A phenomenological stress–strain model for wrought magnesium alloys under elastoplastic strain-controlled variable amplitude loading

International Journal of Fatigue 80 (2015) 306–323

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

A phenomenological stress–strain model for wrought magnesium alloys under elastoplastic strain-controlled variable amplitude loading Johannes Dallmeier a,⇑, Josef Denk a, Otto Huber a, Holger Saage a, Klaus Eigenfeld b a b

Competence Center for Lightweight Design (LLK), Faculty for Mechanical Engineering, University of Applied Sciences Landshut, Germany Foundry Institute, Technical University Bergakademie Freiberg, Germany

a r t i c l e

i n f o

Article history: Received 11 February 2015 Received in revised form 28 May 2015 Accepted 13 June 2015 Available online 24 June 2015 Keywords: Wrought magnesium alloys Strain-controlled hysteresis loops Phenomenological stress–strain model Variable amplitude loading Fatigue modelling

a b s t r a c t Wrought magnesium alloys typically reveal strong basal textures and thus, non-symmetric sigmoidal shaped hysteresis loops within the elastoplastic load range. A detailed description of those hysteresis loops is necessary for numerical fatigue analyses. Therefore, a one-dimensional phenomenological model was developed for elastoplastic strain-controlled constant and variable amplitude loading. The phenomenological model consists of a three-component equation, which considers elastic, plastic, and pseudoelastic strain components with a set of eight material constants. Experimentally and numerically determined hysteresis loops of four different magnesium alloys were compared by means of different examples with constant and variable amplitude. Good correlation is reached and the relevant fatigue parameters like strain energy density were estimated with good accuracy. Applying an energy based fatigue parameter on modelled hysteresis loops, the fatigue life is predicted adequately for constant and variable amplitude loading including mean strain and mean stress effects. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Wrought magnesium alloys possess high potential for lightweight design making them attractive for the automotive industry [1]. The development of an economical twin roll strip casting process for the continuous and semi continuous production of magnesium sheet metals [2,3], e.g. for car body components, was one of the main advancements in the last decade. Magnesium alloys possess a hexagonal close packed crystal structure, indicated in Fig. 1. The most pronounced deformation mechanisms of magnesium single crystals at ambient temperature are basal hai slip and {10–12 }h10–11i extension twinning (e.g. [4,5]). A similar behavior was observed for polycrystalline standard alloys such as AZ31, but for some textures and loading directions, the critical resolved shear stress for prismatic hai slip is exceeded and its portion is large (e.g. [5–7]). Extension twinning is possible at low stresses and enables tensile straining along the c-axis [4]. Thus, it can be activated when a tensile stress is applied parallel or a compressive stress is applied perpendicular to the c-axis as illustrated in Fig. 1 (e.g. [4]). Wrought magnesium semifinished products exhibit strong textures in contrast to magnesium castings, which typically show a random grain orientation (e.g. [3,4,8]). During sheet metal ⇑ Corresponding author at: Am Lurzenhof 1, 84036 Landshut, Germany. Tel.: +49 1704726731. E-mail address: [email protected] (J. Dallmeier). http://dx.doi.org/10.1016/j.ijfatigue.2015.06.007 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.

forming, a basal texture with the c-axis lying almost normal to the sheet plane is developed (e.g. [8]), which results in an asymmetry of the tensile and compressive yield stress (e.g. [9]). In the low-cycle fatigue regime, the cyclic plastic deformation is mainly caused by reiterative twinning and detwinning (e.g. [10]) and sigmoidal shaped stress–strain hysteresis loops can be observed (e.g. [8,10]). Moreover, most of the magnesium alloys show a nonlinear unloading curve in the tensile as well as compressive region, which is called pseudoelastic behavior (e.g. [11–13]). In magnesium alloys, pseudoelastic strain is caused by reversible movements of twin boundaries due to internal driving forces [11]. This strain leads to larger hysteresis loops especially at low stress amplitudes in comparison to material behavior without pseudoelasticity [11]. Several strain-controlled evaluations were carried out on wrought magnesium alloys to predict the fatigue life by the local strain concept (e.g. [9,14–30]). A comprehensive summary of fatigue investigations on wrought magnesium alloys is given in [30]. Accordingly, most low-cycle fatigue investigations [9,14–23] were done using completely reversed strain-controlled conditions, and in [24–30] the influence of the strain ratio was considered. The Manson–Coffin–Basquin approach [31–33] was found to show an adequate correlation with the experiments for completely reversed strain-controlled conditions (e.g. [9,15–19,23]). In addition, energy based fatigue models were shown to give good correlation in cases of variable strain or stress ratios [20,22,26,29,30].

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

307

Nomenclature adown aup CYS CSSC E ED EL K0 mpsel mpl n0 Nb Nf Nf,r Nloop,f P

shape factor for descending reversals shape factor for ascending reversals compressive yield stress cyclic stress–strain curve Young’s modulus extrusion direction tensile elongation cyclic strength coefficient memory factor for pseudoelastic strain component memory factor for plastic strain component cyclic strain hardening exponent number of cycles to break in two pieces number of cycles to failure remaining number of cycles to failure number of full loops to failure material constant, representing the slope of the pseudoelastic strain component PSWT Smith–Watson–Topper fatigue parameter [58] RCS relative coordinate system RD rolling direction Rr ratio between the reduction of both memory factors mpl and mpsel Re strain ratio Rr stress ratio S material constant, representing the slope at the inflection point of the plastic strain component T amount of plastic strain at the inflection point of the plastic strain component TYS tensile yield stress U substitution function UTS ultimate tensile strength D deviation between two compared quantities DWcomb combined strain energy density per cycle DWcomb,ww combined strain energy density per cycle without weighting DWel+ tensile elastic strain energy density per cycle DWpl plastic strain energy density per cycle

Fig. 1. Direction dependency of {10–12}h10–11i twinning.

DWpsel+ positive pseudoelastic strain energy density per cycle D et relative total strain for the actual reversal in the actual RCS/total strain range Deel elastic strain component Depl plastic strain component Depsel pseudoelastic strain component Dr relative stress for the actual reversal in the actual RCS/stress range Drmax maximum stress range of the envelope hysteresis loop e_ strain rate et total strain of the initial loading curve, described by the Ramberg–Osgood equation ea,el elastic strain amplitude ea,pl plastic strain amplitude ea,pl,m1 plastic strain amplitude for assumed linear elastic unloading (method 1) ea,pl,m2 real plastic strain amplitude, half width of hysteresis loop at r = 0 MPa (method 2) ea,pl,m3 half width of hysteresis at the mean stress of the loop (method 3) ea,t total strain amplitude r stress of the initial loading curve, described by the Ramberg–Osgood equation ra stress amplitude rdtw stress at the inflection point of the plastic strain component of the ascending reversal rm mean stress rmax maximum stress rmin minimum stress rp,down pseudoelastic cut-off stress for descending reversals rp,up pseudoelastic cut-off stress for ascending reversals rrp global stress at the beginning of a reversal rtw stress at the inflection point of the plastic strain component of the descending reversal engineering stress or strain (used as index) (e) true stress or strain (used as index) (t)

Zenner and Renner [34] analyzed the shape of hysteresis loops at variable amplitude loading and showed that different magnesium alloys exhibit material memory. Götting and Scholtes [35] investigated the influence of strain-controlled loading history on the stress–strain behavior of AZ31 wrought magnesium alloy and found the shape of the reversals to be strongly influenced by predeformation. The larger the compressive predeformation and thus, the amount of twins, the larger is the deviation from linearity at the beginning of a reversal. It can be seen in both studies [34,35] that the shapes of reversals at variable amplitude loading depend on several parameters such as mean strain, predeformation and strain amplitude. Nevertheless, experimentally determined hysteresis loops were evaluated without consideration of an adequate stress–strain model in [34,35]. Appropriate numerical fatigue analyses require the knowledge of the elastoplastic stress–strain curves of each cycle to provide the required stress, strain, and strain energy density components. In most previous studies, the experimentally determined hysteresis loops were used to obtain the necessary values (e.g. [14–30]), which is not practicable in an engineering fatigue analysis. Therefore, the shape of stress–strain hysteresis loops must be described with an efficient model. For some steel and aluminum alloys, the Ramberg–Osgood equation [36] in combination with the Masing model [37] is an

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adequate and well-known method to describe the shape of stress– strain hysteresis loops at arbitrary uniaxial tensile and compressive loads. This model represents point symmetric and completely convex hysteresis loops. Additionally, the description of pseudoelastic strain is not possible with the Masing model. In contrast, wrought magnesium alloys usually show non-symmetrical stress–strain hysteresis loops with sigmoidal shaped reversals at medium up to large strain amplitudes. In [23,34] is shown that the Masing model [37] fails to describe hysteresis loops of wrought magnesium alloys. For this comparison, completely reversed strain controlled constant amplitude tests on the alloy AZ31 were used in [23] and similar tests on the alloys AE42, AZ31, and AZ80 were used in [34]. Phenomenological formulations to describe the shape of non-symmetrical or sigmoidal shaped material behavior already exist [38,39]. The model in [38] can be used for quasi-static loading without unloading. It does not contain equations and conditions, necessary for modelling hysteresis loops, especially for asymmetric hysteresis loops. In [39], a method to fit sigmoidal shaped hysteresis loops by the use of a hyperbolic tangent function is presented. This phenomenological model is able to calculate symmetric hysteresis loops. Nevertheless, no equations and conditions for the description of asymmetric hysteresis loops were defined in [39]. Furthermore, the models described in [38,39] contain no conditions for variable amplitude loading. Phenomenological models for hysteresis loops with noticeable pseudoelastic strains were developed in [12,13]. These models were developed and evaluated for tensile loading–unloading hysteresis loops and reveal good correlations with experiments. However, the phenomenological equations in [12,13] are not able to calculate sigmoidal shaped hysteresis loops, which occur in a lot of low-cycle fatigue load cases. Several investigations were done recently to develop constitutive models for magnesium alloys (e.g. [40–46]). Some of the models aim on deep drawing simulations for sheet metal forming (e.g. [40,41]) and are able to calculate stress–strain paths considering anisotropic and asymmetric behavior of the material. In [40,41], different loading paths such as tension–compression–tension were used to compare experimentally determined results with numerically determined ones, showing reasonably good agreements. These phenomenological continuum plasticity models for sheet metal forming (e.g. [40,41]) were evaluated by means of loading paths within comparatively large strain ranges (>3%). Wang et al. [44–46] developed crystal plasticity based constitutive models for hexagonal close packed crystals using AZ31 wrought semi-finished products for the verifications. The models described in [44,45] calculate stress–strain paths similar as in [40–43] with the advantage that they are more physics-based and are able to capture the texture evolution during cyclic loading. Furthermore, the models in [44,45] are able to calculate a smooth transition between elastic and plastic deformation instead of a sharp bend as shown in [40,41]. In [46], the pseudoelastic behavior is investigated thoroughly by evaluating a crystal plasticity model. This model suitably captures pseudoelastic behavior during tensile loading and unloading. Behravesh [42] developed a constitutive model with the focus on the fatigue analysis of spot welded magnesium sheets. The stress–strain hysteresis loops, calculated with the constitutive model developed in [42], were compared with experimentally determined stress–strain hysteresis loops and reach good agreements even at lower strain ranges. A numerical fatigue analysis of an automotive substructure was carried out in [43] with constant amplitude loading, using the model described in [42]. The constitutive models [40–43] were developed for 2D and 3D finite element analyses and thus, the simulation time for one cycle is comparatively large. A large computational effort per cycle is

also necessary using crystal plasticity models for polycrystals, which contain large number of orientations at each integration point [40]. As shown e.g. by Heuler and Klätschke [47], typical load sequences for automotive structures reveal a large number of cycles with variable amplitudes. Thus, 2D and 3D finite element based phenomenological continuum plasticity models as well as crystal plasticity models are not adequate for numerical fatigue analyses with large load time functions. To carry out numerical fatigue calculations on wrought magnesium alloys with local strain and energy based concepts in commercial fatigue analysis software (e.g. nCode DesignLife), the most important requirements are:  modelling sigmoidal shaped and asymmetric hysteresis loops,  high numerical efficiency, and  adequate correlation with experiments under variable amplitude loading with large number of cycles. No one of the above-mentioned models fulfills all of these requirements. Thus, a one dimensional phenomenological model for the rapid calculation of stress–strain hysteresis loops at arbitrary mean strains and strain amplitudes by a single set of material constants per alloy is developed within this study. The phenomenological model can be used for numerical fatigue analyses in the elastoplastic range to get the necessary measures for the fatigue model such as stress, strain, and strain energy density components. Completely reversed strain-controlled tests as well as tests with five different complex strain–time functions were used for the determination of the necessary material constants and for the experimental validation of the proposed model. The shapes of the different envelope and inner hysteresis loops are described and evaluated. Furthermore, a calculation of the fatigue life at cases including cyclic stress relaxation is discussed and compared with experimental results.

