Thermodynamic entropy generation model for metal fatigue failure

Transcription

1 Thermodynamic entroy generation model or metal atigue ailure Hossein Salimi a, Mohammad Pourgol-Mohammad *a, Mojtaba Yazdani a a Sahand University o Technology, Tabriz, Iran Abstract: Fatigue damage is comrehensively determined by thermodynamic entroy generation as the damage recursor. For atigue mechanism, time rate o thermodynamic entroy generation is deined as ratio o lastic strain energy density rate er secimen s temerature. Recent researches claim that entroy generation is constant at the time o crack initiation, directly related to the tye o material, indeendent o loading and surrounding conditions. In this study, an analytical solution is roosed to evaluate the temerature o secimen during the atigue test while the Morrow equation emloyed as lastic strain energy density. Result leads to derive new analytical-emirical model or calculating entroy generation. It is shown that temerature obtained rom analytical solution is in good agreement with exerimental data. In the next section, uncertainty and sensitivity analysis are accomlished based on roosed model. Monte-Carlo simulation and sigma-normalized derivative methods are emloyed or uncertainty and sensitivity analysis, resectively. Secimen s diameter, ambient temerature, loading stress level and requency are considered as eective indeendent arameters. Al 04-T4 used as case study. Analytical-emirical model reresents that considered arameters are eective on atigue racture entroy (FFE) and the hyothesis o constant entroy generation is generally unaccetable in the crack initiation time. Keywords: atigue, thermodynamic entroy generation, crack initiation, uncertainty analysis, sensitivity analysis. 1. INTRODUCTION Mechanical methods o lie estimation and reliability analysis have been unchanged or structures or several years while many researches show existence o signiicant uncertainties and conservative actors. Thus, it is necessary to study eective methods with caability o more accurate damage behavior rediction. For lie estimation o comonent, the deinition and evaluation o damage are very imortant. Thermodynamically, all damage mechanisms share common eature o energy dissiation. Dissiation is undamental measure o irreversibility. The irreversibility is quantiied in thermodynamic aroach by entroy generation estimation [1]. Amiri and Modarres [1] state irreversible henomena or dierent mechanical damage mechanisms and Pourgol-Mohammad et al. [] reviewed resent studies in the thermodynamic entroy generation concet or mechanical damage mechanisms. In atigue rocess, crack initiation and growth is an irreversible henomena, thereore, it causes entroy generation in the system [1]. In recent years, several studies are conducted in concet o entroy generation or atigue mechanism [1-]. These studies claimed that the value o entroy generation is constant or a seciic material at the time o ull racture [3, 8-10, 13, 17]. This constant value is indeendent o secimen geometry, loading tye, alied stress level and loading requency. In other word, the necessary and suicient condition or racture o a secimen is that the entroy generation reaches a seciied value named Fracture Fatigue Entroy (FFE) [11]. The hyothesis o constant entroy generation or a seciic material at the time o ull racture led to calculation o FFE or dierent materials as a material roerty. In this regard, the value o FFE = 60 MJ/m 3 K or SS 304 [11, 17] and FFE = 5 MJ/m 3 K or LCS 1018, MCS 1045 and API 5L X5 [8-10] are determined. While, or Al 6061 dierent values o 8 MJ/m 3 K [10] and 9 MJ/m 3 K [8] are calculated. This is in contrast o constant FFE or each material. Recent researches resented that the most art o atigue lie elased u to macro crack initiation [3, 13, 17, 0]. Thus, rom the structural reliability viewoint, there is no sensible dierence between secimen that considered u to crack initiation and the other one which has allowed reaching ull racture [0]. Naderi and Khonsari [13, 17] and Amiri et al. [3] state that the crack initiation oint is * ourgolmohammad@sut.ac.ir

