THERMOMECHANICAL FATIGUE MADE EASY

Transcription

1 THERMOMECHANICAL FATIGUE MADE EASY Darrell Socie * and Benjamin Socie + Thermo Mechanical Fatigue ( TMF ) is a complex subject because o the interaction o several ailure modes including atigue, oxidation and eep with a wide variety o complex thermal and mechanical loads. Many models exist and several commercial sotware codes are available or evaluating the durability o a structure under these conditions. These codes oten require the user to be a atigue specialist with considerable training. In addition, the material properties needed or these calculations are diicult and very expensive to obtain experimentally. In this paper an approximate procedures to obtain estimates o the material properties and a simple sotware system or the non-specialist are desibed. INTRODUCTION TO THERMOMECHANICAL FATIGUE Thermomechanical atigue (TMF) is caused by combined thermal and mechanical loading where both the stresses and temperatures vary with time. This type o loading can be more damaging by more than an order o magnitude compared with isothermal atigue at the maximum operating temperature. Material properties, mechanical strain range, strain rate, temperature, and the phasing between temperature and mechanical strain all play a role in the type o damage ormed in the material. These types o loadings are most requently ound in start-up and shut-down cycles o high temperature components and equipment. Typically design lives are a ew thousand cycles and involve signiicant plastic strains. One o the major dierences between isothermal and thermal mechanical atigue is constraint. When heated, structures develop thermal gradients as they expand. Expansion near stress concentrators is oten constrained by the surrounding cooler material. In this case thermal strain is converted into mechanical strain which causes atigue damage in the structure. Total constraint exists when all o the thermal strain is converted into mechanical strain. Over constraint can occur in a stress concentration where the mechanical strain is greater than the thermal strain. One measure o the degree o constraint is the ratio o the thermal and mechanical strain rates. TMF loading is oten desibed to be in-phase (IP) or out-o-phase (OP). A schematic illustration o the stress-strain response under these two loadings is given in Fig. 1. In IP loading the maximum temperature and strain occur at the same time. In OP loading, the material experiences compression at highest temperature and tension at lower temperatures. IP loading is more likely to cause eep damage during tensile stresses at high temperatures. OP loading is more likely to cause oxidation damage because an oxide ilm can orm in compression at the higher temperature and then rupture during the subsequent low temperature tensile portion o the loading cycle where the oxide ilm is more brittle. *Mechanical Science and Engineering, University o Illinois at Urbana-Champaign, Urbana Illinois, USA + BBM Plus, Saint Joseph Illinois, USA

2 Figure 2 shows early TMF test data rom Jaske[1] or a low carbon steel. Isothermal atigue tests at various temperatures are shown as the lines without symbols. TMF OP tests are shown with the solid symbols in the igure. Fatigue lives or tests conducted with variable strains and temperatures between 93 C and 538 C are more than an order o magnitude shorter than tests conducted at a constant temperature o 538 C. Assuming the worst case isothermal material properties is very non-conservative and a TMF atigue analysis must be employed or TMF assessments. There are many active mechanisms in the TMF process. For discussion it is convenient to consider damage rom three primary sources: atigue, oxidation and eep[2]. Damage rom each process is summed to obtain an estimate o the total atigue lie, N. 1 N = + + (1) N N N atigue oxidation Frequently one o the damage mechanisms is dominant. From a modeling prospective, this suggests that the individual damage models and their associated material properties must be accurate only or those conditions where the lie is dominated by that ailure mechanism. eep Fatigue mechanisms Fatigue damage in the lie regime o interest in TMF is in the orm o nucleation and growth o mioacks. Figure 3 shows an example o the surace damage observed at higher plastic strains[3]. Mioack density will deease at lower strain levels. Many mioacks nucleate on the surace and a ew o them are able to penetrate into the bulk o the material. The process is driven by cyclic plastic strains where oxidation and eep eects are negligible. Fatigue damage will dominate at high strain ranges, strain rates and low temperatures. Oxidation mechanisms Oxidation damage can occur in the orm o an oxide intrusion such as the one shown in Fig. 4 [4]. In OP loading, an oxide layer can orm on the surace when the material is hot and in compression. At the lower temperature the oxide layer becomes brittle. During mechanical straining it then acks to expose new clean metal suraces. This clean metal will rapidly oxidize and the process repeats during the next mechanical strain cycle. Ultimately this will orm a ack which can then grow during the mechanical strain cycle. Oxide acks can also orm during IP loading. In this case, the oxide orms during the hot portion o the loading cycle while the material is in tension. Then upon cooling the oxide ilm undergoes a buckling delamination which ractures the oxide and exposes clean metal suraces. Oxide ormation and rupture during isothermal loading is not the dominant ailure mechanism and is not relected in isothermal test or materials data. Oxide ormation will occur easier and aster at higher temperatures. The tendency o the oxide to develop mioacks will depend on the cyclic strain range, stress does not play a role in the development o oxide induced mioacks.