2. Material and experimental procedure To investigate the quasi-static and cyclic material behavior, a 1.2 mm thick twin roll cast AM50 magnesium alloy sheet metal, provided by the Magnesium Flachprodukte GmbH, was used. The chemical composition, listed in Table 1, was measured via atomic emission spectroscopy and fulfills the tolerances given in ASTM: B951-11. The effective sheet width is 650 mm, and the manufacturing process is described in detail in [3]. Samples for microstructural characterization of AM50 were grinded, polished, and etched with picric acid solution similar to [23,30], and the optical microscope Leitz Laborlux 12ME was used. Extruded ME21 magnesium alloy sheet metal with a thickness of 1.5 mm, provided by the Stolfig GmbH, was used as an additional alloy for the application of the proposed phenomenological model. The chemical composition of ME21 is listed in Table 1 using the values out of [48]. Hysteresis loops of ME21 sheet metals were taken from the raw data of a previous own study [23]. The development of the phenomenological model as well as most of the evaluations and comparisons, presented in Sections 4.2.1–4.2.4, are explained using twin roll cast AM50 sheet metals. Furthermore, all fatigue investigations in Section 5 were carried out on AM50. Extruded ME21 sheet metal is used solely in Section 4.2.4. To evaluate the proposed phenomenological model on other wrought magnesium semifinished products such as extruded rods or tubes, the raw data of Xiong et al. [49] and Li et al. [50] were used in Section 4.2.4: extruded AZ31B in the form of solid rods [49] and extruded AZ61A in the form of tubes [50]. At all four alloys, only results of tests with loading direction parallel to the rolling or extrusion direction were considered. Refer to

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J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323 Table 1 Chemical composition in weight percent of the investigated AM50 and ME21 [48] sheet metals.

AM50 ME21

Al

Mn

Zn

Ce

Si

Fe

Ni

Cu

5.26 60.01

0.37 2.10

0.086 60.015

0.020 60.015

0.0032

0.0012

<0.001

0.70

[23,30,49,50] for more information about the quasi-static and cyclic properties of the used alloys. Standard dog bone mechanical test specimens with dimensions as described in the Stahl-Eisen testing guideline [51] and in [30] were machined via milling from a sheet metal while keeping their nominal thickness. To provide material parameters for fatigue calculations, it is necessary to use the same surface roughness as reached by the original manufacturing process. Thus, the milling process was modified that the surface roughness nearly matches the original sheet metal surface roughness (Rz = 0.6–1.2 lm). Due to the occurrence of compressive loads, an appropriate buckling guide was used, which additionally ensures a precise adjustment parallel to the test rig axis. A 0.2 mm thick PTFE-foil was applied on the buckling guide similar as described in [30] to minimize friction between the specimen and the buckling guide. The ambient temperature during all tests was between 18 °C and 22 °C. At regular intervals, the temperature of the specimens was checked, and it was found that the temperature changes during the tests were negligible. Results of quasi-static tensile and compressive tests were taken from the investigations described in [30] for AM50 sheet metals. They were conducted with an initial strain rate of 103 s1 using a Zwick Z150 universal tensile testing machine. Cyclic strain-controlled tests on AM50 sheet metals were carried out on a servohydraulic test system with a 25 kN cylinder, a 5l pm Moog valve, an Instron Labtronic 8800 controller, and the extensometer Sandner EXA10-0.5x for strain measurement and control. To prevent heating of the specimen, the frequencies for cyclic tests with constant amplitudes and for cyclic stress relaxation tests were chosen between 0.01 Hz and 20 Hz: low frequencies for large strain amplitudes and high frequencies for low strain amplitudes. For tests with variable amplitudes, constant strain rates in the range of 1  103 s1  e_  2  103 s1 were used. The engineering strain was controlled at all tests. All calculations by phenomenological model were done using engineering stress and strain values. This is appropriate, because the typical strain range at fatigue analyses is low and stress as well as strain measurements of fatigue tests usually provide engineering values. The test results and the results of the phenomenological model were converted into true stress (F/actual section, assumed volume constancy) and strain (logarithmic strain) values, which result in more accurate fatigue calculations with energy based fatigue models [30]. The subscript (e) means ‘‘engineering’’ and the subscript (t) means ‘‘true’’. Fatigue life prediction methods such as the local strain approach typically consider the range until the initial macroscopic crack [51]. Thus, only the fatigue life until the initial macroscopic crack is considered within the present study, denoted by ‘‘number of cycles to failure Nf’’. The number of cycles to failure Nf can be assumed from the progression of the maximum stress during cyclic loading, which is suggested and described in [51]. It was calculated using a regression line between the maximum stress value of a hysteresis loop (rmax(t)) at 0.5Nb and 0.75Nb, with Nb as the number of cycles to break in two pieces. Nf is defined as the number of cycles where the relative deviation of the actual maximum stress of a hysteresis loop to the maximum stress of the regression line at the same cycle is 10%. Further details are described in [30,51].

Th

Nd, Pr, Y

0.14

60.065

Other impurities

Mg

<0.1

Bal. Bal.

3. Shapes of experimentally determined hysteresis loops Twin roll cast AM50 sheet metals reveal a strong basal texture with the c-axis lying almost normal to the sheet plane [30]. Thus, {10–12}h10–11i extension twinning can easily be activated with a compressive load parallel to the sheet plane. The sheet metals show a homogenous microstructure with an average grain size of about 5 lm (Fig. 2a), which is similar in all spatial directions [30]. Nearly no twins were detected in the undeformed state (Fig. 2a). Twinning starts at a compressive strain of 0.3% parallel to the sheet plane. Fig. 2b shows the lenticular twinned microstructure at 0.7% and Fig. 2c at 2% compressive strain. With increasing compressive strain, the twin shape changes from narrow twins (Fig. 2b) to relatively wide twins and the twin density increases (Fig. 2c). In Table 2, the quasi-static mechanical properties of twin roll cast AM50 sheet metals and extruded ME21 sheet metals are listed as the arithmetic mean value out of five measurements. In Fig. 3, the tensile and compressive stress–strain curves of AM50 are plotted in detail up to ±3% strain. In tensile direction, the quasi-static plastic deformation in the illustrated range is mainly caused by

Fig. 2. (a) Undeformed microstructure of twin roll cast AM50, (b) microstructure with lenticular {10–12}h10–11i extension twins after 0.7% compressive strain, and (c) higher twin density after 2% compressive strain.

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

Table 2 Quasi-static mechanical properties of twin roll cast AM50 sheet metals in RD and extruded ME21 sheet metals in ED. Values out of

E (GPa)

UTS (MPa)

TYS (MPa)

CYS (MPa)

El(e) (%)

AM50 ME21

[30] [23]

45 ± 1 44 ± 2

292 ± 2 195 ± 3

212 ± 2 121 ± 3

135 ± 4 65 ± 3

27.3 ± 3 19.3 ± 2

1 2 3 4 5

1: extruded ZK60 [52] 2: rolled AZ31 [26] 3: twin roll cast AM50 [30] 200 4: twin roll cast AZ31 [23] 5: extruded ME21 [23] 100

[MPa]

100

tensile branch of the CSSC quasi-static curves

0

-100

example of hysteresis at 0.5Nf a,t(e)=3%

0

stabilized loops R (e)=-1 a,t(e)=0.8%

-200 -1

-0.5

0

true strain

0.5 (t)

1

[%]

-100

(b)

AM50 RD

end of detwinning

-200

AM50 RD compressive branch of the CSSC

-300 -3

-1.5

0 (e)

3

[%]

start of detwinning

basal hai slip, which results in an almost continuous hardening. In contrast, compressive quasi-static plastic deformation is caused by twinning (Fig. 2b and c) with a nearly zero strain hardening rate up to approximately 3% compressive strain (Fig. 3). The twins were found to be {10–12}h10–11i extension twins (e.g. [8,30]). These different deformation modes lead to different values of the tensile and compressive yield stresses (TYS and CYS) with a ratio TYS/CYS of 1.57 for AM50 and 1.86 for ME21. Nearly all stress–strain hysteresis loops of different wrought magnesium alloys show an asymmetrical and sigmoidal shape at strain amplitudes >0.3–0.6% (e.g. [8–10,14–21,23–30]), which is illustrated in Fig. 4a, using the example of five different wrought magnesium semifinished products out of [23,26,30,52] with a strain ratio Re(e) of 1. All shown examples reveal a similar sigmoidal shaped ascending reversal. However, the slopes and the peak stresses are different. Fig. 4b shows strain-controlled hysteresis loops of AM50 at different strain amplitudes ea,t(e). The plastic strain amplitudes are dominated by twinning and detwinning. The twinning and detwinning process is explained as follows: At the descending reversal in the range of CYS, twinning starts, and in the tensile stress region of the ascending reversal, most of the generated twins will be detwinned (e.g. [8,10,30]). If most of the twins are detwinned, the stress is rising with a larger slope (e.g. [10]). This is the reason for an inflection point in the ascending reversals, visible in Fig. 4b. Applying larger strain amplitudes such as ea,t(e) = 3%, an inflection point is also visible in descending reversals [30]. The nonlinear behavior, starting slightly after unloading (Fig. 4b), is called pseudoelastic deformation and is explained in the introduction as well as in [11–13]. Pseudoelastic deformation smoothly changes over to twinning in the descending reversal and to detwinning in the ascending reversal. Ascending reversals typically show smaller curvatures in this transition region (Fig. 4b). In the following paragraphs, the experimentally determined shapes of hysteresis loops under variable amplitude loading are described in Fig. 5a–c using the example of AM50. The load

true stress

Fig. 3. Representative tensile and compressive quasi-static stress–strain curves and the cyclic stress–strain curve CSSC of twin roll cast AM50, RD.

