Finite Element Analysis of Plasticity- Induced Fatigue Crack Closure with Singular Element

Transcription

1 American Journal of Alied ciences 4 (6): , 27 IN cience Publications Finite Element Analysis of Plasticity- Induced Fatigue Crack Closure with ingular Element 1 Rahgozar R., 1 affari H. and 2 Tabatabaei R. 1 Deartment of Civil Engineering, erman University, erman, P. O. Box , Iran 2 Deartment of Civil Engineering, Islamic Azad University of erman, P. O. Box , Iran Abstract: Develoing a new method is resented to comute crack oening values in lanar geometries using the crack surface nodal force distribution under imum loading as detered from finite element analyses (FEM). In this technique, two-dimensional elastic-erfectly lastic finite element analyses of middle-crack tension (MT) geometry is conducted to study fatigue crack closure and to calculate the crack oening values under lane-strain and lane-stress conditions. Triangle singular elements are used to model the head of crack ti, because singular elements would better cature the head of crack ti stress and strain gradients. Mesh refinement studies is erformed on geometry with various element sizes. Next, effect of a highly refined mesh on crack oening values is noted and significantly lower crack oening values than those reorted in literature were found. The calculated crack oening values are comared with values obtained using finite element analysis and more conventional crack oening assessment methodologies. It is shown that the roosed method is indeendent of loading increment, integration method and crack oening assessment location. The comared oening values are exosed in good agreement with stri-yield models and results of other methods. ey words: Crack closure, middle-crack tension, roughness, stri-yield models INTRODUCTION It has been estimated more than fifty ercent of all mechanical failures are due to fatigue, and the majority of these failures are unexected. There are currently many aroaches to fatigue design. If an engineering structure, which can be any load bearing comonent of a comlex assembly, is subjected to reeated or cyclic loading, the structure is inherently accumulating fatigue damage. Wolf Elber quantified and confirmed the imortance of a new fatigue crack growth henomena, crack closure [1]. Based on exerimental results using thin sheets of an aluum alloy, Elber argued that a reduction in the crack ti driving force occurred as a result of residual tensile deformation left in the wake of a growing crack. The residual tensile deformation caused the crack surfaces to close imulsively before imum load was reached. Fig. 1 shows the closure concet. When a secimen is cyclically loaded between and, the crack faces are in contact below oen, the stress intensity at which the crack fully oens. Elber assumed that the ortion of the cycle tress intensity factor oen eff Closure Fig. 1: Definition of effective stress intensity range Time that is below oen does not contribute to fatigue crack growth. Roughness and oxide induced fatigue crack closure are redoating in the near threshold crack growth regime. These two mechanisms are similar to lasticityinduced fatigue crack closure in that the material in the wake region contacts while under tensile loading. Roughness-induced fatigue crack closure occurs when the crack growth is not laner and the mixed-mode loading at the kinked crack ti causes a mismatch of the Corresonding Author: Rahgozar R., Assistant Professor, Deartment of Civil Engineering, erman University, erman, P. O. Box , Iran 357

2 Fig. 2: Three tyes of crack closure mechanisms wake region material. For oxide-induced fatigue crack closure, an oxide film forms on the surface in the wake region and makes contact while under tensile loads. Fig. 2 shows a schematic reresentation of the mechanisms causing fatigue crack closure [2]. Many researchers have roosed finite element analyses simulating lasticity-induced fatigue crack closure, considering different two-dimensional throughthickness cracked configurations under lane-strain or lane-stress conditions. Far fewer efforts have been directed toward the two dimensional roblem. Am. J. Al. ci., 4 (6): , 27 the crack oening and to reresent are emloyed to model the crack oening and to reresent tensile yield in the stri yield zone ahead of the crack. In any case, some of fatigue design methods are simle and inexensive; others are extremely comlex and exensive. If initially an exensive comlete fatigue design rocedure is imlemented, this may lead to lower cost in the long run by reducing failure. In this aer, following the method considering by olanki et al. [4], a 2-D finite element model using triangle