Identification and Validation of a Low Cycle Fatigue

Saeed Masih Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mail: [email protected]

Mohammad Mashayekhi1 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mail: [email protected]

Noushin Torabian Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mail: [email protected]

Identification and Validation of a Low Cycle Fatigue Damage Model for Al 7075-T6 Alloy In this paper, the behavior of 7075-T6 aluminum alloy under low cycle fatigue (LCF) loading is experimentally and numerically investigated using continuum damage mechanics (CDM). An experimental procedure is established to identify the damage parameters for Al 7075-T6. A damage-coupled explicit finite element code is developed using the experimentally extracted damage parameters to study the material behavior under LCF loading. Moreover, fractographic examinations are conducted to identify the fatigue crack initiation locations and propagation mechanisms. The model is employed for lifetime assessment of stringer-skin connection of a fuselage and the results are compared with the data available in the literature. [DOI: 10.1115/1.4028840] Keywords: low cycle fatigue, damage mechanics, Al 7075-T6

1

Introduction

Aluminum alloys have been widely used as structural materials due to their excellent properties such as low density, high strength to weight ratio, and resistance to fatigue. Al 7075-T6, which is an aluminum alloy with primary alloying elements of zinc, magnesium, and copper, has been extensively used for aircraft structural applications such as fuselage, airframe, stiffener, spar, etc. The high machinability of annealed Al 7075-T6 makes it suitable for forming to a variety of geometries. Moreover, Al 7075-T6 is an appropriate option for applications under cyclic loadings with high stress amplitudes [1]; therefore, the structural components fabricated from this alloy are mostly designed based on LCF loading. Explicitly, in order to have a light weight aerospace component, most of part cross sections are designed to be as thin as possible. But this approach consequently leads to increase in stress (more than asymptotic fatigue limit or yield stress) on parts. Quite obviously, there are some exclusive designs cases essentially applied to limited life, e.g., missile design. Such designing are expensive and take too much time. So achieving a predictive model for ductile crack initiation is indispensable for optimizing structural design and reduction of design expenses. One of the efficient tools for predicting material behavior under cyclic loading is CDM. In addition to prediction of material hardening behavior, CDM is able to assess the softening and strength degradation of the material in accordance with experimental observations [2]. Recently, the use of CDM in prediction of the mechanical response and the lifetime of complicated engineering parts, structures [3,4] and even human organs [5], subject to cyclic loading, has become very interesting. Thus, for this purpose, verification of damage approach is necessary. Especially, the effect of multi-axial loading on the fatigue life of components under operational conditions is of highest significance. In the framework of damage mechanics, the effects of material softening and deterioration are incorporated in thermodynamic laws by means of internal state variables. In other words, in this approach, the material degradation is modeled via modeling the nucleation, growth, and coalescence of microvoids and 1 Corresponding author. Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received May 29, 2013; final manuscript received September 8, 2014; published online November 7, 2014. Assoc. Editor: Hanchen Huang.

microcracks by utilizing internal state variables within the framework of thermodynamics. There are various damage mechanics based models developed for LCF problems such as the models proposed by Benallal [2], Doghri [6,7], and Pirondi and Bonora [8]. In the present work, The Lemaitre damage model is employed to investigate the behavior of Al 7075-T6 under LCF loading. Due to the importance of Al 7075-T6 in aerospace industry, numerous studies have been conducted on fatigue behavior of this aluminum alloy [9–15]. Despite the extensive work on Al 7075T6 properties and applications, there is still a need for analysis of this alloy from the LCF damage mechanics point of view. This paper is devoted to present an effective algorithm and validation of the CDM approach by Lemaitre [16] to high strength aluminum 7075-T6 and identifying its material fatigue damage parameters through experimental and numerical investigations. So, the employed LCF damage model is introduced first, and then an efficient algorithm is developed in order that numerically simulate the introduced damage model. In this algorithm, two equations are introduced to update the stresses based on the Newton–Raphson method. After the validation of efficiency of algorithm, an experimental procedure is established to identify the Al 7075-T6 damage parameters under triaxial state of stress. Furthermore, fractographic inspections are conducted on tensile and fatigue specimens in order to determine the active fracture mechanisms and the crack initiation zone. Finally, the damage model is implemented into a finite element code and the numerical predictions for both simple tension and fatigue tests are validated using the experimental observations. Here, prediction of the number of cycles to create microcrack is properly dealt with including the location of crack initiation.