2%

100 a,t(e)=0.4%

(t)

engineering strain

1.5

200

[MPa]

engineering stress

200

(e)

300

true stress

(t)

Alloy

(a) 300

[MPa]

310

0.8% 1.4%

0.26% 0

E

pseudoelastic strain

-100

start of twinning -200 -2

E

-1

0

true strain

E=45GPa R (e)=-1

1

2

(t) [%]

Fig. 4. (a) Comparison of stabilized hysteresis loops, determined on different wrought magnesium alloys by different researchers, loaded in RD or ED, respectively; (b) stress–strain hysteresis loops of AM50 with different total strain amplitudes at 0.5Nf.

sequence of the example is illustrated in Fig. 5a, in which one full loop is indicated. In Fig. 5b the sequence is visible by means of the hysteresis numbers (1, 2, 3, 4, 5, 6, 7, 8, 1, . . .). To quantify the considered load-time functions within this study, one full loop is defined as one envelope hysteresis loop (here: hysteresis number 1) together with its inner hysteresis loops (here: hysteresis numbers 2–8) as shown in Fig. 5b. This full loop was recorded at half of the fatigue life 0.5Nloop,f. The definition ‘‘number of full loops to failure Nloop,f’’ is used to describe the fatigue life in case of variable amplitude loading. It is similarly defined as the number of cycles to failure Nf at constant amplitude tests. The envelope hysteresis loop in Fig. 5b looks very similar to that of a symmetrical test applying the same strain amplitude (Fig. 4b). The inner loops reveal different shapes depending on their mean strains. Inner hysteresis loops with larger positive strains are narrower with a larger stress range Dr, and reveal in some cases a smaller area. The cyclic plastic strain causes a reciprocal obstacle of twin formation and slip, which leads to a rising number of residual twins (undetwinnable twins) (e.g. [10,27]). This effect is most likely the main reason for cyclic hardening in the considered region 3% 6 e(e) 6 3% [10,27]. Cyclic hardening can be seen by means of the cyclic stress–strain curve CSSC in Fig. 3, captured at half

2

1

engineering strain

311

difference between the hysteresis shapes at half number of full loops to failure (0.5Nloop,f) and at the end of life is low. A more detailed investigation on the hardening behavior of AM50 sheet metals at constant amplitude load conditions is given in [30]. The experimentally determined cyclic stress–strain curve CSSC shown in Fig. 3 is used for the phenomenological model described in Section 4. Due to twinning and detwinning, the CSSC is different in tensile and compressive direction (Fig. 3), which was also reported in a previous study [23].

(e)

(a) [%]

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

0

-1

4. Phenomenological model for the description of hysteresis loops

one full loop

4.1. Masing model

. -2 0

100

-3

=2·10 /s

200

300

time [s]

(b)

100

3

5

2

true stress

(t)

[MPa]

hysteresisno.: R (e),envelope=-1 a,t(e),envelope=2% (1) 200 a,t(e),inside1=0.8% (3,5,8) 1 a,t(e),inside2=0.4% (2,4,6,7)

8

  10

6

r r et ¼ þ 0

0

E

4 7

-100

full loop at 0.5Nloop,f AM50 RD

-200 -2

-1

0

true strain

(c)

2

100

(t)

[MPa]

1 (t) [%]

.start and first full loop ..all further loops full loop at ... 0.5Nloop,f

200

true stress

The Ramberg–Osgood equation (Eq. (1)) [36] in combination with the Masing model (Eq. (2)) [37] are typically used for the description of stress–strain hysteresis loops within numerical fatigue analyses with the local strain concept in commercial fatigue analysis software like nCode DesignLife. Within Eq. (1), the three material constants Young’s modulus E, cyclic strength coefficient K0 , and cyclic strain hardening exponent n0 are used for the calculation of the total strain et as a function of the stress r. In Eq. (2), the stress r and total strain et are replaced by the relative stress Dr and relative total strain Det, and the size of the graph of Eq. (1) is doubled within Eq. (2).

0

-100

Nloop,f=40 AM50 RD

-200 -2

-1

0

true strain

1 (t)

2

[%]

Fig. 5. (a) Strain–time function of a variable amplitude strain controlled test, shown in (b) and (c); (b) one full loop at 0.5Nloop,f (AM50); (c) same variable amplitude test, where all loops are shown.

cycles to failure 0.5Nf, and in Fig. 5c. Thus, the slopes of reversals in the twinning and detwinning region are rising with increasing number of cycles. Fig. 5c shows an example with variable amplitude where all loops are shown. It can be seen that the largest difference occurs between the first and the second full loop. The

Det ¼

n

K

  10 Dr Dr n þ2 E 2K 0

ð1Þ

ð2Þ

Fig. 6a shows the experimentally determined cyclic stress– strain curve CSSC. In addition, the tensile and compressive branches of the CSSC were approximated with the Ramberg– Osgood equation (Eq. (1)). The cyclic strength coefficient K0 and the cyclic strain hardening exponent n0 were determined by the method described in [36]. Corresponding values are shown in Fig. 6a. The Masing model as used in commercial fatigue software only allows the usage of point symmetric CSSC without differentiation of the tensile and compressive direction. It can be seen from Fig. 6a that the tension compression asymmetry of magnesium sheet metals leads to different curves for tensile and compressive loading. When using the Masing model, the initial loading cycle is represented by the Ramberg–Osgood equation (Eq. (1)). Every further loading or unloading reversal is determined by Eq. (2). The relative strain Det and the relative stress Dr (Eq. (2)) start at zero at each reversal point of a hysteresis loop. As illustrated in Fig. 6a, the descending reversal, determined by Eq. (2), reveal a large deviation from the CSSC at the opposite site. A similar deviation is reached for the ascending reversal. Experimentally determined hysteresis loops with variable amplitude loading are compared with the Masing model within Fig. 6b. In this case, the values of K0 and n0 were taken from the tensile branch of the CSSC as listed in Fig. 6a. Apart from the right upper reversal point, all other reversal points considerably deviate. In addition, the hysteresis areas and shapes are very poor. Consequently, an alternative phenomenological model is needed for the numerical fatigue analysis of wrought magnesium alloys. 4.2. New model As explained in [34], wrought magnesium alloys exhibit material memory, which must be realized in the new phenomenological

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J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

(a) 300

250

(e)

AM50 RD -300 -2

-1

0

true strain

(b) 300

1 (t)

P,down

E tw

-125

P,up

2 (e)

[%] -250 -2.5

experiment model

ascending RCS -1.25

~P

~S 0

engin. strain

a,t(e),inside=0.8%

200

~S/aup 0

(e)

engin. stress

E = 45 GPa K'= 240 MPa n' = 0.080

-100

125

(e)

0

[MPa]

E = 45 GPa K'= 275 MPa n' = 0.022

100

(t)

true stress

AM50 RD, R (e)=-1 E dtw

-200

1.25 (e)

2.5

[%]

Fig. 7. Visualization of material constants using an experimentally determined envelope hysteresis loop with the corresponding calculated strain components (AM50).

100

(t)

[MPa]

plastic strain component descending RCS (e)

[MPa]

200

true stress

pseudoelastic

elastic experimental CSSC

as well as all further material constants of all investigated alloys are listed in Table 3.

0 -100

DeelðeÞ ðDrðeÞ Þ ¼

-200

R (e),envelope=-1,

-300 -2

-1

AM50 RD a,t(e),envelope=2%

0

true strain

1

2

(t) [%]

Fig. 6. (a) Experimentally determined and with the Ramberg–Osgood equation (Eq. (1)) approximated cyclic stress–strain curve of AM50 for tensile and compressive direction as well as one reversal for each direction, determined by Eq. (2); (b) comparison of the Masing model with experimentally determined variable amplitude hysteresis loops of AM50.

model. Therefore, different conditions were defined within the new model similar to the Masing model [37]. The equations of the model were explained in Section 4.2.1 and programmed within a Matlab routine, explained in Section 4.2.2. 4.2.1. Mathematical formulation for small deformations To represent the deformation mechanisms of magnesium, the total relative strain Det(e) is decomposed into three fundamental components within Eq. (3). These are the elastic Deel(e), the pseudoelastic Depsel(e), and the plastic strain component Depl(e) as a function of the relative stress Dr(e). The relative stress Dr(e) and the relative total strain Det(e) as well as the strain components Deel(e), Depsel(e), and Depl(e) are related to a relative coordinate system RCS. The relative coordinate system represents the origin of each specific reversal similar to the Masing model [37], indicated in Fig. 7.

DetðeÞ ðDrðeÞ Þ ¼ DeelðeÞ ðDrðeÞ Þ þ DepselðeÞ ðDrðeÞ Þ þ DeplðeÞ ðDrðeÞ Þ

ð3Þ

The linear elastic strain component Deel(e) (Eq. (4)) is defined by the Young’s modulus E with the values for the evaluated materials from [23,30,49,50]. Fig. 7 shows a measured stabilized hysteresis loop of AM50 and illustrates the linear elastic component of the descending as well as ascending reversal. The Young’s moduli E

DrðeÞ E

ð4Þ

To represent the pseudoelastic deformation behavior, described in Section 3, a natural logarithm containing a natural exponential function was found to be most suitable. The basic function is introduced in Fig. 8a for an enhanced comprehension of the pseudoelastic strain component Depsel(e). In Fig. 8a, the simplest form y1(x) exhibits a horizontal asymptote y = 0 and an inclined asymptote y = x. Therefore, the graph of the function rises early and converges to a straight line. With a subtrahend behind the variable x, the graph of y2(x) shows how the graph of y1(x) could be shifted along the abscissa. The inclined asymptote of y1(x) and thus the position with the highest curvature shifts too. With a multiplicative value behind the natural logarithm, the function could be stretched in the ordinate direction (y3(x)). Therefore, also the slope of the inclined asymptote increases. Based on the basic function described within Fig. 8a, the pseudoelastic strain component Depsel(e) is described by Eq. (5). The cut-off stresses Dr(e) = rP,up and Dr(e) = rP,down represent the points with the largest curvature and lies in the intersection point of the two asymptotes of Eq. (5) (Fig. 7a). The material constant rP,up is used for ascending reversals and rP,down for descending ones. The designation rP,up/down in Eq. (5) indicates that the relevant constant has to be chosen: rP,up for ascending reversals and rP,down for descending reversals. The pseudoelastic cut-off stresses rP,up/down can be identified as the relative stress in the considered relative coordinate system RCS at which the reversal considerably deviates from linear elastic behavior (about 20%, see Fig. 7). After exceeding rP,up/down, the curve of Depsel(e)(Dr(e)) converges to a straight line, where the slope is proportional to the material constant P (Fig. 7). The determination of P is described in Section 4.2.2. The factor ‘‘50’’ is responsible for the pre-definition of the general curve shape and was found to be valid for all considered Mg alloys. The denominator within the natural logarithm of Eq. (5) ensures that Eq. (5) exactly coincides the origin of the actual relative coordinate system RCS. The memory factor mpsel (Eq. (5)) is used for ensuring the material memory, is explained in detail within Section 4.2.2, and is automatically determined by the routine.