3 Creep mechanisms Creep is essentially a diusion process. At high stresses diusion allows dislocations to climb over barriers. At lower stresses diusion occurs along the grain boundaries. Figure 5 shows a triple point ack that has ormed at the junction between grains[5]. These mioacks orm as a result o grain boundary sliding to accommodate the change in shape o elongated grains. Diusion is highly temperature and time dependant. Maximum stress rather than strain range has a dominant role in the ormation o these mioacks. The interaction o the strain rate and temperature has a strong inluence on the stresses that are observed during cyclic loading. One common eature o all these mechanisms is that they involve the nucleation and early growth o mioacks. There will also be interactions between the mechanisms as well. For modeling, individual components are considered and interactions are ignored. This notation is employed in the ollowing equations: eep, in inelastic, mech mechanical, ox oxidation, and th thermal. The traditional nomenclature is employed or stress, σ, strain, ε, strain rate, ε!, temperature, T, and lie N. Fatigue Damage Model Conventional low cycle atigue damage is a surace phenomena where small mioacks nucleate and grow on the surace o the material. Bulk stresses and strains are employed to desibe atigue damage because the mioacks growth is too complex to desibe in detail. The strain-lie equation is the most common desiption o the process. ε 2 σ = E ' atigue b ' atigue ( 2N ) + ε ( 2N ) c (2) σ b ε c E atigue strength coeicient atigue strength exponent atigue ductility coeicient atigue ductility exponent elastic modulus Oxidation Damage Model The oxidation damage ormulation o Neu and Sehitoglu[6] is employed in this study. Oxide damage will occur when the strain range exceeds a threshold or oxide acking. i ε mech >ε 0 N 1 oxidation H = ox Φ K pe 1 β 2 ( ε ) mech b 1 β ε! β (3) ε o H β b threshold strain or oxide acking constant related to itical oxide thickness mechanical strain range exponent thermal strain rate sensitivity exponent

4 Φ ox = 1 t c t c φ 0 ox dt (4) φ ox 1 ε! = exp 2 th / ε! mech ox ξ (5) ξ ox oxidation phasing constant or thermal and mechanical strains t = c ox H K pe D0 exp dt (6) 0 RT H ox activation energy or oxidation D o scaling constant or oxidation Oxidation damage is a unction o the strain range, strain rate, and temperature. A phasing actor φ ox is introduced to account or the type o oxide acking that occurs in either IP or OP loading. Phasing is represented by the ratio o thermal and mechanical strain rates. Oxidation rate is determined by the eective parabolic oxidation constant, K pe. Creep Damage The eep damage ormulation suggested by Neu and Sehitoglu[6] is also employed in this study. N t m c 1 H α1σ + α 2σh = A eep Φ exp 0 RT (7) K H A m α 1 α 2 activation energy or eep scaling constant or eep eep stress exponent stress state constant hydrostatic stress sensitivity constant Φ = 1 t c t c φ 0 dt (8) φ 1 ε! = exp 2 th / ε! mech ξ 2 1 (9) ξ eep phasing constant or thermal and mechanical strains Creep damage is a unction o the stresses, time and temperature. Miostructural eep damage diers in tension and compression. It is commonly assumed that mioacks do not orm and grow

