Advanced analysis of crack tip plastic zone under cyclic loading

Accepted Manuscript Advanced analysis of crack tip plastic zone under cyclic loading Michael Besel, Eric Breitbarth PII: DOI: Reference:

S0142-1123(16)30246-8 http://dx.doi.org/10.1016/j.ijfatigue.2016.08.013 JIJF 4053

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

30 April 2016 12 August 2016 18 August 2016

Please cite this article as: Besel, M., Breitbarth, E., Advanced analysis of crack tip plastic zone under cyclic loading, International Journal of Fatigue (2016), doi: http://dx.doi.org/10.1016/j.ijfatigue.2016.08.013

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Advanced analysis of crack tip plastic zone under cyclic loading Michael BESEL*, Eric BREITBARTH German Aerospace Center (DLR), Institute of Materials Research, Linder Hoehe, 51147 Cologne, Germany. *[email protected], +49 (0)2203-601-2068 (Telefax +49 (0)2203-696480) [email protected], +49 (0)2203-601-2504

The different types of plastic zones found at a crack tip under cyclic loading of a commercially available aluminium alloy are studied based on 3-dimensional finite element simulations and mechanical testing. During the experiments, the local strain field at the crack tip is computed based on digital image correlation analysis of the specimen’s surface under different load levels. The simulations with an elastic-plastic material model (bilinear isotropic hardening) predict the experimental surface strain field at the crack tip pretty well. The coupling of experimental results and finite element analysis allows for the accurate distinction into monotonic and cyclic plastic zone, and moreover, into backward cyclic plastic zone and forward cyclic plastic zone. Furthermore, it is possible to calculate the energy accumulation ahead of the crack tip which represents a very useful quantitative measure for the analysis of local fatigue damage accumulation inside the plastic zone ahead of a crack tip.

Keywords: Cyclic loading, Crack, Plastic Zone, FEM, Aluminium

1 Introduction Preservation of resources is unquestionably one of the crucial challenges for today’s engineers. One very effective measure is the application of lightweight design. In doing so, an inherent drawback is that lightweight constructions are generally prone to fatigue. Consequently, fatigue cracks and their propagation under cyclic loading need to be studied. As a common matter of fact, linear elastic fracture mechanics (LEFM) is quite often applicable to analyse (fatigue) crack propagation. Nevertheless, the damage process itself usually takes place within the vicinity of the crack tip in the so called (fracture) process zone. In case of (quasi-)brittle materials microcracking or void formation occurs in this process zone ahead of the crack tip [1, 2]. In contrast, ductile materials preferably develop a plastic zone in front of the crack tip [3, 4, 5]. Thus, the study of this plastic zone can be a useful measure to gain insight into the interrelations of material properties, loading conditions, crack closure phenomena, damage evolution and fatigue crack propagation, respectively [5]. Consequently, numerous studies deal with numerical analyses of the plastic zone [4, 6-17]. Especially recent studies based on 3-dimensional finite element (3d-FE) simulations provide better understanding of the characteristics of the plastic zone as they clearly show that the shape of the plastic zone deviates significantly – depending on the specific configuration – from the classical “dog bone” model [12, 13]. It is well known, that the so called primary plastic zone develops during the first loading phase, while a secondary, much smaller, plastic zone develops during the subsequent unloading [15]. Further cycling results in another type of plastic zone, which is called cyclic plastic zone [5, 16, 17]. The cyclic plastic zone again can be subdivided into two more types of plastic zone, i.e. the forward cyclic plastic zone developing during loading and the even smaller backward cyclic plastic zone appearing during unloading, see [18] and the corresponding Figure 1, respectively. While recent studies dealing with the primary plastic zone [10-14] usually apply 3d-FE simulations, those dealing with the cyclic plastic zone are limited to 2-dimensional FE-models [16, 17].

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Figure 1: Schematic representation of different types of plastic zones at a crack tip under fully reversed cyclic loading taking into account hardening effects; this image is basically adopted from [18] and completed mainly by adding FCPZ and BCPZ

To promote the understanding of the plastic zone under cyclic loading, its different types are investigated in the present paper. With respect to Figure 1 and in anticipation of section 2 the following nomenclature and acronyms are used. PZ:

plastic zone, entire region of plastic straining ahead of a crack tip or notch under monotonic or cyclic loading

PPZ:

primary plastic zone, PZ developing during the very first loading phase

PPPZ: persistent part of PPZ, i.e. region of PPZ that does not show repeated plastic straining under cyclic loading CPZ: cyclic plastic zone, entire region of repeated plastic straining during cyclic loading BCPZ: backward cyclic plastic zone, region of plastic straining during unloading FCPZ: forward cyclic plastic zone, region of repeated plastic straining during repeated loading phase Section 2 of this paper provides selected results of extensive numerical studies of all types of PZ and their spatial extensions, respectively. 3d-FE simulations taking into account isotropic hardening (quite a common assumption, see e.g. [11-14]) are used to study the evolution of the elastic-plastic crack tip fields under cyclic loading. This section will focus on the 3-dimensional shape of the PZ. Afterwards, section 3 presents a series of experiments dealing with the (surface) strain field ahead of a crack tip. These experimental results constitute a kind of benchmark, i.e. their comparison with the numerical results yields a first evaluation of the simulations’ validity. On this basis, it is shown how the combination of experimental results (displacement / strain field) and 3d-FE simulations provides a basic quantitative estimate for the energy input into the PZ through plastic strain accumulation under cyclic loading. Section 4 briefly discusses the quantitative results for this energy accumulation for the two different sheet thicknesses used in these studies. Using two different specimen thicknesses with nominal values of 2 mm and 5 mm, to some extent also allows studying the effect of predominantly plane stress or plane strain conditions. The material used for this research is AA2024-T3. It was chosen as it is widely-used in fuselage structures of modern

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passenger planes (thickness range of about 1 to 6 mm), and as fatigue crack propagation is one of the main aspects in their damage tolerance based design.

2 Finite Element Simulations This chapter presents the main results of the 3d-FE simulations performed to study the different types of plastic zones, namely PPZ and CPZ including FCPZ and BCPZ. The first part provides all essential details of the finite element models used to simulate the evolution of the plastic zone(s) in AA2024-T3 under cyclic loading. Afterwards, the corresponding numerical results are briefly discussed. As shown by Camas et al. [13], the crack front curvature can have a pronounced effect on the characteristics of the plastic zone (size, shape) as well as on associated crack closure effects. Nevertheless, in the present study the actual crack fronts did only show minor curvatures which were a little more pronounced near the surface, see Figure 2a. Those pronounced beach marks of the crack front have been achieved by intended variation of the load ratio (1) and cyclic loading with strongly increased KI,max (2), respectively. After about 30 load cycles with KI,max = 30 MPa √m the specimen has been ruptured by monotonically increasing the tensile loading to reveal this fracture surface. Consequently, the slightly darker part on the right hand side of beach mark (2) represents the residual fracture surface.

Figure 2: (a) Fracture surface showing only slightly curved crack fronts in a specimen with t = 4.8 mm, (b) shape of ¼ PZ for curved and straight crack fronts in a sheet with t = 5.0 mm (KI,max = 30 MPa√m), (c) volume distribution of element slices (slice thickness = 0.1 mm) of PZ parallel to the specimen surface

Some corresponding preliminary simulations of different through thickness cracks revealed, that the plastic zone of an only slightly curved crack front does hardly deviate from that one found at a straight crack front. The overall shape of the PZ is mainly affected by a kind spatial transformation of the plasticized volume due to the curved crack front itself, see Figure 2b. As a result, the corresponding volume distributions along the crack front in slices of constant thickness (tslice = 0.1 mm) parallel to the specimen surface basically share the same characteristics with about 10 % (centre plane slice, z = 0.05 mm) to 30 % (surface slice, z = 2.45 mm) higher volumes for the curved crack front, see Figure 2c. In the case of t = 5.0 mm and KI,max = 30 MPa√m the difference in the total volume of the PZs is only about 13 % (volume of PZstraigth = 21.42 mm³, volume of PZcurved = 24.14 mm²). At first glance, the difference in the volume of the surface slices of about 30 % might be unacceptable. Nevertheless, the admittedly very rough estimation that this factor 1.3 (30 % difference) for the surface near volume of the PZ transforms into about √(1.30) = 1.14 (constant thickness of thin slice) for the surface extension of the bounding rectangle of the PZ in x- and y-direction, respectively. In the

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present study this deviation is thought to be acceptable. Furthermore, as a straight crack front obviously facilitates meshing as well as spatial quantification of the PZ, all simulations here are performed with a straight (through thickness) crack front. Simulation Model The essential details of the material model are summarized in Table 1. All material properties are taken from literature [19], [20]. Consequently, all simulation results presented in this work are actual predictions as neither parameter fitting nor any additional adjustment of the model (e.g. to improve matching with the experimental results) has been performed. A constitutive bilinear isotropic hardening model was used (commands/settings: TB,BISO,1,1; TBDATA,1,345; TBDATA,2,984). The elastic behavior was completely isotropic (commands/settings: MP,EX,1,73100; MP,NUXY,1,0.33).The details of the finite element simulation are summarized in Table 2. Table 1: Details of material model for FE-simulation of AA2024-T3 [19], [20] Elastic-plastic model E-Modulus Poisson Ratio  Yield Stress Hardening Modulus

Bilinear isotropic hardening 73.1 GPa 0.33 345 MPa 984 MPa

Table 2: Details of FE-simulations for plastic zone (Commercial) FE-Code Element Type Formulation Meshing Stress-Strain Data NLGEOM Poisson ratio for plastic deformation Symmetry Plastic zone

Loading Plate thickness

ANSYS R15.0 SOLID185 Linear, enhanced strain Mapped True stress, logarithmic strain On 0.5 (assumption: isochoric, i.e. constant volume) 1/8 - model: MT specimen containing ¼ PZ 100,000 – 150,000 elements Element size: 0.02 - 0.04 mm Element Aspect Ratio AR: 0.5 < AR < 2.0 KI,max = [10, 15, 20, 25, 30, 35, 40] MPa√m 30 load cycles –> 60 load steps (each 4 sub steps) Ratio R = [0.0, 0.1] t = [0.5, 1, 2, 3, …, 15] mm

Figure 3 shows the 3d finite element model of the centrally cracked panel used to study the different types of plastic zones (a), the refined mesh around the crack tip (b), and subfigure (c) schematically illustrates the cyclic loading sequence. As the cracks in these simulations do not propagate, the constant stress amplitude in the cyclic loading sequence is equivalent to a constant nominal KI,max. The initial element size around the crack tip has been defined based on estimations of the size of the PPZ rPPZ and CPZ rCPZ according to the LEFM based approximate solutions provided by Rice [15]. Furthermore, the effect of element size on the 3d-shape of the PPZ has been studied, and if necessary, the mesh has been refined. To obtain a mapped meshed volume, the volume was separated into several blocks and the fragmentation for each line/edge of these blocks was specified (command: LESIZE). To achieve such a systematically refined/coarsened mesh as shown in Figure 3b the line divisions must fit a given ratio that can be found in the ANSYS Help. After meshing (command/setting: VMESH,ALL) the number and size of elements have been checked. The element size in the PZ with the regularly meshed volume in the inner part was reduced if the total number of elements exceeded 256.000 to ensure computational efficiency. This was automatically done taking into account the element aspect ratio, see Table 2. After initial computation with a very fine mesh (element size = 0.02 mm) the element size was successively increased until the desired number of elements has been reached. Lower

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order hexahedron elements (type: SOLID185) were used to obtain a suitable trade-off between computation time and spatial resolution. To avoid the well-known numerical artefacts like “shear locking” and “hourglassing” the elements were extended by an enhanced strain formulation. Load was introduced as a surface load (command/settings: SFA,ALL,,PRES, ) at the top surface. For symmetry reasons the nodes at the left surface were constrained in x-direction (command/setting: D,ALL,UX,0). The nodes of the rear surface were fixed in z-direction (command/setting: D,ALL,UZ,0). Symmetry conditions along the crack plane were used as all nodes ahead of the crack front got a fixed displacement boundary condition in y-direction (command/setting: D,ALL,UY,0), see Figure 3a. Auto time stepping was used for all computations (command/setting: AUTOTS,ON) resulting in three to five sub steps per load step (command/setting: NSUBST,4,5,3). Within each load step the load was applied as ramp (command/setting: KBC,ON). Also large-deflection effects were taken into account (command/setting: NLGEOM,ON). Consequently, the computing time (Intel desktop PC, 8 x 3.40 GHz, 8 GB RAM) for the elastic-plastic simulation of 30 load cycles (i.e. 60 load steps) varied between approximately 35 h and 60 h. As it is questionable, in how far an even slowly (KI,max ≈ const.) propagating crack tip affects the characteristics of the PZ, this situation is studied for t = 2 mm. Therefore, in another simulation the PZ ahead of a crack tip propagating by one element (i.e. 0.05 mm) per load cycle is computed as illustrated in Figure 4. In this case, as the crack length changes during the propagation, the load level is stepwise reduced to keep the KI,max constant, i.e. the only difference between this simulation procedures and that one shown in Figure 3 is the fact that the crack tip moves in steps of one element size per cycle.

Figure 3: Finite element model for 3d-analysis of plastic zone, centrally cracked panel: (a) ¼-PZ-model and boundary conditions, tensile load is applied to the upper side in y-direction, (b) fine mesh around crack front and the expected area of plastic zone, (c) cyclic loading sequence

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Figure 4: Procedure for the simulation of the effect of a propagating crack on size of the plastic zone

During this crack propagation simulation, the nodes (I) to (V) are disassociated successively after each full load cycle, i.e. node (I) between load steps 2 and 3, node (II) between 5 and 6, and so on. This disassociation was conducted in two steps: firstly all corresponding nodes were selected, and secondly, their initially fixed displacement boundary conditions were deleted (command/setting: DDELE,ALL,UY). Keeping KI,max constant reveals the interdependencies between existing PZ and “newly developing” one during further cycling. Crack propagation is simulated until Δa = 0.5 rpl = 1.2 mm, i.e. until the crack propagated through half of the initial PZ with a radius of rpl = 2.4 mm Thus, a step size of 0.05 mm results in a simulation of 24 load cycles. Figure 5 shows the comparison of the outer boundaries of the initial PZ and the final PZ after cyclic crack propagation of Δa = 0.5 rpl under a constant load amplitude with KI,max = 30 MPa√m and R = 0. The crack propagates in x-direction, while the z-coordinate represents the thickness direction. As a ¼PZ-model is used, z ranges from 0 mm (centre of sheet) to 1 mm (surface). The image in Figure 5a shows the outer boundaries corrected for translation due to crack (tip) propagation. Although such static “3d-diagrammes” are always somehow difficult to deal with it can clearly be seen that the shapes of both PZs are very similar. This first impression is confirmed in the diagram in Figure 5b which shows the spatial distances (i.e. their size difference) between both PZs’ boundaries in crack propagation direction x. The differences in the crack plane (i.e. xz-plane) are mostly below 0.1 mm, i.e. less than 5 % of the PZ’s extension in x-direction.

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Figure 5: Results of ¼-model of plastic zone in AA2024-T3 (t = 2 mm, KI,max = 30 MPa√m, R = 0): (a) outer boundary of PZs before and after crack propagation, (b) size difference between initial and final (“primary”) plastic zones after crack propagation of Δa = 0.5 rpl = 1.2 mm

Based on these results it is concluded that a small change of the position of the crack tip as typically found during stable fatigue crack propagation does hardly affect the (spatial) development of the plastic zone under cyclic loading. Furthermore, the expected crack propagation rate in the alloy under consideration for K = 30 MPa√m is only about 0.0027 mm/cycle [21]. Thus, the maximum change in the crack length during 30 load cycles should be less than 0.1 mm on each side of the crack, which is only about 0.5 % of the initial overall crack length (ai = 20.0 mm). This justifies both the assumption of a constant KI,max during cyclic loading and the simplification using a quasi-static crack tip. Consequently, the following simulations dealing with PZ / PPZ and CPZ under cyclic loading are performed with a quasi-static, i.e. non-propagating crack tip. This simplification reduces the computational effort significantly, and it is still expected that the main characteristics of PZ / PPZ and CPZ will be computed with sufficient accuracy. Results and Discussion This section briefly summarizes the main findings of the simulations and of the measuring of the plastic zones. For the sake of brevity, only selected figures and diagrams are shown here to document the conclusions drawn based on these investigations. If not stated otherwise, all other corresponding diagrams and figures not shown in this paper corroborate the conclusions in the same manner. Primary / monotonic Plastic Zone The PPZs in plate with two different thicknesses are shown in Figure 6. As can easily be seen, the overall size of the PPZ strongly increases when the load is increased such that the maximum mode I stress intensity factor (SIF) KI,max rises from 20 to 30 MPa√m. This “growth” of PPZ is more pronounced perpendicular to the loading direction, i.e. the increase in crack propagation direction is greater than that one perpendicular to the crack plane. As also observed by other authors [12], [13], [14] before, Figure 6 clearly shows that the maximum spatial extend of the plastic zone that is found in x-direction is not located on the surface but inside the bulk. Furthermore, its position relative to the surface changes with increasing load. While subfigure (a) shows two maxima (see red arrows) located approximately 0.6 mm below each surface the higher load in subfigure (b) results in only one

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maximum in the centre of the sheet, i.e. 1.0 mm below the surface(s). In this case this maximum is located in the crack plane, i.e. xz-plane. As mentioned in the introduction, while t = 2 mm is assumed to represent a predominantly plane stress state the results in Figure 6 (c) and (d) for t = 5 mm should to a greater extent reflect a plane strain condition. In both load cases the maximum extension of the PPZ in x-direction (red arrows) in principle resembles the characteristics of the PZ in the thinner sheet, i.e. increasing load results in a shift of the maximum extension from surface near areas into the centre of the bulk. Such similarities were observed for a wide range of load cases and sheet thicknesses (not shown here) leading to the assumption that specific combinations of sheet thickness t and maximum loading condition KI,max will basically result in PPZs with similar characteristics.

Figure 6: ½ primary plastic zone in terms of equivalent plastic (von Mises) strain for t = 2 mm: (a) KI,max = 20 MPa√m, (b) KI,max = 30 MPa√m; and for t = 5 mm: (c) KI,max = 20 MPa√m, (d) KI,max = 30 MPa√m

For the sake of brevity and without neglecting the 3-dimensional aspects of these investigations, in this paper the related considerations will be limited to the volumes of the PPZ and its corresponding CPZs. Consequently, Figure 7 represents the volume of the PPZ and the corresponding standardized volume (i.e. divided by sheet thickness t), respectively, for maximum loads KI,max ranging from 10 to 40 MPa√m, and different sheet thicknesses t ranging from 0.5 to 15.0 mm. The volume of the PPZ in subfigures (a) and (b) shows the self-evidently expected characteristics, i.e. the increase in volume with increasing maximum load KI,max is (naturally) more pronounced for higher thicknesses, see Figure 7a. This increase follows an almost linear trend as shown in subfigure (b).

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(a)

Volume of PPZ vs. KI,max

(b) t [mm]

t = [0.5,..., 15.0] mm

Standardized Volume of PPZ vs. KI,max

KI,max [MPa√m]

KI,max = [10,..., 40] MPa√m

KI,max [MPa√m]

(c)

Volume of PPZ vs. t

Sheet thickness t [mm]

(d) t [mm]

Standardized Volume of PPZ vs. t

KI,max [MPa√m]

t = [0.5,..., 15.0] mm KI,max = [10,..., 40] MPa√m

KI,max [MPa√m]

Sheet thickness t [mm]

Figure 7: Effect of loading KI,max and thickness t on volume of PPZ: (a) volume of PPZ as a function of KI,max, (b) volume of PPZ as a function of t, (c) and (d) analogous with volume of PPZ standardized for thickness t

Standardization of the volume of the PPZ with the sheet thickness t, i.e. dividing by t, reveals some interesting findings. For all sheet thicknesses ranging from 0.5 to 15.0 mm the same standardized volume of PPZ is developed under different stress intensity factors that do not differ more than about 3 MPa√m, see Figure 7c exemplarily between 15 and 20 MPa√m or between 35 and 40 MPa√m. The smallest standardized volume is in principle developed in the thinnest sheets, i.e. t = 0.5, 1.0, and 2.0 mm, respectively. But plotting the same set of data with a different abscissa, i.e. standardized volume over t, provides another insight into the effect of the sheet thickness t, see Figure 7d. What firstly appeared as an almost linear trend on the volume of PPZ in subfigure (b) transforms into a mostly non-linear relationship for the PPZ’s standardized volume shown in subfigure (d). It can clearly be seen that the different thicknesses must provide different inherent constraints for the plastic flow at the crack tip, especially for higher values of KI,max. The maximum standardized volume for KI,max = 20 MPa√m is found in the sheet with a thickness of t = 2 mm, but with increasing loads the

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maximum occurs in thicker sheets, i.e. {25 MPa√m, 3 mm}, {30 MPa√m, 4 mm}, {35 MPa√m, 6 mm}, and {40 MPa√m, 9mm}, respectively. So basically, the commonly accepted idea that a plane stress condition (i.e. thinner sheets) results in a larger PPZ than a plane strain condition (i.e. thicker sheets) is not that strongly reflected in these volume data mainly as they actually incorporate the nonlinear effects of 3-dimensional plastic material flow. Generally, all regions of plastic straining during fatigue cycling should be located well within the region of the PPZ, and the overall volume of the PZ (i.e. all areas of plastic straining) under cyclic (constant amplitude) loading should not change. This assumption is clearly confirmed by the diagram in Figure 8 showing the behavior of the PZ under further cycling with a constant stress intensity factor of 20 MPa√m. Only due to the tiny ranges of the ordinates both curves show a pronounced oscillation as result of the elastic deformation during cycling. Their amplitudes clearly reflect the elastic volume change due to  = 0.33 (see Table 1). The observable changes of the volumes due to plastic deformation (i.e. the changes in the mean values of both graphs) are in the range of up to 0.05 %. The corresponding total change of the mean volumes until reaching a constant mean value (e.g. after 20 Load steps for t = 5.0 mm) is about 0.3 %. This is in the same order as the elastic volume change during cycling. Consequently, it can be concluded, that the overall volume of the PZ is nearly constant and equal to that of the PPZ. Furthermore, the regions outside the cyclic plastic zones can consistently be called persistent part of PPZ as suggested in Figure 1.

Figure 8: Volume of PZ under cyclic loading with KI,max = 20 MPa√m, R = 0: (a) t = 2 mm (left abscissa), (b) t = 5 mm (right abscissa)

Secondary / Cyclic Plastic Zone As a well-known matter of fact, the so called cyclic plastic zone (CPZ) shows continually significant plastic straining under cyclic loading. Figure 9 illustrates the different regions of the plastic zone under cyclic loading. The PPZ develops during the very first loading phase, see also Figure 1, and its initial total volume consists of the sub-volumes (I), (II), and (III), as illustrated in Figure 9. Already during the first unloading phase repeated plastic straining occurs in a prism-like shaped region in the vicinity of the crack tip reaching from surface point B through the whole thickness of the plate. This region represents the BCPZ (II). The subsequent loading phase of the second load cycle results in more pronounced plastic activity in the original region of the BCPZ (II) and additionally in the more complex shaped region (III) forming the so called FCPZ, please see points A, B and C (centre of plate) in Figure 9. Consequently, the total volume of the FCPZ is the sum of sub-volumes (II) and (III), and equals the volume of the CPZ. The extension of the FCPZ in crack propagation direction is about six times that of the BCPZ. As already shown above, the rest of the original PPZ does hardly undergo any changes during further cycling, and is therefore in this paper called persistent part of PPZ (PPPZ).

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Figure 9: Results of ¼ -FE-model after 30 load cycles, KI,max = 30 MPa√m, R = 0, t = 5 mm: Decomposition of plastic zone (PZ) into (I) its persistent part of PPZ (i.e. PPPZ), (II) the backward cyclic plastic zone (BCPZ), and (III) the forward cyclic plastic zone (FCPZ)

Of course, the existence of the CPZ is mainly the result of the material’s hardening behavior (see Figure 1), and the associated evolution of high local residual stresses in the vicinity of the crack tip. Figure 10 shows the residual stresses (in the crack plane) at the tip of a crack in a sheet with t = 5 mm after 60 load cycles with KI,max = 30 MPa√m and R = 0. Compressive residual stresses up to several hundred MPa (σxx,min ≈ -580 MPa, σyy,min ≈ -850 MPa, σzz,min ≈ -530 MPa) dominate very close to the crack tip, while tensile residual stresses up to about 120 MPa (σxx,max ≈ 120 MPa,

σyy,max ≈ 70 MPa, σzz,max ≈ 80 MPa) are found in distances up to about 2 mm ahead of the crack tip for σyy,max. Basically, the residual stresses shown in Figure 10 superimpose the cyclic loading, and thus, the tensile residual stresses easily explain the evolution of the FCPZ during the next loading phase. Consequently, this effect does not need to be discussed in detail here. In contrast, the corresponding compressive residual stresses do not directly explain the BCPZ as their values will obviously change with the evolution of the FCPZ during the next loading phase. Thus, their effective magnitude during the unloading phase is basically unknown. Nevertheless, as the material model does not provide any kind of softening mechanism (see Table 1) it is unquestionable that their evolution during cycling (in connection with local hardening) must be the explanation for the existence of the BCPZ.

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Figure 10: Results of ¼ -FE-model, KI,max = 30 MPa√m, R = 0, t = 5 mm: Residual stresses ahead of crack tip after 60 load cycles

As the term residual stress basically reflects an inner stress state in the absence of any type of external loading (e.g. mechanical loading, thermal loading, etc.), the more suitable term of instantaneous residual stresses (IRS) shall be introduced here, and it is defined as follows: The plastic flow at a crack-like notch or defect (or an existing, plastic-zone-free crack tip) during the loading phase, i.e. the evolution of the primary plastic zone results in a localized, heterogeneously distributed concentration of elastic energy (i.e. stress, strain). This elastic energy field and its associated stresses allow for the evolution of a backward cyclic plastic zone during unloading. The underlying mechanisms are basically the same as during the evolution of conventional plastic-flow associated residual stress fields. Furthermore, the local material behavior under subsequent mechanical unloading is affected in the same way. Nevertheless, as this specific stress field continuously changes during the unloading phase, the corresponding stresses shall be called instantaneous residual stresses (IRS) and instantaneous residual stress field (IRS-field), respectively. Only such type of IRS-field can explain the evolution of the BCPZ, as the energy needed for the local plastic flow during unloading can only be provided by the elastic energy stored in this instantaneous residual stress field during the loading phase. The effect of the IRS-field on the locally acting cyclic loadings is briefly discussed for t = 5 mm, see Figure 11. The provided diagrams show the local stress evolution during fatigue cycling. These local stress values represent the superposition of outer loading conditions and internal stresses of the IRSfield. Their evolution during cycling basically reflects both, the stress redistribution due to plastic straining and the local residual stresses themselves. The experiments presented in this paper provide only surface strains (based on surface observations). Consistently, here the discussion of the numerical results is limited to four points located on the (traction free) surface of the FE-model. Point 1 is directly located at the crack tip, and during the first load step the local tensile stress in mode I direction, i.e. σyy reaches about 400 MPa while σxx is about 40 MPa. During the first unloading phase the local stress values change to σyy ≈ -530 MPa, and σxx ≈ -240 MPa, respectively. Consequently, the corresponding local stress range in y-direction is 930 MPa. During the next four load cycles this local value under minimum loading changes stepwise to about σyy,min ≈ -550 MPa (see load step 8), while it takes about six load cycles, i.e. load step 13, to stabilize the maximum local tensile loading to σyy,max ≈ 550 MPa. Consequently, the effective local stress range under cyclic loading changes to about 1100 MPa. Point 2, located about 0.2 mm ahead of the crack tip, see Figure 11, shows a slightly different behavior. The maximum local stress decreases during the first eight load cycles from σyy,max ≈ 370 MPa at load step 1 down to σyy,max ≈ 300 MPa, while the minimum local stress value increases within the first four load cycles from σyy,min ≈ -310 MPa to about σyy,min ≈ -350 MPa (see load step 8). The minimum and