2

Low Cycle Fatigue Damage Model

Continuum damage theory is a powerful tool for modeling material failure under various loading conditions. The modeling strategies as well as the damage variable (scalar or tensor) are determined by the nature of the mechanical process and the material. In CDM method, softening effects related to material damage are taken into account by means of thermodynamic internal state variables. Damage variable, D, is defined as the surface density of the microcracks and intersections of microvoids lying on a plane cutting the representative volume element [17]. By incorporating the damage variable in the material constitutive equations, it is

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possible to predict material failure under various loading conditions including fatigue loading. According to thermodynamics of damage, the damage evolution law can be derived by assuming the existence of a potential of dissipation, w which is a scalar convex function of the state variables. This potential function can be decomposed into the plasticity loading function, wf the nonlinear kinematic hardening related component wkin and the damage potential function wD as

(a) At first, the imposed load increment is assumed to be purely elastic and the trial strain and stress tensors are computed as follows: eetrial nþ1 ¼ en þ De retrial nþ1

ðrD  bÞeq

 ið RÞ  ry0 ð 1  DÞ b wkin ¼ b : b 2a  sþ1 r Y wD ¼ ðs þ 1Þð1  DÞ r

(1)

ðÞnþ1 ¼ ðÞtrial nþ1

Y¼

r2eq r2H þ 6G 2K

1 ð1  DÞ2

!

ðrD nþ1  bnþ1 Þeq ð1  Dnþ1 Þ

@w ¼ D_ ¼ k_ @Y

Y r

s

k_ 1D

@w ¼ k_ R_ ¼ k_ @i @w e_p ¼ k_ ¼ Nk_ @r @w ¼ k_ðaN  bbÞ b_ ¼ k_ @b

bnþ1  bn  DkðaNnþ1  bbnþ1 Þ ¼ 0  s Dk Y Dnþ1  Dn  ¼0 1  Dnþ1 r

(3)

k_  0;

wf k_ ¼ 0;

3 rD  b 2 ð1  DÞðrD  bÞeq

(11) (12) (13) (14)

rD nþ1

þ ð1 

(15)

Dnþ1 ÞKeetrial vnþ1

where eev denotes the volumetric elastic strain. (5)

(6)

(7)

2.1 Damage Model Implementation. The constitutive equations’ variables can be calculated by integrating Eqs. (4) and (5); given the mechanical state rn , en , Dn , and bn at tn , and the imposed total strain increment De such as De ¼ enþ1  en , the mechanical state rnþ1 , enþ1 , and bnþ1 can be calculated at time increment tnþ1 . The integration scheme, which is based on return mapping algorithm, follows the steps: 011004-2 / Vol. 137, JANUARY 2015

eed ¼ eetrial dnþ1  DkNnþ1 Rnþ1 ¼ Rn þ Dk rnþ1 ¼

In addition, N is the vector normal to the yield surface defined as

N¼

(10)

(c) After solving Eqs. (11)–(14) simultaneously and computing Nnþ1 and Dk, it is possible to evaluate the deviatoric elastic strain tensor, eed , and update Rnþ1 and rnþ1 as follows:

(4)

where k_ is the plastic multiplier parameter which is subject to the loading/unloading criterion (Kuhn–Tucker conditions) as [16] wf  0;

 iðRn þ DkÞ  ry0 ¼ 0

etrial rD nþ1  ð1  Dnþ1 Þ2Gðenþ1  DkNnþ1 Þ ¼ 0

(2)

i1 and c are material constants related to the isotropic hardening, and req are rH the hydrostatic and the equivalent stresses, respectively, and finally G and K are the shear and bulk module. According to the so-called normality rule of generalized standard materials, the damage evolution law as well as the evolution laws of other internal variables can be derived as [16] 