5 in compression. I no eep damage occurs in compression α 1 = 1/3 and α 2 = 1. Here, K is the drag stress which will be deined in the next section. A phasing actor φ is also introduced to account or dierent eep damage mechanisms such as intergranular or transgranular acking. Constitutive Equation Model A uniied constitutive model irst suggested by Bodner and Partom[7] is employed to compute the stresses. The combined eects o both eep and plasticity are treated as inelastic strains. At lower stresses, time dependant eep dominates the behavior. Plasticity dominates at higher stresses. A drag stress, K, is introduced into the ormulation. The drag stress is an internal state variable that is related to the strength o the material. It is the stress that deines the transition rom eep to plasticity dominated deormation. It is not constant but depends on the temperature. n1 in σ H σ A 0 exp 1 K RT K in ε! = (10) n2 σ in H σ A 0 exp 1 exp 1 K RT K H in A o n 1 n 2 activation energy or inelastic deormation scaling constant or inelastic deormation exponent or eep dominated deormation exponent or plasticity dominated deormation A linear temperature or the drag stress is oten employed. Other orms are possible. K = K 0 K1T K 0 K 1 back stress back stress temperature dependence A linear dependence or the elastic modulus is also requently employed. E = E 0 E1T E 0 E 1 elastic modulus elastic modulus temperature dependence The thermal expansion coeicient is also needed in the analysis to determine thermal strains. α thermal expansion coeicient

6 These TMF equations represent a model or steady state deormation and require a total o 27 material modeling constants. SIMPLIFIED MATERIAL PROPERTIES TMF mechanisms are complicated and inluenced by the material miostructure, environment and external loading and a complete set o material data is preerred. Twenty seven material modeling constants means that only a ew materials have been ully characterized. Yet, lie assessments must requently be made in the early stages o design beore a complete set o materials data is available. As a result, there is a need to make estimates o the material properties rom other more readily available data. In this paper, simpliied material properties are employed. They are based on a classiication system, low carbon steels, alloy steels, aluminum etc. because a complete set o data is available or only a ew materials. These materials will be designated as reerence materials. Properties rom these materials will be modiied to account or miostructural dierences between materials within a given class. Fatigue constants are always needed or the analysis. Both eep and oxidation damage models have a phasing actor. The phasing actor is shown in Fig. 6 as a unction o the thermal and mechanical strain ratio. This actor determines the dominant ailure mechanism, eep or oxidation. Creep will dominate IP TMF loading. Oxidation will dominate or both isothermal and OP TMF. It should be noted that many TMF problems involve constrained heating and cooling which result in out-o-phase loading. In this case, knowledge o the eep properties are unnecessary because only the oxidation and atigue behavior is important. Fatigue Constants Many correlations between atigue and tensile properties have been proposed. Many o them have been validated only or steels. Muralidharan and Manson s[8] method has been validated or wide range o materials. It is based on the elastic modulus, E, ultimate strength, S u, and true racture strain, ε. ε 2 = Su E u ( 2N ) ( ε ) ( 2N ) S E (11) Oxidation Constants Oxidation is dominated by the matrix material rather than miostructure in most alloys. As a irst approximation, alloys o a similar matrix are expected have the same oxidation behavior. No adjustments are needed or miostructure and all materials o the same class will have the same behavior. Creep Constants Creep is a process that is driven by diusion either in the bulk material or along the grain boundaries. Creep damage should be directly related to the eep rate and rupture lie. Equation 7 or eep damage can be divided into two terms, one or temperature and one or stress dependence. Figure 7 shows a Larson-Miller plot or various alloys. Two materials with the same value o the

