottom to the top of the crack surface, along the lower (-) and upper (+) sides of the x-axis and encloses the crack tip at x = a, the J-integral and the cyclic J-integral become J=

s s

c,o+ au a,

c,O-

dy ay

c,o+

AJ* =

c,o-

s

a
’

a O"

ho 3(Au,) dy -

ay

ax

dx

’

’ Wu; - uy I dx (I ax s ‘0

(25)

in which c represents the point at which r traverses the strip yield zone. From eq. (7), the y-direction displacements u,(x) of the upper (+) and lower (-) crack and yielded strip can be expressed as

(26)

for a <[xI
the crack tip opening displacement 6,(a) can be obtained from (26) and (27) to be a ln{sec[~(/~+~5?)]~

S,(a)=u~(a,O)-u,b:(a,O)=~~

/_+!+L%

.

On the other hand, the residual crack tip opening displacement obtained as [in(l 6:(a) = 6,(a) -As,(a)

.,),*n(l

6:(a)

(28)

after unloading can be

+g)]

= iy

/W+L$$

(29)

where As,(a) is the cyclic crack tip opening displacement and rPcis the cyclic plastic zone size given by eq. (13) in the case of SSY.

12

C. L. CHOW

and T. J. LU

L

AE

2

0

Fig. 7. Power law work-hardening

cyclic stress-strain

curve.

From (27), (28) and (29), J and AJ of (25) can then be rewritten as J = cr$6(c) + a,*[6(a) - 6(c)] = a$6(a) = 0$6,, i AJ* = rr,*AS(c) + o$[A6(a) - AS(c)] = a,*A6(a) = o,*A6,.

(30)

Therefore, the magnitudes for J and AJ are independent of the point c at which r traverses the strip yield zone. On the other hand, some workers[26,34] erroneously demonstrated that J and AJ,,,, of (14) are path-dependent, due to their failure in recognizing the fact that the terms Jr II/ dy of eq. (24) and jr ticyc,dy of eq. (14) do not vanish identically “when evaluated on any path r for which the total excursion in y approaches zero”[33], because II/ and ticyc, are densities of stress-working and are Dirac-singular at points along the Dugdale strip yield zone. Due to the same reason, many formulations and subsequent conclusions of Ref. [26] for the original version of the cyclic J-integral, i.e. AJ,,,, of (14), are misleading. ANALYSIS OF R-INFLUENCED FATIGUE THRESHOLD AND FATIGUE CRACK GROWTH (a) An assumption on the fatigue threshold Fatigue crack extension is basically a quasi-statically fracturing process with non-continuous, incremental, finite-sized advancements which is remarkably different from that under monotonic loading, regardless of whether the original material is brittle or ductile in nature (Fig. 8). The specific work of fracture per unit crack surface area, usually called the crack growth resistance R, depends on the crack propagation rate as well as the load ratio, namely R = R(da/dN, R, other parameters). Before any fracture event occurs during cyclic loading, we can only define an energy

CYCLES Fig. 8. Quasi-static

OF STRESS

nature

of fatigue

N crack

propagation

-Cyclic J-integral

13

Fig. 9. Schematic illustration of the R-curve influenced by load ratio R during cyclic loading.

release rate term (not necessarily the release rate of elastic strain energy), AJ, being also the generalized force just attached to the crack tip. At a local critical condition, AJ is equal to the current growth resistance R (with negligible kinetic energy and temperature variation) and the crack advances an increment, Au, and then arrests to wait for the establishment of a new energy balance condition corresponding to the new crack growth resistance IW(a+ Aa). For constant amplitude cyclic loading, [wis a monotonically increasing function of crack length u with the smallest and largest values of R corresponding to the threshold and final failure respectively (Fig. 9). Referring to the assumption made at the end of the Introduction for fatigue threshold, it follows that R,, = AJlh is a material property not affected by load ratio R. Since we are concerned mainly about fatigue threshold behavior associated with low stress levels under SSY, it is justified to assume that AJth = AG,,,. Consequently, we have AGfh(Rz)=

LS'G,I(R,) = AG,,,,,

&

' 4

Under SSY, R = P,i”lP,,, = KminlK,,,,, SO that AG,,,,, = (1 + R)K,,,,,AK,, relation must hold (Fig. 1): A&

-C A&R,)

< AK,,,,,,

&

' 4

(31)

' 0.

and hence the following

' 0.