singular element for the fatigue crack closure is resented. There is clearly a need for a comrehensive investigation of the closure behaviour of real three-dimensional cracks and to comare this to the redictions of simlified two-dimensional models. Proence is focused on the difficulties in modeling with high oinion to mesh modification level, crack oening assessment location and crack shae rogress techniques. Furthermore, comaring two dimensional finite element models of a center-cracked late with a simle stri yield model of closure under lane stress and lane strain. Two dimensional finite element mesh modification: McClung [6] have resented general reviews in their Finite element analysis of lasticity-induced fatigue resective aers. Dougherty et al. [9] erformed 2-D crack closure is theoretically simle. A mesh is created analyses of the comact tension CT and MT geometries with an initial crack, and the mesh is loaded by slightly under lane-strain, and demonstrated a good alied tractions. For constant amlitude loading, the comarison between redicted closure levels and loading is cycled between a imum alied stress exerimental results. Their finite element meshes were comosed of four-noded and eight-noded quadrilateral and a imum alied stress.during the elements. Ashbaugh et al. [1] erformed a study similar cyclic loading the crack is advanced in some mode, to that conducted by Blom and Holm [11], focusing on leading to the develoment of a lastic wake behind the finite element analysis of lasticity-induced crack crack ti. This modeling concet is simle; however, closure in the CT secimen under lane-strain there are several results from which must be addressed conditions. In their analyses four noded quadrilateral during the fatigue crack growth simulation. elements were used, and their results indicated that closure does occur in lane-strain. Again, their results Crack organization: All over this aer, references are are also susect due to a lack of mesh refinement and made to secific regions around the crack front. Fig. 3 otential lane-strain locking. A 2-D lane-stress and shows the regions of attention. In communication of lane-strain model of the CT secimen was constructed these regions, they are described with reference to the location of the crack front. For instance, the area to the with constant strain triangular (CT) elements. Under right of the crack ti in this figure will be referenced as lane-strain conditions closure was observed, and the "ahead of the crack ti". imilarly, the region to the left lane-strain closure levels were smaller than those for is "behind the crack ti". Also, throughout resent study lane-stress. Their results are also questionable due to a are references to the crack ti lastic zones. Normally, relatively coarse mesh and the use of element tye the crack ti lastic zone refers to the crack forward which is rone to lane-strain locking [12]. olanki et al. lastic zone. This is the region of yielded material roosed, a detailed and comrehensive review ahead of the crack ti at imum load. The reverse considering the finite element analysis of lasticity- lastic zone is the region of material ahead of the crack induced fatigue crack closure using linear elements for ti that yields in comression at the imum load. both two and three-dimensional geometries [4]. The twodimensional When referring to lastic zone sizes in this study, this is lane stress stri yield model used is the size of the lastic zone on the crack lane. Further, essentially hysically similar to that suggested by as the crack rogresses through the initial lastic zone, Newman [8]. In stri yield model, Dislacement yielded material with residual stresses is left behind the discontinuity boundary elements are emloyed to model crack ti; this region is referred to as the lastic wake. 358

3 Am. J. Al. ci., 4 (6): , 27 Fig. 3: Plastic deformation in the region of a growing crack Fig. 5: Bilinear lasticity model secifically investigated, an elastic erfectly lastic material is assumed, with H=. In all cases, the von- Mises yield criterion is used with the associated flow rule. When strain hardening is resent, kinematics hardening is used. Fig. 4: 2D singular element and element division around a crack ti Element tye: Early two-dimensional analyses were erformed with constant strain triangle elements [8]. But, higher order elements would better cature the near crack ti stress and strain gradients more recently, researchers commonly use linear four noded quadrilateral elements or quadratic eight noded quadrilateral elements, but would result in a higher bandwidth [3,4]. In the resent study, 2D singular element is emloyed. Because, in analysis of