(9)

However, if wtrial > 0, a radial plastic correction is f required. In this case, in order to determine the unknowns rD nþ1 , bnþ1 , Dnþ1 , and Dk, it is necessary to solve the following system of equations, including the equation of yield surface (Eq. (11)), the plastic corrector (Eq. (12)), the kinematic hardening evolution law (Eq. (13)), and the equation of damage evolution (Eq. (14)):

where rD represents the deviatoric stress tensor, b is the back stress tensor and i(R) denotes the isotropic hardening stress variable which is a function of the accumulated plastic strain R. ry0 is the initial yield stress, a and b are material constants related to the kinematic hardening function, r and s are material parameters, and Y refers to the strain energy density release rate. The isotropic hardening stress and the energy release rate can be defined as [16] ið RÞ ¼ i1 ð1  ecR Þ

(8)

where E is the standard isotropic elasticity tensor. (b) The trial yield function wtrial ¼ ðrDtrial nþ1 ; bn ; Rn ; Dn Þ is evalf uated. If wtrial  0, then the load increment is efficiently f elastic and no plasticity and damage growth has occurred, therefore, the solution is updated as follows:

w ¼ wf þ wkin þ wD wf ¼

¼ ð1  DÞE :

eetrial nþ1

The procedure of solving Eqs. (11)–(14) and updating Eq. (15) is presented in the Appendix in detail; in this approach Eq. (3) is combined with Eqs. (11)–(13) resulting in Eq. (A11). This equation along with Eq. (14) is solved via backward Newton–Raphson method.

3

Experimental Procedure

Among aluminum alloys from 7xxx series, Al 7075-T6 is well suited for application in aerospace, automotive and marine industries due to its high strength to weight ratio and high fracture toughness [1]. The material composition of Al 7075-T6 alloy is presented in Table 1 [18]. It should be mentioned that the T6 heat treatment consists of solution treatment followed by artificial aging. In this section, the mechanical properties and damage parameters of Al 7075-T6 are extracted from appropriate experiments. 3.1 Material Specification. Standard tensile specimens were made according to ASTM-E8 [19]. All specimens were machined from an 8 mm thick Al 7075-T6 plate in the L-T direction according to ASTM-E399 Standard [20]. Due to the effect of surface layers on material strength, the specimens were polished Transactions of the ASME

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Table 1 Chemical composition of Al 7075-T6 alloying elements [18] Element Zinc Magnesium Copper Chromium Iron Silicon Manganese Titanium Other elements, each Other elements Total aluminum

Min%

Max%

5.1 2.1 1.2 0.18 — — — — — —

6.1 2.9 2 0.28 0.5 0.4 0.3 0.2 0.05 0.15 Remainder

according to ASTM-E606 to achieve a mirror-like finish [21]. Four specimens were tested and he obtained mechanical properties are presented in Table 2. The following expressions were considered for the evolution of isotropic and kinematic hardenings [17]: i ¼ i1 ð1  ecep Þ

(16)

b ¼ b1 ð1  ebep Þ

where b1 ¼ 3a=2b in which a and b are material constants introduced previously. The stress–plastic strain diagram obtained from the tensile test was fitted by the following equation to extract the material hardening parameters which are listed in Table 2: r ¼ ry0 þ i þ b

(17)

The hardening coefficients are found by using the nonlinear least square method minimizing the norm of the residual between the tensile experimental curve and the stress–strain curve. 3.2 Identification of the Material Damage Parameters. Employing the Lemaitre damage model requires identification of five material parameters, namely, epD and m which are related to damage nucleation and initiation, r and s which corresponds to damage evolution and the critical damage value, Dc , which corresponds to macroscopic crack initiation state. Since damage cannot be measured directly, damage evaluation was performed based on the effect of damage on measurable physical parameters. It is well known that the elasticity modulus decreases as damage progresses; therefore, damage parameter can be quantified based on its influence on elasticity modulus variations, as follows [17]: Di ¼ 1 