(32)

A plausible interpretation of AG,*(,, is that it is the smallest amount of energy required to produce crack advance equal to a lattice spacing (in the AK characterization of FCG, some workers[35] suggest that AKrh is the smallest value of AK required to produce crack advance equal to a lattice spacing, which is not very convincing from the present energy balance considerations). Since comparisons amongst AK,,,,,,, AKrhcR,)and AKrhcRt)etc. are carried out on an equal da/dN basis (Fig. l), one concludes that the number of cycles required for the crack to advance one lattice spacing, d, is independent of load ratio R. Consider, for instance, the energy changes during crack advancement from initial crack length a0 to a, + d under zero load ratio and positive load ratio R, P

t

a0 I

a,,+d

hink)

Prninlo)

0

Fig. 10. Quasi-static energy analysis of deeply-cracked CT specimen in the threshold region of fatigue crack growth.

14

C. L. CHOW and T. J. LU

respectively, we obtain

as shown in Fig. 10. For the special geometry

of compact

tension specimen,

1 AreaOAB d = g

Q/z,,

(33)

1 Area CDEF

A&,,,, = B

i

d

3

which results in the following relation because of the assumption of AGfhh(0) = AG,,,(Rj, Area OAB = Area CDEF. Together with Fig. 10, straightforward

(34)

algebra shows that (34) can be reduced to

PLx(R) -

PL(R)

=

(35)

Pfnax(*) 5

or alternatively (36) On the other hand, since K$$, = AKIA(Rj/(I- R) and AK$& = RAK,,(,,/(l -I?), reduced to the next final form

(36) is further

The threshold-load ratio relationship as expressed by (37) is then used to fit the available experimental results for several metallic materials. Figure I;1 shows a comprison of the predictions of eq. (37) with test data of Ti-6AL4V obtained by McEvily and Groeger[36] in both air (50% relative humidity) and vacuum (lo-’ Pa) conditions, from which fairly good agreement between experiment and theory can be seen. For two steels on which the most extensive threshold measurements were made[37-391, the A& vs R dependence at room temperature can also be fitted closely by eq. (37), as shown in Fig. 12. The steels investigated are of either mixed ferrite-pearlite or futly pearlite microstructure or microstructure consisting predominantly of tempted martensite. values for two aluminum alloys (2024-T3 and 7075”T6) are tabulated Finally, the calculated A~~*~~~ in Table 1 together with the values measured by Mackay[O], and reasonable agreement is obtained. A&R, data for both materials were determined in laboratory air at a relative humidity of 45%, and at load ratios of R = 0.05, 0.2, 0.4, and 0.6. These results indicate that fatigue threshold behavior depends on load ratio in a predictable manner even though further study is necessitated to substantiate the assumption made in deriving eq. (37). To close this section, we notice that, based on the assumption that the threshold level corresponds to a constant crack opening displacement range which is independent of load ratio R, McEvily and Groeger[36] also deduced the relation of (37) but without detailed physical reasoning or mathematical analysis.

56

A

VACUUM

,

AIR A’%h(o)

:

I

0

I

I

0.2

0.1 LOAD

t

0.6 RATIO R

I

0.6

I*

1.0

Fig. I 1. Dependence of fatigue threshold AK,,ot on load ratio R for Ti-6AI-N alloy in air (50% relative humidity) and in vacuum (lOwsPa) (data reproduced from ref. [36] with those predicted).