crack roblems, it is known that the dislacements near ahead of the crack ti vary as r, where r is the distance from the crack ti. The stresses and strains are singular at the crack ti, varying as1 r. To resolve the singularity in stain, the crack faces should be coincident, and the elements around ahead of the crack ti should be quadratic, with the midside nodes laced at the quarter oints. uch elements are called singular elements. Figure 4 shows a 2-D triangle singular element. MATERIAL AND METHOD Non-linear material roerties must be used to model lasticity-induced closure. For simlicity, a bilinear stress strain curve (Fig. 5) is used for all the models in the resent study. With excetion to the cases where the effects of material hardening are being Mesh refinement: ufficient mesh refinement is always a matter when conducting finite element analyses. The idea is to have enough refinement to confine all strain gradients of imortance, but to avoid excess refinement, which can lead to unnecessarily long run-times. For 2D dimensional lane-strain closure analyses when a stress ratio R =, it has been suggested that the mesh should be refined such that there are aroximately ten elements contained in the forward lastic zone. Also, it has been observed that for crack oening level stabilization to take lace, the crack must be advanced comletely through the initial forward lastic zone [6]. This means that too much refinement contained in the model increases the execution time by two means: first, the number of nodes is increased which increases the time required to solve each load ste; secondly, the number of load cycles required for crack oening level stabilization is increased, which increases the number of load stes required. Because of this, it is essential that the coarsest ossible mesh be used. ince lastic zone sizes are not known before the analyses, an aroximation for the lastic zone size must be used to estimate an aroriate mesh size. The equation develoed by Irwin's Theory is used, 2 1 π r = sec 1 (1) π 2α where, r is the crack forward lastic zone size, is the amlitude stress of remote cyclic loading, is the imum stress intensity factor, material yield stress. Considering the crack ti oening dislacement can be written in a slane roblem [5], 359

4 Am. J. Al. ci., 4 (6): , 27 Number of Elements in Plastic Zone Ahead of Crack = 1.7 mm R =, =.3 lane stress forward lastic zone -M(T) lane stress reverse lastic zone -M(T) lane stain forward lastic zone -M(T) lane stain reverse lastic zone -M(T) Normalized Element size, a Fig. 6: Distinction in crack ti lastic zone size with mesh distinct for MT r = 1 π 2 β (2) Where β is.5 and.167 for lane stress and lane strain, resectively. The mesh density ahead of the crack on Middle crack tension MT geometry is then created with elemental length a.1r, where a is both the element size and the amount of crack extension simulated er load cycle. The imum load is then alied statically to the model and the actual lastic zone size is checked to make sure adequate refinement. Fig. 6 is suggested that an adequate mesh refinement for the reversed lastic zone. Finite element analysis: In the revious sections, all of the comonents of crack closure model are defined. Now, the lasticity-induced fatigue crack closure with cyclically alying loads is modeled by rogram using the ANY Parametric Design Language (APDL). At the first, a mesh is created with an initial crack, and the mesh is loaded by remotely alied tractions. For constant amlitude loading, the loading is cycled between a imum alied stress and a imum alied stress. Initially a large load increment is used to save execution time. After the first node on the crack surface oens. a smaller load increment is used until all the nodes on the crack surface is oen. A larger load increment is then used until the imum load is reached, at which oint crack advance takes lace. The first load ste of crack advance, the nodal fixities on the crack front are removed and are relaced by a force equal to 5% of the node reaction forces. The forces are then reduced over three additional load stes when they become near r P =69 Ma 2a=8 mm 2W=8 mm 2H=8 mm a=4 mm Initial crack ti W=4 mm H=4 mm Fig. 7: Tyical MT crack model with values for numerical study zero, after which they are comletely removed. The entire crack front has now advanced one elemental length erendicular to the crack front. Unloading then takes lace. imilar to the loading, a large increment is used initially, which is decreased when the crack begins to close and is increased again after the entire crack surface has closed. These load cycles are reeated several times until the crack oening levels reach