Assessing Plastically Dissipated Energy as a Condition for Fatigue Crack Growth

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1 Cleveland State University Mechanical Engineering Faculty Publications Mechanical Engineering Deartment 7-9 Assessing Plastically Dissiated Energy as a Condition for Fatigue Crack Growth D. Cojocaru University of Delaware Anette M. Karlsson Cleveland State University, a.karlsson@csuohio.edu Follow this and additional works at: htts://engagedscholarshi.csuohio.edu/enme_facub Part of the Mechanical Engineering Commons How does access to this work benefit you? Let us know! Publisher's Statement NOTICE: this is the author s version of a work that was acceted for ublication in International Journal of Fatigue. Changes resulting from the ublishing rocess, such as eer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for ublication. A definitive version was subsequently ublished in International Journal of Fatigue,, 7, (7--9);.6/j.ijfatigue.8.9 Original Citation Cojocaru, D., and Karlsson, A. M., 9, "Assessing Plastically Dissiated Energy as a Condition for Fatigue Crack Growth," International Journal of Fatigue, (7). -6. This Article is brought to you for free and oen access by the Mechanical Engineering Deartment at EngagedScholarshi@CSU. It has been acceted for inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of EngagedScholarshi@CSU. For more information, lease contact library.es@csuohio.edu.

2 Assessing lastically dissiated energy as a condition for fatigue ack growth D. Cojocaru, A.M. Karlsson * Deanmenr of Mechanicol fnginuring. University of Delawo,.,., 6 Sencer taborn/ary. Nrwork. DE 976. United SIOles. Introduction Fatigue of materials has imosed limitations on engineering structures since the beginning of the industrial revolution ilnd even though tremendous research efforts have been aimed towards understanding and redicting fatigue. a reliable method to redict fatigue ack growth still does not exist [ - 6]. Currently. there are several avenues roosed for modeling fatigue ack growth in order to obtain life rediction models, including using damage mechanics [7.8[, stress intensity factors [9- J.and energy iteria [- 8] as rediction tools for ack roagation. Some of these aroaches are enhanced with incororating various forms of "cycle-jum" techniques, to accelerate the numerical simulations [9!. In this work, we investigate the concet of using lastically dissiated energy as a iterion for ack roagation due to cyclic loading. The lastically dissiated energy can be directly linked to the accumulation of lastic strain, which in turn can for examle be linked to accumulation of dislocation in metals. Rice [[ suggested fatigue ack growth using lastic dissiation as a iterion in 967. Turner roosed using the rate of energy dissiation for defining ductile tearing resistance [.6[. Using the finite element (FE) method, Klingbeil [7[ roosed a technique (based on the work of Bodner et al. [[ ) for redicting fatigue ack growth in terms of the er-cycle rate of lastic energy dissiated in the reversed lastic region (defined in Fig. ). The aroach is based on evaluating the conditions around the ack ti for a non-roagat Corresonding author. Tel.: +l 8l 67: f~)(: +) 8 l 6l9. E mail addrm; karlsson@udel.edu (A.M. Karlsson ). ing ack. Klingbeil! 7[ initially used the technique for stationary acks under mode I loading and the technique was later alied by Daily and Klingbeil [,[ for stationary acks under mixed mode loading conditions. Cojocaru and Karlsson jl8[ resented a numerical scheme (a modeling frame) where cyclic ack growth can be simulated using finite element analysis (FEA ). The modeling frame allows the ack to roagate once a user-defined roagation iterion is satisfied. The total (i.e. accumulated ) lastically dissiated energy in a redefined region in front of the ack-ti was suggested as one of many ossible roagation iteria for FE simulation of fatigue ack growth. The roosed concel is based on that. from a continuum ersective. the fatigue ack advances by cyclic material degradation in the rocess zone (Fig. ) associated with the ackti (see. for examle, Ref. ). If the material is ductile, then the degradation of the material in the rocess region is accoman ied by significant lastic deformation (e.g.. ductile metals and olymers), For metals. lastic deformation is associated with dislocation motion, which is associated with fatigue [. Therefore, the lastically dissiated energy in the ack-ti vicinity may be a suitable measure for evaluating ack roagation. In this comutational scheme. the er-cycle ack roagation rate is not resibed, but automatically obtained by robing the lastically dissiated energy in a user-defined domain ahead the current osition of the ack-ti I8[. The FE simulations are conducted cycleby-cycle. i.e. the entire cyclic loading ath is simulated. Even though cycle-by-cycle simulations may be comutationally rohibitive for a comlete simulation of high-cycle fatigue, exlicitly studying the evolution of the ack due to cyclic loading is imortant in many cases, including obtaining ack roagation rates for