Cyclic J-integral

e

En3A[38]

e

En24[39]

ferrite-pearlits

I 0.4

1 0.2

0

15

marhsitlc

steel

I 0.6

I 0.0

steela

I 1 .o

LOAD RATIO R

Fig. 12. Dependence of fatigue threshold AK,,,, on load ratio R for three steels: comparison between test data and theoretical predictions. Table 1. Comparison of measured and predicted fatigue threshold values between two aluminum sheet alloys in laboratory air AK,,, ,

MPafi

AK,,,, MPafi (7075-Tb, clad)

(2024-T3) Load ratio R

Measured [40]

Predicted

Measured [40]

0.0 0.05 0.2 0.4 0.6

3.52 2.75 2.31 1.87

3.70 3.52 3.01 2.42 1.85

2.75 2.42 1.93 1.50

Predicted 2.90 2.75 2.37

1.90 1.45

(b) Fatigue crack propagation characterized by AJ For a fatigue crack to start propagation, two criteria must be satisfied. The first requires that there exists a certain physical process which establishes the necessary condition of producing atomic bond breakage in the immediate vicinity of the crack tip, even though the applied K,, level is much lower than the critical value KCrequired for crack initiation under monotonic loading. This physical process, as we have discussed before, occurs within the cyclic crack tip plastic zone, being unique for a crack under fatigue stressing. The energy balance conditions at both global and local levels provides the other criterion for fatigue crack growth which, in the situation of monotonic stressing, resembles that of the instability criterion of “maximum free energy change” for a Griffith crack. When approached in a similar way as that in refs [41-431, consideration of the global nature of the Griffith fracture theory will result in the next expression of energy balance for a fatigue crack (38) for which it has been assumed that the crack propagates at the maximum level of P or PG) (see Fig. 6) and the effects of body forces and kinetic energy are negligible. Equation (38) can readily be reduced as da WP-R (39

dN = A - Jmax

where n represents specific fracture energy per unit crack advance, J/VP’

o,,de,,dV

=

II/(As,,)dV sV

C. L. CHOW and T. J. LU

16

denotes the total hysteretic energy per cycle in the material volume V within which all dissipative processes occur, and R signifies heat generated per cycle due to cyclic crack tip plastic deformations. The FCG law as expressed by (39) can also be derived from a quite different approach. This is accomplished by considering fracture as an irreversible process which, according to the theory of continuum thermodynamics, must be associated with entropy production, or expressed in an alternative way, the entropy content of a separating body is increased as a result of irreversible crack propagation. The cracked surface area, taken as an internal state variable in the thermodynamic theory for irreversible processes, should then be associated with a generalized thermodynamic work conjugate force. The work conjugate force for FCG rate, da/dN, has been identified as JmaX- n [44], and hence, for the problem considered (Fig. 2), the rate of entropy production due to fatigue crack propagation is

(40) in which T denotes temperature, d~ldN represents entropy production per cycle owing to irreversible crack propagation and other dissipative mechanisms and dD/dN = Wf’ is energy dissipated per cycle. A part of WP is converted into heat R not available for crack propagation and the remaining part, WY - 0, is the essential energy dissipated on fatigue crack extension per cycle. Consequently, eq. (40) can be rewritten as

-A)$+

wp-R

(41)

where (dr]/dN), represents the entropy production per cycle due to fatigue crack propagation. As a crack generally propagates along a direction in which the energy is dumped at the maximum rate or the energy of the body is decreased most rapidly by the real crack path which also coincides with maximum entropy production, for slow crack extension, the r.h.s. of eq. (41) can be set equal to zero resulting in exactly the FCG law of (39). Assuming that the fraction of Wp converted into heat is (1 - r), eq. (39) then reduces to da
(42)

A, being a characteristic property of the material, represents resistance to crack extension and may change with crack velocity, temperature, environment, etc. For high cycle fatigue fracture in the usual sense, A = G, and J,,,,, = G,,,. For J-dominated crack extension where the SSY assumption is invalidated, _4 may be identified as J, so that eq. (42) can be rewritten as da

CWP

(43)

dN=:J,-.