As shown in Fig. 2, a 6  2.6 mm strain gauge, TML: FLA-6350-23, with the initial resistance of 350 X was precisely attached to the minimum cross section of the specimen using epoxy glue, CN Cyanoacrylate adhesive. The specimen was then loaded under displacement control conditions using an Instron-5585 testing machine with a load capacity of 100 KN (Fig. 3). The rate of displacement was adjusted to 0.1 mm/min. The tests were carried out with a series of partial unloading–reloading to measure the change in the elastic slope while the strain increases. The obtained engineering stress–strain diagram is shown in Fig. 4. The tests were continued until the deformation limit for the strain gauge was reached. Then the specimen was completely unloaded and removed from the testing machine and a new strain gauge was attached, as required. According to Fig. 4, the damage variable can be obtained from Eq. (18) and the variations of damage versus plastic strain can be plotted as depicted in Fig. 5. As shown in this figure, the point where the diagram cuts the plastic strain axis introduces the damage threshold, epD , and the maximum value of damage variable is considered as the critical damage parameter, Dc . 3.3 Fatigue Test. Fourteen fatigue specimens were made according to ASTM-E606 standard [21]. The specimens’ geometry and dimensions are shown in Fig. 6. The fatigue tests were performed at three different levels of cyclic load, using an Instron-MHF25 machine with a load capacity of 2.5 ton and a load frequency of 5–10 Hz (Fig. 7). Table 3 lists the test results for the three loading conditions. In order to extract other material damage parameters (r, s and m), Eq. (4) is rewritten as follows: r¼

r2 1

2Eð1  DÞ2 ½dD=dep s

(19)

dD=dep can be calculated from the slope of the line fitted to D  ep data in Fig. 5. By assuming the damage value to remain constant in a loading cycle, integrating Eq. (4) produces [22] "

#s   r2eq max Rv pD 2Dep N  2Er 2Dep  2 2 rH Rv ¼ ð1 þ Þ þ 3ð1  Þ 3 req  m ru  r1 f pD ¼ epD ðreq:max þ req:min Þ=2  r1 f D¼

Ei E0

(18)

where Ei , the effective elasticity modulus of the damaged material, is derived from measurements during unloading and E0 is the Young’s modulus of the virgin material. In this work, in order to quantify damage under triaxial state of stress, a flat rectangular notched tensile specimen with the notch radius of 2.1 mm was tested (Fig. 1). Table 2

Fig. 1 Geometry of the notched specimen (all dimensions are in mm and t ¼ 5 mm)

(20)

Mechanical properties of Al 7075-T6

Young’s modulus, E (GPa) Yield stress, ry (MPa) Ultimate stress, ru (MPa) True fracture strain, eu Poisson’s ratio,  i1 (MPa) c a (MPa) b

71.77 444 545 0.1327 0.33 2000 0.8 2000 20 Fig. 2 Strain gauge attached to the notched tensile specimen

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Fig. 6 Dimensional details of the fatigue specimen (all dimensions are in mm and t ¼ 5 mm)

Fig. 3

Experimental setup

Fig. 7 The experimental setup for fatigue test

Fig. 4 Engineering stress–strain diagram for the tensile specimen

where N denotes the number of loading cycles and D is the damage value. Dep denotes the plastic strain for a loading cycle, req:max and req:min are, respectively, the maximum and minimum equivalent stresses in cyclic loading. r1 f represents the asymptotic fatigue limit, which has been considered as 159 MPa for a minimum fatigue life of 5  108 cycles [18]. Moreover, for uniaxial tensile test: Rv ¼ 1. Table 4 presents the calibrated damage parameters m, r, and s, extracted from Eqs. (19) and (20) and the fatigue test results.