3 Plastic wake behind the ack-ti Reversed lastic region Forward lastic region Process Region (cyclic failure) Fig.. Plastic regions develoing during cyclic loading around a ack-ti and ensuing ack roagation (assuming maoscoic length scales). The forward lastic region forms during loading and the reverse lastic region develos when the load is reversed. The lastic wake develos when the ack roagates through the lastic region. high-cycle fatigue and for cases where the growth of the ack influences the overall resonse (e.g., debonding of coatings). The roosed aroach was shown to be able to cature both ack acceleration and ack retardation [8]. We will here qualitatively investigate if well established exerimentally observed trends under selected forms of fatigue loading including various load ratios and overloads can be catured by a roagation iterion founded on the (accumulated) lastically dissiated energy. In this work, we focus on assessing this qualitatively via the numerical simulations. A quantitative evaluation can only be done by comaring with carefully develoed exerimental investigations. This will be done in a later study. The arameters introduced in Section of this study (the integrated domain and itically dissiated energy ) are material arameters and thus will deend on the material investigated. The numerical simulations are based on the comutational scheme develoed reviously [8]. The scheme uses a modeling frame for conducting FE simulations, where the ack roagation is simulated via a node-release technique. The main characteristics of this modeling frame are outlined in the next section.. A numerical framework for simulating cyclic ack growth.. General concets The modeling frame used in this work was resented in Ref. [8] and the main concets will be summarized here for clarity. The unique feature of this aroach is that the ack extension er load cycle (da/dn) is not resibed but is the outut from the simulations. The ack roagation is achieved by releasing (reviously constrained) nodes along a redefined roagation ath. Node-release techniques have frequently been used for FE simulations of fatigue ack growth, e.g. Ref. [6 9]. In addition to its concetual simlicity for simulating ack roagation along redicted aths under cyclic loading, the node-release technique has the major advantage of avoiding time consuming re-meshing rocedures. This is itical for simulating a large number of load cycles. Furthermore, the node-release aroach allows using contact formulations for the ack surfaces. The develoed modeling frame uses ABAQUS Siting Interface [], where a FE model is generated automatically. The fundamental modeling idea is that any two-dimensional (D) structure can be decomosed in two sets of (data) entities: (i) a set of continua and (ii) a set of continuum interfaces. A continuum reresents a sub-domain having its own constitutive resonse, tyically coded in a searate constitutive subroutine. There are two tyes of continuum interfaces desibing (i) the interactions between two adjacent continua, or (ii) between a continuum and the exterior. Central to simulating cyclic ack growth is the reresentation of the interface between two continua. This tye of interface is reresented as an arbitrary sequence of (i) failed (i.e. searated) segments (modeling acks) and (ii) intact segments (modeling otential ack extension aths). The end vertices of the intact segments are otential ack-tis and can be assigned roagation iteria based on any quantity available in the FE analysis. An iterative evaluation rocedure is called at the end of user resibed intervals (measured in number of load cycles) to assess the roagation iteria and to advance the ack-tis when a roagation iterion is fulfilled. This modeling frame is a convenient research tool for modeling cyclic ack growth in a variety of cases. Alications of cyclic ack roagation in fracture mechanics secimens, multi-layered systems and structures with geometrically modeled mio-features are illustrated in Ref. []... Crack roagation The ack roagation iterion investigated in this work is based on using the total (i.e. accumulated) lastically dissiated energy in a redefined region in the vicinity of the ack-ti. Evaluating the lastically dissiated energy only in the vicinity of the ack-ti (as oosed to, for examle, evaluating the lastic work for the entire structure) is motivated by the concet of the rocess region (Fig. ). The roosed aroach allows the ack roagation to be assessed indeendently (for each ack-ti) in the case of multile acks. This was illustrated in Ref. [8] for a bi-layer structure containing two acks. The accumulated lastically dissiated energy is determined by evaluating this quantity in a disete domain (the integrated domain, D) in front of the ack-ti. Here, the disete domain is defined as a set of elements, E D, aroximating a theoretical semi-disk domain of radius r, centered at the current ack-ti, Fig.. The lastically dissiated energy in the integrated domain P is defined by W ðdþ ¼ ee D w e, where w e reresents the lastically dissiated energy within one element, e, (comuted by Gaussian Crack-ti Theoretical semi-disk r =.. domain - Element.. size h = - e Predicted Crack Path Fig.. The integrated (semi-disk) domain, D, used by the roagation iterion, and the reversed lastic region after the first cycle (gray elements). Darker shades of gray indicate higher lastically dissiated energy in the reversed lastic region.