The hysteretic plastic energy Wp depends generally on operational conditions including J,,,,, , R, T, f (frequency), etc. and an exact functional expression of Wp is still to be developed due to the lack of a rigorous analytic solution of the stress and strain distributions within the monotonic plastic zone r,,(e) as well as the cyclic plastic zone r,,(8) under mode I loading. Rice[45] in 1967 gave an analytic solution for the anti-plane shear (mode III) loading under the SSY assumption and McClintock[46] noticed that, for the case of SSY, there is an analogy between mode I and mode III. Assuming that the material under consideration obeys the following Ramberg-Osgood stress-strain relation

0=

‘I0 go-

&

so

N 5

(44) for g 3 a,;

17

Cyclic J-integral

[ 1

where c,, = Es,, and N’ is the work hardening exponent. Using the analogy relation between mode I and mode III, the stress and strain components within r,(e) can be given as[47] a mn =

ao

+yfmn(& N') 1N//Cl +wgm(& N') 1l/Cl

J (1 + N’)7caoeor

for SSY, (45) J EInn= Co[ (1 + N’)7caoeor wheref,,(& N’) and g,,,,,(e, N’) are angular distribution of stress and strain components dependent upon whether plane strain or plane stress is assumed. It is noted that eqs (45) are very similar to the well-known HRR fields (Hutchinson-Rice-Rosengren) [48,49] for non-linear elastic materials conforming to deformation plasticity theory. For the situation of cyclic loading where a cyclic plastic zone r,,(0) well contained in the monotonic plastic zone r,(e) occurs, we can choose a circular contour r of radius r for the evaluation of the cyclic J-integral such that eq. (21) becomes AJ -_= r

n

(46) ti [A&,,(r, e)]cos 8 si-n Since all terms within the integrand are expressed in terms of stress times strain increment, one tends to conclude that there is a l/r singularity in the product of stress and strain increment during the change of external loading, namely r-0 a function of 8 and AJ

ti (b,m ) =

r

(47)

In view of the above relation and as a first approximation, it may be assumed that Wp is a power function of AJ only, which leads to the next expression for a specific FCG formulation da

dN =

A*

(AJY” z=LJ,,,’

We notice that the use of AJ (or AG under SSY) other than AK as the major parameter characterizing dN has the advantage of automatically accounting for the effect of Young’s modulus E, and avoiding the quite arbitrary practice of dividing AK by E to get a parameter AK/E with ambiguous physical significance. In addition, applications of any FCG formulation assuming a functional relationship between da/dN and AK are limited to certain restricted cases. Specimen geometries, crack configurations and loading conditions of relatively complicated nature, for example, are untenable due to mathematical difficulties involved in the analytical solutions of the stress intensity factors. Determination of the range of energy release rate AJ nevertheless requires no stress field solution at the crack tip and hence makes it possible to use a test specimen of any shape with arbitrary configurations and loading conditions. Experimentally, the hysteretic energy per cycle as obtained from the load-displacement-crack area diagram (P-u-A diagram), which can be readily provided by the autographic plotting of load-displacement from the test machine, offers a convenient and yet accurate measure for the magnitudes of AJ[27]. The applicability of eq. (48) for characterizing FCG has been examined by Chow and co-workers for aluminum alloys (2024-T3 and 7075T6)[50], mild steel[51], PMMA[SO], PVC[52] and a short fiber composite (SFC)[53]. Here we want to state that the captioned FCG law is capable of placing all of the above-mentioned fatigue data from both metals and non-metals in a narrow scatter band, as shown in Fig. 13, in which FCG data of two structural adhesives (adhesive no. 16 and adhesive no. 21) from ref. [54] are also included. It can be concluded that the unified FCG formulation (47) can characterize both metals and non-metals with a master FCG diagram, despite the fact that the mechanical properties of the materials may differ by one order of magnitude (Table 2). No other FCG law proposed hitherto as reviewed by us in ref. [55] possesses such privileges. The merit of the unified approach also lies in the fact that AJ and J,,,,, are macroscopic parameters which do not necessitate any detailed quantitative microscopic models to be derived for the individual fracturing mechanisms like cleavage and shear decohesion. They are applicable as long as they can characterize the stresses and strains at the potential site of fracture.