313

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323 Table 3 Values of the model input parameters (material constants) for all evaluated alloys. Acronym

Unit

Expected range

AM50

ME21

AZ31B [49]

AZ61A [50]

E P

GPa – MPa MPa – – MPa –

40–46 5  104–5  103 50–300 50–300 0.01–0.1 15–60 (50)–(200) 0–1

45.0 1.40  103 125 175 0.0370 35.0 170 0.600

44.0 2.50  103 50.0 100 0.0400 25.0 77.0 0.0500

44.8 2.00  103 75.0 225 0.0400 50.0 155 0.800

43.3 1.50  103 75.0 125 0.0500 50.0 135 0.500

rp,up rp,down T S

rtw Rr

(a) 6

y2(x)=ln(exp(x-3)+1) y1(x)=ln(exp(x )+1) asymptotes

(e) [MPa]

200

engin. stress

ordinate -y

4

descending reversals of loops with R (e)=-1, AM50 RD

(a) 300

y3(x)=ln(exp(x-3)+1) 2

2

plastic portion

(mpl=1)

0 tw

-100 -200

~S T

-300

0 -3

0

3

twin exhaustion

6

-400

abscissa -x

8

6

4

relative total engin. strain

(b) 2 (b) 1

2

0

t(e) [%]

ascending reversals of loops with R (e)=-1, AM50 RD

300

(e) [MPa]

200

0

engin. stress

ordinate -y

pl(e)

100

y6(x)=0.5 (tanh((x-4)/2)+1)

y5(x)=0.5 0.5 (tanh((x-4)/2)+1) y4(x)=0.5( tanh((x-4)/2)+1) -1 -3

0

3

6

9

100 0 -100

slope at

-200

dtw=- tw·aup

-300

abscissa -x Fig. 8. Basic functions of the (a) pseudoelastic and (b) plastic strain component.

dtw (~S/aup)

-400 0

2

4

relative total engin. strain

DepselðeÞ ðDrðeÞ Þ ¼ ln



 expððDrðeÞ  rP;up=down Þ=50 MPaÞ þ 1  P  mpsel expðrP;up=down =50 MPaÞ þ 1 ð5Þ

At wrought magnesium alloys, cyclic plastic strain is mainly caused by twinning and detwinning (e.g. [8,10]) and is represented by the plastic strain component Depl(e). This component has to describe the sigmoidal shape, which is provided by a hyperbolic tangent. The basic function is introduced within Fig. 8b. By the graph of y4(x), the point symmetric and sigmoidal shape, extending from 1 < y < 1, can be seen. The curve progression is stretched or shrinked in the abscissa direction by multiplying or dividing the variable x, and is shifted in the ordinate direction by adding a value to the hyperbolic tangent function (y5(x)). By subtracting a value from the variable x the graph is shifted in the abscissa direction (y6(x)). Multiplying the hyperbolic tangent function with 0.5 leads to a sigmoidal shaped function with values from 0 to +1 (y6(x)).

6

8 t(e) [%]

Fig. 9. Visualization of material constants and factors using (a) descending reversals and (b) ascending reversals of experimentally determined hysteresis loops, which were horizontally aligned (AM50); a calculated plastic strain component is additionally plotted in (a).

The corresponding equations for Depl(e) are Eqs. (6) and (7). Eq. (7) specifies the substitution function U(Dr(e)), which is included in Eq. (6). The subtraction of U(Dr(e) = 0) within Eq. (6) ensures that this strain component Depl(e) exactly starts at zero. This is necessary, because Eq. (7) converges rapidly to zero, but did not reach it. The three material constants T, S, and rtw are necessary to model the plastic strain component. To visualize these constants, descending reversals of strain controlled hysteresis loops with a strain ratio Re(e) of 1 were horizontally aligned within Fig. 9a, so that every reversal starts at the relative total engineering strain Det(e) = 0. The illustrated reversals were taken from hysteresis loops with total strain amplitudes between 0.3% 6 ea,t(e) 6 4%,

314

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

including the loops shown in Fig. 4b. It can be recognized from Fig. 9a and [30] that twinning occurs in a range around a specific stress level. In [30] is explained that a distinct start and end of twinning exist and that the associated global stress level is equal for one alloy, independent from the strain ratio Re or strain amplitude ea,t(e). This stress level is expressed by the material constant rtw and is defined as the stress at the inflection point of the plastic strain component of the descending reversal (Figs. 7 and 9a). The maximum amount of Depl(e) is represented by the material constant T (Eq. (6)) as shown in Fig. 9a. The half of T is reached at the inflection point of the plastic strain component. It is proven by [30,52,53] that twinning exhausts, which is also indicated in Fig. 9a. The material constant S (Eq. (7)) is proportional to the slope at the inflection point of the plastic strain component (Figs. 7 and 9a) and therefore, it stretches or shrinks the curve of Depl(e) in the stress direction with the inflection point as the center. The value rrp(e) (Eq. (7)) is the global stress of the last reversal point at the origin of the actual RCS. The memory factor mpl (Eq. (6)) is used for ensuring the material memory, is explained in detail within Section 4.2.2, and is automatically determined by the developed Matlab routine. The plastic strain component Depl(e)(Dr(e)), plotted in Fig. 9a, is valid for the alloy AM50, where a memory factor mpl of 1 is chosen in order to visualize the material constant T.

  DeplðeÞ ðDrðeÞ Þ ¼ UðDrðeÞ Þ  UðDrðeÞ ¼ 0Þ  T  mpl

ð6Þ



   DrðeÞ  jrrpðeÞ j þ aup=down  rtw 1 tanh UðDrðeÞ Þ ¼  aup=down þ 1 2 S ð7Þ The shape factors aup and adown are necessary for modelling the distinct differences between ascending and descending reversals and are written as aup/down within Eq. (7). The indices ‘‘up’’ and ‘‘down’’ of the shape factors aup/down in Eqs. (7) and (8) indicate the use for ascending or descending reversals, respectively. Apart from material scatter and independent from the strain ratio Re, the inflection points of descending reversals lie on the same global stress level and the slopes at these points are equal for all reversals (Fig. 9a) [30]. Thus, the shape factor adown is always equal one (Eq. (8)). Fig. 9b contains the ascending reversals of the corresponding descending reversals, illustrated in Fig. 9a. For the ascending reversals, the global stress level of the inflection points rdtw (Fig. 7) is proportional and the slope at these points is inversely proportional to the plastic strain component of the envelope hysteresis loop (Fig. 9b). Both proportionalities are modelled by the shape factor aup, defined by Eq. (8). The upper inflection point is expressed by the stress level (rdtw = rtw  aup) with a slope proportional to S/aup (Eq. (7)). In Eq. (8), the representative value aup between 0 and 1 for the maximum occurring plastic strain component is determined. Therefore, the variable Dr(e) is replaced by the value Drmax(e), which represents the maximum stress range of the envelope hysteresis loop. This maximum stress range Drmax(e) is determined from the CSSC. Because the descending reversal of the envelope hysteresis loop is considered for the calculation of aup with Eq. (8), rrp(e) represents the global stress of the upper reversal point on the CSSC. The shape factor aup is automatically precalculated by the developed Matlab routine and is constant over the whole simulation.

adown ¼ 1; aup     DrmaxðeÞ  jrrpðeÞ j þ 1  rtw 1 tanh ¼ 1 þ1 2 S

ð8Þ

4.2.2. Simulation procedure The developed mathematical formulations were programmed in a Matlab routine. At each reversal point, a new relative

coordinate system RCS is placed, so that the following reversal extends into the first sector of the new RCS (Fig. 7). At the beginning of a reversal, Dr(e) and Det(e) are zero. Afterwards, Det(e) is increased gradually following the strain–time function and every associated Dr(e) is calculated. To represent the material behavior and fulfill the material memory (e.g. [34]), the reversal points of the envelope hysteresis loop must coincide with the cyclic stress–strain curve CSSC as shown in Fig. 10a and b. In standard fatigue software, the Masing model [37] typically uses point-symmetric laws like the Ramberg–Osgood equation [36] with an identical shape of every reversal. Thus, modelled hysteresis loops are closed and coincide with the CSSC automatically. In contrast, wrought magnesium alloys reveal asymmetric hysteresis loops and the shape of reversals at envelope hysteresis loops with different strain amplitudes as well as various inner hysteresis loops differ distinctly. Furthermore, ascending and descending reversals reveal different shapes. Therefore, additional conditions must be included in the routine to enable arbitrary loadings and to follow the material memory. To fulfill material memory, one condition is that the stress–strain curve follows the superior hysteresis loop after completing an inner loop. A further condition is that all three strain components must be equal for corresponding ascending and descending reversals, which represents a cyclic stable material. Satisfying these conditions, closed loops without gaps or overlaps are produced. The conditions can be concluded as follows: i. envelope hysteresis loop must coincide with the CSSC ii. stress–strain curves must follow superior reversals after completing inner loops iii. the three strain components must be equal for corresponding ascending and descending reversals To fulfill the conditions i–iii, the memory factors mpsel and mpl are used (Eqs. (5) and (6)). Both factors are determined automatically by the Matlab routine and are different for each specific reversal. The load definition in terms of a strain–time function is read out by the Matlab routine and the maximum and minimum strain values are filtered. With these values, the first calculated reversal is the descending reversal of the envelope hysteresis loop. Thus, the calculation of this reversal represents a special case, where both memory factors have to be changed relative to the start value 1. To ensure that the lower reversal point of the envelope descending reversal coincides with the CSSC (condition i), mpl and mpsel are reduced or increased, and thus, the appropriate strain components (Eqs. (5) and (6)) are adjusted. The relation between the change of mpl and mpsel within the envelope descending reversal is defined by the ratio Rr (Eq. (9)). The ratio Rr enables the correct adaption of the memory factors, so that the shape of the stress–strain curves coincides with experiments most accurately. In detail, if mpl is changed by x%, mpsel will be changed by Rr  x% (Eq. (9)). To fulfill condition i, this procedure is only necessary once and just for the descending reversal of the envelope hysteresis loop. Considering the conditions ii and iii, mpl and mpsel are determined automatically by the Matlab routine for all further reversals.

Rr ¼

1  mpsel 1  mpl

ð9Þ

To illustrate the determined memory factors, Fig. 10a–c show three different hysteresis loops with and without the application of the memory factors. The hysteresis loops in Fig. 10a and b are envelope hysteresis loops and in Fig. 10c, an inner hysteresis loop with the strain ratio Re(e) = 0 is evaluated. Different values of both memory factors mpl and mpsel were determined for the descending reversals shown in Fig. 10a and b

315

el(e)=0.98%,

psel(e)=0.67%, model with mpl=mpsel=1 final model experiment ascending reversal: 125 m pl =0.64 mpsel =0.76 aup =0.77

pl(e)=2.35%

(e)

engineering stress

(e)

[MPa]

250

0

AM50 RD CSSC descending reversal: mpl =0.83 mpsel =0.90 Rr =0.60

-125

R -250 -2.5

(e)=-1,

a,t(e)=2%

-1.25

0

1.25

engineering strain

(b)

(e)

el(e)=0.87%,

psel(e)=0.36%, model with mpl=mpsel=1 final model experiment ascending reversal: 125 m pl =0.11 mpsel =0.49 aup =0.30

2.5

[%] pl(e)=0.37%

engineering stress

(a)

[MPa] hysteresis

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

el(e)

A

psel(e)

pl(e)

B C 200 100

B

0 C

-100

A

a,t(e),C=1% a,t(e),B=1.5% a,t(e),A=

-200 -2

-1

0

engineering strain

1

2%

2

(e) [%]

Fig. 11. Visualization of material memory using the example of numerically determined hysteresis loops of AM50; strain components for the ascending and descending reversals are indicated above the hysteresis loops.

engineering stress

(e)

[MPa]

250

0

AM50 RD

R -250 -2.5

(e)=-1,

a,t(e)=0.8%

-1.25

0

engineering strain

(c)

125

engineering stress

(e)

[MPa]

250

descending reversal: mpl =0.33 mpsel =0.60 Rr =0.60

CSSC

-125

1.25

2.5

(e) [%]

model with mpl=mpsel=1 final model ascending reversal: mpl =0.16 mpsel =0.74 aup =0.77 AM50 RD

0

descending reversal: mpl =0.83 -125 mpsel =0.90 Rr =0.60 R (e),envelope=-1, a,t(e),envelope=2%, a,t(e),inner=1% -250 -2.5 -1.25 0 1.25 2.5

engineering strain

(e)

[%]

Fig. 10. Illustration of the effect of the memory factors mpl and mpsel as well as the ratio Rr at AM50; (a) using the example of an envelope hysteresis loop with ea,t(e) = 2% and (b) with ea,t(e) = 0.8%; (c) example of an inner hysteresis loop with ea,t(e),inside = 1% within an envelope hysteresis loop with ea,t(e) = 2%.