4 quadrature using the density of lastically dissiated energy). The density of the lastically dissiated energy is calculated by the constitutive subroutine at each integration oint. The disete roagation rate er cycle after the Nth load cycle, Daj N, is established by an iterative comutation of the lastically dissiated energy over the integrated domain (here: the semi-disk domain). That is, the roagation rate is a result from the simulations and it is not resibed. The rate DajN is a disete aroximation of the continuous roagation rate da/dn. The integrated domain is translated with one element length along the roagation ath and the rocedure is reeated until the value of the lastically dissiated energy in the current integrated domain, W (D), is less than a user-secified itical value, W. The iterative rocedure for establishing the disete roagation rate, Daj N, is desibed in Ref. [8,]. In this work, we assume that the itically dissiated energy, W, is constant, since there is (to the knowledge of the authors) no exerimental data available to quantify such arameter. Moreover, the size of the integrated domain (i.e. radius r) is based on the size of the reversed lastic region. Investigations on the shae, size and osition of the integrated domain with resect to the ack-ti, and the aroriate value for the itically dissiated energy will be address in a future study where numerical studies are combined with extensive and careful exerimental work.. Model desition In this study, we simulate fatigue ack growth in a lane strain secimen with a center ack. All dimensions are normalized, since the aim of the study is to qualitatively asses the usefulness of the roosed method. Thus the half width of the secimen is assumed B/ = and the initial half length is a =., Fig. A. Only the right side of the secimen is modeled since symmetry along the central vertical axis is assumed, with aroriate boundary conditions imosed. The boundary conditions are shown in Fig. A. Linear elastic, erfectly lastic constitutive resonse is used for all simulations, with the normalized yield strength r ys =, the normalized elastic modulus E =, and Poisson s ratio v =.. Bi-quadratic lane strain elements with reduced integration were used for the entire model. The J comutational lasticity theory [,] is used to redict the inelastic material resonse including the density of the lastically dissiated energy. The secimen is subjected to cyclic stress alied to the to and bottom edges (Fig. ). During each cycle, the stress is varied linearly from an initial minimum value, r min, to a maximum value, r max, and back to the initial value, r min. Here, r max ¼ =6 unless otherwise secified. An imortant arameter used to characterize the cyclic load is the so-called load ratio, defined as R ¼ r min =r max [6]. In the benchmark cases considered in the following, we use R =(r min ¼ ), thus the load cycle has only a tensile character, and R = -, where the load is fully reversed from tensile to comressive during one load cycle (r min ¼-=6). The ack roagation is simulated using the rocedure summarized in Section [8]. A refined structured mesh is used in a rectangular region (. x - by. x - ) along the simulated ack ath, Fig. A. In the refined mesh, all elements have the same size. In the current simulations, the element length in this structured mesh is h e =. x -. This choice of element length concurs with the suggestion of Solanki et al. [8], who recommended to include at least elements along the reversed lastic region (which will be verified later in the discussion). As shown in Ref. [8], the disete roagation rate incurred after the Nth load cycle, assumes values of the form Daj N ¼ kh e ; k ¼ ; ;..., where k is an integer resulting from the iterative udating rocedure. The roagation rate, Daj N, is the dis- Fig.. (A) Finite element model of the investigated secimen where a refined structured mesh