Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 1, pp. 69-82, Warsaw 2004 A NON-LOCAL FATIGUE CRACK GROWTH MODEL AND ITS EXPERIMENTAL VERIFICATION1 Andrzej Seweryn Adam Tomczyk Faculty of Mechanical Engineering, Białystok University of Technology e-mail: seweryn@pb.bialystok.pl Zenon Mróz Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw e-mail: zmroz@ippt.gov.pl The present paper is concerned with the modelling of fatigue crack ini- tiation and propagation by applying the non-local critical plane model, proposed by Seweryn and Mróz (1996, 1998). Using the linear elastic stress field at the front of a crack or sharp notch, the damage growth on a physical plane is specified in terms of mean values of the stress and strength function. Themodel is applied to study crack propagation under cyclically varying tension-compression conditions.The predictions are compared with experimental data. Key words: fatigue, damage accumulation, crack propagation 1. Introduction Most engineering elements subjected to variable loads experience multia- xial stress and strain states forwhich the principal stress vary in time.Usually, the elements contain stress concentrators (notches, holes, joints), which am- plify nominal stresses and generate fatigue cracks. In most cases of combined loads the notch tip stress and strain fields do not vary proportionally, and multiaxial fatigue parameters should be introduced to provide crack initiation and propagation conditions.Most fatigue data in the form of S-N curves have 1 Thispaperwas presentedonSymposiumDamageMechanics ofMaterials and Structures, June 2003, Augustów, Poland, 2003 70 A.Seweryn et al. been generated for uniform specimens under uniaxial loading and next used to predict the fatigue life for notched specimens in terms of the local stress and strain amplitudes. The critical plane approaches have been widely used in correlating fatigue data and in formulating fatigue conditions. This approach is natural since the plane crack initiation and growth is dependent on the surface traction components, and the resulting crack openingand shearprovidedamage strains associated with the crack surface. Consideraphysicalplane inamaterial element specifiedby theunitnormal vector n. The plane traction vector and its components are T =σn σn =(n ·σn)n τn =(I−n⊗n)σn (1.1) Similarly, the surface strain components are Γ = εn εn =(n ·εn)n γn =(I−n⊗n)εn (1.2) where I is the identity tensor. The critical plane can be assumed in advance as a representative plane on which the critical condition is satisfied. It was first Findley et al. (1956) was the first to postulate that the representative plane is themaximum shear plane with both shear and normal strain amplitudes spe- cifying the damage parameter. A particular form of the mentioned condition was proposed by Brown and Miller (1973). McDiarmid (1991) provided an alternative stress condition expressing the fatigue parameter in terms of shear and normal stress amplitudes on the maximal shear planes. Other criteria of this type combine the shear strain amplitude and the maximal normal stress acting on themaximal shear plane, cf. Socie (1993). These conditions can be easily applied to the case of proportional loading. However, for non-proportional loading, the properdefinition of stress and stra- in amplitudes should be generated. Furthermore, experimental observations indicate that cracks do not develop on the maximum shear planes for all me- tals. Amore consistent approach is not obtained by specifying the critical plane approach in advance but requiring themaximumof the failure condition to be reached with respect to all orientations, thus max n F(σn,τn,εn,γn)=Fc (1.3) where Fc represents the critical value reached by the failure condition. The present definition provides the critical plane which is also the extremal plane, so that the critical condition is not violated on other potential failure planes. A non-local fatigue crack growth model... 