Preprint 2010 Low Cycle Lifetime Assessment of Al2024 alloy t h c A. Vyshnevskyy, S. Khan and J. Mosler a h t s e e G e r t n s e c C i n a h h c r c a e e M s e s R l a This is a preprint of an article accepted by: S i r S e International Journal of Fatigue (2010) t K a G M LOW CYCLE LIFETIME ASSESSMENT OF AL2024 ALLOY A. Vyshnevskyy1, S. Khan2 and J. Mosler3 1,2,3 GKSS Research Centre Geesthacht GmbH, Max-Planck-Strasse 1, 21502, Geesthacht, Germany Abstract: The 2024-T351 aluminum alloy is extensively used for fabricating aircraft parts. This alloy shows a relatively low ductility at room temperature and is generally heat treated in various conditions to suit particular applications. The present study experimentally and numerically analyzes the damage mechanism of an Al2024-T351 plate (short transverse direction) subjected to multi-axial stress states. The purpose of this work is to predict the cyclic lifetime of the considered alloy, based on the local approach of damage evolution using continuum damage modeling (CDM). The experimental program involves different kinds of specimens and loading conditions. Monotonic and cyclic tests have been conducted in order to measure the mechanical response and also to perform micromechanical characterization of damage and fracture processes. The cyclic plasticity behavior has been characterized by means of smooth cylindrical specimens. For analyzing the evolution of plastic deformation and damage under multi-axial stress conditions, cyclic loading tests in the low cycle regime have been conducted on different round notched bars. The predictions of the CDM were compared to the experimentally observed mechanical response and to the micromechanical characterization of damage. Emphasis was placed on the prediction of the number of cycles to failure. 1 INTRODUCTION In contrast to high cycle fatigue (HCF), low cycle fatigue (LCF) is generally characterized by failure in less than 104 cycles showing a pronounced plastic mechanical response. These loading conditions can occur in airplane structures during or after unpredictable mechanical impacts, e.g., hard landing, bad weather conditions, operational errors, failure of structural integrity, etc. Today’s airplane design is based on numerous tests on specimens of various sizes for the estimation of the residual strength of certain components such as the fuselage. These test programs are time consuming and expensive. The development of predictive models for ductile crack initiation is thus essential for improving structural design and maintenance, and it accelerates the introduction of new materials or assembling methods. Crack initiation as the result of LCF is a complex process that is influenced by many factors: stress (strain) history, rate of loading, environmental influence, temperature, hold times, etc. In addition to these extrinsic factors, numerous features of the micro-structure of the material undergoing cyclic loading affect crack initiation as well. Low cycle fatigue damage in ductile metals and alloys is usually associated with macroscopically observed plastic deformations. Therefore, it is reasonable to examine stress singularities showing material yielding which leads finally to crack initiation. In the present paper, the aforementioned physical processes are analyzed by means of continuum damage modeling (CDM). The original version of the CDM approach used here was proposed by Lemaitre (1985). This particular method falls into the range of phenomenological macroscopic material models. Hence, the complexity of the damage processes on the micro-scale is not approximated in detail, but only in an average sense (macroscopically). This is in sharp contrast to the majority of steels which are frequently modeled by means of micro-mechanically sound porous metal plasticity. Unfortunately, the damage mechanisms in highly inhomogeneous and anisotropic aluminum alloys are significantly more complicated that those in steels and hence, porous plasticity theory is not well suited. The CDM approach considered in the present paper approximates the aforementioned physical phenomena by means of plastic deformations and by 2 damage mechanisms. Conceptually, damage affects the mechanical response through the degradation of the stiffness. Clearly, the application of CDM to the prediction of the mechanical response and the lifetime of complex engineering components and structures subjected to cyclic loading is highly challenging. In this case, the design and the validation of damage evolution laws is required. Particularly, the influence of multi-axial loading on the fatigue life of materials and structures under operational conditions is of utmost importance. This paper is devoted to the extension and validation of the CDM approach by Lemaitre and Desmorat [1] to high strength aluminum alloys. A very common aluminum Al2024 alloy used for years in the aerospace industry is studied here as a reference. This alloy is preferred because of its good damage tolerance. The work includes both experimental and simulation parts. The experimental program involves different specimen geometries showing various multi-axial stress states. Furthemore, the testing conditions range from monotonic to cycling loading within the LCF regime. The S-direction (direction perpendicular to the plate) was selected for the current investigation. This is related to previous experimental and modeling activities in our working group [2]. Within the cited work, it was shown, that despite the strong anisotropy of the micro-structure caused by the manufacturing process, the mechanical properties are almost identical in any material direction. The major difference in the mechanical response is the fracture resistance. In this connection, the L- and T-directions are significantly stronger than the S-direction. Furthermore, if loading is applied in S-direction, a fracture surface being perpendicular to the loading direction is observed. As a consequence, the S-direction shows a relatively simple fracture mechanism and thus, it is convenient for further analysis. The present paper is concerned with the comparison of the results predicted by the aforementioned CDM approach to experiments. Special emphasis is placed on the prediction of the number of cycles to final failure including the locations of crack initiation. 2 MODELING Axi-symmetric FE models of the specimen geometries were developed using 4-noded standard elements (standard integration). Symmetry conditions are imposed in order to increase computational efficiency. The mesh size in the region of expected failure is of the same order as a representative volume element in metals [1]. The simulations are run by using ABAQUS v6.7 software with Z-Mat as material library. The constitutive behavior of the material was modeled by means of the CDM approach of Desmorat-Lemaitre [3]. This CDM model is based on rate- independent elasto-plasticity (isotropic and kinematic hardening) combined with anisotropic damage evolution (with micro-defect closure effects). Further details can be found in [3]. For the sake of comprehensibility, an overview of the governing equations is presented in the next sections. 2.1 Elasto-plasticity The model is based on the assumption of additive decomposition of the strains into elastic and plastic parts. More precisely, (cid:2763)(cid:3557)(cid:4662) (cid:3404) (cid:1777):(cid:2749)(cid:4662)(cid:1805) (cid:3404) (cid:1777):(cid:4666)(cid:2749)(cid:4662):(cid:2749)(cid:4662)(cid:2926)(cid:4667), (1) where (cid:2763)(cid:3557)(cid:4662) - is the effective Cauchy stress tensor (rate) and (cid:1777) - is the 4-th order stiffness tensor. 3 Yielding is governed by the von Mises-type yield function (cid:1858) (cid:3404) (cid:4666)(cid:2763)(cid:3557)(cid:3398)(cid:1798)(cid:4667) (cid:3398)R(cid:3398)σ , (2) (cid:2915)(cid:2927) (cid:2935) with (cid:1798) - being the back stress tensor, R -representing a stress-like internal variable associated with isotropic hardening and (cid:2026) - denoting the (initial) yield stress. Here and henceforth, (cid:4666)·(cid:4667) is (cid:3052) (cid:3032)(cid:3044) the equivalent stress measure of von Mises-type. The evolution of isotropic hardening and that of its kinematic counterpart are governed by an exponential law and an Armstrong-Frederick-type relation, respectively [1, 3]. More precisely, (cid:1844) (cid:3404) (cid:1844) (cid:4666)1(cid:3398)exp(cid:4666)(cid:3398)(cid:1854)(cid:1870)(cid:4667)(cid:4667), (3) (cid:2998) 2 (cid:2180)(cid:4662) (cid:3404) C(cid:2239)(cid:4662)(cid:2198)(cid:3398)(cid:2011)(cid:2180)(cid:1870)(cid:4662) (4) 3 Here, R and b - are parameters defining the isotropic hardening (saturation stress and saturation (cid:2998) rate) and r - is a strain-like internal variable (the equivalent plastic strain). Kinematic hardening is described by the material parameters C and γ . 2.2 Damage evolution For the description of anisotropic damage evolution, a 2-nd order damage tensor D is introduced. Without going too much into details, the effective stresses are computed from (cid:1731)(cid:2763) (cid:1732) (cid:1731)(cid:3398)(cid:2763) (cid:1732) (cid:2252)(cid:3557) (cid:3404) (cid:4666)(cid:1782)(cid:1816)(cid:2763)(cid:1778)(cid:1782)(cid:1816)(cid:4667)(cid:1778)(cid:3397)(cid:4666)(cid:1782)(cid:1816)(cid:2763)(cid:1778)(cid:1782)(cid:1816)(cid:4667)(cid:1778)(cid:3397) (cid:4680) (cid:1782) (cid:3398) (cid:1782) (cid:4681) (cid:1783) (5) (cid:2878) (cid:2879) 1(cid:3398)(cid:2015)(cid:1830) 1(cid:3398)(cid:2015)(cid:1860) (cid:1830) (cid:1782) (cid:3028) (cid:1782) (cid:1782)(cid:1790) (cid:3404) (cid:4666)(cid:2778)(cid:3398)(cid:1778)(cid:4667)(cid:2879)(cid:2869)(cid:3415)(cid:2870) and(cid:1782)(cid:1814) (cid:3404) (cid:4666)(cid:2778)(cid:3398)(cid:1860) (cid:1778)(cid:4667)(cid:2879)(cid:2869)(cid:3415)(cid:2870) (6) (cid:3036)(cid:3037) (cid:3036)(cid:3037) (cid:3036)(cid:3037) (cid:3028) (cid:3036)(cid:3037) where (cid:2763)(cid:1778), (cid:2763)(cid:1778) - are the positive and the negative part of the deviatoric stresses (based on (cid:2878) (cid:2879) eigenvalues); (cid:2763) - is the hydrostatic stress; (cid:1731)·(cid:1732) - are McCauly brackets; (cid:2015) and (cid:1860) - are the closure (cid:1782) (cid:3028) effect and hydrostatic damage sensitivity parameters, respectively. The damage evolution law is assumed to be a function of the plastic strain rate (cid:2749)(cid:4662)(cid:2926) and the effective elastic energy density (cid:1851), i.e., (cid:3046) (cid:1851) (cid:1830)(cid:4662) (cid:3404) (cid:4678) (cid:4679) |(cid:2749)(cid:4662)(cid:2926) | (7) (cid:3036)(cid:3037) (cid:1845) (cid:3036)(cid:3037) Clearly, (cid:1851) depends implicitly on the stress state and hence, a tri-axiality dependence is included. An important consideration within the aforementioned damage framework is a threshold value of damage initiation. It is assumed, that damage starts only after the plastic stored energy [4] (cid:3047) (cid:1827) (cid:4666)(cid:2869)(cid:2879)(cid:3040)(cid:4667) 3 (cid:1875) (cid:3404) (cid:3505)(cid:1844) (cid:4666)1(cid:3398)exp (cid:4666)(cid:3398)(cid:1854)(cid:1870)(cid:4667)(cid:4667) (cid:1870) (cid:3415)(cid:3040)(cid:1870)(cid:4662)(cid:1856)(cid:1872)(cid:3397) (cid:1798)(cid:2870) (8) (cid:3046) (cid:2998) (cid:1865) 4(cid:1829) (cid:2868) reaches the value (cid:1875) and continues as long as the eigenvalues of the damage tensor (cid:1778) are less (cid:3005) than the critical damage (cid:1830) . In Eq. (8), (cid:1870) is the accumulated plastic strain, (cid:1798) is the back stress (cid:3030) 4 tensor, (cid:1827),m are parameters defining the plastic stored energy function, and (cid:1844) ,(cid:1854),(cid:1855) are additional (cid:2998) parameters describing the cyclic plasticity model (see Eqs. (3)-(4)). The mechanical response during the strain controlled symmetric cyclic loading can be divided into two stages: (cid:190) Initially, a gradual increasing of the stress peaks is observed (see Fig. 1b). It