is used along the simulated ack ath and (B) schematics of the load cycle. ete equivalent of the fatigue ack growth rate da/dn. The ack roagation rate is determined according to the following scheme (Section.) [8]: A disete semi-disk domain (the integrated domain), D, of radius r, (Fig. ) is defined in front of the ack-ti. The lastically dissiated energy is evaluated in this domain after each cycle, W (D). Crack roagation ensues when a itical value, W, is reached, W ðdþ W. Since this is a new iterion, there is no rocedure for selecting the radius, the shae and/or the osition of the integrated domain with resect to the ack-ti. The goal of this aer is to assess the suitability of using the lastically dissiation energy as a roagation iterion from a qualitative ersective. Further numerical and exerimental work is necessary to calibrate such a iterion. Since both W and D are material arameters, these will deend on the material investigated. The size and the shae of the integration domain considered here are obtained from reliminary analytical and numerical evaluations. In this initial study, we correlate the size of the semi-disk domain to the size of the reversed lastic region (Fig. ) where significant lastic deformation occurs during both loading and unloading. The Integrated domains of more comlex shaes can be develoed using other models for the lastic region if desired.

5 size and shae of the reversed lastic region can be estimated based on one of the various analytical models existing in the literature, or from numerical simulations. For examle, a simle analytical aroximation of the size of the cyclic lastic region (in the direction of roagation) for a linear elastic, erfectly lastic material, subjected to mode I loading, can be obtained using the well-known relation for lane strain conditions (see for examle Ref. [,6]): r c ¼ DK I r ys where r c is the size of the reversed lastic region (measured from the ack ti), r ys is the material yield strength, and DK I ¼ K I ðr max Þ-K I ðr min Þ.K I (r) is the stress intensity factor (SIF) for mode I of ack oening at the stress r. For the secimen considered here (Fig. A), the mode I SIF can be comuted using the formula due to Feddersen []: ffiffiffiffiffiffi ( a ) = K I ðr; a; BÞ ¼r a sec ðþ B In equation Eq. (), r is the alied stress, a is the half length of the ack and B is the secimen width. For r min ¼ ; r max ¼ =6; a ¼ : and B = we obtain DK I ¼ :. Assuming r ys =, the size of the reversed lastic region in front of the ack according to the equation Eq. () is r c =.867 x -. Alternatively, one can easily select the value of the radius based on a reliminary analysis with a single load cycle. For the load conditions listed above, we found that a semi-disk of radius r =. x - covers most of the reversed lastic region in front of the ack ti, generated during the desibed load cycle for the considered initial ack length (i.e. a =.). The value obtained from the numerical analysis is close to the analytical value, and we will use the value obtained from the numerical aroach. The integrated domain is desibed as the set of elements that have at least one node within the theoretical semi-disk domain, Fig. [8]. During cyclic loading, art of the ack surfaces (defined as the failed segments above) may be under contact. Thus, contact between the surfaces must be considered in the model. Contact occur not only for negative load ratios (i.e., r min < ), but can occur for cases where the load ratio vanishes or is ositive (i.e., r min ). For these latter cases, contact is tyically induced by what is commonly referred to as lasticity induced ack closure (PICC) and is associated with the lastic zone, the reverse lastic zone, and the lastic wake. Many authors have studied this henomenon. It is generally agreed uon that PICC and the lastic zone directly affect the ack roagation rate; in articular it aears to