(condition i) to coincide with the CSSC. The memory factors of the ascending reversals were calculated, so that all strain components are equal in comparison to that of the corresponding descending

reversals (condition iii). The descending reversal of the inner hysteresis loop in Fig. 10c is coincident with the descending reversal of the envelope hysteresis loop, which is why its memory factors are identical to those of Fig. 10a. Fig. 10a–c list also the shape factors aup. To illustrate the conditions ii and iii, an example of numerically calculated hysteresis loops is shown in Fig. 11. The envelope hysteresis loop A and two inner loops B and C are shown with its corresponding strain components. At the reversal point at e(e) = 1% the Matlab routine uses the upper reversal point of the superior loop A (envelope loop) for the determination of the memory factors. As second information, the Matlab routine uses the strain component values of the corresponding descending reversal. This input is sufficient to determine both memory factors mpsel and mpl. Starting at the upper reversal point e(e) = 1% (loop C, Fig. 11), the routine uses the lower reversal point e(e) = 1% as the first input and the stored strain component values between e(e) = 1% and e(e) = 1% as the second input to calculate the descending reversal. The examples with a total strain amplitude ea,t(e) of 2% (Fig. 12a–c) and ea,t(e) of 0.8% (Fig. 12d–f) were chosen to illustrate the curve progressions of the different strain components. Fig. 12a and b show the strain components as a function of the relative stress Dr(e) of a modelled completely reversed test with a strain amplitude ea,t(e) of 2%. Fig. 12d and e illustrate a similar example with ea,t(e) = 0.8%. Fig. 4b and Fig. 9a and b contain the corresponding measured hysteresis loops. The summations of the three strain components are shown in Fig. 12c and f. The start point of each reversal was drawn in the origin of the relative coordinate system (RCS). It can be seen that ascending and descending reversals differ distinctly, but coincide at the reversal points. This is in accordance with the material memory and the experiments as shown in Section 3. If the total strain amplitude is large (e.g. ea,t(e) = 2%), the plastic strain component dominates compared to the elastic or pseudoelastic one (Fig. 12a and b). The elastic component becomes more and more dominant for smaller strain amplitudes ea,t(e) (Fig. 12d and e). The individual strain component Depsel(e) or Depl(e) exhibits a different curve progression for an associated ascending and descending reversal, but its amount is finally equal (Fig. 12a, b, d, and e). Viewing the plastic strain components of the descending reversals, shown in Fig. 12b and e, it is visible that these components are not exhausted at the endpoints. This is in

200

=0.8%, R (e)=-1 ascending reversal a,t(e)

elastic c.

(e)

el(e)

pseudoelastic 0.5 c. psel(e) plastic comp. pl(e)

0 0

100

200

relative engin. stress

300 (e)

200

relative engin. stress 1

elastic c.

(e)

(f)

pl(e)

0 200

[%]

[MPa]

pseudoelastic 0.5 c. psel(e) plastic comp.

100

descending reversal ascending 2 reversal reversal start point AM50 RD

0 200

[MPa]

400

relative engin. stress

(e)

[MPa]

=0.8%, R (e)=-1

1.5

a,t(e)

descending reversal 1 ascending reversal reversal 0.5 start

point AM50 RD

0 0

300 (e)

=2%, R (e)=-1

a,t(e)

0

el(e)

relative engin. stress

4

400

=0.8%, R (e)=-1 a,t(e) descending reversal

0

[MPa]

(e)

0 0

relative engin. strain

(e)

[%]

1

elastic comp. el(e)

400 (e) [MPa]

relative engin. strain

0

1

[%]

el(e)

pseudoelastic component psel(e)

(e)

elastic comp.

(c)

=2%, R (e)=-1 descending reversal 2 plastic comp. pl(e) a,t(e)

relative engin. strain

pl(e)

pseudoelastic 1 comp. psel(e)

relative engin. stress

relative engin. strain

(e)

relative engin. strain

plastic component

[%]

2

0

(d)

(b)

=2%, R (e)=-1 ascending reversal a,t(e)

(e)

relative engin. strain

(e)

(a)

[%]

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

[%]

316

100

200

relative engin. stress

300 (e)

[MPa]

Fig. 12. Elastic, pseudoelastic and plastic strain components of two numerically determined stress–strain hysteresis loops of AM50; (a and b) ea,t(e) = 2% and (d and e) ea,t(e) = 0.8%; (c) and (f) illustrate the summation of the respective strain components.

agreement with the descending reversals, shown in Fig. 4b and Fig. 9a. In contrast, the ascending reversals of the plastic strain component (Figs. 12a and d) reveal the inflection point similar to the experimentally determined hysteresis loops as shown in Section 3 (Fig. 4b). To visualize the workflow of the developed Matlab routine, Fig. 13 shows a flowchart of all relevant steps. First, the material constants as listed in Table 3 and the load definition must be given by the user. The Matlab routine does not consider an influence of the waveform type such as sinusoidal or multilinear waveform since creep effects are negligible at ambient temperature [54]. During the pre-calculation step, a strain vector with a user-defined increment is created out of the strain–time function. The strain increment was set to 0.05% to provide an adequate resolution. During the next pre-calculation step, mpl and mpsel are determined, so that the reversal points of the envelope hysteresis loop coincide with the cyclic stress–strain curve CSSC. After the pre-calculation, the calculation of the stress–strain curve is done, considering the conditions i, ii, and iii. Within the post-calculation, the result can be compared with experimentally determined data and all relevant hysteresis loop measures like strain energy density components or various stress or strain values can be saved. The values of all material constants are bounded within Table 3 to provide meaningful values for the simulation of further alloys. The material constants can be manually adapted until an adequate correlation between experiments and the model is reached. With a few steps, the material constants can be fixed. Overall, there are eight material constants, which have to be given by the user. Five constant amplitude tests in the strain range 2% 6 e(e) 6 2% were found to be enough to determine the material constants.

4.2.3. Derivation of fatigue parameters from hysteresis loops For a fatigue life calculation, different fatigue parameters can be derived from hysteresis loops. In Fig. 14, an overview of general definitions is shown. Stress, strain, and strain energy density components as indicated in Fig. 14 can be measured from hysteresis loops and used for the definition of a fatigue parameter. In many completely reversed strain-controlled evaluations, the Manson–Coffin–Basquin approach [31–33] shows an adequate correlation with the experiments (e.g. [9,15–19,23]). Elastic and plastic strain components were used to predict the fatigue life in such cases. To decompose the total strain amplitude ea,t into an elastic component ea,el and a plastic component ea,pl, different methods were defined, which results in quite different values as explained by Kandil [55]. Three methods are indicated in Fig. 14, where their differences can be seen. The mostly used method ea,pl,m1 = ea,t  ra/E with E = E1 = E2 as the Young’s modulus and ra as the stress amplitude (e.g. [9,15,16,19,23]) is described in the Stahl-Eisen testing guideline SEP1240 [51], and was also described in the original investigations of Manson and Coffin [32,33]. The standard ASTM: E606/E606M-12 uses a similar equation. The plastic strain amplitude ea,pl,m2 represents the real plastic strain amplitude at 0 MPa stress. In [56,57], the plastic strain amplitude ea,pl,m3 is defined as half width of the hysteresis loop at the mean stress rm (Fig. 14). More comprehensive explanations of the described methods are given in [18,30,55]. The Manson–Coffin–Basquin approach works well at completely reversed strain-controlled conditions, but at load conditions with different mean stresses, a correction is necessary. There exist different stress and strain based methods such as the Smith–Watson–Topper model [58], evaluated e.g. in [14,21,29]. In [29] is shown that an energy based fatigue model correlates better with experiments compared to the Smith–Watson–Topper

317

pre-calculation

input

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

input of the eight material constants

load definition: strain-time curve

fragment strain-time curve into single strain increments

adapt mpl and mpsel so that the envelope hysteresis agrees with the CSSC

select the subsequent strain step

material memory?

yes

leave the completed loop and engage in the superior one

calculation

no calculate the stress increment to the corresponding given strain increment

reversal point? no

yes realign the RCS and adapt mpl and mpsel for the next branch

post-calculation

end of strain-time curve ? yes evaluation and storage of stress, strain, and energy density components

no

comparison of experiment and model data yes

content with result?

no

Fig. 13. Flowchart of the developed Matlab routine for the phenomenological model.

reached in several studies (e.g. [20,22,26,29,30]) by using energy based fatigue models. To develop an energy based fatigue parameter, different strain energy density components can be defined (Fig. 14). The plastic strain energy density per cycle DWpl(t) and the tensile linear elastic strain energy density per cycle DWel+(t) are the most important ones [30]. Values of the tensile pseudoelastic strain energy density per cycle DWpsel+(t) (Fig. 14) are generally very small in comparison to DWpl(t) and DWel+(t), which is why they were neglected completely [30]. It was found that fatigue crack initiation is mainly caused by dislocation slip at low strain amplitudes (ea,t(e) < 0.3%) and by reiterative twinning and detwinning at high strain amplitudes (ea,t(e) > 0.3%) [18,59]. After exceeding the compressive yield stress, the plastic strain increase rapidly due to twinning, resulting in large hysteresis areas [30]. Thus, the influence of the twin formation on the fatigue live is considered by the plastic strain energy density per cycle DWpl(t). Twinning and detwinning lead to asymmetric stress–strain hysteresis loops with tensile mean stresses. It was found in [60] that a tensile mean stress is harmful to the fatigue behavior by accelerating the crack initiation and crack propagation. This effect is represented by DWel+(t) as explained in [26,29,30]. For all evaluations within this paper, strain energy density components are numerically integrated with the trapezoidal rule, based either on experimentally determined (crosses in Fig. 14) or on calculated hysteresis loops. During the calculation of DWel+(t) in the range of Rr(t) 2 [0;1], the portion of positive rmin(t), defined as r2min(t)/(2E), has been subtracted [30]. In most of the previous energy based fatigue investigations, the combined strain energy density without weighting DWcomb,ww(t) (Eq. (10)), also called total strain energy density, was used [20,22,26,29,30]. The combined strain energy density DWcomb(t) (Eq. (11)) was developed in [30] to improve the correlation especially for a wide range of load conditions. A 25% weighting of the tensile elastic strain energy density per cycle DWel+(t) achieved the maximum correlation of predicted and experimentally determined cycles to failure Nf. It was found out that the 25% weighting of DWel+(t) represents the material specific mean stress sensitivity. To investigate fatigue life prediction for tests with variable amplitudes, three different fatigue parameters were chosen: the two energy based fatigue parameters described by Eqs. (10) and (11) and the Smith–Watson–Topper model [58] (Eq. (12)). As explained in [29,30] the regression functions show a bilinear behavior in the logarithmic scale. Eqs. (10)–(12) list the corresponding regression functions for all three evaluated fatigue models. Material constants shown in Eqs. (10) and (11) were taken from [30]. The regression functions of the Smith–Watson–Topper parameter PSWT(t) with the corresponding material constants were calculated similar as described in [29], but for true stress and strain values.

(

DW comb;wwðtÞ ¼ DW elþðtÞ þ DW plðtÞ ¼

364N0:891 f

if Nf 6 1124

3:25N0:223 f

if Nf > 1124 ð10Þ

, AM50 RD (

DW combðtÞ ¼ 0:25DW elþðtÞ þ DW plðtÞ ¼

453N 0:959 f

if Nf 6 2204

3:02N0:308 f

if Nf > 2204 ð11Þ

Fig. 14. General definitions of hysteresis loop measures, example of AM50.

model. Park et al. [26] claimed that an energy based fatigue model predicts the fatigue life most accurately. Good correlations were

PSWTðtÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rmaxðtÞ ea;tðtÞ E ¼

(

1754N0:315 f

if Nf 6 1114

380N0:097 f

if Nf > 1114

ð12Þ

318

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

4.2.4. Comparison of the numerical model with experiments For a comparison with experimentally determined hysteresis loops, recordings at a nearly stabilized state at half of the fatigue life (0.5Nf or 0.5Nloop,f) were used. A comparison of calculated and experimentally determined hysteresis loops is given in Fig. 15a for AM50 sheet metals using

(a)

the example of completely reversed strain-controlled conditions with three different strain amplitudes ea,t(e). The examples show a good congruence. In Fig. 15b–f, comparisons of five different strain-controlled tests with variable amplitude loading can be viewed. The hysteresis numbers indicate the test sequence (see e.g. Fig. 5a and b). In all five examples, each hysteresis loop occurs

(d)

experiment model

200

[MPa]

1.8% Nf=96

0 1% Nf=209 -100

-1

0

1

true strain

30 25

[MPa]

10 15

AM50 RD Nloop,f=58 R (e),envelope=-1

21

(t)

2

-2

-1

[%]

0

1

true strain

2

(t) [%]

(e)

(3,5,8) hysteresis no.: (2,4,6,7) a,t(e),envelope=2% (1) 1 experiment model 2 a,t(e),inside1=0.8% a,t(e),inside2=0.4%

5

(t)

true stress

5

-200 -2

100

3

0

-100

-200

200

2

4 35

AM50 RD R (e)=-1

(b)

39

100

(t)

a,t(e)=0.4% Nf=5300

true stress

true stress

(t)

[MPa]

200

100

experiment hysteresis no.: model a,t(e),inside=0.1% (2-39) 1 a,t(e),envelope=2% (1)

8

3 6

0 4

7

-100

AM50 RD Nloop,f=40 R (e),envelope=-1

-200 -1

0

true strain

(c)

18

100

16

15

(f) 150 2

100

19

17

4

-100

5

11 10

8

9

7

6

AM50 RD Nloop,f=57 R (e),envelope=-1

-200 -2

-1

experiment hysteresis no.: model a,t(e),inside=0.1% (2-4) 2 1 a,t(e),envelope=0.4% (1)

50

(t)

3

12

0

true stress

(t)

14 13

true stress

2

experiment hysteresis no.: model a,t(e),inside=0.2% (2-19) 1 a,t(e),envelope=2% (1)

200

[MPa]

1 (t) [%]

[MPa]

-2

0

true strain

1 (t) [%]

2

0

3

-50 4

AM50 RD Nloop,f=1570 R (e),envelope=-1

-100 -150 -0.5

-0.25

0

true strain

0.25 (t)

0.5

[%]

Fig. 15. Comparison between experimentally and numerically determined hysteresis loops of AM50 sheet metals; (a) completely reversed strain-controlled tests; (b–f) examples with variable strain amplitudes; results at 0.5Nf or 0.5Nloop,f; hysteresis numbers indicate the test sequence.

319

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

5. Fatigue investigations at variable amplitude strain-controlled loading In the following, all fatigue investigations were carried out considering AM50 sheet metals.

(a) 150

experiment model

50

true stress

(t)

[MPa]

100

a,t(e)=0.4% Nf=5100 a,t(e)=1.6% Nf=89

0 a,t(e)=0.8% Nf=504

-50

ME21 ED R (e)=-1 -100 -2

-1

0

true strain

(b) 300

1 (t)

2

[%]

experiment model a,t(e)=2%

100

true stress

(t)

[MPa]

200

a,t(e)=0.4%

0

a,t(e)=1%

-100

AZ31B ED R (e)=-1

-200 -2

-1

0

true strain

(c)

[MPa] (t)

100

1 (t)

2

[%]

experiment model

200

true stress

once in one full loop until the same sequence starts again (1, 2, 3, 4, . . ., 1). The congruence between the phenomenological descriptions and the experimentally determined curves is high for the envelope hysteresis loop as well as for the inner loops (Fig. 15a–f). A slight deviation can be seen at hysteresis number 5 and 8 with a strain amplitude ea,t(e) = 0.8% (Fig. 15b). The maximum and minimum stresses rmax(t) and rmin(t) are adequately represented in all cases. The largest deviation was observed at hysteresis number 6 with about 20 MPa at rmin(t), but its area fits well (Fig. 15b). Fig. 15c shows an example with 18 inner hysteresis loops with a small strain amplitude ea,t(e) of 0.2%. The inner hysteresis loops reveal a small hysteresis area, which is rising slightly with decreasing mean strain. This behavior is adequately reproduced by the model. Furthermore, most of the peak stresses rmin(t) and rmax(t) are nearly lying upon each other. A similar example is represented in Fig. 15d with 38 inner hysteresis loops with ea,t(e) = 0.1%. All inner hysteresis loops reveal a nearly linear elastic behavior with a very low hysteresis area. The envelope hysteresis loop as well as all inner loops of the model fit well and all peak stresses match the experimental results adequately. Fig. 15e and f show examples with smaller envelope hysteresis loops, where the model can predict the experimental determined behavior quite good. In addition, the phenomenological model was applied on extruded ME21 sheet metals, extruded AZ31B massive rods [49], and extruded AZ61A tubes [50]. All corresponding material parameters are listed in Table 3. A comparison of calculated and experimentally determined hysteresis loops is given in Fig. 16a–c using the example of completely reversed strain-controlled conditions with three different strain amplitudes ea,t(e) per alloy. The general shape of the three alloys appears to be similar to AM50, but the stress values are quite different. The numerically determined hysteresis loops reveal a good congruence for ME21, AZ31B, and AZ61A as well, which can be recognized in Fig. 16a–c. To quantify the correlation between the model and the experiments, the combined strain energy density without weighting DWcomb,ww(t) (Eq. (10)) was evaluated for the example of AM50 (Fig. 17a–f). Fig. 17a shows a comparison of DWcomb,ww(t) for completely reversed strain-controlled hysteresis loops. The maximum deviation between the phenomenological model and the experimental result is 9%. All envelope and inner hysteresis loops of variable amplitude tests, shown in Fig. 15b–f, are evaluated in Fig. 17b–f, where the respective hysteresis numbers are identical. The maximum deviation D is 16% in the example with large inner hysteresis loops (Fig. 17b). A similar good correlation with D = 15% is reached by the example shown in Fig. 17e, where the evaluated hysteresis loops are also comparatively large. At variable amplitude tests with very small inner hysteresis loops (Fig. 17c, d, and f), a larger deviation occurs at some of the inner loops. A small absolute deviation of a hysteresis area causes a large relative deviation for such small inner loops. In Fig. 17c and d, the largest deviation occurs when the tensile elastic strain energy density per cycle DWel+(t) is zero, which is when the maximum stress is negative. The maximum deviation, shown in Fig. 17d is 152%. Nevertheless, such small values of DWcomb,ww(t) cause a negligible low macroscopic damage. As shown in [30], values of DWcomb,ww(t) smaller than 101 lead to a number of cycles to failure Nf greater than 106. Thus, deviations as shown in Fig. 17d of such small absolute values (<102) are negligible.

a,t(e)=0.5% a,t(e)=0.7%

0 a,t(e)=1%

-100

AZ61A ED R (e)=-1 -200 -1

-0.5

0

true strain

0.5 (t)

1

[%]

Fig. 16. Comparison between experimentally and numerically determined hysteresis loops of (a) extruded ME21 sheet metals, (b) extruded AZ31B solid rods [49], and (c) extruded AZ61A tubes [50] at completely reversed strain-controlled tests Re(e) = 1; stabilized loops.

5.1. Experiments with inner hysteresis loops Using Eqs. (10)–(12), the fatigue life of the variable amplitude tests, shown in Fig. 15b–f, were evaluated based on the numerically described hysteresis loops. The results are listed in Table 4.

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

10

0

10

-1

total strain ampl.

W comb,ww(t) [MJ/m³]

(d) 10 1 10

AM50 RD

a,t(e)

0

=152% 10

-1

10

-2

10

-3

1

8

15

22

29

=16%

10

10

0

(c) 10 1

-1

2

1

[%]

experiment model

experiment model

W comb,ww(t) [MJ/m³]

AM50 RD R (e)=-1

AM50 RD

3

4

5

6

7

=65%

10

-1

10

-2

1

36

hysteresis no. as in Fig. 15d

AM50 RD

experiment model =15%

10

0

10

-1

10

-2

1

2

3

4

hysteresis no. as in Fig. 15e

5

9

13

17

hysteresis no. as in Fig. 15c

hysteresis no. as in Fig. 15b

(e) 10 1

experiment model

AM50 RD

0

10

8

(f) 10 W comb,ww(t) [MJ/m³]

1

(b) 10 1

W comb,ww(t) [MJ/m³]

10

experiment model

=9%

3.0 2.6 2.2 1.8 1.4 1.0 0.8 0.6 0.4 0.3

W comb,ww(t) [MJ/m³]

(a) 10 2

W comb,ww(t) [MJ/m³]

320

0

AM50 RD

10

-1

10

-2

10

-3

experiment model =49%

=35%

1

2

3

4

hysteresis no. as in Fig. 15f

Fig. 17. Comparison between experimentally and numerically determined combined strain energy densities without weighting DWcomb,ww(t) of AM50; (a) envelope hysteresis loops including the loops shown in Fig. 15a; (b–f) hysteresis loops as shown in Fig. 15b–f.

Table 4 Fatigue life evaluation at tests with inner hysteresis loops (AM50).

For each example, two specimens were tested and the listed experimentally determined number of full loops to failure Nloop,f is the mean value. The examples shown in Fig. 15b–d reveal comparatively large envelope hysteresis loops (ea,t(e) = 2%), which are mainly responsible for the short fatigue life. As seen in Table 4, all three fatigue parameters calculate adequate results, if the fatigue life is short (examples of Fig. 15b–e). The model with the best result for each example is shaded grey at all further evaluations. The Smith–Watson–Topper parameter PSWT(t) is not defined if the maximum stress rmax(t) of a hysteresis loop is in the negative range. In such cases, PSWT(t) of the specific inner loop was set to zero. At medium fatigue lives (example of Fig. 15f), DWcomb(t) reaches the best result. In [29] was shown that the correlation of the Smith–Watson–Topper fatigue model is the higher the lower the number of cycles to failure Nf at tests with constant amplitudes. The results listed in Table 4 indicate a similar behavior for tests with variable amplitudes. 5.2. Cyclic stress relaxation The aforementioned variable amplitude strain-controlled tests contain each envelope and inner hysteresis loop equally often. Such tests deliver no information about the influence of stress

relaxation. It was observed by several researchers for different magnesium alloys that stress relaxation occurs at asymmetrical strain-controlled tests (e.g. [26,30,61]). Four specimens were tested at different conditions as shown in Fig. 18a and b. The developed model considers generally no cyclic stress relaxation as well as hardening, which is why the hysteresis shape at 0.5Nf or 0.5Nloop,f is calculated. It can be seen in Fig. 5c that the shape of hysteresis loops changes at most at the first few cycles and after that, the change is low. At all four specimens, evaluated in Fig. 18a and b, ten completely reversed cycles with a total strain amplitude ea,t(e) of 2% were applied at the beginning to reach a stabilized state. The tenth cycle of each specimen is shown in Fig. 18a and b with a dashed line. After that, cycles with a smaller strain amplitude were applied at a reversal point until failure occurs. Inner hysteresis loops lie on the most left or most right side of the envelope hysteresis loop. The total strain amplitude ea,t(e) is 0.1% at specimen 1 as well as 2 and 0.3% at specimen 3 as well as 4. The cyclic change of the stress values was evaluated between the first inner cycle and the hysteresis loop determined at 0.9Nf,r. Nf,r is the number of remaining cycles to failure for the small strain amplitude after the first ten cycles with ea,t(e) = 2%. Hysteresis loops with ea,t(e) = 0.1% reveal in both cases (specimen 1 and 2) a nearly linear elastic behavior with negligibly low plastic strain energy density DWpl. Specimen 1 tolerates more than 106 remaining cycles to failure Nf,r, after which the test was stopped. The reason, why no fracture was detected at specimen 1 is that the hysteresis loop is located in the compressive stress region. This region is less harmful for the specimen, which was also observed in [30]. As shown in Fig. 18a, the maximum stress rmax(t) of the left inner loops is rising about 40 MPa within 106 cycles. Specimen 2 endures only 34,000 remaining cycles to failure Nf,r with ea,t(e) = 0.1% and the minimum stress rmin(t) of the inner loops is decreasing about 50 MPa. It can be recognized from Fig. 18a that at such small strain amplitudes, the cyclic mean stress relaxation is more pronounced in the tensile region.