71 A particular form of Eq. (1.3) is obtained by applying the strain energy density associated with the amplitudes of stress and strain components acting on the critical plane, cf. Glinka et al. (1995). This parameter represents only a fraction of the strain energy. However, it does not account for the effect of the mean stress. An alternative energy condition was proposed by Chu (1995) by combining the maximum normal and shear stresses with the strain amplitudes. The formulation of Seweryn and Mróz (1996, 1998) followed the idea of the non-local stress or strainmeasures on the critical plane area of size d0 × d0. In the present paper, we shall develop this critical plane model for the prediction of fatigue crack propagation under uniaxial loading conditions. The predictions are compared with experimental data. 2. Basic assumptions To illustrate applicability of the model, consider a plate of uniform thick- ness (Fig.1a)with an edge crack of the length l, loaded by a cyclically varying stress σ of the amplitude ∆σ andmean value σm =∆σ/2 (Mróz et al., 2000). Thematerial is assumed tobe linear elastic but exhibiting aprocess or damage zone Ω of the length d0 at the crack tip (Fig.1b). Fig. 1. (a) Polar coordinate system connected with the tip of the edge crack, (b) scheme of the damage zone propagation The existence of the localized damage zone is usually assumed for the cohesive crackmodelwithanadditional rule relating the stress todisplacement discontinuity. Here, however, the stress distributionwill be treated within the 72 A.Seweryn et al. linear elasticity but the existence of the process zone will be accounted for using the non-local damage rule discussed in the previous section. It is assumed that damage growth occurs only in the damage zone and is specified by themean value ωn affecting the critical stress σc. Themean value of the normal stress in the zone Ω equals σn = 1 d0 d0 ∫ 0 σn dr= 2KI√ 2πd0 (2.1) where KI is the stress intensity factor formode I. Let us note that σn is twice as large as the stress value at the end of the damage zone. 3. Damage accumulation and crack growth in one cycle of loading We divide the cycle of fatigue loading into four stages (Fig.2). Fig. 2. Consecutive stages in one loading cycle When the stress in a cycle increases from zero, then in stage I there is no damage growth as σn < σ0 and KI < KIth where σ0, and KIth are the damage initiation threshold values. Let us note that both σ0 and KIth depend on the damage state, thus σ0 =σ ∗ 0(1−ωn) p KIth =K ∗ Ith(1−ωn) p (3.1) where σ∗0 and K ∗ Ith are the respective values for the undamagedmaterial. A non-local fatigue crack growth model... 73 In the second stage σ0 < σn < σc, the damage growth occurs in the zone Ω, according to the following rule dωn =A (σn−σ0 σc−σ0 )n dσn σ∗c −σ ∗ 0 dσn > 0 σn >σ0 (3.2) and we have σc =σ ∗ c(1−ωn) p KIc =K ∗ Ic(1−ωn) p (3.3) where σ∗c and K ∗ Ic are the critical values for the undamagedmaterial. Let us note that in Eq. (3.2) the stress difference (σ∗c −σ ∗ 0) in the denomi- nator occurs. This differs from the original formulation of Seweryn and Mróz (1996) where the form (σc−σ0) was used. Introduce the ratio σ∗0/σ ∗ c = η and assume that η= σ∗0 σ∗c = σ0 σc = KIth KIc = K∗Ith K∗ Ic (3.4) In view of Eqs (3.1), (3.3) and (3.4), the damage evolution rule takes the form dωn = A (1−η)n+1 ( KI K∗ Ic (1−ωn)p −η )ndKI K∗ Ic (3.5) Athis stage, the stress value σn increases but thevalues of σ0 and σc