slow the ack growth rate down, since PICC effectively reduces the stress intensity and may blunt the ack. This is automatically catured in our model (within the context of the formulation). For a review ertaining to modeling asects on the subject, see Solanki et al. [], and more recent work can be found for examle in Refs. [6,7]. In our model, the normal, frictionless contact formulation in ABAQUS [8] was emloyed. Cyclic ack growth is simulated for cycles. Effects of (i) roagation iterion, (ii) load ratio, (iii) single overloads, and (iv) reeated overloads on the simulated fatigue ack growth, based on A similar relation but for lane stress conditions was initially edited to Rice []. The density of lastic dissiation (and so the equivalent lastic strain) has its absolute maximum within the first two elements immediately behind the ack-ti. Because these elements articiate in ack-ti blunting, the dissiated energy within these elements may exhibit mesh deendency due to excessive distortion. Therefore, we osition the integrated domain so that only the elements in front of the ack-ti are included. Simulating more cycles is comutationally rohibitive on our single rocessor PC work station, due to the refined mesh used along the roagation ath. ðþ Δ a ( - ).6... W =. } } W =. R = - R= - R= R= Fig.. The change in half ack length, Da, as a function of the number of cycles, N, for two load ratios and two itical values of the lastically dissiated energy. The dashed lines indicate the reduction in the lifetime when R = - comared to R =. lastically dissiated energy evaluated in a semi-disk domain in front of the ack ti (r=. x - for all simulations) are resented next.. Numerical simulations: results and discussion.. The effect of the roagation iterion First, we investigate the effect the roagation iterion has on the ack roagation, considering two magnitudes of the roagation iterion: W ¼ : and W ¼ :. We note that due to the lack of suitable exerimental data, these values were selected based on reliminary numerical investigations aiming to achieve a reasonable roagation rate within the context of the resent urose. Thus, we do not claim that these are hysically realistic values, but are selected to demonstrate the concets investigated here. In ongoing studies, realistic values are extracted. The load ratio R = with r min ¼ =6 is assumed. Ongoing exerimental investigations will rovide realistic values of both the roagation iterion in conjunctions with the size and shae of the integrated domain. The resulting ack roagation is almost linear for both cases (Fig. ). As exected, the ack roagates faster for the lower itical value of the lastic dissiation. The ack advances in inements of or elements er cycle when W ¼ : and in inements of or elements er cycle when W ¼ :, Fig.. The roagation rate, Daj N, deeases after the initial 6 cycles when W ¼ : and after 9 cycles for W ¼ :. We believe this deease is caused by the formation of the lastic wake behind the ack-ti (associated with PICC as discussed above). The lastic wake forms as the ack roagates through the lastic region that has develoed around the ack-ti during the loading unloading sequence (Fig. ). The lastic wake can effectively reduce the ack driving force for fatigue ack growth, see for examle the review by Solanki et al. []. In contrast, one can observe an inease in roagation rate after 7 for the case W ¼ : and after 77 cycles for W ¼ :. As the ack advances, the size of the lastic region, and energy dissiated er cycle in the vicinity of the ack-ti inease. This is catured by the iterative evaluation rocedure of the roagation iterion and it results in larger roagation inements (i.e. ack acceleration). When assessing the evolution of the disete roagation rate at a secific cycle number N, one should consider the disete value at that cycle but also a sequence of several values occurred before, at and after N.

