P, BroZ Department of Engineering Structures, Klokner Institute of the Czech Technical Prague, Czech Republic. Abstract.

Transcription

1 Crack characteristics of residual stress fields P, BroZ Department of Engineering Structures, Klokner Institute of the Czech Technical Prague, Czech Republic Abstract A finite element algorithm implemented in plane stress condition utilizing isoparametric elements may predict fatigue crack closure behaviour of fatigue cracks in residual stress fields; the finite element methodology in elastic-plastic formulation employing high order elements is established. The crack opening and closing characteristics via a compressive residual stress field is considered to be affected by the magnitude of applied load and the crack tip position. By dint of a compressive residual stress field, three variant types of crack opening behaviour, namely normal, unsymmetric partial and symmetric partial ones are studied. At the same time, the partial crack opening stress intensity factor inclusive of the opening effect is suggested to predict the fatigue crack propagation. The boundary element method of modelling crack propagation in the presence of residual stress fields is demonstrated. The approach shown is an elastoplastic fracture mechanics concept for crack growth simulation with a nonlinear J-integral being employed in the fatigue life analysis. The eventuality when the pre-stress is applied to specimens pre-cracked is also encompassed. 1 Introduction When used an unloading elastic compliance method to crack closure measurements, Kikukawa, Jono and Tanaka [1] revealed that in the transition zone the dependence of the load x displacement is of a shape out of the common run. That is, the loading curve has the normal hi-linear form whereas it is possible to approach the unloading upper curve being a straight-line transition linking up the initial and final straight lines of the unloading attribute, It was also devise that the unusual shape of the curve could be applied to the partial crack

2 154 DamageandFractureMechanics VII opening process and the model was postulated for the crack opening and closing characteristics in the transition zone. The partial phenomenon afore said is reflected to be narrqwly referred to the unusability of the effective stress intensity factor range. On top of that, so as to expound crack propagation characteristics in the transition zone, the partial crack opening point is qualilled. Herein, the elastic-plastic finite element calculation with high order elements is performed to examine drack closure characteristics in residual stress fields with the effect of the partial crack opening and closing simulation being checked. There are important structural constituents, eg perforated (hollow-web) girders in the fields of civil and mechanical engineering. For these problems, it is decisive the crack propagation model with a residual stress field being on the scene, produced, for the most part, by a prestressing manner. At the same time, the components weakened by fastener cut-outs are used to going through a mechanical treatment. A dual boundary element statement is hereafter employed in accordance with Cisilino [2], to model fatigue crack propagation by means of an elastoplastic fracture mechanics method. Concurrently, a nonlinear J-integral is applied, with both a primarily plain specimen and a precracked sample being investigated. 2 Residual stress and crack behaviour 2.1 Method of determining crack closure The theory of incremental rate-development classical plasticity and the Von Misses criterion are used. Use is made of the kinematic and isotropic hardening to model Bauschinger s effect connected with reserve yielding, conforming to Choi and Song [3]. Being a bilinear law, the stress-strain is simulated. After numerical integrating the constitutive equation in conformity with a mean normal method, the nonlinear equations are tackled employing the Newton-Raphson method being based on the tangent stiffness matrix. The opening and closing conditions of the crack surface are recognized by means of watching all nodal displacements and reaction forces along the crack at each load increment and subsequently the boundary conditions are altered. If the vertical nodal displacement becomes zero, the crack surface is circumscribed as closed and then the corresponding node is fixed being a roller support. If the nodal reaction force turns zero, the crack surface is demarcated as open and next the corresponding is released. Each load increment belongs to (1/1000) CSY, where OYmeans the yield stress of material. The crack tip opening and closing load levels are specitled being the grades at which the nearest mode behind the crack tip begins to open or close. Crack extension is made every cycle by moving forward the crack tip node at the minimum load. By virtue of the outcomes from the residual stress-free specimen, fatigue crack closure characteristics are decided in residual stress fields on. To compare analytical computations with results of experiments, the initial residual stress field, indicated in Fig. la, is utilized. In analysis, the following material properties were held:

3 Damage and Fracture Mechanics VII 155 Elasticity modul E = 210 GPz yield stress OY= 330 MI % and the plastic modulus of the linear strain hardening characterization H = (da/de) = 0.01, E = 2100 MPa. 2.2 Characteristics of the crack closure over a compressive residual stress field Figure 1: Diagram of residual stresses for CCT and SEN specimens The crack in the single edge-notched sample possessing the initial residual stress distribution demonstrated in Fig. lb gows over a compressive residual stress field. The position Tat x = 25 mm in Fig. lb is the transition point at which the residual smess alteres from compression to tension. The residual stress intensity factor K, in the present case is always negative in the extent of crack length considered (a <30 mm) and it has a minimum value at a = 24 mm, The crack propagates from an initial length of ~ = 10 mm that belongs to the initial notch length of the specimen, a = mm. The calculation is carried out for R = Cracked specimen analysis Surface treatment methods are growingly employed by manufactures to improve the fatigue performance of critical components of structures. These techniques originate residual stress fields in the parts mentioned. In civil and mechanical engineering and also in the aircraft industry, coldexpansion is a cornmordy used mechanical treatment to increase the fatigue life of components with circular cut-outs, such as fastener holes, see examples demonstrated in Figs. 2-4, according to Bro.Z [4]. The cold-expansion process consists of pulling an over-sized mandrel through a hole thus deforming the surrounding material. If the degree of expansion is suftlcien~ permanent (plastic)