7 Larson-Miller parameter, P LM, will have the same time and temperature dependence. Note that the lines desibing the material behavior are nearly parallel or a given class o materials. For example, the dierence between low carbon and Cr-Mo steel is essentially a shit in stress level. This observation allows us estimate the behavior o other materials in the same class. The activation energy or eep, H, is directly related to the sel-diusion activation energy which will depend on the matrix material not the miostructure. The stress constants, α 1 and α 2, and the phasing constant, ξ, can also be considered unctions o the matrix material and as a irst approximation will not change with the miostructure. Exponents such as m are an indication o the mechanism and are not expected to change. This leaves a single constant, A, that will depend on the miostructure. As a irst approximation, this constant can be scaled rom the reerence material data. Two materials with the same P LM will have the same eep damage and the integrand o Eq. 7 must also be the same or both materials. In Fig. 7 consider low carbon and C-Mo steel. The allowable stresses or C-Mo steel are shited up by about 25%. Let the eep strength be denoted as S and the eep strength o the reerence material as S. Manipulating Eq. 7 and eliminating the re temperature terms results in an approximate expression or the constant A. A S S re m = A (12) re K re Miostructural eects are indirectly included in the eep damage ormulation by normalizing the stress by the drag stress K. Drag stress will also be directly related to the materials eep strength since the drag stress represents the transition between eep and plasticity dominated behavior. Then Eq. 12 can be urther simpliied to A = A. The inal result is that no adjustments are re needed or the eep constants. Dierences between materials within a class are modeled by changes in the drag stress. K Constitutive Equation Constants Stresses are not included in the atigue and oxidation models, only the eep damage model. In the uniied constitutive models no attempt is made to separate the eep and plasticity strains, they are all considered as inelastic strains. The exponential terms in Eq. 10 are related to high temperature eep deormation. Following the same arguments used or determining eep damage constants, the activation energy and exponents are expected to be a constant or alloys within a class o materials. The remaining two constants A o and K will depend on the material miostructure. Drag stress should scale directly with the materials strength, either eep strength or yield strength. The constant A o is assumed to remain constant. A very simple model or approximating TMF material constants is proposed, simply adjust the material drag stress in proportion to the material strength. Creep strength is preerred but room temperature yield strength could be employed when eep data is not available.

8 FATIGUECALCULATOR To encourage more widespread use o atigue analysis, a website has been developed to perorm routine atigue damage calculations. The website is ree and open to everyone to use. TMF analysis is included with an appropriate material database o TMF constants or various materials. The TMF portion consists o three sections, deinition o the duty cycle, material properties and results. Figure 8 shows an example o the duty cycle input. It is in the orm o a list o endpoints or each loading segment in the duty cycle or the strain, time and temperature. The loading may be speciied as mechanical strains, total strains or stress or strain hold times. The materials database includes several materials and the user can enter their own properties. Results o the analysis are shown in Fig. 9. In addition to the computed atigue lives, several plots o stresses and strains are provided. COMPARISON WITH EXPERIMENTS Two o the datasets analyzed are presented here. The basis o comparison is the computed TMF lives because our objective is to make estimates o atigue lives not reproduce material modeling parameters. The irst dataset analyzed is that shown in Fig. 2 or 1010 steel. The reerence data employed was 1070 steel[6]. Creep strength data was not available or both materials so the yield strength was used. Fatigue constants were obtained rom the SAE handbook[10] or 1010 steel. Yield strengths or 1010 and 1070 steel were obtained rom It lists the yield strengths as 300 and 580 MPa or 1010 and 1070 steel respectively. Beore perorming the calculations the drag stress constants were reduced by a actor o No other changes in the oxidation, eep or constitutive equation constants were made. Results o the analysis are given in Table 1. In addition to the experimental and calculated atigue lives, the relative contribution o each o the three damage mechanisms is given. TABLE Steel results rom Jaske [1] Mechanical Experiment Analysis Fatigue Oxidation Creep Strain Range T max Cycles Cycles Damage Damage Damage OP % 40% % 49% % 57% % 70% % 6% % 10% % 17% % 1% % 1% - IP % 32% 13% % 18% 59% % 8% 7% % - - A second set o test data [11] or a Ni based superalloy IN 738LC was also analyzed. The reerence material selected was another Ni based superalloy Mar M247[12]. Creep strength o IN