6 A - ) ( Δa N ( - ) Δa N... W = Further insight to the ack roagation can be obtained by studying the normalized dissiated energy, Fig. 6, which is robed after the end of each cycle (during the first iteration of the iterative scheme [8]) in the disete semi-disk domain, D, ahead of the ack-ti. The normalized dissiated energy is defined by W ðdþ ¼W ðdþ=w ðþ Thus, the roagation iterion is fulfilled for any load cycle with W nðdþ :, and the ack roagates at least one element. The oscillations in the lastically dissiated energy corresond to a ack roagating with an alternating number of elements for consecutive cycles. Thus, the ack roagates with k elements after cycle N, with k + elements after cycle N + and again with k elements after cycle N +. In these transitory regimes, due to the disete nature of roagation associated with the finite element, the integrated domain maybe advanced too far. Therefore, the lastically dissiated energy needs to accumulate over the next cycles until the iterion is fulfilled again. For examle, in Fig. A for the cycles between and 9, the ack roagates in one cycle with one element length and in the following cycle with elements. This is manifested with the oscillation in Fig. 6A. The smooth ortions in the evolution of the dissiated energy are secific to the ack advancing at constant rate. This can be clearly seen by correlating the evolution of the disete roagation rate (Fig. ) to that of the dissiated energy for R = (Fig. 6). Number of elements Number of elements A Normalized dissiated energy Normalized dissiated energy R = R = W =. B. 6 W =. 9 N (Cycles) B.8 6 W =..6.. R =..8.6 R =. Fig.. Disete roagation rate, Daj N, with the corresonding number of elements. (right ordinate), as a function of the number of cycles, N, for (A) W ¼ :, and (B) W ¼ :. (the element length is h e =. x - ). N (Cycles) Fig. 6. Normalized dissiated energy in the semi-disk domain, W nðdþ, after each comleted load cycle, as a function of the number of cycles, N, for R = and R = - when (A) W ¼ :, and (B) W ¼ :. For all cases the maximum stress is the same, r min ¼ =6. and therefore the load for R = - is twice the load range than for R =. For negative load ratios, R = -, the energy dissiated in the integrated domain is higher than for the cases with R = (Fig. 6). The inease in lastically dissiated energy for R = - leads to higher roagation rate, Daj N, and consequently shorter ack growth life than for R = (Fig. ). This agrees with exerimental observations, which associate negative load ratios with higher fatigue ack growth rates and shorter lifetimes. For examle, Stehens and co-workers [9] found that the fatigue ack growth life when R = - is about 88% of the ack growth life for R =, for modified comact secimens made from -T and 77-T6 aluminum alloys. In our simulations, for W ¼ :, the ack length after cycles for the case R = is achieved in about 8 cycles when R = -. Similarly, for W ¼ : the final ack length for the case R = is obtained in about 77 cycles when R = - (Fig. ). Thus, our numerical results agree qualitatively very well with the exerimental observations, clearly showing the same tendencies... Effect of a single overload.. Effect of the load ratio Next, we investigate the effect of the load ratio on ack roagation. It is established from exerimental work that load cycles with negative load ratio are associated with faster fatigue ack growth, e.g., [9]. We verify here that a roagation iterion based on lastically dissiated energy can cature this effect. We comare the two cases of itical values W ¼ : and W ¼ : for two load ratios: R = (i.e., r min ¼ ) and R = - (r min ¼-=6). In this section, we investigate how the iterion based on lastically dissiated energy catures the effect of a single tensile overload. The magnitude of the overload is characterized by the overload factor (or overload ratio ), f ¼ r overload =r max : It is well established exerimentally that tensile overload cycles induce ack retardation, where the lastic deformation including ack ti blunting is commonly considered the major cause of ðþ

7 da/dn initial acceleration caused by overload delayed retardation Delay cycles, N D maximum retardation end of retardation cycles Fig. 7. Schematic reresentation of the ack roagation rate as a function of load cycles showing the effect of single tensile overloads on the ack growth rate, based on Refs. [,6]. served exerimentally, see for examle Ref. []. The overload mark may contribute to a discontinuous closure of the ack as the ack roagates []. The normalized dissiated energy in the disete semi-disk domain, W nðdþ (robed after each comleted load cycle), is resented in Fig. 9A, for various overload factors. When the overload is alied, the lastically dissiated energy deeases significantly in the cycles that follow. The lastically dissiated energy deeases with ineasing overload factor. Overload factors higher than. result in intermittent ack roagation, i.e. the ack does not roagate after each cycle. (When W nðdþ < : the ack does not roagate and the lastically dissiated energy accumulates over several cycles before ack roagation occurs.) The numerical results also clearly indicate that the overloads cause ack retardation, Fig. 9B. Ineasing the overload factor deeases the ack roagation rate which is in agreement with many exerimental observations, see for examle Ref. []. The effect of a single tensile overload can further be analyzed by studying the disete ack roagation rate, Daj N,(Fig. ) and comaring to the absence of overloads (Fig. B). In the cycles fol lowing the overload, the ack growth rate first resumes to the value receding the overload and then deeases. The maximum ack retardation does not occur immediately after the overload, but after some additional cycling, N D. This delay in attaining maximum deease in ack growth rate has been observed exerimentally, as discussed above [,6, ]. Thus, the simulations show that the features associated with ack retardation (Fig. 7) are catured during the simulation, including the delay, N D, and the inease of N D with ineasing overload factor. The delay in attaining maximum deease in ack growth rate is best visible for f =. (Fig. A). However, the maximum ack retardation the slow down [,6, ]. Interestingly, the maximum ack retardation does not occur immediately after the overload, but after some additional cycling, N D, [,6, ]. A schematic reresentation of the effect of a single tensile overload on the fatigue ack growth rate following those resented in the literature (e.g., Refs. [,6]) is shown in Fig. 7. Verma and Pandey [] found from exeriments on center ack tension anels of -T Aluminum-alloy, that the number of delay cycles due to overload, N D,(Fig. 7) ineases significantly with the inease of the overload factor, f. Vardar [] erformed overload tests on 77-T6 Alclad sheets and found an exonential influence of the overload factor on the fatigue ack retardation. We will here investigate if the roosed roagation iterion can cature these exerimentally observed results. The simulations resented here are conducted for the case when the itical lastic dissiation is W ¼ :, the load ratio is R = with r max ¼ =6, and we will robe the results for single overloads of overload factors ranging from f = to. The single overload is alied during the th load cycle. As discussed in sub-section., for this case (W ¼ :), a sufficient lastic wake forms dur ing the first 9 cycles leading to a stabilized disete rate of roagation. Crack rofiles after cycles are comared in Fig. 8, for A. Normalized dissiated energy.8 fo =..6. fo =. fo =.. fo =. f o= f =. and f =.7 showing the integrated domain at the final.6 fo =.7 ack-ti osition. In this lot, the marks roduced by the overloads can be clearly distinguished. Overload marks can also be ob- f o =. B Overload marks Δ a ( - ).6 fo =.. fo =. fo =. fo =.7 fo =.. fo = f o =.7. Initial ack-ti osition Deformation factor x Fig. 8. Crack rofiles after cycles showing the marks eated by a single Fig. 9. Effect of a single tensile overload, f, of various magnitudes, on (A) overload alied during the th cycle. The disete semi-disk domain is centered normalized dissiated energy, W nðdþ, after each comleted cycle; and (B) change at the final ack-ti osition. (The deformations are drawn to scale (W ¼ :). in half ack length, as functions of the number of cycles, N (W ¼ :).

8 A. Tyical evolution f o =. 9 of ack growth rate 8 observed during 7 exeriments. with single overloads 6. ( - ) Δa N B ( - ) Δa N C Δa ( - ) N. f o = f o = Fig.. Disete roagation rate, Daj N, as a function of the number of cycles, N, with the corresonding number of elements (right ordinate), when the overload factor is (A) f =., (B) f =. and (C) f =.. (The element length is h =. x -.) The dashed line in (A) indicate a tyical evolution of the ack growth rate observed during exeriments with single overloads [,6] (W ¼ :). is not attained during the simulated cycles when f P., indicating that the delay time ineases with the overload factor. Thus, the ack retardation behavior as catured by the roosed roagation iterion based on the lastically dissiated energy is in agreement with exerimental observations, [,6, ]... Effect of reeated overloads Lastly, we investigate the effect of reeated overloads, where we first consider the resonse due to reeated overloads alied at different time intervals, DN, and then the effect of itically dissiated energy associated with the roagation iterion. When investigating the reeated overloads at different time intervals, a constant overload factor, f =., is used. As for the revious cases, the itical lastic dissiation is W ¼ :, and the load ratio is R = with r max ¼ =6. The first overload is alied Number of elements Number of elements Number of elements A Normalized dissiated energy B Δ a ( - ) ΔN= ΔN= ΔN= Single overload ΔN= ΔN= ΔN= Fig.. For reeated overloads with f =.: (A) normalized dissiated energy in the semi-disk domain, W nðdþ, after each comleted load cycle, and (B) change in half ack length, a; as functions of