3.4 Fractographic Examination of Fatigue Fracture Surfaces. In order to detect fatigue crack initiation location as well as the active fracture mechanisms, fatigue facture surfaces of the specimens were examined using scanning electron microscopy (SEM) according to ASTM-E112-96 and the grain size was measured as 44.9–63.5 lm [23]. The macrofractographic features of the specimen indicate the formation of a flat surface as well as a surface with shallow striations; Fig. 8 shows the macroscopic photograph of the fracture surface, which comprises of three distinct regions. The first area, illustrated in Fig. 8 as A1 , introduces the fatigue crack nucleation site. SEM micrographs of this area are shown at two different magnifications in Fig. 9. In this area, A1 , some large separate “crater” areas (due to Intermetallic Particle) are formed near or right at the edge of the fracture surface (Fig. 9). This is a good reason for fatigue cracks initiation and propagation from the surface close to corner of the Table 3 The fatigue test results req;min  req;max (MPa)

Average number of cycles to failure, N

129–434 131–490 133–540

29,670 12,000 6820 Table 4 Damage parameters for Al 7075-T6

m

r (MPa)

s

Dc

epD

4

2.175

2

0.37

0.01

Fig. 5 Damage versus plastic strain

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Fig. 8 Macrofractograph of the fatigue specimen fracture surface

specimen [24]. Subsequently, with increasing accumulation of plastic strain, dense dimple structures appear. Finally, the distributed damaged zones (“crater” areas) connect to each other due to ductile cracks propagation and coalescence. “Fatigue striations” are easy to see around the separate “crater” areas. The second area, A2 , which is shown in a further magnification in Fig. 10, represents the stable fatigue crack growth region. Investigating this area, which is the most important region for fatigue behavior study, reveals the present crack growth mechanisms in the specimen. The detected active fracture mechanisms are dimpled ductile fracture, brittle “cleavage” which is limited to

Fig. 10 The stable crack growth region, A2, accompanied by cleavage (23703 magnification)

a small region and fatigue striations which stem from the plastic strains at the fatigue crack tip. Moreover, a micrograph of the final fracture zone, A3 , is shown in Fig. 11. In this region, sudden rupture occurs due to the reduction of the cross-sectional area, caused by crack growth. The presence of dimpled morphology confirms the ductile nature of fracture in this region [25]. Generally, the areas of the stable and unstable crack development regions change by loading conditions. Evidently, for high loading amplitude, the stable crack development region is smaller compared to lower amplitude. Generally, there is no defined boundary between stable/unstable crack development regions and trans/intergranular crack propagation. Failure happens often in a mixed mode depending on many factors, e.g., grain morphology. The SEM micrographs show that crack growth is due to nucleation, growth and coalescence of voids, since the formation of dimples in both sides of fracture surface is evident. The shape of dimples in some parts of the opposing fracture surfaces is the same, which shows that fracture is resulted from tensile stresses. In some limited parts of fracture surfaces, dimples are stretched which is due to plastic deformation behind crack tip. Additionally, the extension of stretching is visible in both opposing fracture surfaces. Therefore, it can be concluded that damage in Al 7075 under low cycle fatigue loading depends on the stress field and plastic deformation of the material and consequently the selected model is capable of modeling the behavior of the material.

4

Results

In order to assess the performance of the employed damage model and the proposed integration algorithm, a three-

Fig. 9 SEM micrographs showing the fatigue crack nucleation site in the region A1 at (a) 503 magnification and (b) 1723 and 6133 magnification

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Fig. 11 SEM micrograph showing dimpled ductile rupture at the final fracture zone, A3 (11503 magnification)