2 almost about 90% o atigue lie. They also reorted that there is almost linear relationshi between N normalized entroy generation and normalized alied cycle ( ). Based on hyothesis o N constant entroy generation at ull racture time and by considering o 90% atigue lie or crack initiation lie, it leads to 0.9 FFE or entroy generation at the time o crack initiation. It means that entroy generation is also constant or a seciic material in the crack initiation time. However, Ontiveros et al. [18-0] disuted the concet o constant entroy generation in the crack initiation. They reorted that the hyothesis o constant entroy accumulation (or constant energy accumulation) is still valid or atigue crack initiation and suggested that more studies are needed or the claim to be aroved [18]. Yousei Faal et al. [] calculated reliability u to the atigue crack initiation time based on numerical and theoretical analysis and they conirmed Ontiveros reort. In this study, calculation o entroy generation in the crack initiation time is o interest or metal materials. All resented researches are mainly based on exerimental basis, while analytical analysis is considered in this study. The entroy generation is determined by calculating secimen temerature while the emirical relations (such as Morrow equation) are emloyed as lastic strain energy density. A new analytical method roosed or temerature analytical solution and new analytical-emirical model derived or calculating entroy generation in the crack initiation time.. THEORETICAL BACKGROUND Entroy exresses a variation o energy associated with a variation in the temerature [3]. Considering irreversible rocess, second law o thermodynamic reresent [1]: ds des d i S, d i S 0 (1) where d e S is the entroy exchange (low) with the surrounding and d i S is entroy generation inside the system (Figure 1). Figure 1 Entroy balance in a system [1] Under the hyothesis o local equilibrium, the entroy balance (Eq. (1)) is exressed in local orm as: ds. Js () dt where: S sdv (3) V ds e s. d dt J Ω (4) ds i dv dt (5) V Here, the symbols are: s the entroy er unit mass, ρ the density, J s the total entroy low er unit area and unit time, dω the element o surace area (Figure ), and denotes the entroy generation er unit volume er unit time.

3 Figure Entroy Generation and Entroy Flow or a System [1] The Clausius Duhem inequality states that all the deormations cause ositive entroy generation rate in solids with internal riction [3]: 1 q σ: ε AV k k. 0 (6) T T where is the time rate o entroy generation, and σ,, A k, V k, T and q are stress tensor, rate o lastic strain, thermodynamic orces related to internal variables, internal variable evolution, temerature and vector o thermal gradation, resectively. Subscribe o k denotes number o internal variables [3]. Entroy generation in Eq. (6) has three dissiated terms: lastic dissiation ( σ: ε ), dissiation related to internal variables ( σ: ε AV k k ) and thermal dissiation due to conduction ( ε q. T ) [4]. AV k k reresents the nonrecoverable energy stored in the material. For metals, this is the energy o the ield o the residual micro stresses, accomanying the increase in the dislocation density. It reresents only 5-10% o the term and is oten negligible [3] so: AV k k 0 (7) In rocesses involving low-cycle atigue, the entroy generation due to lastic deormation is dominant and the entroy generation is negligible due to heat conduction. The corresonding entroy generation starts slowly and it grows while the entroy generation due to lastic deormation remains to be dominant throughout the rocess. Thereore, Eq. (6) reduces to [13]: 1 σ: ε (8) T T Thereore, the total entroy generation can be obtained by integration o Eq. (8) u to the time t when macro crack initiation occurs: t W dt (9) T 0 where γ is the total entroy generation at the onset o macro crack initiation. Heat equation or atigue mechanism resented as [3]: σ e k T CT σ : ε T : ε (10) T where denotes Lalacian oerator and C is Seciic Heat Caacity o material. Eq. (10) indicates balance between 4 terms: Conductivity heat transer ( k T ), retardation eect because o heat inertia ( CT ), internal heat generation due to lastic deormation ( σ: ε ), which is resonsible or increase σ in mean temerature, and thermoelastic couling term ( T : ε e ) [11]. Temerature luctuation due to T thermoelastic couling is small in comarison with the mean temerature and it is neglectable [5]: CT k T W (11) In Eq. (11) lastic strain energy density aect as thermal source which is uniormly distributed in secimen s gage section [6]. Figure 3 resents tyical evolution o temerature during atigue test. For stresses more than atigue endurance, there is 3 hase in Figure 3. Phase 1 (initial hase) which temerature is gradually raised and that has greater initial sloe or higher stress level. Phase (steady state) which there is balance between generated hysteresis energy and dissiated heat energy. Phase 3 (crack initiation hase) which temerature suddenly increased beore inal ruture [6]. W