continues till a maximum value. In line with other metallic materials, stabilization of the cyclic mechanical response is observed. At the end of this stage (cid:1875) (cid:3404) (cid:1875) is (cid:3046) (cid:3005) fulfilled. The respective number of cycles is denoted as (cid:1840) (cid:3005) (Fig. 1b). (cid:190) During the second stage, the stress peaks start to decrease and mechanical degradation is observed (Fig. 1b). This stage is associated with the initiation and progress of damage (0 (cid:3409) (cid:1830) (cid:3409) (cid:1830) (cid:4667). The number of cycles during this stage is (cid:3030) denoted as (cid:1840) . (cid:3006) a) 600 Stress, b) 600 [MPa] a] Δε P p M 400 s,[ NeDxp NeExp ak Nexp -0.01 0 0.01 e R p Strain, [-] s 200 Δε es tension (experiment) Str compression (experiment) CDM (simulation) -600 0 0 50 100 150 Number of cycles, [-] Figure 1: (a) Mechanical hysteresis and (b) Experimental vs. simulated absolute maximum value of tension and compression stresses for a particular strain loading amplitude, DLC-specimen (∆(cid:2013)(cid:3404)0.02 ; ∆(cid:2013) (cid:3404) (cid:3043) 0.0063 ; (cid:1840)(cid:3032)(cid:3051)(cid:3043) (cid:3404)138(cid:4667). (cid:3019) Beyond, these stages a (meso-) macro-crack in the critical region will be initiated. However, the modeling of crack extension was not considered in the present framework. Summarizing the aforementioned stages, rupture occurs at (cid:1840) (cid:3404) (cid:1840) (cid:3397) (cid:1840) (9) (cid:3019) (cid:3005) (cid:3006) number of cycles (Fig. 1b). 2.3 Modeling with the Coffin-Manson relation A standard representation of LCF lifetime for uni-axial cyclic loading is based on the relation between plastic strain amplitude ∆(cid:2013) and number of cycles to rupture (cid:1840) (in logarithmic (cid:3043) (cid:3019) scale).This results in the so-called Coffin-Manson relation given by (cid:1829) (cid:3082)(cid:3262)(cid:3252) (cid:1840) (cid:3404) (cid:4678) (cid:3014)(cid:3004)(cid:4679) (10) (cid:3019) ∆(cid:2013) (cid:3043) 5 where (cid:1829) and (cid:2011) are material constants [4, 5]. (cid:3014)(cid:3004) (cid:3014)(cid:3004) 3 EXPERIMENTS The material under consideration is Al2024-T351. It was supplied in form of a 100 mm thick rolled sheet in the T351 temper (solution heat treated, air-quenched, stress-relieved by cold stretching). The main alloying element in the 2XXX series is copper, while magnesium and manganese are also introduced in order to improve the quenching properties. The material’s chemical composition (as specified by the supplier Alcoa Inc., cf. [6]) is given in Table 1 below. Table 1: Chemical composition limits of Al2024 alloy. Maximum value is given, if range is not shown. Element Si Fe Cu Mn Mg Cr Zn Others, Al total Wt.% 0.50 0.50 3.8-4.9 0.30-0.9 1.2-1.8 0.10 0.25 0.15 balance 3.1 Mechanical testing Two types of specimens were tested (see Fig. 2). A so-called Damage Low Cycle specimen (DLC, smooth round bar) was designed for tension-compression tests showing a stress state close to the uni-axial one (see Fig. 2a). The specimen’s dimensions were taken from the ASTM E 606 standard test method, with the following modifications: (i) a shorter gauge length to prevent buckling due to high compression straining; (ii) design of a new gripping for tension- compression testing. Additionally, circumferential Round Notch Bars (RNB) with various radii were designed to achieve a variation of the stress state in the middle cross section of the specimen (see Fig. 2b). The geometry of the RNB specimens was designed similar to that in [7]. All specimens were cut from the same plate of Al2024-T351 in S-direction and they were mechanically treated with a turning machine. The length of the specimens is equal to the thickness of the plate. Hence, the effective measuring length is associated with the mid-plane of the plate under investigation. The surfaces of the specimens were ground and mechanically polished in the direction of loading. The final polishing was performed