stabilized values. Numerical study: A 2D-dimensional finite element analysis of comact tension MT geometry is shown in Fig. 7. The area is modeled using singular noded elements around ahead of the ti crack and four-noded quadrilateral elements. The material is assumed to be elastic erfectly lastic with modulus of elasticity E =2 GPA and material yield stress =23 MPa. In all cases, the von-mises yield criterion is used with the associated flow rule. Most of the revious finite element analyses reorted in the literature utilized only with four-noded quadrilateral elements. These elements generally do not meet the incomressibility requirement associated with lastic strains as shown by Nagtegaal et al. and are thus disosed to lane-strain locking. The imum normalized stress intensity factor = 1. 7, and therefore aroximately related initial forward lastic region sizes. Fatigue crack growth is modeled by reeatedly loading the geometry, advancing the crack, and then unloading. A large amount of crack growth may be required before stabilized crack-oening values are generated. (Loading increments is.125 ). After each loading amlify, the crack ti node is released, allowing the crack front to advance one elemental length a cycle. The alied load is then incrementally lowered until the imum load is achieved. Considered crack values by resent study are 36

5 Am. J. Al. ci., 4 (6): , 27 Normalized Oening Value, Normalized Element size, a r P Finite Element Method: olanki et al. [4] Present tudy tri-yield Method: Daniewicz [7] Newman [8] Fig. 8: Comarison of calculated crack values for numerical study comared with all other methods, and the result is shown in Fig. 8. CONCLUION Finite element analyses are frequently used to model growing fatigue cracks and the associated lasticity-induced crack closure. Present study is develoed in the finite element analyses to model lasticity induced fatigue crack closure. Desite other conventional methods which use linear elements, the near crack ti in current study is defined by triangular singular elements. Because higher order elements would better cature the head of crack ti stress and strain gradients. Furthermore, the equation develoed by Irwin's Theory is used, for making the lastic zone. The functionality of this scrit is tested by comaring redicted crack oening levels. This verification included a two-dimensional center-cracked geometry. Results have been obtained for a range of different ratios alied stress levels and hardening models. Proence is focused on the difficulties in modeling with high oinion to mesh modification level, crack oening assessment location and crack shae rogress techniques. Finally, comaring two dimensional finite element models of a center-cracked late with a simle stri yield model of closure under lane stress and lane strain. These results in this case in more accurate results comared to result of lane strain stri yield method and results of other technique ublished. 2. Anderson, T.L., Fracture Mechanics: Fundamentals and Alications, 2nd Edition, CRC Press, Boca Raton. 3. olanki,.,.r. Daniewicz and Jr.J.C. Newman, 23. Finite Element Modeling of Elasticityinduced Crack Closure with Emhasis on Geometry and Mesh Refinement Effects. Engineering Fracture Mechanics, 7: olanki,.,.r. Daniewicz and Jr.J.C. Newman, 24. Finite Element Analysis of Plasticityinduced Crack Closure: an Overview. Engineering Fracture Mechanics, 71: Chang, T. and W. Guo, Effects of train Hardening and tress tate on Fatigue Crack Closure. Intl. J. Fatigue, 21: McClung, R.C. and H. ehitoglu, The Finite Element Analysis of Fatigue Crack Closure. Engineering Fracture Mechanics, 33: Daniewicz,.R., A Closed-form mall-scale Yielding Collinear tri Yield Model for train Hardening Materials. Engineering Fracture Mechanics, 49: Newman, Jr.J.C., A Crack Closure Model for Predicting Fatigue Crack Growth under Aircraft ectrum Loading, PA: American ociety for Testing and Materials; : Dougherty, J.D., J. Padovan and T.. rivatsan, Fatigue Crack Proagation and Closure Behavior of Modified 171 teel: Finite Element tudy, Engineering Fracture Mechanics, 56: Ashbaugh, N.E., B. Dattaguru, M. hobaib, T. Nicholas, R.V. Prakash, T.. Ramamurthy, B.R. eshadri and R. under, Exerimental and Analytical Estimates of Fatigue Crack Closure in an Aluum-coer Alloy, Fatigue & Fracture Engineering Materials & tructures, 2: Blom, A.F. and D.. Holm, An Exerimental and Numerical tudy of Crack Closure. Engineering Fracture Mechanics, 22: Pommier,., 21. Plane train Crack Closure and Cyclic Hardening, Engineering Fracture Mechanics, 69: REFERENCE 1. Elber, W., 197. Fatigue Crack Closure under Cyclic Tension. Engineering Fracture Mechanics. 2: