Numerical Simulation of Temperature-Dependent, Anisotropic Tertiary Creep Damage

Transcription

1 47th AIAA Aerosace Sciences Meeting Including The New Horizons Forum and Aerosace Exosition 5-8 January 009, Orlando, Florida AIAA Numerical Simulation of Temerature-Deendent, Anisotroic Tertiary Cree Damage Calvin Stewart, Ali P. Gordon, and David W. Nicholson 3 Deartment of Mechanical, Materials, and Aerosace Engineering, University of Central Florida, Orlando, FL Directionally-solidified (DS) Ni-base sueralloys are commonly alied as turbine materials to rimarily withstand cree conditions manifested in either marine-, air- or landbased gas turbines comonents. The thrust for increased efficiency of these systems, however, translates into the need for these materials to exhibit considerable strength and temerature resistance. Accurate rediction of crack initiation behavior of these hot gas ath comonents is an on-going challenge for turbine designers. Aside from the sectrum of mechanical loading, blades and vanes are subjected to high temerature cycling and thermal gradients. Imosing reeated start-u and shut-down stes leads to cree and fatigue damage. The resence of stress concentrations due to cooling holes, edges, and sites sustain foreign object damage must also be taken into account. These issues and the interaction thereof can be mitigated with the alication of high fidelity constitutive models imlemented to redict material resonse under given thermomechanical loading history. In the current study, the classical Kachanov-Rabotnov model for tertiary cree damage is imlemented in a general-urose finite element analysis (FEA) software. The evolution of damage is considered as a vector-valued quantity to account for orientation-deendent damage accumulation. Cree deformation and ruture exeriments on samles from a reresentative DS Ni-base sueralloys tested at temeratures between 649 and 98 C and three orientations (longitudinally-, transversely-oriented, and intermediately oriented). The damage model coefficients corresonding to secondary and tertiary cree constants are characterized for temerature and orientation deendence. This advanced formulation can be imlemented for modeling full-scale arts containing temerature gradients. Nomenclature A,n = secondary cree constants B,l,k = tertiary cree constants I = identity tensor n = first rincial direction of the effective stress tensor s% = deviatoric stress of the effective stress tensor α,β = weighing factors in the Hayhurst stress γ = isotroic/anisotroic control variable ε = strain tensor σ = stress tensor σ% = effective stress tensor eq σ Ω = Hayhurst (triaxial) stress of the effective stress tensor σ% = first rincial stress of the effective stress tensor σ% = hydrostatic (mean) stress of the effective stress tensor H σ% VM = von Mises stress of the effective stress tensor Φ = damage alied ω = damage Ω = damage tensor Ω & = damage rate tensor Graduate Student, cmstewar@mail.ucf.edu, Member AIAA. Assistant Professor, agordon@mail.ucf.edu, Member AIAA. 3 Professor, nicholsn@mail.ucf.edu, Member AIAA. Coyright 009 by Stewart et al. Published by the, Inc., with ermission.