with a diamond paste showing 1μm grains. a) 100 b) 100 27.5 R 26 10 R 26 32.5 35 M 20 Ø 18 M 20 Ø 6.5 R Figure 2: Specimen geometries: (a) DLC-specimen; (b) RNB specimen with circumferential notches R = 10mm and R = 2mm. The entire experiments were conducted on a 160kN servo-hydraulic testing machine Schenck H160. Specially designed gripping of the testing setup and clamping threads of the specimens allow for tension-compression reversal loading. The cyclic tests in all cases were performed displacement-controlled. The displacement was measured using an extensometer attached to the working distances in the case of the DLC-specimen in Fig. 3a. For the RNB-specimen in Fig. 3b and 3c, the extensometer was attached to the specially manufactured grooves. A symmetric cycle (cid:4666)(cid:1844) (cid:3404) (cid:3398)1(cid:4667) based on a triangular wave shape was imposed with various displacement amplitudes ∆(cid:1864). The time period was not varied ((cid:1846) (cid:3404) 100(cid:1871)(cid:4667). All tests were conducted until complete failure 6 (separation of the specimen into two parts). Time-force-displacement responses during the tests were recorded with a digital acquisition device. a) b) c) Figure 3: Attachment of the strain gage: (a) DLC-specimen; (b, c) RNB specimens with circumferential notches R = 10mm and R = 2mm. Within the present paper, focus is on the application of the CDM approach to the analysis of LCF in Al2024. Thus, it is not intended to conduct as many experiments as necessary for certain statistics. Nevertheless, some of the experiments were repeated with identical loading conditions to check the reproducibility of the mechanical response. For the modeling proposes only selected experiments have been used. The whole experiments are summarized in Tab. 2. Table 2: Matrix of LCF experiments. Specimen Notch. mm Gage Range ofΔl, Number of length, mm mm specimens DLC - 10 mm 0.01-0.4 14 RNB 2mm 25 mm 0.025-0.1 10 RNB 10mm 25 mm 0.075-0.1875 6 3.2 Micro-structure investigations In order to successfully predict the lifetime of a material, it is important to understand its damage evolution. Damage in Al-alloys usually nucleates at large and brittle intermetallic particles or coarse precipitates, and sometimes also at smaller particles such as dispersoids or precipitates [8- 10]. In the specific case of fracture in high strength Al-alloys, the intergranular damage is induced by growth of voids around coarsened precipitates in ductile Precipitate Free Zones (PFZ) [11]. Optical microscopy (OM), scanning electron microscopy (SEM) and X-ray micro- tomography (XTM) have been used in our working group in the past to obtain quantitative information about micro-structural characteristics [12]. For fracture and LCF damage, the analysis of big intermetallic particles (equivalent diameter 23.5 + 16.7 µm) aligned in a network- like structure (clusters or Precipitate Free Bands (PFB)) which separate the matrix into domains is of utmost importance. The majority of them are Al CuMg particles appearing in SEM pictures 2 (BSD mode) in Fig. 5 a and Fig. 5b in white color. According to XTM investigations, the domains have a disc-like shape, with the S-direction as the shortest axis and almost identical dimensions in L- and T-directions. This microstructure results in higher damage rates in the PFB, if the material is loaded in S-direction. When damage evolves further, the most probable macroscopic crack is perpendicular to the loading direction with “shearing jumps” between parallel PFBs [2]. To systematically characterize fatigue fracture, fracture surfaces corresponding to monotonic loading were compared to their cyclic counterparts. 2D and 3D snapshots of the micro-structure have been extracted. 