decrease, according to Eqs (3.1) and (3.3).When σn reaches the critical value σn =σc and KI =KIc, the crack growth process occurs, so that the condition Fc =σn−σc =σn−σ∗c(1−ωn) p =0 or (3.6) Fc =KI−KIc =KI−K∗Ic(1−ωn) p =0 and dl> 0 is satisfied. The consistency condition for the growth crack is dFc = dσn−dσc =0 dFc = dKI−dKIc(ωn)= 0 (3.7) Let us note that KI =KI(σ,l), so we have dKI = ∂KI ∂σ dσ+ ∂KI ∂l dl (3.8) In most cases, the first term dominates as the crack growth value dl/dN is small. Then dKI ∼=Mk √ πl dσ (3.9) where Mk depends on the geometry of the plate. 74 A.Seweryn et al. The damage growth during the propagation stage is decomposed into two terms dωn = dωn1+dωn2 dωn1 > 0 dωn2 < 0 (3.10) where the first term is associated with the loading increment and the second is associated with the damage zone propagation, so dωn = 2AdKI√ 2πd0(σ ∗ c −σ ∗ 0) −ωn dl d0 or (3.11) dl d0 =− [ 1+ pA (1−η) (1−ωn)p−1 ]dωn ωn Integrating Eq. (3.11), we obtain ∆l d0 =− ωnp ∫ ωnk [ 1+ pA (1−η) (1−ωn)p−1 ]dωn ωn (3.12) where ωnk and ωnp denote the damage values at the beginning and the end of propagation stage III. Noting that K∗I =K ∗ Ic(1−ωnk) p KImax =K ∗ Ic(1−ωnp) p (3.13) where K∗I denotes the stress intensity factor at the beginning of the propaga- tion stage, relation (3.12) can be rewritten in the form ∆l d0 = 1 p KImax ∫ K∗ I [C(KI) ( 1 p −1 ) +BCp 1−C(KI) 1 p ] dKI (3.14) where C = ( 2 σ∗c √ 2πd0 ) 1 p B= pA 1−η Relations (3.12) and (3.14) specify the crack growth during one cycle, so that ∆l = dl/dN. Consecutive stage IV corresponds to elastic unloading, so that dσn < 0 dKI < 0 dωn =0 dl=0 (3.15) Using the double logarithmic scale, the crack propagation curves are shown in Fig.3, for varying exponents n and for varying values of the damage A non-local fatigue crack growth model... 75 Fig. 3. Crack propagation curves log(∆l/d0) versus log(KImax/K ∗ Ic ): (a) dependence on the exponent n, (b) dependence on the parameter A Fig. 4. The effect of a single overloading cycle on crack propagation rates: (a) single overloading cycle KImax/K ∗ Ic =0.9 and subsequent cycles KImax/K ∗ Ic =0.5; (b) single overloading cycle KImax/K ∗ Ic =0.42, 0.6, 0.8, 0.9 with subsequent cycles KImax/K ∗ Ic =0.4 76 A.Seweryn et al. growth parameter A. The curves can be compared with the usual diagrams dl/dN = f(∆KI) available in literature. It is seen that the crack propagation curves correspondqualitativelywell to experimental curves.When KI tends to K∗Ic, the crack propagation rate tends to infinity, when KI tends to KIth, the propagation rate tends to zero (or the logarithmicmeasure tominus infinity). Figure 4 illustrate the effect of overloading on the subsequent crack pro- pagation rate for a single overloading cycle and different values of A and overloading amplitude. Themodelpresented in this section considers the translation of thedamage zone at the propagating crack tip. It is possible to propose an alternative approach considering both: motion of the damage zone and growth of this zone. 4. Unstable crack growth condition Let us note that when KI tends to K ∗ Ic (or σn tends to σ ∗ c), then the case of brittle fracture occurs. Let us remind that the first term of Eq. (3.8) dominates for the stable crack growth, and the second term is greater for the unstable growth. To formulate the brittle fracture condition, we can disregard the first term of Eq. (3.8) because ∂KI ∂σ dσ≪ ∂KI ∂l dl so dKI ≈ ∂KI ∂l dl (4.1) which is justified in the case of the load control (then dσ/dl ­ 0). When a kinematic control occurs, we have dσ/dl < 0 and it is necessary to consider the complete form of (3.8). Rearranging Eq. (3.7), we can formulate the brittle fracture criterion in the following form: — crack propagation condition KI =K ∗ Ic(1−ωn) p (4.2) — unstable crack growth condition ∂KI ∂l ­ pK∗Ic(1−ωn) pωn d0 [ 1−ωn+ pA 1−η (1−ωn)p ] (4.3) A non-local fatigue crack growth model... 