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dimensional damage-coupled finite element analysis was conducted, using ABAQUS/EXPLICIT commercial software. The Lemaitre damage model, introduced in Sec. 2, was implemented and utilized by developing a user defined subroutine (VUMAT). The primary advantage of using such a subroutine incorporated in this commercial software is the availability of a variety of elements, material models, and other features such as efficient solvers. The mesh dependency phenomenon is mainly attributed to a consequence of the strain softening induced by the presence of damage. The prediction of crack initiation, crack growth, and the ultimate rupture is sensitive to the size of finite element discretization. There are a variety of approaches to reduce the mesh dependency, such as employing singular elements, introduction of strainrate dependence, introduction of couple stress, employing nonlocal theory, and limitation of element size [26]. The regularization method used in this paper consists in calculating the damage law coefficients by taking into account the mesh size of the elements [27]. In this work, the simulations were carried out for simple and notched tensile specimens and the fatigue specimen as well as tension–compression test and the results are presented in the following sections. 4.1 Results of the Simple Tensile Test Simulation. Due to symmetry, only one-eighth of the geometry was modeled. The model was discretized into a mesh of eight-node brick elements. Since the damage mechanics analysis results depend on the element size in general, various finite element models with different mesh sizes were constructed and examined to minimize the effect of mesh dependency. The numerical predictions are compared to the experimental results in Figs. 12 and 13. It is clear from these figures that there is a good agreement between the numerical and experimental results and the maximum error equals to 5%. The numerical predictions of damage distribution through the simple and notched specimens are illustrated in Figs. 14 and 15, respectively. Due to symmetry, only an eighth of the specimens are shown in these illustrations. From these figures, it is clear that the numerical predictions conform to the fractography results; for the simple tensile specimen, as Fig. 14 illustrates, maximum damage was predicted at the center of the specimen while for the notched specimen the critical element was detected at the notch root (Fig. 15). The critical damage value (DC) equals to 0.37, according to Table 4. The elements for which damage parameter has reached this critical value cannot resist the load and have experienced failure. Therefore, these elements introduce the crack initiation location.

Fig. 13 Experimental and numerical load versus displacement for notched specimen

Fig. 14 Damage distribution through the simple tensile specimen, fracture initiates from the center of the specimen

examined under tension–compression test. Figure 16 compares the numerical results with the experimental data obtained from Ref. [28]. From this figure, it is clear that there is a good agreement between numerical and experimental stress versus axial strain diagram which confirms the validity of the calculated damage parameters and hardening coefficients. However, the numerical simulations predict the material to reach the yield point at a higher rate comparing to the experimental results.

4.2 Tension-Compression Test Simulation. After validating the model under monotonic loading, the model performance was

4.3 Fatigue Loading Simulation. In order to simulate the fatigue specimen, only a quarter of the geometry around the

Fig. 12 Experimental and numerical stress versus strain for simple tensile specimen

Fig. 15 Damage distribution through the notched specimen, fracture initiates from the notch root

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calibration technique, i.e., for calibrating the material parameters, the moment and location of crack initiation is necessary (the distinctive border point between damage and fracture mechanics). But this point is not defined exactly by the performed experiments (certain assumptions are necessary).

5 Application of the Low Cycle Fatigue Damage Model

Fig. 16 Stress–strain diagram for a tension–compression cycle

net-section was modeled for the sake of computation time reduction. A sinusoidal loading was applied in the range of 133–540 MPa and the numerically obtained number of cycles to failure was 6300 cycles. Moreover, as shown in Table 3, the experimental average number of cycles to failure for this loading condition was 6820 cycles, indicating an error of 7.6% in the numerical predictions. Therefore, the proposed model predicts the number of cycles to failure with an acceptable accuracy. Figure 17 illustrates the damage distribution through the specimen after 4000 and 6300 cycles. The results indicate a competition between the stress triaxiality within the specimen center and the plastic strain in the fillet root for dominating the crack initiation site. Finally, the element located in the flat area, adjacent to the fillet root, reaches the critical damage value prior to the central element and fracture initiation occurs in this zone (Fig. 17(b)). According to Fig. 17(c), the predicted fracture location conforms to the experimental observations. Even though the simulation results are satisfactory, slight discrepancies occurred compared to the experimental data. The reason that CDM underestimates the number of cycles to failure is mainly attributed to definition of failure in CDM in which microcrack initiates when D ¼ Dc . However, immediately after crack initiation the problem becomes a mechanical fracture one and the actual failure point happens when crack reaches the critical value. In addition, One other reason for this difference is the used