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maximum values of σxx reach their approximately constant values after about 24 load cycles showing σxx,min ≈ -280 MPa, and σxx,max ≈ 200 MPa, respectively. (*)Origin located at crack front in centre plane of plate

y z

x (*)

Local stress xx

Local stress yy

Figure 11: Results of ¼ -FE-model, KI,max = 30 MPa√m, R = 0, t = 5 mm: Plastic zone and its corresponding local cyclic loading ranges (in terms of maximum and minimum local stresses)

Point 3 (see Figure 11), 1 mm ahead of the crack tip, is already well outside the PZ on the surface, but still within its subsurface extension. Consistently, an only slightly pronounced effect of the IRS-field on its local stress value σxx can be seen, while σyy shows constant values of σyy,max ≈ 330 MPa and σyy,min ≈ 20 MPa from the very beginning. The maximum value of σxx changes slightly within the first four load cycles from 195 MPa to 200 MPa, the corresponding minimum value σxx,min takes about 15 load cycles to change from about -10 MPa to about 0 MPa. Point 4 is located well outside the PZ with a distance of 2 mm to the crack tip, see PZ in Figure 11. From the very beginning nearly constant local stress values of about 40 MPa and 290 MPa for σyy, and of about 35 MPa and 180 MPa for σxx are observed.

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The stress evolution in the thinner plate t = 2 mm shows a very similar behavior, see Figure 12. The characteristics of the different stress evolutions are very comparable to those for the thicker sheet but with deviating absolute values.

y z

x (*)

Point 1 Point 2 0,4mm

Point 3 Point 2 Point 1

Point 3

0.2mm 1mm

Point 4

Point 4 2mm

Local stress xx

Local stress yy

Figure 12: Results of ¼ -FE-model, KI,max = 30 MPa√m, R = 0, t = 2 mm: Plastic zone and its corresponding local cyclic loading ranges (in terms of maximum and minimum local stresses)

At point 1 σxx ranges from about -190 MPa to 190 MPa, while σyy reaches values between -490 MPa and 500 MPa. The average stress values at point 2 are higher for the thinner sheet ranging from 370 MPa to 290 MPa for σxx and from -320 MPa to a maximum of 395 MPa (first loading) for σyy. The location of point 3 with respect to the PZ is somehow comparable for both sheet thicknesses (please compare Figure 11 and Figure 12). This is also reflected in the nearly constant minimum and

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maximum stresses for t = 2 mm with values of -10 / 260MPa for σxx and -30 / 370 MPa for σyy, respectively. While point 4 in Figure 11 is fully outside the plastic zone, in case of t = 2 mm the PZ under the surface extends well beyond this point. Nevertheless, also in this case the associated stress ranges are nearly constant with minimum and maximum values of about 80 MPa / 250 MPa for σxx and 80 MPa / 370 MPa for σyy, respectively. In comparison with t = 5 mm the thinner sheet shows clearly higher stresses at point 4 which is very consistent with the higher total strain caused by the PZ directly located under the surface below point 4. In the case of the thinner sheet (t = 2 mm), the evolution of the minimum and maximum values of σyy at Points 1 and 2 during the first five to ten load cycles is clearly reflected in the corresponding volumes of the BCPZ and FCPZ, respectively, see Figure 13 showing the relative volumes as fraction of the total volume of the PPZ. Keeping in mind that the overall volume of the PZ (≈ PPZ) is approximately constant, the volume of the BCPZ for t = 2 mm shows its strongest increase during the first four load cycles, i.e. load step 8 in Figure 13a, while its maximum value of more than 4.05 % is reached between load steps 32 and 52 (i.e. after 16 to 26 load cycles). Afterwards, the relative volume of BCPZ starts to slightly decrease. The overall change of the volume of the BCPZ for t = 2 mm is only about 0.37 percentage points, which is approximately 10 % of its initial volume.

Figure 13: Results of ¼ -FE-model, KI,max = 30 MPa√m, R = 0: Evolution of relative volumes of (a) BCPZ and (b) FCPZ

The evolution of the BCPZ in the thicker sheet (t = 5 mm) basically shares the same characteristics. Beside a small decrease around load step 12 the relative volume of the BCPZ reaches a maximum of about 3.05 % of the volume of the PPZ in load step 20 followed by a monotonic decrease down to 2.65 % in the final load step 60. Anyhow, also these values show that the total volume of the BCPZ during cyclic loading only changes by +5% and -9% of its initial volume. Based on this observation it can be concluded that for the given load cases the volume of the BCPZ during cyclic loading is approximately constant. Anyhow, as the evolution of the plastic zone strongly depends on load level, crack length, sheet thickness, sheet width and ligament ahead of the crack, respectively, the validity of these observations is limited to the specific configuration considered here. The evolution of the volume of the FCPZ in the thinner sheet t = 2 mm, see Figure 13 (b), more clearly reflects the change in the local stresses at Points 1 and 2. Its maximum value is found directly during the second loading phase, i.e. load step 3. At the beginning the volume of the FCPZ is about 39 % of that of the PPZ. Further cycling results in a strong decrease of the FCPZ reaching its minimum value of about 12 % at load step 11 (i.e. during the sixth load cycle). Afterwards the volume of the FCPZ gradually increases during further cycling and reaches its final value of about 15 % after a total of 29 load cycles (i.e. load step 59). In contrast, the volume of the FCPZ in the thicker sheet t = 5 mm

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remains comparably constant under cyclic loading as it deviates only between about 29 % and 32 % of the volume of the PPZ. Based on these observations it is concluded that the actual damage accumulation under fatigue loading is limited to a comparably small fraction of the initial PPZ covering between 40 % (propagating crack) and 12 to 15 % (non-propagating crack) of its volume for t = 2 mm and between 29 % and 32 % for t = 5 mm. Taking into account that the different types of plastic zones discussed here extend through the whole plate thickness direction, the average extension of the damage contributing cyclic plastic zone at a non-propagating crack tip in x- and y-direction is in the range of (t = 5 mm) or well below (t = 2 mm) one tenth of the extension of the PPZ. 3 Experimental Program and Its Computational Analysis The aluminium alloy AA2024-T3 has been ordered from a commercial aluminium supplier, i.e. it basically reflects the mechanical properties of technical alloys as actually used for aircraft structures. MT-specimens (middle-tension, 400 x 160 mm²) with a crack-like machined notch in their centre were used to study the different PZs under pure mode I loading, see (4) in Figure 14. All specimens were coated with two different stochastic black and white patterns (2), a coarse one on their back sides (observed with a stereo-camera-system (1)) and a finer one on their front sides (observed with a microscope (3)). This b/w-pattern is used for surface strain analysis based on digital image correlation (DIC, see e.g. [22] for a detailed introduction into image-based measuring techniques) performed with the commercial software code ARAMIS [23].