77 An alternative form of (4.3) is ∂KI ∂l ­ KI d0 p [ 1− ( KI K∗ Ic ) 1 p ] ( KI K∗ Ic ) 1 p + pA 1−η KI K∗ Ic (4.4) Figure 5 shows a graphic illustration of these equations. Fig. 5. Dependence of the critical crack length on KI for the unstable crack growth (a) p=1, η=0.1, (b) A=50, η=0.1 5. Experimental verification An experimental program was executed by plane testing specimens of PMMA with edge notches. The selection of the material was motivated by its linear elastic response and the possibility of visual observation of the crack tip. The tests were carried out using theMTS tensile machine, and the crack growthmeasurement was realised bymeans of a spiral microscope (VEBCarl Zeiss Jena) with accuracy of the order of 0.001mm. Fig.6 and Fig.7 present the experimental data of the crack growth measurement and the predicted values pertaining to the present model. The comparison with the prediction resulting from Paris (1963) equation dl dN =C(∆K)m (5.1) was also presented. 78 A.Seweryn et al. Fig. 6. Diagrams of fatigue crack growth: comparison of experimental data with model prediction: (a) specimen 14A, (b) specimen 14B Five types of specimenswere used (12A, 12B, 12C, 14A, 14B) in the tests. The parameter specification for the Paris model is presented in Table 1, and of the present model in Table 2. It is seen that the parameter values are scattered, which typical for fatigue tests forPMMA.Thepresentmodelpredictsmuchbetter fatigue crackgrowth, especially for high values of KImax close to K ∗ Ic. We note that the value of the parameter p equals one, and then the specification of damage zone growth is essentially simplified. It is also interesting to note that the maximum size of the damage zone d0 =0.16mm was confirmed from both static and cyclic A non-local fatigue crack growth model... 79 Fig. 7. Diagrams of fatigue crack growth: comparison of experimental data with model prediction: (a) specimen 12A, (b) specimen 12B, (c) specimen 12C 80 A.Seweryn et al. loading tests, providing good correlation with the experimental data. Biaxial stress programs are being currently tested, and the prediction of crack paths will be compared with measurements. Table 1.Parameter specification for Paris equation Sample No. C m 12A 6.19×10−7 3.31 12B 7.09×10−7 3.55 12C 6.65×10−7 3.78 14A 1.52×10−6 5.21 14B 8.66×10−7 4.43 Table 2.Parameter specification for present model Sample No. A n p K∗Ith K ∗ Ic d0 [MPa √ m] [MPa √ m] [mm] 12A 0.009 9 0.5 1.2 12B 0.013 12.5 0.6 1.2 12C 0.007 7 1 0.55 1.2 0.16 14A 0.024 7 0.6 1.35 14B 0.017 8 0.6 1.35 Unstable crack growth condition (4.4) for loadingmode Iwas also verified. Let us remind that for each sample, the parameter p equals 0, so equation (4.4) can be rewritten in the form ∂KI ∂l ­ KI d0 1− KI K∗ Ic KI K∗ Ic ( 1+ A 1−η ) (5.2) To verify the unstable crack growth condition, the critical values lc were calculated according to equation (5.2) and presented in Table 3. The valu- es lcexp were measured after experimental tests from the surface of fatigue fractures. It is possible that the values lc are a little underrated because it was assumed that the decohesion process occurs when KI = KImax. In re- ality, a crack could start propagate unstable in the last cycle of the loading for KI < KImax – which corresponds with a greater value of the damage measure ωn. A non-local fatigue crack growth model... 81 Table 3. Critical values of the crack length and corresponding values ∂KI/∂l Sample No. ∂KI ∂l [ MPa√ m ] lcexp [mm] lc [mm] 12A 51.949 12.63 14.588 12B 51.943 13.19 14.587 12C 51.948 13.33 14.589 14A 59.317 13.17 13.757 14B 59.320 13.22 13.754 6. Concluding remarks Thepresent paper provides amodel for the analysis of crack initiation and propagation formonotone and variable loadings. A damage zone of a constant length is introduced with averaged measures of the stress and damage within the zone.Thezone is assumed topropagatewith the crack tipwhen the critical propagation condition is reached. Stable and unstable growth stages can be treated. The model proposed enables calculation of the crack growth in a linear elastic material and analysis of the effect of overloads on the crack growth rate. The analysis is confined to asymptotic stress fields near the crack tip. However, it can be extended to more complex descriptions containing more terms of asymptotic expansions or generated by approximate methods. The analysis can also be extended to two dimensional stress states and the associa- ted damage zones. Furthemore the damage initiation function and the stress failure function can be then introduced to describe damage accumulation and crack propagation processes in a cycle of the loading. Acknowledgement The investigation described in this paper is a part of the research project No. 8T07A04921 sponsored by the State Committee for Scientific Research. References 1. Brown M.W., Miller K.J., 1973, A theory for fatigue failure under mul- tiaxial stress-strain conditions, Proc. of the Institute of Mechanical Engineers, 187, 745-755 82 A.Seweryn et al. 2. ChuC.C., 1995, Fatigue damage calculation using the critical plane approach, J. Engng Mat. Techn., 117, 41-49 3. Findley W.N., Coleman J.J., Handley B.C., 1956, Theory for combined bending and torsion fatigue data for 4340 steel, Proc. Int. Conf. Fatigue of Metals, The Inst. Mech. Eng., 150-157 4. GlinkaG., ShenG., PlumtreeA., 1995,Amultiaxial fatigue strain energy density parameter related to the critical plane,Fatigue and Fract. Engng Mat. Struct., 18, 37-46 5. McDiarmid D.L., 1991, A general criterion for high-cycle multiaxial fatigue failure,Fatigue Fract. Engng Mater. Struct., 14, 429-453 6. Morrow J., 1965, Cyclic plastic strain energy and fatigue of metals, Internal Friction Damping and Cyclic Plasticity, ASTM STP, 378, 45-87 7. Mróz Z., Seweryn A., Tomczyk A., 2000, A non-local critical plane mo- del for the analysis of fatigue crack propagation, In: Continuous Damage and Fracture, Ed. Benallal A., Elsevier, 373-384 8. Paris P., ErdoganF., 1963,A critical analysis of crack propagation laws, J. Basic Engng, Trans. ASME, 528-534 9. Seweryn A., Mróz Z., 1996, A non-local stress failure and fatigue damage accumulation condition, In: Multiaxial Fatigue and Design, Eds: Pineau A., CailletaudG., Lindley T.C., 259-280,Mech. Eng. Publ., London 10. SewerynA.,MrózZ., 1998,On the criterionofdamage evolution for variable multiaxial stress state, Int. J. Solids and Struct., 35, 1599-1616 11. Socie D.F., 1993, Critical plane approaches for multiaxial fatigue damage assessment,Advances in Mechanical Fatigue, ASTM STP, 119, 7-36 Nieokalny model wzrostu szczeliny zmęczeniowej i jego doświadczalna weryfikacja Streszczenie Praca dotyczy modelowania inicjacji i propagacji szczeliny zmęczeniowej przy wykorzystaniu nielokalnego podejścia związanego z płaszczyzną krytyczną (Seweryn iMróz, 1996, 1998).Wykorzystując liniowo sprężyste pola naprężeń przed wierzchoł- kiem szczeliny opisano kumulację uszkodzeń i wzrost szczeliny na płaszczyźnie fizycz- nej. Do rozważańprzyjęto uśrednionąw strefie kumulacji uszkodzeńmiarę uszkodzeń oraz uśrednione naprężenia. Wyniki obliczeń numerycznych porównano z wynikami badań doświadczalnych. Manuscript received October 1, 2003; accepted for print October 20, 2003