In order to investigate the applicability and proficiency of the model and the proposed algorithm, a complex engineering structure is simulated. It is a stringer–skin connection of a fuselage with a pre-existing crack. The external body of an aircraft consists of stiffening panels and is divided to some cells by frame and stringer. The stiffened skin is prone to damage during its service life. For instance, a crack can exist in the longitudinal direction. It is assumed that the crack length equals to the maximum value which is not detectable during investigations (adet). It is so important to predict the number of cycles remained for the crack to reach the critical value (aCr). The whole visible part of an aeroplane is constituted of stiffened panels (Fig. 18), i.e., an outer thin sheet (skin) with generally orthogonally arranged integral or fastened stiffeners (stringers and frames in the fuselage shell are sketched in the Fig. 18). The fastened stiffeners can be riveted, adhesive bonded, or welded to the skin [29]. These aeronautical stiffened panels are prone to the in-service formation of damage. An example for the final stage of damage represented by a through-the-skin crack is sketched in Fig. 18. In the past, different designers have made different assumptions about the initial damage. Today, a crack extending into two frame cells and also cutting the central frame is assumed. This throughthe-skin crack can propagate along the fuselage shell and if it is not detected before becoming unstable (at the critical crack length) it can cause a catastrophic failure. For this reason, the maximum undetectable crack length (adet ) must be long enough to permit the establishment of a practicable inspection plan capable of detecting the crack before it reaches its critical length (aCr), during the operative life of the aircraft. The design of such structural components such as a stringer– skin connection requires the estimation of their lifetimes especially in the case of arrangements for overhaul schedules.

Fig. 17 (a) Damage distribution through the fatigue specimen after 4000 cycles. (b) Damage distribution through the fatigue specimen after 6300 cycles (at failure). (c) The fatigue test specimen after failure.

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Fig. 18 Fuselage stiffened panel with a longitudinal skin crack over a broken frame and configuration and loading condition [29]

Fig. 19 Geometrical characteristics of a part of the simulated panel [30]

For simulating, the structure is loaded by prescribing the symmetric displacement condition (R ¼ 1), at the boundary. Further details for geometry are shown in Fig. 19. Although the stringer is bonded to the skin, the whole assembly is considered as homogeneous and no special contact conditions have been defined within the respective finite element simulations. For modeling a preexisting crack as considered within the respective experiments, an elliptical through-thickness notch is taken, see Figs. 19 and 20. The assumption of such a blunt notch reducing stress singularities is very practical from a modeling point of view [29]. A mesh of C3D8R elements was generated with very fine elements around the elliptical notch. The material parameters used in the numerical analyses were extracted from Refs. [24,29] and are presented in Table 5. During the computations, one loading cycle is discretized by one time step. The results of the finite element simulation are shown in Fig. 21. The damage model predicts microcrack initiation in the assembly after 56 cycles. It is clear that the results of the present work agree with the results reported in the literature. However, the model

Fig. 20 Damage distribution in the stringer–skin connection of a fuselage. (a) Crack front at the elliptical notch (top view). (b) Profile of damage distribution in semi side of the notch (side view).

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Table 5 Elastoplastic-damage parameters of Al 2024-T3 Elasticity

E (GPa) 

70 0.3

Plasticity

ry (MPa) i1 (MPa) c b a (MPa) r (MPa)

284 150 4 80 2=3  17;000 1.3

s Dc

1 0.23

Damage

bn aDk Nnþ1 þ 1 þ bDk 1 þ bDk ¼ ð1  Dnþ1 Þ2Gðeetrial nþ1  DkNnþ1 Þ

bnþ1 ¼ rD nþ1

(A1)

Then, the tensor Znþ1 is defined as the subtraction of the kinematic hardening tensor from the deviatoric stress tensor Znþ1 ¼ rD nþ1  bnþ1

(A2)

According to Eq. (A1), Znþ1 can be rewritten as Znþ1 ¼ Ztrial nþ1  gðDk; Dnþ1 ÞNnþ1