Figure 14: Experimental setup and details of MT-specimen with central crack (2a = 40 mm) used for mechanical testing

The stereo-camera-system is operated with a comparably large field of view covering the complete centre area of the specimen. The recorded images are mainly intended to verify the actual loading conditions during the experiment in terms of 3-dimensional displacements. In the case of single-edgenotched-tension (SENT) specimens such verification is crucial as the asymmetric loading conditions generally result in comparably large errors if not properly taken into account, see [24]. In the given case, using the MT-specimens resulted in nearly perfectly uniform displacements, i.e. the loading / boundary conditions of the specimen are very close to those when using an ideal test-rig of unlimited stiffness. Only the characteristics of the specimen geometry itself dominated its slightly inhomogeneous displacement field. Consequently, the corresponding images do not need to be discussed here.

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For the local DIC observations of the PZ a Zeiss STEMI 2000-C microscope with a mounted Canon EOS 1000D reflex camera were used, see Table 3. With a 1,6x magnification the microscope observed an area of about 6.8 x 4.5 mm² containing the crack tip including its different types of plastic zones. The camera’s CMOS sensor has 2592x3888 pixels. For the DIC analysis a facet size of 60x60 pixels and a facet distance of 30 pixels were chosen. Consequently, the local resolution in this setup is 52.5x52.5 μm². This cannot be achieved with the stereo-camera setup at the rear side of the specimen with a spatial resolution of ~1.1x1.1 mm² which is almost half the size of a plastic zone. Before the first loading several images were taken along the expected crack path needed as reference images for the DIC analysis. Based on such image series, the different types of plastic zones will be discussed. The details of the experimental program shown in this paper are summarized in Table 3. Table 3: Details of Experimental program Test machine

2D displacement & strain “measurement” for plastic zone

3D displacement & strain “measurement” for rear side

Specimen-type Specimen thickness t Fracture mechanical loads Cyclic loading

Servo-hydraulic, standard-type DIC-based, ARAMIS system Microscope: Zeiss STEMI 2000-C (1.6x) Camera: Canon EOS 1000D Sensor: 22,2x14,8mm² CMOS with 10,1Megapixel (2592x3888 pixel). Image section: 6.8x4.5mm² Facet size: 60 pixel / 104.9μm Facet distance: 30 pixel/ 52.5 μm Filter/kernel/repeats: Mean value/5 x 5/1 GOM Aramis 12M Lens: 50mm Slider distance: 240mm Measuring distance: 630mm Measuring volume: 250 x 190 x 44 mm³ Facet size: 19 pixel / 1.1mm Facet distance: 15 pixel/ 0.9mm MT (Nominal: 2.0 mm) Actual value: 2.0 mm (Nominal: 5.0 mm) Actual value: 4.8 mm KI,max = [20, 25, 30] MPa√m f = 30 Hz, R = 0.0

While the thinner specimens satisfy the nominal thickness of 2.0 mm, the thicker specimens slightly deviate with an actual thickness of 4.8 mm. This deviation is about 4 % and has been taken into account for the calculations of stress intensity factors. In contrast, as such small deviation does hardly affect the numerical results for the different types of plastic zones, the time-consuming elastic-plastic FE-simulations have not been repeated. Nevertheless, it will be shown that also in the case of the thicker specimens the numerical results match pretty well with the experimental surface observations. Fatigue crack propagation from the starter notch has been achieved using maximum stress intensity factors well below 10 MPa√m. Consequently, the initial crack tips before increasing the mechanical loads showed only very small PPZs that could be neglected during the subsequent study of the different types of plastic zones developed under much higher loads. The crack propagation rates even under maximum cyclic loadings were only in the range of up to 2 µm per load cycle. Thus and as already indicated in section 2, simulations with a non-propagating crack tip should be suitable to investigate and analyze the different types of plastic zones. Nevertheless, to preserve accurate usage of terms in the following sections only the term monotonic PZ will be used to describe that region of a plastic zone that does no longer change after the first load cycle of a series with constant amplitude loading. Primary / monotonic Plastic Zone The image series in Figure 15 shows the monotonic PZs (in unloaded specimen) in terms of equivalent strain (von Mises) after different crack tip loads KI,max of 20, 25, and 30 MPa√m, respectively. The

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maximum value of the scale has been chosen with respect to the definition of the 0.2 % (i.e. technical) yield strength. Consequently, the PPZs should be adequately represented by the red regions, as they indicate regions with plastic strains ≥ 0.2 %. It can be clearly seen for both sheet thicknesses that the size of their monotonic PZs increases with increasing load level. In the case of t = 4.8 mm increasing the crack tip load by 50 % (i.e. from 20 to 30 MPa√m) doubles the height of the monotonic PZ from about 1.8 to 3.6 mm.

Figure 15: Monotonic plastic zones in terms of local equivalent strain (von Mises) in unloaded specimen

In contrast to the volume data of the PPZs, see Figure 7, the effect of the predominating mechanical condition is clearly reflected in the shape of the monotonic PZs. The thinner specimen (t = 2.0 mm, plane stress) shows a nearly circular-shaped monotonic PZ while that one of the thicker specimen (t = 4.8 mm, plane strain) is butterfly-shaped. The effect of plane stress and plane strain is also clearly seen in the smaller height of the monotonic PZ under plane stress condition. Increasing the load KI,max from 20 to 30 MPa√m again nearly doubles the height from 1.5 to 3.0 mm (t = 2.0 mm). Consequently, the height of the monotonic PZ in the thinner plate under the same maximum load is about 20 % smaller than that in the thicker plate. In contrast, as can easily be seen in Figure 15, the specimen thickness does hardly affect the horizontal extension of the monotonic PZ. Figure 16 compares experimental and numerical results and provides a simple basis for a first estimation of the validity of the 3-dimenional elastic-plastic simulations, and the applied material model, respectively. The experimentally determined strain field is overlain by the isolines of the corresponding numerical results. The strain values of the different isolines coincide with the colorscaling as indicated by several asterisks “ * ”. It can be seen that especially the results for the thicker sheet (t = 4.8 mm) match very well. The results for the t = 2.0 mm sheet show some deviation that is

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slightly more pronounced for higher strains, i.e. in the centre of the PZ. Nevertheless, experimental as well as numerical results reflect the butterfly-shape of the monotonic PZ and its spatial extension (on the surface) pretty well. Taking into account the natural shortcomings of such kind of experimental results (e.g. scatter in local material properties, different types of measurement errors, etc.) it can be concluded that the simulations seem able to adequately predict the plastic straining ahead of the crack tip. Consequently, they should be suitable to study and evaluate the different types of plastic zones as will be presented in the following subsection. DIC [%] 0.700*

FEM

0.600* 0.525 0.450* 0.375 0.300* 0.225 0.150 0.075 0.000

PZ’s outer contour Figure 16: Comparison of experimental and numerical results for monotonic PZ (KI,max =30 MPa√m ) in terms of equivalent strain (von Mises); DIC: total strain after unloading, FEM: contour lines (i.e. strainisolines) for the corresponding plastically strained regions

Cyclic Plastic Zone As shown above the PPZ or, more precisely, the monotonic PZ could easily be recognized in the remaining (i.e. mainly plastic) strain field determined based on the DIC analysis of the specimens after different maximum loadings. In contrast the visualization of the CPZ, and its BCPZ, and FCPZ, respectively, turned out to be much trickier. At first glance, measuring the BCPZ just means to do DIC with the specimen in loaded and consecutively unloaded condition. And the reverse should hold for the FCPZ. The difficulty with this approach is that in both cases the characteristics of the computed strain field are mainly dominated by the pronounced elastic part of the crack tip field. This elastic part does not only reflect the theoretical strain concentration at a crack tip as could be easily calculated based on LEFM, but also stress redistribution effects due to local plastic flow, and the resulting local hardening effects. Unfortunately, with the given experimental (and especially optical) setup it has not been possible to clearly distinguish BCPZ and FCPZ, and consequently, those results are not shown here. Instead, to overcome all these difficulties a more advanced approach will be presented, that consists of several steps that are summarized as follows. (1) Computation of displacement field ahead of the crack tip (in each case) based on two consecutive load steps with digital image correlation (here: ARAMIS).

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(2) Mapping of the experimental displacement field onto the mesh of a 2d-FE-model with higher order PLANE182 elements and plane stress assumption (here: ANSYS). (3) The (spatial) derivative of this displacement field represents the associated total strain field and is directly accessible via the postprocessor of the FE-software. (4) Based on a suitable material model (here: elastic-plastic, bilinear, isotropic hardening, see Table 1): Computation of the corresponding stress and strain fields for each load step. This allows for the decomposition of the total strain field into its elastic and its plastic part. (5) Selection of all finite elements undergoing plastic straining and plotting of the results of interest. One inherent drawback of this approach is that the material model itself is needed for the identification of the areas of plastic strain. Consequently, the comparison of those experimental results with the “pure” FE-results should not be regarded as basis for a substantial validation. Nevertheless, such comparison will at least serve as first estimate for the suitability of the simulation to deal with the CPZ. Figure 17 shows a selection of numerical and experimental results for the BCPZ and FCPZ, respectively. As already explained above, the experiment started with a long crack propagation phase at very low maximum loads, i.e. the initial crack before monitoring the different types of PZs shown here had a very small plastic zone. The monotonic PZ shown here was recorded after the first load cycle with an increased maximum load of KI,max = 30 MPa√m. The overall extension of this monotonic PZ (grey) is about 2.5 x 4.2 mm² (width x height) in case of the FE-simulations, and 3.3 x 4.2 mm² (including the small, isolated areas) in case of the experimental results. The size of the BCPZ, see subfigures (a) and (d), differs slightly more with approximately 0.4 x 0.8 mm² for the FE-results, and about 1.0 x 1.5 mm² in the experimental observation. In contrast, the spatial overall extension of the FCPZ matches pretty well, as can easily be seen by comparing subfigures (b) and (e). Also the much greater size of the FCPZ if compared to the BCPZ is similarly reflected in both, experimental and numerical results. Furthermore, the decrease of the FCPZ (compare also Figure 13b) with increasing number of load cycles is clearly observed. While the initial FCPZ in subfigures (b) and (e) covers most of the monotonic PZ, after five load cycles a much smaller FCPZ is found in subfigures (c) and (f). Beside these size comparisons, Figure 17 offers additional information about the local energy input during fatigue cycling. The color scale reflects the accumulated plastic strain energy density, i.e. it shows the specific local energy accumulation ahead of the crack tip. Of course, most energy is accumulated in the vicinity of the crack tip. While the outer shape of the CPZs held in dark blue contains energy densities up to 20 N/mm², the areas colored in dark green (i.e. > 80 N/mm²) accumulate more than four times that energy. In the case of the experimental results, this area of high energy accumulation is initially only about 0.2 x 0.3 mm² for both, BCPZ and FCPZ (subfigures (d) and (e)). After five load cycles this area of high energy increases to about 0.3 x 0.5 mm², see subfigure (f). The FE-simulation qualitatively shows the same tendency, but the region of high energy is always significantly smaller. In case of subfigure (b) it consists of only two elements, while after five load cycles it contains eight elements, see subfigure (c).

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FE-results for CPZ (a) BCPZ after first unloading

(b) FCPZ after second loading

(c) FCPZ after five load cycles 𝑁

𝑤𝑝𝑙 𝑖=1

[N/mm²]

1 mm

Experimental results for CPZ (d) BCPZ after first unloading

(e) FCPZ after second loading

(f) FCPZ after five load cycles 𝑁

𝑤𝑝𝑙 𝑖=1

1 mm

[N/mm²]

Figure 17: Comparison of experimental and numerical results for CPZ (t = 2.0 mm, KI,max = 30 MPa√m) in terms of accumulated plastic energy density. (a)-(c) FE-results, (d)-(f) experimental results

Obviously, a more accurate comparison of these regions of high energy accumulation demands a much higher resolution in the experimental results (i.e. higher magnification with finer b/w-pattern for DIC) as well as a much finer finite element mesh in the simulations. Nevertheless, and consistent to common knowledge about fatigue crack propagation (in ductile materials) these results clearly confirm that the main damage accumulation process takes place inside a very small region of the CPZ, i.e. in a limited area in front of the crack tip that is much smaller than the easily observable monotonic PZ. As the numerical simulations reflect the experimental surface observations under fatigue cycling very well, the following section will briefly discuss some (preliminary) energy considerations based on these FE-simulations. 4 Energy Accumulation under Cyclic Loading In this section the energy accumulation inside the PZ under cyclic loading is studied in terms of the accumulated plastic energy. As plastic strain is basically isochoric (see also Table 2) this energy can easily be calculated by integrating the accumulated plastic energy density (see Figure 17) over the volume of the PZ. In the given case a 3-dimensional finite element model is used for this evaluation simplifying this calculation to equation (1) which represents the summation over all elements of the PZ as exemplarily shown for the specimen surface in Figure 17 (a), (b), and (c), respectively.

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𝑤

(1)

with

𝑤

Based on Figure 18 the energy accumulation during cyclic loading will briefly be discussed for the different sheet thicknesses. Subfigure (a) shows the individual energy inputs during fatigue cycling separately for loading (i.e. in the FCPZ) and unloading (i.e. in the BCPZ) phase, respectively. These values for energy input are consistent to the results discussed in section 2, i.e. the quantitative differences between t = 2.0 mm and t = 5.0 mm correlates pretty well with the volume of the PZ and can be mainly attributed to the sheet thickness itself. (a)

Plastic energy input per load step

(b)

Accumulated plastic energy input

Figure 18: Plastic energy input per cycle and plastic energy accumulation in PZ for KI,max = 30 MPa√m

In the beginning the energy input into the FCPZ (loading) is about 20 % higher than that into the BCPZ (unloading). All curves monotonically decline during the first ten load cycles, i.e. until load step 20. While the graphs for the BCPZs continue declining down to about 0 mJ after 29 load cycles, the values for the FCPZs start increasing again. After ten load cycles the energy input into the FCPZs is always more than double of that into the corresponding BCPZs which clearly documents that even under such quasi-static condition (i.e. without crack propagation) the main damage accumulation must occur during the loading phase. Subfigure (b) provides the corresponding graphs for the accumulated energies. The strong initial decrease of all graphs in subfigure (a) is clearly reflected by the slightly decreasing gradient of the curves in subfigure (b). Their initial values of about 19 mJ (t = 2.0 mm) and 38 mJ (t = 5.0 mm) mainly reflect the energy stored inside the PPZ. After five load cycles these values increase to about 25 mJ and 50 mJ, respectively. As shown before, this energy accumulation takes solely place inside the CPZ whose corresponding volume after more than five load cycles is always less than 15 % of the PPZ’s volume, see Figure 13b. Further cycling results in a nearly proportional increase of the accumulated plastic energy reaching final values of about 2.6 times their initial values, please see values for load steps 1 and 60, respectively.

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These observations bring about and confirm several conclusions. The energy input ahead of a crack tip under cyclic loading is highest during the loading phase. Nevertheless, during the first ten load cycles even the unloading phase significantly contributes to the energy accumulation with percentages of about 40 % to 30 % of the total energy input, see Figure 18a. A non-propagating, i.e. an arrested crack under cyclic loading will approximately double the plastic strain energy accumulated inside its PZ during the first subsequent 15 load cycles. Furthermore, as this energy accumulation is limited to the CPZ, the local energy density in the vicinity of the crack tip will increase by factor ≈ 8…10. Consequently, the dominating part of the PZ which should also mainly determine its dependency on the local microstructure ought to be the CPZ. 5 Summary and Conclusions A 3-dimensional finite element simulation using an elastic-plastic material model with bilinear, isotropic hardening has been used to study the different types of plastic zones under cyclic loading in a commercial aluminium alloy with the designation AA2024-T3. Several experiments were performed with MT-specimens of different thicknesses t = 2.0 mm and t = 4.8 mm, respectively. Based on digital image correlation the local strain field ahead of the crack tip provides quantitative information about the different types of plastic zones. Furthermore, coupling of experimental and numerical results allows a basic quantitative estimation of the local energy input ahead of a crack tip under cyclic loading. The main results and conclusions can be summarized as follows. General conclusions: 





 



Both, experimental and numerical results clearly confirm the existence of three different types of plastic zones at a crack tip under cyclic loading: A primary plastic zone (PPZ) developing during the first loading, a backward cyclic plastic zone (BCPZ) found during unloading phases, and a forward cyclic plastic zone (FCPZ) found during subsequent loading phases. FCPZ and BCPZ form the so called cyclic plastic zone (CPZ). The FCPZ is always much larger than the BCPZ. The comparison of numerical and experimental results clearly shows that 3d-FE simulations with bilinear, isotropic hardening model are capable to predict the surface strain fields of the PPZ pretty well. The combination of experimental data and 3d-FE simulation allowed for the detailed study of the characteristics of BCPZ and FCPZ. Furthermore, it was possible to calculate the energy input ahead of the crack tip in terms of accumulated plastic strain energy. The energy input ahead of a crack tip under cyclic loading is highest during the loading phase. The damage dominating part of the plastic zone which should also mainly determine its dependency on the local microstructure ought to be the CPZ. Consequently, this region needs to be studied to understand the interrelations between local microstructure, crack tip field and crack propagation under cyclic loading. The main energy accumulation ahead of the crack tip during cyclic loading is limited to a very small fraction of the CPZ.

Conclusions drawn for all sheet thicknesses (t = 0.5 – 15 mm) numerically studied:  



The relative volume of the BCPZ is always well below 5% of the PPZ’s volume. The simulations of different sheet thicknesses show, that if the volume of the PPZ is standardized, i.e. divided by the corresponding sheet thickness, it does no longer reflect the effects of plane stress (thinner sheet) and plane strain (thicker sheet) condition. During the first ten load cycles even the unloading phase significantly contributes to the energy accumulation with percentages of about 40 % to 30 % of the total energy input.

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Conclusions drawn for t = 2mm: 

 

The spatial extension of the FCPZ strongly decreases with further cyclic loading. Especially during the first five load cycles the FCPZ’s volume changes from about 40 % of the PPZ’s volume to about 12 %. The effect of plane stress is experimentally as well as numerically observed as a circular shape of the PPZ at the specimen’s surface It was shown that a non-propagating crack under cyclic loading will approximately double the plastic strain energy accumulated inside its plastic zone during the first subsequent 15 load cycles. The local energy density in the vicinity of the crack tip will increase by factor ≈ 8…10.

Conclusions drawn for t = 5 mm:    

Especially in the case of the 5 mm sheet the curvature of the crack front hardly influences the main characteristics of the PPZ. The volume of the FCPZ remains comparably constant under cyclic loading as it deviates only between about 29 % and 32 % of the volume of the PPZ. The predominating effect of plane strain results in a butterfly-shape of the PPZ at the surface of the specimen. The simulations of a cyclically loaded non-propagating crack in a sheet with t = 5 mm yield the same results as those for t = 2.0 mm: During the first 15 load cycles the plastic energy accumulated ahead of the crack tip is approximately doubled. This accumulated energy for t = 5.0 mm is always about twice as much as that one in a sheet with t = 2.0 mm, which can be mainly attributed to the difference in sheet thickness (factor 2.5).

Although it is obvious that more detailed investigations, especially in terms of optical resolution for DIC-analyses, of the plastic straining ahead of the crack tip might be advisable, this paper clearly shows that the introduced approach of coupling experimental and numerical results provides a very useful basis for the quantitative, energy based analysis of the damage accumulation process ahead of a crack tip in metallic materials under cyclic loading. Acknowledgement This work has been supported by the German Federal Ministry for Economic Affairs and Energy (BMWi) through the project MetLife embedded in the German aeronautic research fund LuFo 20142017 (code 20W1302B). The authors would like to thank H. Hinderlich for preparing the specimens and E. Dietrich for conducting and supporting the experimental investigations.

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[22] P. K. Rastogi (Ed.): Photomechanics. Springer, Topics in Applied Physics Volume 77, 2000 [23] Homepage of company developing and providing ARAMIS: http://www.gom.com/ [24] J. Schwinn, M. Besel, U. Alfaro Mercado: Experimental determination of accurate fatigue crack growth data in Tailored Welded Blanks. Engineering Fracture Mechanics, 2016, DOI: 10.1016/j.engfracmech.2016.07.006

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Highlights • Quantitative experimental and numerical study of plastic zones under cyclic loading • Experimental observation of forward and backward cyclic plastic zones • Effect of sheet thickness on evolution of different types of plastic zones • Procedure for quantitative estimation of energy accumulation inside plastic zones