18

C. L. CHOW and T. J. LU Eoo 1.00

(AJ)~~

da SC, dN

Jc-Jmax

0.00

E Z

r

-1.w

Y 0

2_

-2.00

z

::

E 7 5 :%I

-3.00

z s -4.w

-5.00

0.1 s -0.6 -0.4 -0.2

0.0

0.2

LOG(A J),

0.4

0.6

0.8

1.0

1.2 EOl

N/mm

Fig. 13. Fatigue crack propagation of aluminum alloys (2024-T3 and 7075T6), mild steel, PMMA, PVC, SFC (short fiber composite) and adhesives using an energy formulation.

Table 2. Mechanical properties of several metallic and polymeric materials Yield stress (MN/m*)

Ultimate tensile strength (MN/m*)

Fracture toughness

Material

Young’s modulus @N/m’)

2024-T3 7075T6 Mild steel PMMA PVC SFCt Adhesive no. 16 Adhesive no. 21

67.85 67.85 217 2.36 3.35 7.93 6.6 2.2

290 469 300 35.76 41.40 39.41 -

434 538 470 48.94 55.18 135.86 49.6 29.4

50.22 25.08 236 0.44 26.87 15.25 1.21 13.50

&J/m*)

$Short glass fiber reinforced injection-moulded nylon 6.6.

CONCLUSIONS 1. It is elucidated, based on a modified Dugdale model, that cyclic plastic deformations ahead of a crack tip under fatigue stressing depend not only on the magnitude of load fluctuations but also on the mean stress level at which the load is being cycled, implying that a single parameter like AK alone is insufficient if complete description of fatigue crack propagation is to be expected. 2. The original definition of cyclic J-integral being found inaccurate, a new cyclic J-integral is introduced which is identified as the crack driving force attached to a fatigue crack tip and also as the source supplier for the essential energy dissipated on fatigue crack propagation; under SSY conditions, AJ reduces to the range of elastic energy release rate AG.

Cyclic J-integral

19

3. The R-dependent fatigue threshold behavior is analysed with the help of the energetic property of the cyclic J-integral; predicted values of fatigue threshold A& as a function of load ratio R are correlated closely with available experimental results for several metallic materials. 4. AJ and .I,,,, are considered as better fracture-controlling macroscopic parameters than AK of a unified FCG law, the validity of which has been checked on both and L,, in the fo~ulation metals and non-metals including mild steel, aluminum alloys, PMMA and PVC.

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[21] A. P. Kroufri and K. J. Miller, Crack separation energy rates for crack in a biaxial stress field on an elastic--plastic material. Multiaxial Farigue, ASTM STP 853, 85-107 (1985). [22] N. E. Dowling and J. A. Begley, Fatigure crack growth during gross plasticity and the J-integral. Mechanics of Cruck Growth, ASTM STP 590, 82-105 (1976). [23] N. E. Dowhng, Geometry effects and the J-integral Fracfure, ASTM STP 601, 19-32 (1976).

approach to elastic-plastic fatigue crack growth. Cracks and

[24] H. S. Lamba, The J-integral applied to cyclic loading. Engng Fracwe Mech. 3, 693-703 (1975). [25] C. Wiithrich, The extension of the J-integral concept to fatigue cracks. Int. J. Fracture 20, R35-R37 (1982). [26] K. Tanaka, The cyclic J-integral as a criterion for fatigue crack growth. Inf. J. Fracture 22, 91-104 (1983). [27] C. L. Chow and T. J. Lu, On the cyclic J-integral applied to fatigue cracking. Inf. J. Fracture 40, R53-R57 (1989). 1281 S. Sadanada and P. Shahinian, Elastic-plastic fracture mechanics for high temperature fatigue crack growth. Fracture ~echanj~s: Twe&h Conjerence, ASTM STP 700, 152-163 (1980). [29] S. Sadanada and P. Shahinian, A fracture mechanics approach to high temperature fatigue crack growth in Udiment 700. Engng Fracfure Mech. 11, 73-86 (1979).