4 156 DamageandFractureMechanics VII deformation occurs. Once the mandrel is removed it is not possible for the material to return to its original position, and a residual stress field remains. The residual stresses are ~ompressive around the rim of the hole and tensile in the outer field. The residual stress fields influence upon the fatigue life of the components; compressive stresses generally prolong fatigue life by reducing crack growth. rates (and possibly by delaying crack initiation), whereas tensile stresses generally redute fatigue life. It is important to be able to predict the effect of the surface treatment on the fatigue life of the components as the rate of growth of a crack will determine when the component must be repaired of replaced. Cold-expension utilization is restricted to repair and refit applications. Where replacement of components is not possible, repairs are carried out which usually consist of the removal of fatigue damage by reaming and subsequent hole coldexpansion. The critical locations where repairs are necessary, occur at highly stressed regions such as the ligament between a fastener hole and the edge of a bolted component. It may not always be feasible to use the conventional repair technique described above in these situations as the remaining ligament between the hole and the component edge may not be large enough to sustain the applied loading or prevent bulging during the cold-expansion process. In such situations, where replacement is not thriftily practicable, the alternatives are to cold-expand the hole with existing damage or to remove some or all of the damage prior to cold-expansion. However, to avoid ligament bulging, it may be necessary to use a cold-expansion level lower than that generally recommended. In this contribution, we concentrate on the effect of the processing given on fatigue crack growth in these highly loaded low edge margin locations, with particular emphasis on the degree of cold expansion and the amount of residual fatigue damage when this treatment is performed. Realistic assessment study requires an elastic-plastic mechanics concept. Characteristic circumstances for the application of EPFM approaches are those being pertinent to high levels of residual stresses produced by mechanical or thermal treatments embracing plastic deformation, eg in welded constituents or members subject to life improvement methods, namely prestress and cold expansion. Two EPFM parameters, the J integral and the CTOD are proper for numerical evaluation applying BEM. To boot, the J integral, by virtue of an energy concept, eliminates the need to solve local crack tip fields precisely, as if the integration domain is defined over a comparatively large part of the mesh, an accurate modelling the crack tip is unneces,wuy for the crack tip fields contribution to J is unessential. Three basic charts have formed for the computation of the J- integral in three dimensions: virtual crack extension techniques, generalization of Rice s contour integral and domain integral methods. Three= dimensiona.l J-integml parameters J(q), have been worked out to determine the intensity of stresses and strains along crack fronts. Domain integral methods are derived by applying the divergence theorem to Rice s J integral, that forms an integral defined over a finite volume specified some part of the crack ffont.

5 Damage and Fracture Mechanics VII 15 7 Domain integral techniques being more suitable for numerical analysis than contour integral procedures, are equivalent to the virtual crack extension systems. Example Figure 4: Girder web with openings A centre cracked panel (CCP) specimen in plane strain condition was investigated subjected to an uniaxial remote tension a in conformity with Fig. 5. Normal displacements of version faces were limited so that modelled the plane

6 158 Damage and Fracture Mechanics VII strain condition, It was substituted: crack length a = 10 mm, the specimen width b=2a, and the model thickness t =:. For the problem, one can assume that the material possesses elastic-perfectly plastic characteristics, with yield stress ay= 1000 MPa. Four rings of internal cells with radii reaching from 20 A to 75 A of the crack length were employed for the interpretation of J. Remote uniaxial tension was applied in twelwe increments as far as a load level that forced the crack plastic zone to go round to an extent of 75 Aof the crack length. There are J values in Table 1, computed for four load levels and normalized with regard to the plane strain value commensurable to K = my&. Remote load is demonstrated being a fraction of yield stress CY.The lowest load level, a/oy = 0,21, is analogous to load at fwst yield, whereas the largest, cr/ay= 0.21, represents the maximum applied load. The independence of the value of J with respect to the integration domain is considered to be convenient, as approximately 2 A for the lowest load level and 5?4. in the case of the largest grade, The development of J with increasing load is sketched in normalized form in Fig. 6, in company with reference issues, Outcomes are drawn proportionally to W. Very satisfactory agreement is found between computed values and ones determined applying the Irwin concept, the maximum difference between them is loosely 3O/. at the maximum load level, Likewise, it is possible to notice that K values stipulated employing the Dugdale model have a tendency to be overestimated, (J Figure 5: Cracked plate: geometry and loading