4 Figure 3 Tyical temerature evolution or a atigue test [3] 3. ANALYTICAL SOLUTION Figure 4 resents schematic shae o secimens in atigue test (ASTM E466 and ASTM E606). They could have rectangular or round cross section. Figure 4 Schematic shae o atigue test secimens [7, 8] In these secimens, arts with larger cross section area (arts with l length at the end o secimen) be hold in testing machine gri. A samle o atigue test setu resented in Figure 5. Figure 5 A atigue test setu or uniaxial loading [8] Testing machine gri due to larger volume in comarison with secimen s end section act like a heat sink. Because o high conductivity o metal secimens, the end section o secimens which are in contact with testing machine gri could be considered as constant temerature in equilibrium with gri temerature (T ). Exerimental result conirms this assumtion (Figure 6). Thereore, temerature gradient occurs in middle section o secimen (section with L length). In this middle section, lastic strain energy act on gage section (section with l 1 length) and the rest section (section between l 1 and L) is stable. In metallic material that usually adoted in engineering structures the heat transer caacity by convection is much lower than the heat transer caacity by conduction and thereore, the

5 temerature variation in the secimen cross section can be neglected. This act is demonstrated by the very low Biot numbers (much lower than 1) that can be calculated or secimens which commonly used in atigue tests [5]. Figure 6 Temerature contour in a) MCS 1045 in 55% o atigue lie under low cycle atigue with σ max = 500 MPa, requency 10 Hz and load ratio o -0.6 [8], b) ULTIMET suer alloy in (let) and 9550 cycle (right) under high cycle atigue with σ max = 700 MPa, requency 0 Hz and load ratio o 0.05 [6] (a) (b) Thereore, heat equation and boundary and initial conditions (Figure 7) becomes: T CT k W x (1) T 0, t 0 x T L, t T (13) T x,0 T i T (14) Figure 7 Boundary conditions with 1-dimensional heat transer assumtion Beore exression o analytical solution or Eq. (1) with boundary and initial conditions o (13) and (14), let us consider second hase o temerature evolution (Figure 3). In this hase which is the most art o atigue lie, temerature is in steady state: (15) Thereore Eq. (1) becomes: dt T 0 dx k (16) l1 For very simle case o 1 Eq. (16) results: L W dt T x x C1x C x 0 0, T x L T k dx (17) W T x L x T k By utilizing Fourier law, heat transer by conductivity at each oint o x becomes: W

6 dt q Cond x ka AW x (18) dx As result, the heat transer at the end o secimen becomes: q Cond x L AW L (19) By considering: T Lc q Cond, R (0) R ka where R is heat resistance and L c is eective length, results: W L ka W L L L T x L T0 T, q Cond x L W A Lc (1) k Lc k Lc This means or secimens with uniorm internal heat source, we can used Eq. (0) or heat transer amount at the end o secimen while eective length o secimen was considered as L/. For l 1 L 1 we need to calculate eective length. In this condition and or section with no internal heat source, because o energy stability we have (Figure 8): q Cond q Cond 1 () Figure 8 Heat balance in atigue secimen Eq. () results: T T1 T0 T T0 T1 l1 T0 T ka ka ka ka Lc (3) Lc Lc1 Lc l1 T0 T1 In Eq. (3) term o T 1 aears that needs to be removed. By considering heat balance or section with no internal heat source (Figure 9): q Cond 1 q Cond (4) Figure 9 Heat balance in section with no internal heat source which leads: T T T T T T l L l ka ka ka ka T T T (5) L L l L l L l l c1 c By substituting Eq. (5) in Eq. (3) we have: L l1 Lc (6) Thereore, heat transer at the end o secimen becomes: L l1 R (7) ka T0 T T0 T q Cond ka (8) R L l 1