[30] A. A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. R. Sot. A211, 163-197 (1921). [31] J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. .J. uppl. Mech. 35, 379-386 (1968). 1321 K. G. Broberg, Crack-growth criteria and non-linear fracture mechanics. J. Mech. Phys. Solids 19, 407-418 (1971). 1331 J. R. Rice, Discussion: “The path dependence of the J-contour integral”, by G. G. Chell and P. T. Heald. Inr. L Fracture 11, R352-R353 (1975). [34] G. G. Chell and P. T. Heald, The path dependence of the J-contour integral. In!. J. Fracture 11, R349-R351 (197.5). [35] A. G. Atkins and Y. W. Mai, Elastic and PIustic Fracture, p, 557. Ellis Horwood, Chichester (1985). [36] A. J. McEviiy and J. Groeger, On the threshold of fatigue-crack growth, in ICF4. Univ. Waterloo Press, Waterloo, Canada (1977). [37] R. J. Cooke and C. J. Beevers, Slow fatigue crack propagation in pearlitic steels. Mater. Sci. Engng 13,201-210 (1974). [38] K. Jerram and E. K. Priddle, System for determining the critical range of stress-intensity factor necessary for fatigue crack propagation. J. me& Engng Sci. 15, 271-273 (1973). [39] R. J. Cooke, P. E. Irving, G. S. Booth and C. J. Beevers, The slow fatigue crack growth and threshold behavior of a medium carbon alloy steel in air and vacuum. Engng Fracture Mech. 7, 69-77 (1975). [40] T. L. Mackay, Fatigue crack propagation rate at low AK of two aluminum sheet alloys, 2024-T3 and 7075-T6. Engng Fracture Mech. 11, 753-761 (1979). [41] T. S. Short and D. W. Hoeppner, A global/local theory of fatigue crack propagation. Engng Fracture Mech. 33, 175-184 (1989).

20

C. L. CHOW and T. J. LU

~421K. N. Raju, An energy balance criterion for crack growth under fatigue loading from consideration of energy of plastic deformation. ht. J. Fracture 8. I-14 (19721. [43] Y. Izumi, M. E. Fine and T. Mura, l&erg; considerations in fatigue crack propagation. Znt. J. Fracrure 17, 15-25 (1981). (441 A. Chudnovsky and A. Moet, Thermodynamics of translational crack layer propagation. J. Marer. Sci. 20,630-635 (1985). [45] f. R. Rice, Stress due to a sharp notch in a work-hardening elastic-plastic material loaded by lon~tudinal shear. J. appf. Mech. 34, 387-398 (1967). [46] F. A. McClintock, Discussion of fracture testing of high strength sheet materials. Mater. Res. Srandards I, 367-378 (1969). [47] D. Kujawski and F. Ellyin, A fatigue crack growth model with load ratio effects. Engng Fracture Me&

28, 367-378 (1987). 1481J. W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13-31 (1968). [49] J. R. Rice and G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material. J. Meek. Phys. Solids 16, l-12 (1968). [SO] C. L. Chow and C. W. Woo, A unified formulation of fatigue crack propagation in aluminum alloys and PMMA. Engng Fracture Mech. 21, 589-608

(1985).

[Sl] C. L. Chow, C. W. Woo and K. T. Chung, Fatigue crack propagation in mild steel. Engng Fracture Mech. 24,233-241 (1986). [52] C. L. Chow and K. H. Wong, A comparative study of fatigue crack propagation models for PMMA and PVC. Thear. appl. Fracture Mech. 8, 101-106 (1987). [53] C. L. Chow and T. J. Lu, Characterization of fatigue crack propagation in short fiber reinforced thermoplastics-a unified approach. J. Reinforced Plasr. Compos. 10, 58-83 (1991). (541 J. Luckyram and A. E. Vardy, Fatigue performance of two structural adhesives. J. Adhesion 26, 273-291 (1988). [SS] C. L. Chow and T. J. Lu, A unified approach to fatigue crack propagation in metals and polymers. J. Mater. Sci. Lett. 9, 1427-1430 (1990). (Received 28 December

1989)