7 For monotonouss loading, the fracturee surface shhows a dennse dimple structure, which is charaacteristic off ductile failure (Fig. 4aa). However,, the picturee provides oonly two-dimmensional inforrmation (the depth is miissing). Moree details aboout the depthh will be givven in the suubsequent paraggraphs. Somme multiple ccracks are allso present aat the surfacce of the nottch. Slight nnecking in the nnotch root is observed. a) b) Figure 4: SEM of fractuure surface cloose to the nottch root of (a) monotonicallyy loaded specimmen, and (b) cycclically loadedd specimen (703 cycles) In cyyclic loadingg (Fig. 4b), llarge single “craters” (ddue to particlle embeddinng) appear cllose to or direcctly at the eddge of the frracture surface. This is aa strong indicator for faiilure originatting from the surface. Fatiggue lines aree highly visibble around thhe single “crraters”. By wway of contraast, in the monootonic case, the appearaance of multtiple brokenn particles juust under thee fractured ssurface in the mmiddle of specimen (Figg. 5a) is eviddent. This is not the casee in the regiion close to the notch root (Fig. 5b). HHence, the ccrack starts in the midddle of the sspecimen annd propagattes to the surfaace. The speccimens in Fiigs. 5a,b havve been cut uund polishedd in the longiitudinal direcction. a) b) Figure 5: SEM of longitudinal micro--sections close to the fracturee surface of thhe monotonicallyy loaded RNB specimen (a) middle of thee specimen, (bb) region close too the notch rooot Imagging softwarre (Alicona) combined wwith SEM waas used to sttudy the 3D shape of thee fracture surfaaces. In Fig. 6a, it can be seen that tthe fracture initiates at vvarious heigght levels, crreating an unevven structuree. This kindd of ductile damage occcurs at various stages wwithin the mmaterial’s layerrs. Finally, thhe scattered micro-crackks coalesce aand form a mmacro-crack. The bright planes in Fig. 6a representt the shearinng ductile ffracture. Acccordingly, sso-called sheear lips (smmoothened dimpples due to shear deformmation) are observable (instead of dimple struuctures). In the LCF- regimme (Fig. 6b), the mechanisms changge to mixed--mode failurre. Consequuently, uneveen ductile failurre areas, fattigue lines aand shear lipps are visiblle. Failure starts within clusters of particles. Subssequently, wwith increasing accumullation of plaastic strain, dense dimpple structurees appear. Finallly, the distrributed damaaged zones connect to eeach other ddue to ductille cracks proopagation and coalescence (fatigue linnes). Howevver, shearingg ductile fraccture as obsserved for mmonotonic loadiing, can be sseen as well. 8 a) b) Figure 6: 3D re-constructed fractured surface (a) monotonous loading and (b) cyclic loading Evidently, a complete understanding of the aforementioned mechanisms requires particularly a careful analysis of damage evolution at the micro-scale. Consequently, non-destructive micro- structural techniques such as synchrotron X-ray micro-tomography have also to be applied. During the last 10 years, significant progress has been made in terms of resolution. More precisely, the spatial resolution is now close to that of an optical microscope (2-3 μm). In the field of damage/fracture characterization, micro-tomography has been used for visualizing cracks in metals also for Al-alloys [13-16]. a) b) c) Figure 7: Section of reconstructed XTM 3D-image: (a) as a full solid, (b) as a semi visible volume with cracks, and (c) projection of the crack front on the plane perpendicular to the axis of specimen In Fig. 7, results obtained from XTM are shown. The geometry of the notch specimen is slightly modified to adjust the cross section size to the XTM penetration capability. E.g., the inner diameter is 4mm and the notch radius is 4mm as well. This is geometrically equivalent to the 10mm notch geometry used before. With the help of SEM and XTM micrographs, the results of the CDM reported in the next sections can be proven (particularly, the evolution of damage), cf. [17]. So far, the XTM measurements have been performed after loading the specimen. In the future, true in-situ investigations will be conducted. By doing so, the mechanisms of damage evolution can be investigated at a fine temporal and spatial resolution. 