(A3)

where etrial Ztrial nþ1 ¼ 2Gð1  Dnþ1 Þenþ1 

bn 1 þ bDk

gðDk; Dnþ1 Þ ¼ 2Gð1  Dnþ1 ÞDk þ

aDk 1 þ bDk

(A4)

(A5)

According to the definition of Znþ1 , the equation of the yield surface normal vector (Eq. (5)), as well as the yield surface equation (Eq. (11)), can be rewritten as follows: rffiffiffi 3 Znþ1 (A6) Nnþ1  2 ð1  Dnþ1 ÞkZnþ1 k Fig. 21 Damage evolution during cyclic loading

underestimates the failure point compared to test results which indicates conservativeness of the proposed approach. This discrepancy is mainly attributed to the definition of failure in CDM in which microcrack is initiated when D ¼ Dc . However, right after crack initiation the problem lies within mechanical fracture theory and the actual failure point happens when crack reaches the critical value. Generally, the results reveal the capability of the presented model in prediction of failure in engineering problems.

6

Conclusions

In this paper, the behavior of AL 7075-T6 was experimentally and numerically investigated under tension, tension–compression and fatigue loading. The mechanical properties and damage parameters of the material were identified through experiments. A damage-coupled elastic–plastic model was developed using the identified material parameters to investigate the behavior of Al 7075-T6 under low cycle fatigue loading. The numerical predictions such as the fatigue life estimation and fracture zone prediction were validated according to standard experiments as well as fractographic analysis and the results indicated an acceptable accuracy. The employed approach is appropriate for simulation of a variety of problems with nonlinear geometries and loadings. Moreover, the proposed algorithm is capable of solving the respective nonlinear set of equations with acceptable accuracy. As a representative example, the model was applied to service-life estimation of stringer–skin connection of a fuselage with a preexisting crack and the results showed satisfactory agreement with the data reported in the literature.

Appendix In order to simultaneously solve the integrated equations, Eqs. (12) and (13) are rewritten as follows: Journal of Engineering Materials and Technology

kZnþ1 k  iðRnþ1 þ DkÞ  ry0 ¼ 0 ð1  Dnþ1 Þ

(A7)

By extracting kZnþ1 k from Eqs. (A6) and (A7) and equating the results, Znþ1 is obtained as a coefficient of Nnþ1 rffiffiffi   2 ð1  Dnþ1 Þ2 iðRn þ DkÞ þ ry0 Nnþ1 Znþ1 ¼ (A8) 3 Moreover, according to Eqs. (A3) and (A6), it can be shown that Ztrial nþ1 is a coefficient of Znþ1 , thus based on the definition of Nnþ1 , from Eq. (A6), it is concluded that Nnþ1 ¼ Ntrial nþ1 , consequently, Ztrial nþ1 can be derived from Eq. (A6) as follows: rffiffiffi 2 trial Znþ1 ¼ ð1  Dnþ1 Þ Ztrial (A9) nþ1 Nnþ1 3 According to the above-mentioned relation for Ztrial nþ1 , Eq. (3) can be rewritten as

Znþ1

rffiffiffi 2 ð1  Dnþ1 Þ Ztrial ¼ nþ1 Nnþ1  gðDk; Dnþ1 ÞNnþ1 (A10) 3

where Ztrial nþ1 can be calculated from Eqs. (A8) and (A10), while Ztrial nþ1 is obtained from Eq. (A4) trial   Z  ð1  Dnþ1 Þ iðRnþ1 þ DkÞ þ ry0  nþ1

rffiffiffi 3 gðDk; Dnþ1 Þ ¼0 2 ð1  Dnþ1 Þ (A11)

By numerical solution of Eq. (A11) and Eq. (14), Dnþ1 and Dk can be obtained. The backward Newton–Raphson method was employed to solve the mentioned equations. The proposed solution algorithm is presented in Fig. 22. JANUARY 2015, Vol. 137 / 011004-9

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Fig. 22 The proposed integration algorithm

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