7 Damage and Fracture Mechanics VII ~ BEM,.. Dugdale / --- Irwin / J. -n l>y 0.20 /s.x //d,.s /9 F / ;*.,p /, /8 / m /6 /,, / (s/(3 Y Figure 6: J-integral course in the CCP specimen Table 1. Normalized J outcomes for the CPP specimen rla slay , Average , ,0386 L , , , , Prestress effects on fatigue fracture The Elastoplastic Fracture Mechanics formulation is used, consisting of the traction boundary integral equation in company with the displacement boundary integraj equation, (in the work by Leitao et al [5]); the calculation procedure overcomes the necessity for subdomains in general mixed-mode EPFM problems. The von Mises yield condition and the elastoplastic dependence,

8 160 DamageandFractureMechanicsVII according to Mendelson, of increments of equivalent plastic strain owing to a given load increment upon equivalent total strains, were taken over. An elestoplastic modpl is applied in numerical analysis in a manner that the calculation is repeated for the category of initial cracks. Semi-discontinuous boundary elements and cells are used in the zone adjoining to the crack-tip to get out of inconveniences concerning the modelling of the singularity of the tractions and stresses at this point. When performing an incremental crack extension analysis, at each new position of the crack-tip, stres~ intensity factors are calculated in the new conilguration. By applying proper standards, the direction of the crack path at each increment may be redefined. In the presence of residual stresses, crack growth is modelled by the crack to be propagated by a fixed quantity (defined by the dimension of the elements on the crack faces) for a constant load applied at the limit of the structural member; this load accords with the maximum value of the fatigue load, At each stage of the crack growth, nonlinear J-integrals (ie T*-integrals) are obtained. Atluri formulated the T* integral in the form, as follows where W is the strain energy density, p denotes the radius of a contour surrounding the crack-tip. After applying Green s theorem to the foregoing equation we get T*= Jr (w, tiut,,) dr -lii[on (1) Jw-t oui ldn 2) where Q means the region within general contour r and QP is the domain within rp, Afterwards, equivalent stress intensity factors (K: =@E, K~=O) are calculated and next the fatigue crack propagation parameters AK and R can be specitled to employ them in a crack growth prediction. Hereafter, in compliance with [5], some results are in Figs. 7-8 demonstrating the influence of prestressing on phenomena of both the residual stress and cracking,

9 Damage and Fracture Mechanics VII 1fj 1 z i 200 j ;100 * : m o z 4 $100 & Soo o distance trom edge of hole (mm) 5 Conclusion Figure 8: Residual stress distributions owing to the prestressing The elasto-plastic FEM algorithm using the plane stress formulation and 8-node isopammetric elements may predict to good effect fatigue crack closure behaviour over residual stress fields likewise in a residual stress-free specimen.

10 162 DamageandFractureMechanicsVII A fatigue crack opens and closes in a usual way in a tensile residual stress field similar to occurrences in a residual stress-free specimen, In the present case, the value of KOPimperat$e to determine AKefffor the prediction of fatigue crack growth rate, can be simply stipulated from the crack-tip opening load. The crack opening and closing behaviour over a compressive residual stress field is complex and proves to be affected by the relative magnitude of the applied load to the compressive residual stress distribution and the stress gradient in the transition zone from residual compression to tension, Herei% the crack opening characteristics over a ~wnpressive residual stress field may be categorized into the three main ~es, viz. normal, unsymmetrical partial and symmetric partial crack opening behaviour, The partial crack opening K value, KPm on that was put into practice to expound fatigue crack growth in the transition zone, is derived to go along well with the crack-mouth closing K value predicted by the FEM calculation. The approximate quantity of KPti,oPbeing inclusive of the partial crack opening influence may be simply determined through the FEM procedure. This evaluated KPm.P can be suggested for the prediction of fatigue crack growth as it heads for to specify conservative prognoses of service life, The J-integral computation procedure has been embraced being a postprocessing algorithm so it maybe used to the issues from a particular model. The EPFM methodology seems to be superior to the other techniques, in particular for the highest pre-stress level and the shortest cracks when plastic deformation is presumably to be most substantial. Acknowledgement The author gratefully acknowledges the financial support of the presented research by the Grant Agency of the Czech Republic (project No. 103/00/0897) and by the grant CEZ J04/98: References [1] Kikukawa, M., Jono M. and Tanaka, K. Fatigue crack closure behaviour at low stress intensity levels. Proc.ICA42, pp , [2] Cisilino, A. Linear and Nonlinear Crack Growth Using Boundary Elements, WIT Press: Southampton and Boston, pp ,2000. [3] Choi, H.C. and Song, J.H. Finite element analysis of closure behaviour of fatigue cracks in residual stress field. Fatigue ffact. Engng Mater. Strucf. 18(1), pp , [4 Bro2, P. Fatigue Prediction of Metals in Practical Terms. CTU Reports, Czech Technical University in Prague, 4(2), pp ,2000. [5 Leitao, V.M,A., Aliabadi, M.H,, Rooke, D.P, & Cok, R. Analysis of crack growth in residual stress fields (Chapter 1). Nonlinear Fracture & Damage Mechanics, ed. M.H. Aliabadi, WIT Press: Southampton and Boston, pp. 1-30,2001,