4 SIMULATION RESULTS 4.1 Material parameters of the plasticity model For the simulation of the cyclic mechanical response, a combined isotropic-kinematic hardening (cid:3032)(cid:3051)(cid:3043) plasticity model is used. Since no damage occurs within the first (cid:1840) cycles (see Fig. 1a), the (cid:3005) material behavior is purely elasto-plastic (without damage). The material parameters were adjusted using experimental cyclic stress-strain curves of DLC-specimens for ∆(cid:2013) (cid:3404) 0.02. The resulting material parameters are summarized in Table 3. 9 Table 3: Elasto-plastic parameters model Elasticity Plasticity (cid:1831) (cid:4670)(cid:1839)(cid:1842)(cid:1853)(cid:4671) (cid:2029) (cid:2026) (cid:4670)(cid:1839)(cid:1842)(cid:1853)(cid:4671) (cid:1844) (cid:4670)(cid:1839)(cid:1842)(cid:1853)(cid:4671) (cid:1854) (cid:2011) (cid:1829)(cid:4670)(cid:1839)(cid:1842)(cid:1853)(cid:4671) (cid:3052) (cid:2998) 7·105 0.3 284 150 4 80 1.7·104 Since a von Mises yield function is adopted, the tension-compression asymmetry is not captured. However, this can easily be improved by using a more realistic yield function. 4.2 Material parameters for the damage model The material parameters are calibrated starting with those for the evolution and the threshold value of the stored plastic energy function, cf. Eq. (8). For that purpose, an analytical method related to the underlying model is frequently applied, cf. [18]. Within this method it is assumed, that the isotropic hardening function reaches saturation prior to damage initiation. This is a common observation in ductile steels. Unfortunately, Al2024-T351 does not exhibit such a stabilization and thus other methods have to be used. In the present work, the objective function [19] (cid:3014) (cid:4668)(cid:1827),(cid:1865),(cid:1875) (cid:4669)(cid:1499) (cid:3404) argmin (cid:3437)(cid:3533)(cid:4672)log(cid:1840)(cid:3032)(cid:3051)(cid:3043) (cid:3398)log(cid:1840)(cid:3046)(cid:3036)(cid:3040) (cid:4666)(cid:1827),(cid:1865),(cid:1875) (cid:4667)(cid:4673)(cid:3441) (11) (cid:3005) (cid:3005) (cid:3036) (cid:3005) (cid:3036) (cid:3005) (cid:4668)(cid:3002),(cid:3040),(cid:3050)(cid:3253)(cid:4669)(cid:1488)(cid:3019)(cid:3126)(cid:3119) (cid:3036)(cid:2880)(cid:2869) is minimized, where (cid:1840)(cid:3032)(cid:3051)(cid:3043) and (cid:1840)(cid:3046)(cid:3036)(cid:3040) are the experimentally observed cycles before damage (cid:3005) (cid:3036) (cid:3005) (cid:3036) initiation and the numerically computed counterparts (see Fig. 1b). (cid:1839) is the number of experiments (with different loading amplitudes). The problem is solved by implementing an algorithm in MATLAB. The whole set of material parameters is summarized in Tab. 4. Table 4: Parameters of the model governing the evolution and the threshold values of the stored plastic energy and the damage evolution Stored plastic energy Damage (cid:1865) (cid:1827) (cid:1875) (cid:4670)(cid:1839)(cid:1836)/(cid:1865)(cid:2871)(cid:4671) (cid:1845)(cid:4670)(cid:1839)(cid:1836)/(cid:1865)(cid:2871)(cid:4671) (cid:1871) (cid:2015) (cid:1860) (cid:1830) (cid:3005) (cid:3028) (cid:3030) 3.9 0.0113 0.897 1.3 1.0 2.8 0 0.1 The choice of the damage parameters is motivated by the macroscopic material behavior. In several experiments with DLC-specimens no degradation of the mechanical properties was observed. The decreasing of stresses showed a rather abrupt or “quasi-brittle” character. A fast evolution of damage and a small value of critical damage (cid:1830) suit such a material behavior well. (cid:3030) The parameters for abrupt fracture and gradual degradation of the material behavior (e.g. Fig. 1b) are given in Table 4. The damage parameter was calibrated, without an optimization technique. 4.3 Simulation of fatigue lifetime of DLC-specimens Fig. 8 shows the results of FEM simulations performed on one single 3D finite element with the set of CDM-parameters taken from Tab. 3 and 4. For the sake of comparison, experimental