9 738 was taken rom [13] as 210 MPa at 870 C. Similar data or Mar M247 was obtained rom [14] as 280 MPa. Again the drag stress constants or Mar M247 were reduced by 0.75 to obtain an estimate o the IN 738 constants. No other adjustments were made to the Mar M247 oxidation, eep or constitutive equation constants were made. Strain lie data was taken rom reerence 11. Results o the analysis are shown in Table 2. TABLE 2 IN 738LC results rom Fleury and Ha [11] Mechanical Experiment Analysis Fatigue Oxidation Creep Strain Range Cycles Cycles Damage Damage Damage OP % 91% % 99% - IP % - 97% % - 98% % - 99% Results in Tables 1 and 2 show that the simple approximate method can be expected to produce atigue lives that are within an order o magnitude o the experimental lives. SUMMARY A simple method or determining material modeling constants or TMF damage calculations has been proposed. The method consists o using reerence materials or which the constants are known and then adjusting the drag stress to obtain constants or other similar materials. A web based analysis procedure was also desibed. REFERENCE LIST (1) Jaske, C.E. Thermal Fatigue o Materials and Components, ASTM STP 612, 1976, pp (2) Sehitoglu, H. Advances in Fatigue lietime Predictive Techniques, ASTM STP 1122, 1992, pp (3) Stolarz, J, Ecole Nationale Superieure des Mines, Presented at LCF 5 in Berlin, 2003 (4) Viswanathan, R. and J. Stringer, J. o Engineering Materials and Technology, Vol. 122, July 2000, pp (5) Morris, D.G. and D.R. Harries, J. o Materials Science, Vol. 12, No. 8, 1977, pp (6) Neu, R.W. and H Sehitoglu, Metallurgical Transactions A, Vol. 20A, 1989, pp (7) Bodner, S.R. and Y. Partom, J. o Applied Mechanics, Vol. 46, 1979, pp

10 (8) Muralidharan, U and S.S. Manson, J. Engineering Materials and Technology, Vol. 110, 1988, pp (9) Dowling, N.E. Mechanical Behavior o Materials, Prentice Hall, 1998 (10) SAE Handbook, Society o Automotive Engineers, 2003 (11) Fleury E. and J.S. Ha Materials Science and Technology, Vol. 17, 2001, pp (12) Sehitoglu, H. and D.A. Boismier, J. Engineering Materials and Technology, Vol. 112, 1990, pp (13) High Temperature High Strength Nickel Base Alloys, International Nickel Company, 3 rd Edition, 1977 (14) Superalloys A Technical Guide, ASM International, 2 nd Edition, 2002 Out-o-Phase In-Phase Figure 1 Load and Temperature Phasing Mechanical Strain Range ε T max = 538 ε T max =538 ε T max =427 ε T max = 316 ε T max =316 T = 21 T = 316 T = 427 T = Fatigue Lie, Cycles Figure 2 TMF Test Results or 1010 Steel [1]

11 surace view oss section view FIGURE 3 Formation o Surace Cracks in Austenitic Steel [3] Figure 4 Oxidation Damage in Steel [4]

12 Figure 5 Wedge ack nucleation in Type 316 stainless steel [5] Out-o-Phase TMF φ 1 In-Phase TMF φ ox φ ε! th ε! mech Free Isothermal Free Contraction Fatigue Expansion Figure 6 Phasing Constants or Oxidation and Creep

13 Figure 7 Larson-Miller Plot [9] Figure 8 TMF Duty Cycle Deinition

14 Figure 9 TMF results