7 Metallic material that adoted in atigue test usually have high thermal conductivity (k) and short length (L) which results small heat resistance (R) and thereore it can be assumed that heat transer to the gri at the end o secimen at any instant is: T t T q Cond t ka (9) L l1 Integration o Eq. (1) on control volume and alying boundary conditions results: T T 4 4 W dv C dv k. nds h T T ds cv n T T ds ir t x (30) V Al1 V LA Scd Scv S ir where h is convection heat transer coeicient, κ surace emissivity, σ n Stehan-Boltzmann constant 8 W which is equal to and S 4 cd, S cv and S ir are conduction, convection and radiation m K suraces, resectively. As mentioned beore in metallic secimens the heat transer by conduction is major heat dissiation mechanism and the other two terms could be neglected. By substituting Eq. (9) in Eq. (30) becomes: dt t T t T VC V W (31) dt R In 1965 Morrow roosed that the lastic strain energy density er cycle was relativity constant throughout the atigue lie and or constant loading conditions under a ully reversed atigue loads, lastic strain density could be resented as [9-31]: dw 1 n bc 4 N (3) dn 1 n where σ is atigue strength coeicient, b atigue strength exonent, ε atigue ductility coeicient, c atigue ductility exonent, n cyclic strain hardening exonent and N inal number o cycles. Another equation that relates atigue lie to stress amlitude (σ a ) searately roosed by L. F. Coin and S. S. Manson which generally called Coin-Manson relationshi [3, 33]: b N (33) a By considering Morrow equation as lastic strain density and solving dierential equation o (31), secimen s temerature at midoint becomes: t dw V C 1 V C 1 VC W C1 T t T e T i T, L l1 dn VC VC (34) ka where is loading requency. By substituting Equations (3) and (34) in Eq. (9) we have: W t t W V L c N ln 1 e e t, T, t T (35) i ka 3.1. Validation Figure 10 shows the temerature evolution u to crack initiation time calculated rom analytical solution and exerimental data measured by Jiang et al. [6].

8 Figure 10 The temerature evolution obtained rom exerimental and analytical solution or σ a = 703 MPa and 76 MPa. Figure 10 reresents that the analytical results are in very good agreement with exerimental data. 4. CASE STUDY As a case study, two atigue regime (Low Cycle Fatigue and High Cycle Fatigue) are considered searately. For both analysis Aluminum 04-T4 is used which has common use in airrame constructions. Table 1 and Table reresent the hysical, mechanical and thermal roerty and the atigue roerty o Al 04-T4, resectively. Table 1 The hysical, mechanical and thermal roerties o Al 04-T4 [34] ρ (Kg/m 3 ) Sy (MPa) Sut (MPa) Se (MPa) k (W/m.K) C (W/Kg.K) Table The atigue roerties o Al 04-T4 [35] σ (MPa) ε b c n Figure 11 shows the schematic o low cycle atigue and high cycle atigue secimens. These secimens are based on ASTM E606 and ASTM E466 standards. Middle art cross section diameter selected as d = 7mm or LCF and d = 15 mm or HCF. Figure 11 Schematic o secimen or a) LCF, b) HCF [7, 8] (a) (b) Loading condition was selected as constant amlitude loading under a ully reversed atigue loads with loading requency o =.75 Hz and six dierent alied stress levels (σ a = 35, 350, 375, 400, 45 and 450 MPa) or LCF and = 55 Hz and seven dierent stress level selected (σ a = 150, 175, 00, 5, 50, 75 and 300 MPa) or HCF. Initial temerature was equal to ambient temerature and that was K. Figure 1 resents Fatigue Failure Entroy (FFE) or dierent atigue lie in LCF and HCF.

9 Figure 1 Fatigue Failure Entroy (FFE) or dierent atigue lie in a) LCF, b) HCF a b It is evident that or both LCF and HCF regimes FFE is greater or higher atigue lie. In the other words low alying stress results higher FFE. 5. UNCERTAINTY ANALYSIS For uncertainty analysis, arameter uncertainty is erormed based on Eq. (35) and accuracy o measurement equiment such as load cell (±1%) and IR sensor (±1.5% ± K) emloyed as uncertainty o inut variables. For this urose, inut variables are assumed in orm o a normal distribution, and Monte Carlo simulation is used to roagate their uncertainties throughout the model. Since almost all mechanical roerties in engineering structures are obeyed normal distribution. However, goodness o it test would be emloyed or the better it to the available data. The convergence o Monte Carlo simulation is achieved in reetitions. Table 3 and Table 4 resent the mean and standard deviation o inut arameters in LCF and HCF regimes, resectively. In Table 4 variables with similar quantity to Table 3 are avoided. Table 3 Uncertainty arameters o inut variables in LCF Parameter Normal Distribution Parameters (Mean, STD) Frequency = N (.75, 0.075) Hz Ambient Temerature T = N (96.37, 6.445) K Amlitude Stress level σ 1 = N (35, 3.5), σ = N (350, 3.5), σ 3 = N (375, 3.75), σ 4 = N (400, 4.0), σ 5 = N (45, 4.5), σ 6 = N (450, 4.5) MPa Density ρ = N (780, 7.8) Kg/m 3 Thermal Proerties C = N (875, 8.75) W/Kg.K, k = N (11, 1.1) W/m.K Fatigue Proerties σ = N (194, 1.94) MPa, ε = N (0.37, ), b = N (-0.14, ), c = N (-0.645, ), n = N (0.08, ) Table 4 Uncertainty arameters o inut variables in HCF Parameter Normal Distribution Parameters (Mean, STD) Frequency = N (55, 0.55) Hz Amlitude Stress level σ 1 = N (150, 1.5), σ = N (175, 1.75), σ 3 = N (00,.0), σ 4 = N (5,.5), σ 5 = N (50,.5), σ 6 = N (75,.75), σ 7 = N (300, 3.0) MPa Figure 13 a and b resent the uncertainty quantiication result or dierence stress level in LCF and HCF, resectively. In these igures the standard deviation (measure o uncertainty) are lotted versus atigue test time. It is shown that uncertainty is increased by atigue cycle increment and the uncertainty growth is more signiicant or high stress level. This is because o uncertainty accumulation.

10 Figure 13 Inut uncertainty growth throughout model or a) LCF, b) HCF a b 6. SENSITIVITY ANALYSIS As mentioned beore, Figure 13 resents FFE in the crack initiation time is not constant and could not considered as material roerty. Thereore, sensitivity analysis and investigation o more eective arameter on FFE are needul. In this section sensitivity analysis is erormed or both LCF and HCF based on dierent oeration conditions. At irst, dierential method emloyed or sensitivity analysis and in the next art based on common selection range or inut variables, Sigma-normalized derivative method is used. This method considers usual range or inut variables and resents a measure or rank the inut variables Dierential method Table 5 resents sensitivity analysis result or LCF in oeration condition o d = 7 mm, =.75 Hz, T = K and σ a = MPa and or HCF d = 15 mm, = 55 Hz, T = K and σ a = 5 MPa. It is worth noting these conditions selected according to standards constrains, usual conditions in laboratory and loading limits in each atigue regime (LCF and HCF). Table 5 Sensitivity analysis based on dierential method or LCF and HCF Inut variable (Zi) Z i or LCF (MJ/m 3.K) Z i or HCF (MJ/m 3.K) Diameter (d) (-1.15%) (-1.59%) Frequency () (-1.46%) (-0.%) Ambient Temerature (T ) (-0.3%) (-0.30%) Amlitude Stress (σ a ) (-0.44%) (-0.96%) As it is recognizable almost all o the 4 arameters are eective on FFE and FFE is sensitive to all our arameters. This method states ambient temerature and loading requency have lowest eect on FFE or LCF and HCF, resectively. 6.. Sigma-normalized derivative method Standards, testing equiment limitation, laboratory conditions, testing constrains and some other actors cause atigue variables are selected in seciic ranges which these ranges could be dierent or LCF and HCF. For examle, ASTM E466 limits selection range o secimen s midsection diameter to 5.08 u to 5.4 mm, or or LCF alied stress level must be more than yield stress and less than ultimate stress. Thereore, Sigma-normalized derivative method emloyed to sensitivity analysis or LCF and HCF. Normal distribution assumed or selecting oeration oint and Monte Carlo simulation used to calculate disersion o model outut. Sigma-normalized derivative measure deines as [36]: Z i S Z (36) i Z i