the number of cycles when the overload is reeated at selected intervals (W ¼ :). tion of the ack half length, a, is resented in Fig. B. For the cases studied, we found that when the overload is alied after each cycles (i.e. DN = ), the ack retardation is most noticeable. The cases with DN = and DN = are characterized by longer ack lengths than DN =, Fig. B. When DN =, the ack roagation rate is influenced by that the overload is alied too frequently and each overload cycle results in a higher roagation inement. For DN =, the retardation effect induced by one overload diminishes before the following overload is alied. Δ a ( - ).6... W =. } } W =. fo =. fo =. fo =. fo =. during the th cycle. Following the initial overload, we study the resonse ensuing when overloads of the same magnitude (f =.) are reeated at intervals of, and cycles. The nor- Fig.. Change in half ack length, Da, as a function of the number of cycles, N, for malized dissiated energy (obtained during the first iteration of reeated overloads with f =. and f =. shown for W ¼ : and the iterative rocedure) is shown in Fig. A, whereas the evolu- ¼ : when the overload is reeated after each cycles. W

9 However, in all cases with reeated overloads, the final half ack length is less than for the case when only the initial overload is alied (Fig. B). Finally, we investigate the effect of the itical lastically dissiated energy, considering the cases W ¼ : and W ¼ :, for the cases of reeated overloads of f =. and f =.. The overload is reeated after each cycles and the load ratio is R = with r max ¼ =6. The evolution of the half ack length is shown in Fig.. Crack retardation is catured for all cases, but is more noticeable when W ¼ :. As exected, the higher overload factor (f =.) deeases the ack roagation rate more than the case of the lower overload factor (f =.).. Concluding remarks A roosed condition for simulating ack roagation due to cyclic loading through a numerical scheme has been investigated qualitatively by emloying the finite element method. The roagation iterion is based on a condition that relates the lastically dissiated energy to a itical value. To this end, the accumulated lastically dissiated energy is integrated over a disete semi-disk domain in front of the ack-ti and the ack roagates when the iterion is fulfilled. Thus, the roagation rate is not secified, but results from an iterative evaluation of the roagation iterion. A higher value of the itical lastically dissiated energy results in a slower ack roagation rate. To investigate if a roagation iterion based on the lastically dissiated energy in a domain defined in front of the ack ti is a viable condition, the effects of load ratio (R ¼ r min =r max ) and of tensile overloads (with overload factor f ¼ r overload =r max ) on the fatigue ack growth rate were studied. Based on exerimental observations, it is well established that a negative load ratio may inease the ack roagation rate. Contrarily, tensile overloads tend to deease the ack roagation rate. The numerical results resented here suggest that the roosed scheme qualitatively catures these rate changes very well. Thus, the roosed roagation iterion aears to be a viable aroach for numerically simulating ack growth due to cyclic loading where the roagation rate automatically results from simulations. In the current form, the aroach resented in this work can be used to establish relative ack roagation tendencies for different load cases. For absolute values ( real values), calibration with hysical exeriments are needed. The shae and size of the integrated domain and the itically dissiated energy are material arameters in the roosed scheme. These arameters can in the simulations be made deendent (in addition to the material selected with associated roerties) on any measurable quantity in the FE-model, such as (but not limited to) ack length, cycle number and time. The values used in this resentation were selected to show that the roosed concet is viable. For examle, the value of the itically dissiated energy was selected solely with the urose of achieving a reasonable roagation rate with resect to the numerical simulations and will not give a correct ack growth rate for a real system. A quantitative develoment and calibrating the roosed iterion requires significant exerimental work, where the size and shae of the integration domain along with aroriate values for the itically dissiated energy are established. Moreover, further numerical investigations are necessary for calibrating the mesh size along the redicted ath of ack roagation. All these are the toics for future work. Acknowledgment The authors would like to acknowledge funding from NSF DMR 7 and the University of Delaware. References [] Ellyin F. Fatigue damage, ack growth and life rediction. New York: Sringer;. [] Newman Jr JC. The merging of fatigue and fracture mechanics concets: a historical ersective. NASA Langley Research Center; 997. [] Ritchie RO. 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