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

125

specimen 1: 10·(0±2%)+ 6 10 ·(-1.9±0.1%); 6 Nf,r>10 1 N2%=10

true stress

0

6

N0.1%=10

103 104 4 3.1·10

5

N0.1%=10 , 10

1

N2%=10

40MPa -125

specimen 2: 10·(0±2%)+ Nf·(1.9±0.1%); Nf,r=34,000

0 4 10 , 10

-250

AM50 RD -2

-1

0

1

true strain

(b) 250

specimen 3: 10·(0±2%)+ Nf·(-1.7±0.3%); Nf,r=48,000 1 N2%=10 4 4.3·10

true stress

(t)

[MPa]

125

N0.3%=100 40MPa 2

4

10 103

10 103 0 N0.3%=10

-125

-1

5.5·10

3 1

N2%=10 specimen 4: 10·(0±2%)+ Nf·(1.7±0.3%); Nf,r=6,150

0MPa AM50 RD -2

2

(t) [%]

20MPa

60MPa

0

-250

Table 5 Fatigue life evaluation at tests with variable amplitudes, considering stress relaxation (AM50).

0

(t)

[MPa]

(a) 250

321

0

true strain

1

2

(t) [%]

Fig. 18. Cyclic stress relaxation of AM50 at different mean strains until fracture; specimens with ea,t(e) = 2% for the envelope loop and (a) ea,t(e) = 0.1% as well as (b) ea,t(e) = 0.3% for the inner loops.

At larger inner hysteresis loops with ea,t(e) = 0.3%, a noticeable hysteresis area and accordingly plastic strain energy density DWpl can be observed in the tensile as well as compressive region at the first inner loops (Fig. 18b). At specimen 3, a peculiar cyclic stress relaxation behavior was observed. The maximum stress rmax(t) rises about 60 MPa within 43,000 cycles and the minimum stress rmin(t) stays nearly constant. A similar behavior was found out in [27,30] at asymmetrical strain-controlled tests in the compressive region. Yu et al. [27] found on tests with a strain ratio Re = 1 that the asymmetrical cyclic hardening and accordingly the asymmetrical cyclic stress relaxation is caused by a larger number of residual twins (undetwinnable twins) compared to more tensile strain ratios such as Re = 1 or Re = 0. Due to the cyclic hardening, also the hysteresis area is decreasing with rising number of cycles N at specimen 3. The asymmetrical cyclic stress relaxation is less distinct in the tensile region at specimen 4 (Fig. 18b). The relaxation of the minimum stress rmin(t) is about 40 MPa and the relaxation of the maximum stress rmax(t) is about 20 MPa. In addition, the hysteresis area is decreasing with rising N. The four specimens in Figs. 18a and b show that the cyclic stress relaxation behavior and accordingly the cyclic hardening behavior depend on several conditions. A consideration of cyclic stress

relaxation was not implemented in the model. Only the first inner hysteresis loops are taken into account adequately. To evaluate the influence of cyclic stress relaxation on the results of a fatigue analysis, Table 5 lists calculated remaining cycles to failure Nf,r using Eqs. (10)–(12). To get the necessary stress, strain, and strain energy density components used in the fatigue models, the first inner hysteresis loops and the hysteresis loops at 0.9Nf,r of experimentally determined hysteresis loops were evaluated. When calculating the remaining cycles to failure Nf,r of the small inner hysteresis loops, the damage portions of the first ten cycles with ea,t(e) = 2% were added with linear damage accumulation. As seen in Table 5, the Smith–Watson–Topper model fails to predict the fatigue life in all cases. At specimen 1, where the maximum stress of the inner hysteresis loops lie at every cycles below zero, the Smith–Watson–Topper fatigue parameter PSWT(t) is not defined. The results, determined by both energy based fatigue parameters (Eqs. (10) and (11)) are more adequate. Independent of the considered inner hysteresis loop, the calculated remaining fatigue life Nf,r is >106, which is in agreement with the experimental result (Table 5). At specimen 2, the combined strain energy density without weighting DWcomb,ww(t) (Eq. (10)) calculates more adequate results compared to the combined strain energy density DWcomb(t) (Eq. (11)). Considering the hysteresis loop of the first inner cycle, the number of remaining cycles to failure Nf,r is calculated too small and considering the loop at 0.9Nf,r, Nf,r is calculated slightly too high. Using DWcomb(t), a similar tendency can be observed for specimen 2, but Nf,r is calculated too high for both evaluated loops. As seen in Table 5 at specimen 3 and 4, DWcomb,ww(t) achieves in all cases large deviations. Using the combined strain energy density DWcomb(t), the numbers of cycles to failure Nf,r are too small at both specimens when evaluating the first inner hysteresis loops, and slightly too high when evaluating the inner loop at 0.9Nf,r. On average, DWcomb(t) calculates the most adequate results. Nevertheless, a consideration of cyclic stress relaxation in the model can improve the fatigue analysis. Therefore, more efforts are necessary. 5.3. Experiments with multiple inner hysteresis loops The tests illustrated in Fig. 15b–f as well as in Fig. 18a and b are extreme examples. Load-time functions, which occur e.g. at car body structures reveal a more random behavior, where small loads appear very often and large loads appear rarely (e.g. [47]). The strain–time function of an idealized example is illustrated in Fig. 19a and the corresponding hysteresis loops are shown in

322

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

[%]

(a) 2

.

Table 6 Fatigue life evaluation at tests with multiple inner hysteresis loops (AM50).

-3

=2·10 /s =2·

N0.3%=20

engineering strain

(e)

1

6. Conclusions

0

one full loop

-1

N0.8%=6

-2 0

2500

5000

time [s]

hysteresis no.: (1) (4,6,9) 1 a,t(e),inside2=0.4% (2,3,5,7,8,10) exp. (full loop no. 3) 4 model a,t(e),envelope=2%

a,t(e),inside1=0.8%

125

true stress

2

6

9

(t)

[MPa]

(b) 250

7

10

0 3 5

8

-125

35MPa

AM50 RD

37MPa Nloop,f=6.0 R (e),envelope=-1

-250 -2

-1

0

true strain

1 (t)

2

[%]

Fig. 19. (a) Strain–time function of an example with multiple inner hysteresis loops, shown in (b); (b) comparison of experimentally and numerically determined loops of AM50.

Fig. 19b. The large inner hysteresis loops (ea,t(e) = 0.8%) occur six times at each of the three positions and the small inner hysteresis loops (ea,t(e) = 0.3%) occur twenty times at each position (Fig. 19a). This distribution enables the inner hysteresis loops to account for a large damage in comparison to the envelope hysteresis loop. Two specimens were tested with the same strain–time function (Fig. 19a) and the fatigue life was 6.0 full loops for the first specimen and 6.2 full loops for the second specimen. Fig. 19b shows the experimentally determined stress–strain curve of one full loop at 0.5Nloop,f and the corresponding calculated hysteresis loops. The experimentally determined hysteresis loops reveal a cyclic stress relaxation behavior similar as in Fig. 18a and b. When changing to the next superior hysteresis loop after twenty small cycles with ea,t(e) = 0.3%, the cyclic stress relaxation is compensated. This means that the stress values coincide with the original envelope loop as if no cyclic stress relaxation has occurred. Due to this phenomenon, the numerically determined hysteresis loops adequately fit, although no cyclic stress relaxation was considered. The results of a fatigue calculation with the numerically determined hysteresis loops are listed in Table 6. With the combined strain energy density DWcomb(t), the best result for the example, shown in Fig. 19a and b, is reached.

A phenomenological model for the calculation of uniaxial stress–strain hysteresis loops at elastoplastic strain-controlled variable amplitude loading was developed for wrought magnesium alloys. This model enables numerical fatigue analyses with variable strain amplitudes. All relevant measures from a hysteresis loop such as stress, strain, and strain energy density components can be determined. For a verification of the model, 1.2 mm thick twin roll cast AM50 magnesium alloy sheet metals, provided by the Magnesium Flachprodukte GmbH were used in most cases. Additionally, the phenomenological model was evaluated with extruded ME21 sheet metals, extruded AZ31B solid rods, and extruded AZ61A tubes. The following conclusions can be drawn: (1) Hysteresis loops of the considered magnesium alloys at strain-controlled tests in the elastoplastic region with a strain amplitude >0.3–0.6% reveal a sigmoidal and asymmetrical shape. (2) Twinning and detwinning plays a dominant role in the elastoplastic strain range under constant as well as variable amplitude loading. (3) The shape of inner hysteresis loops at strain-controlled tests with a variable amplitude depends on the shape of the envelope hysteresis loop and the mean strain. (4) A three-component equation with an elastic, pseudoelastic, and plastic strain component was developed to describe hysteresis loops at constant and variable amplitudes. (5) Eight material constants were defined for the phenomenological model, which can be read out of experimentally determined hysteresis loops. It is recommended to validate the material constants with five constant amplitude tests in the strain range 2% 6 e(e) 6 2%. (6) The model assumes a cyclic stabilized state. For comparisons, hysteresis loops at half of the fatigue life were used. Cyclic hardening or softening as well as cyclic stress relaxation was not considered within the model. A deviation between model and experiments exists at constant amplitude cyclic loading with large positive or negative mean strains due to cyclic stress relaxation. (7) A good correlation between model and experiments is reached for constant and variable amplitude straincontrolled loads. An example with multiple inner hysteresis loops reveals an adequate correlation as well. (8) The model is able to provide fatigue parameters such as different strain energy density as well as stress and strain components out of the modelled hysteresis loops. The proposed model can be integrated in numerical fatigue analysis software. (9) At load conditions, where predominant large strain amplitudes occur and the fatigue life is low, all three models calculated an adequate fatigue life. With rising fatigue life, the energy based fatigue models show better results compared to the model of Smith–Watson–Topper. Considering all investigated examples, the combined strain energy density DWcomb(t) reaches the best correlation between calculations and experiments on average.