11 where Z i and are standard deviation o inut variables and model, resectively. Table 6 resents mean and standard deviation o each inut variable or LCF and HCF. Table 6 Normal distribution arameters o each inut variable or LCF and HCF Inut variable (Zi) Normal Distribution Parameters Normal Distribution Parameters (Mean, STD) or LCF (Mean, STD) or HCF Diameter (d) d = N(7, 1) mm d = N(15, 3.33) mm Frequency () = N(.75, 0.75)Hz = N(55, 15)Hz Ambient Temerature (T ) T = N(96.37, 5) K T = N(96.37, 5) K Amlitude Stress (σ a ) σ a = N(387.5, 0.83) MPa σ a = N(5, 5) MPa Monte Carlo simulation calculates MJ/m 3 K or mean value and MJ/m 3 K or standard deviation o FFE in LCF. Similarly, and MJ/m 3 K are calculated as mean value and standard deviation o FFE in HCF, resectively. Results o Sigma-normalized derivative sensitivity analysis resent in Table 7 or LCF and HCF. Table 7 Sensitivity analysis based on Sigma-normalized derivative method or LCF and HCF Inut variable (Zi) or LCF or HCF S Z i Diameter (d) Frequency () Ambient Temerature (T ) Amlitude Stress (σ a ) As it is recognizable rom Table 7 in LCF all o the our inut variables are eective on FFE. However, in HCF FFE almost is not sensitive to ambient temerature. Thereore, ambient temerature has low cororation in FFE and is not imortant actor or HCF regime. For both LCF and HCF amlitude stress level is the most imortant actor and FFE is the most sensitive to amlitude stress level. 7. CONCLUSION In this study, hyothesis o constant entroy generation is investigated at the time o crack initiation in atigue rocess. For this urose, a new analytical solution is carried out to evaluate the temerature o secimen during the test while the common emirical relation (e.g., Morrow equation) is emloyed as lastic strain energy density. Result leads to derive new analytical-emirical model or calculating entroy generation at the time o crack initiation. It is shown that temerature obtained rom analytical solution is in good agreement with exerimental data. In the next section, uncertainty and sensitivity analysis accomlished based on roosed model. Monte-Carlo simulation is emloyed or uncertainty analysis. Sensitivity analysis erormed using dierential and Sigma-normalized derivative methods. Several actors such as secimen s diameter, ambient temerature, loading stress and requency are considered as eective indeendent arameters. Al 04-T4 used as case study. Analytical-emirical model reresents that considered arameters are eective on atigue racture entroy (FFE) and the hyothesis o constant entroy generation is generally unaccetable in the crack initiation time and it cannot be assumed that it is only related to the tye o material. Sensitivity analysis shows in LCF all o these our arameters are eective on FFE. However, in HCF between these arameters, ambient temerature has low eect on FFE and FFE is not sensitive on that. For both LCF and HCF amlitude stress level is the most imortant actor and FFE is the most sensitive to amlitude stress level. Also observed that other arameters were imortant and could not be ignored. Reerences [1] M. Amiri, M. Modarres, "An entroy-based damage characterization", Entroy, Vol. 16, No. 1, , (014). S Z i

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