J. Dallmeier et al. / International Journal of Fatigue 80 (2015) 306–323

Acknowledgement The authors acknowledge the financial support of the Federal Ministry of Education and Research (BMBF) within the funding program FHprofUnt in the project ‘‘MagFest’’ under the contract number 03FH015PX2.

References [1] Agnew SR. Wrought magnesium: A 21st century outlook. JOM 2004;56(5):20–1. [2] Park SS, Park WJ, Kim CH, You BS, Kim NJ. The twin-roll casting of magnesium alloys. JOM 2009;61(8):14–8. [3] Kawalla R, Oswald M, Schmidt C, Ullmann M, Vogt HP, Cuong ND. New technology for the production of magnesium strips and sheets. Metalurgija 2008;47:195–8. [4] Agnew SR, Duygulu Ö. Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B. Int J Plasticity 2005;21:1161–93. [5] Hutchinson WB, Barnett MR. Effective values of critical resolved shear stress for slip in polycrystalline magnesium and other hcp metals. Scripta Mater 2010;63:737–40. [6] Koike J, Ohyama R. Geometrical criterion for the activation of prismatic slip in AZ61 Mg alloy sheets deformed at room temperature. Acta Mater 2005;53:1963–72. [7] Agnew SR. Plastic anisotropy of magnesium alloy AZ31B sheet. In: Kaplan HI, editor. Magnesium Technology 2002. Seattle (WA), Warrendale (PA): The Minerals, Metals & Materials Society; 2002. p. 169–74 (2002 February 17–21). [8] Lou XY, Li M, Boger RK, Agnew SR, Wagoner RH. Hardening evolution of AZ31B Mg sheet. Int J Plasticity 2007;23:44–86. [9] Noster U, Scholtes B. Isothermal strain-controlled quasi-static and cyclic deformation behavior of wrought magnesium alloy AZ31. Z Metallkd 2003;94:559–63. [10] Wu L, Jain A, Brown DW, Stoica GM, Agnew SR, Clausen B, et al. Twinning– detwinning behavior during the strain-controlled low-cycle fatigue testing of a wrought magnesium alloy, ZK60A. Acta Mater 2008;56:688–95. [11] Cáceres CH, Sumitomo T, Veidt M. Pseudoelastic behaviour of cast magnesium AZ91 alloy under cyclic loading–unloading. Acta Mater 2003;51:6211–8. [12] Mann GE, Sumitomo T, Cáceres CH, Griffiths JR. Reversible plastic strain during cyclic loading–unloading of Mg and Mg–Zn alloys. Mater Sci Eng, A 2007;456:138–46. [13] Min J, Lin J. Anelastic behavior and phenomenological modeling of Mg ZEK100O alloy sheet under cyclic tensile loading-unloading. Mater Sci Eng, A 2013;651:174–82. [14] Hasegawa S, Tsuchida Y, Yano H, Matsui M. Evaluation of low cycle fatigue life in AZ31 magnesium alloy. Int J Fatigue 2007;29:1839–45. [15] Chen L, Wang C, Wu W, Liu Z, Stoica GM, Wu L, et al. Low-cycle fatigue behavior of an as-extruded AM50 magnesium alloy. Metall Mater Trans A 2007;38A:2235–41. [16] Begum S, Chen DL, Xu S, Luo AA. Strain-controlled low-cycle fatigue properties of a newly developed extruded magnesium alloy. Metall Mater Trans A 2008;39A:3014–26. [17] Lin XZ, Chen DL. Strain controlled cyclic deformation behavior of an extruded magnesium alloy. Mater Sci Eng, A 2008;496:106–13. [18] Matsuzuki M, Horibe S. Analysis of fatigue damage process in magnesium alloy AZ31. Mater Sci Eng, A 2009;504:169–74. [19] Huppmann M, Lentz M, Brömmelhoff K, Reimers W. Fatigue properties of the hot extruded magnesium alloy AZ31. Mater Sci Eng, A 2010;527:5514–21. [20] Albinmousa J, Jahed H, Lambert S. Cyclic axial and cyclic torsional behaviour of extruded AZ31B magnesium alloy. Int J Fatigue 2011;33:1403–16. [21] Yu Q, Zhang J, Jiang Y, Li Q. An experimental study on cyclic deformation and fatigue of extruded ZK60 magnesium alloy. Int J Fatigue 2012;36:47–58. [22] Shiozawa K, Kitajima J, Kaminashi T, Murai T, Takahashi T. Low-cycle fatigue deformation behavior and evaluation of fatigue life on extruded magnesium alloys. Proc Eng 2011;10:1244–9. [23] Dallmeier J, Huber O, Saage H, Eigenfeld K, Hilbig A. Quasi-static and fatigue behavior of extruded ME21 and twin roll cast AZ31 magnesium sheet metals. Mater Sci Eng, A 2014;590:44–53. [24] Renner F. Einflussgrößen auf die Schwingfestigkeit von Magnesium-Guss- und -Knet-Legierungen und Lebensdauerrechnung. Dissertation, Technische Universität Clausthal, Papierflieger-Verlag Clausthal-Zellerfeld; 2004. [25] Begum S, Chen DL, Xu S, Luo AA. Low cycle fatigue properties of an extruded AZ31 magnesium alloy. Int J Fatigue 2009;31:726–35. [26] Park SH, Hong SG, Lee BH, Bang W, Lee CS. Low-cycle fatigue characteristics of rolled Mg–3Al–1Zn alloy. Int J Fatigue 2010;32:1835–42. [27] Yu Q, Zhang J, Jiang Y, Li Q. Effect of strain ratio on cyclic deformation and fatigue of extruded AZ61A magnesium alloy. Int J Fatigue 2012;44:225–33. [28] Geng CJ, Wu BL, Du XH, Wang YD, Zhang YD, Wagner F, et al. Low cycle fatigue behavior of the textured AZ31B magnesium alloy under the asymmetrical loading. Mater Sci Eng, A 2013;560:618–26. [29] Dallmeier J, Huber O, Saage H. Numerical fatigue analysis for twin roll cast magnesium sheet metal structures. Adv Mater Res 2014;891–892:1021–6.

323

[30] Dallmeier J, Huber O, Saage H, Eigenfeld K. Uniaxial cyclic deformation and fatigue behavior of AM50 magnesium alloy sheet metals under symmetric and asymmetric loadings. Mater Des 2015;70:10–30. [31] Basquin OH. The exponential law of endurance tests. Am Soc Test Mater Proc 1910;10:625–30. [32] Coffin LF. A study of the effect of cyclic thermal stresses on a ductile metals. Trans ASME 1954;76:931–50. [33] Manson SS. Fatigue: a complex subject-some simple approximation. Exp Mech 1965;5:193–226. [34] Zenner H, Renner F. Cyclic material behaviour of magnesium die castings and extrusions. Int J Fatigue 2002;24:1255–60. [35] Götting M, Scholtes B. Influence of tension–compression loading history on plastic deformation of Mg wrought alloy AZ31. Int J Mater Res 2006;97:1378–83. [36] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters. Technical note no. 902, National Advisory Committee for Aeronautics, Washington, DC; 1943. [37] Masing G. Zur Heynschen Theorie der Verfestigung der Metalle durch verborgene elastische Spannungen. Wissenschaftliche Veröffentlichungen aus dem Siemens-Konzern 1924;3:231–9. [38] Hertelé S, Waele WD, Denys R, Verstraete M. Full-range stress–strain behaviour of contemporary pipeline steels: Part I. Model description. Int J Pres Ves Pip 2012;92:34–40. [39] Awrejcewicz J, Dzyubak L, Lamarque CH. Modelling of hysteresis using Masing–Bouc-Wen’s framework and search of conditions for the chaotic responses. Commun Nonlinear Sci Numer Simul 2008;13:939–58. [40] Lee MG, Wagoner RH, Lee JK, Chung K, Kim HY. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets. Int J Plasticity 2008;24:545–82. [41] Li M, Lou XY, Kim JH, Wagoner RH. An efficient constitutive model for roomtemperature, low-rate plasticity of annealed Mg AZ31B sheet. Int J Plasticity 2010;26:820–58. [42] Behravesh SB. Fatigue characterization and cyclic plasticity modeling of magnesium spot-welds. Dissertation, University of Waterloo, University of Waterloo Electronic Theses and Dissertations; 2013. [43] Behravesh SB, Jahed H, Lambert S, Chengji M. Constitutive modeling for cyclic behavior of AZ31B magnesium alloy and its application. Adv Mater Res 2014;891–892:809–14. [44] Wang H, Wu PD, Tomé CN, Wang J. A constitutive model of twinning and detwinning for hexagonal close packed polycrystals. Mater Sci Eng, A 2012;555:93–8. [45] Wang H, Wu PD, Wang J, Tomé CN. A crystal plasticity model for hexagonal close packed (HCP) crystals including twinning and de-twinning mechanisms. Int J Plasticity 2013;49:36–52. [46] Wang H, Wu PD, Wang J. Modeling inelastic behavior of magnesium alloys during cyclic loading–unloading. Int J Plasticity 2013;47:49–64. [47] Heuler P, Klätschke H. Generation and use of standardised load spectra and load–time histories. Int J Fatigue 2005;27:974–90. [48] Huppmann M, Gall S, Müller S, Reimers W. Changes of the texture and the mechanical properties of the extruded Mg alloy ME21 as a function of the process parameters. Mater Sci Eng, A 2010;528:342–54. [49] Xiong Y, Yu Q, Jiang Y. Multiaxial fatigue of extruded AZ31B magnesium alloy. Mater Sci Eng, A 2012;546:119–28. [50] Yu Q, Zhang J, Jiang Y, Li Q. Multiaxial fatigue of extruded AZ61A magnesium alloy. Int J Fatigue 2011;33:437–47. [51] SEP 1240, Testing and documentation guideline for the experimental determination of mechanical properties of steel sheets for CAE-calculations, 1st ed.; 2006. [52] Xiong Y, Yu Q, Jiang Y. An experimental study of cyclic plastic deformation of extruded ZK60 magnesium alloy under uniaxial loading at room temperature. Int J Plasticity 2014;53:107–24. [53] Hong SG, Park SH, Lee CS. Role of 10–12 twinning characteristics in the deformation behavior of a polycrystalline magnesium alloy. Acta Mater 2010;58:5873–85. [54] Yamada T, Kawabata K, Sato E, Kuribayashi K, Jimbo I. Presence of primary creep in various phase metals and alloys at ambient temperature. Mater Sci Eng, A 2004;387–389:719–22. [55] Kandil FA. Potential ambiguity in the determination of the plastic strain range component in LCF testing. Int J Fatigue 1999;21:1013–8. [56] ISO/DIS 12106. Metallic materials – fatigue testing – axial-strain-controlled method; 1998. [57] BS 7270:1990. Metallic materials – constant amplitude strain-controlled axial fatigue – method of test; 1990. [58] Smith KN, Watson P, Topper TH. A stress–strain function for the fatigue of metals. J Mater 1970;5:767–78. [59] Li Q, Yu Q, Zhang J, Jiang Y. Effect of strain amplitude on tension–compression fatigue behavior of extruded Mg6Al1ZnA magnesium alloy. Scripta Mater 2010;62:778–81. [60] Lin YC, Chen XM, Liu ZH, Chen J. Investigation of uniaxial low-cycle fatigue failure behavior of hot-rolled AZ91 magnesium alloy. Int J Fatigue 2013;48:122–32. [61] Begum S, Chen DL, Xu S, Luo AA. Effect of strain ratio and strain rate on low cycle fatigue behavior of AZ31 wrought magnesium alloy. Mater Sci Eng, A 2009;517:334–43.