2 I. Introduction Cree is the inelastic deformation of a material due to high temerature. As temerature increases the cree strain rate increases accordingly. Early work in the field of cree continuum mechanics focused solely on modeling the isotroic cree strain rate. One such examle is the Norton ower law for secondary cree. Further develoment, geared towards including the non-linear strain exerience in the tertiary cree regime caused the inclusion of the concet of damage. This is where the cree strain rate is influenced by the degraded or damage state of the material at each time ste. It results in a damage evolution equation being couled with the cree strain rate equation. Isotroic cree models work best when the mechanical load exerienced by a comonent is similar to what is alied during uniaxial cree tests. Such models are inaroriate for modeling comonents which undergo comlex states of stress. In most cases, the off-axis comonents of a stress tensor in the vicinity of stress raisers, cracks, and notches have non-zero values and the stress state is more comlex than that which can be achieved in conventional cree testing. Moreover, the deendence of cree rate is strongly non-linear with stress; therefore, cree damage may also deend on the direction of the stress vector as well as on the absolute value of its comonents. Additionally, isotroic cree models are unable to model the orientation deendence of material roerties. An isotroic-scalar comliance term cannot relicate anisotroic comliance tensor in symmetry classes such as triclinic, orthotroic, transversely isotroic materials, and so on. Drives by the aerosace and ower generation industries to increase turbine efficiency have roduced a massive influx in the develoment of anisotroic alloys since the 980 s. Through the rocess of directional solidification, material manufacturers can directly control the alignment of grain boundaries. Such alloys are design to roduce enhanced strength, stiffness, and/or cree resistance in a articular orientation. When used on turbine blades, they are manufactured so that the enhanced material direction is aligned with the direction where the most mechanical load is exerienced. This rocess roduces gas turbine comonents which have longer lives. Of articular imortance to both aerosace and the ower generation industries is the temerature gradient exerienced by turbine comonents. In ower generation, gas turbines exerience cyclic thermomechanical loading occurs due to eak ower requirements. Reeated startu and shutdown stes lead to cree and fatigue damage. In the aerosace industry the thermal gradient across the turbine changes as a function of flight history and maneuvers. These changes affect the rate at which cree occurs across the comonent. Both industries deal with thermal variations which necessitate a cree model which includes temerature-deendence. An anisotroic tertiary cree damage model is imlemented in the ANSYS finite element software. The cree material roerties are otimized over a range of temeratures and material orientations. Through regression analysis, temerature-deendent and orientation-deendent functions are develoed for each material roerty. Comarison between available cree deformation data and the simulations is conducted to verify the accuracy of the anisotroic tertiary cree damage material model. z y L-oriented (0 ) x Off Axis (45 ) T-oriented (90 ) 3 µm Figure. Structure of GTD- (a) Image of microstructure. Dark areas are the bimodal γ reciitated articles (b) Schematic of grain structure

3 Table. Nominal chemical comosition of GTD- sueralloy 3 Cr Co Al Ti W Mo Ta C Zr B Fe Si Mn Cu P S Ni Bal Bal. II. Material The material under consideration is GTD-, a directionally-solidified (DS) Ni-base sueralloy commonly used in gas turbine blading. Directional solidification is a technique by which the grain structure of the material can be aligned in a articular orientation. This roduces a material that has enhanced stiffness, strength, and cree ruture roerties in one or more local coordinate directions. Alloy DS GTD- was develoed in the 980 s and is a modification of the GE sueralloy Rene 80. It is a transversely isotroic material where solidification occurs in the <00> longitudinal (L) orientation while the transversely (T) orientations <00> and <00>, are uncontrolled. Microstructurally, GTD- is formed of a nickel austenite (γ) matrix, bimodal gamma rime (γ ) reciitated articles, γ γ eutectic, carbides and small amounts of toological close-acked hases σ, δ, η and laves. 3,4 The matrix and (γ ) reciitated articles are observed in Fig. a. It has a high volume fraction of gamma rime (γ ) reciitated articles, (>60%) which imart enhanced imact strength, high temerature cree and fatigue resistance, and imrove corrosion resistance. This microstructure causes difficulties when considering comonent reair but a number of methods are under investigation to mitigate this roblem.,5 The nominal chemical comosition can be found in Table. Due to the directionally solidified nature of the material, the cree strain rate is deendent on orientation. This is due to the grain structure being aligned in the <00> direction. Figure b deicts where grains are longitudinally (L) and transversely (T) oriented. Of articular interest is the cree resonse at off axis locations (such as 45 ) where an intermediate resonse between L and T is exected to occur. 50 µm 500 µm Figure. Grain structure of GTD- (a) T-oriented Secimen (b) L-oriented Secimen Through otical and scanning electron microscoy investigations, a closer look at the grain structure of the material was resolved. Figure shows the grain structure of both L and T-orientations. It shows that in the T- oriented directions long grain boundaries are found. In the L-oriented direction a textured attern is observed. The material behavior in the transverse directions <00> and <00> is similar to that of olycrystalline (PC) materials. Cree deformation and ruture tests were erformed on L and T-oriented secimen. A arametric study including temeratures ranging from 649 to 98 C and varying stress levels was imlemented to determine the cree resonse of the material over a wide range of conditions. Figure 3 demonstrates the cree resonse at 87 C. This collected data was used to comare the exerimental results with those that are numerically simulated. 3

4 3. Strain, ε/ε ref Figure 3. Cree deformation of L-oriented (unfilled symbols) and T-oriented (filled symbols) GTD- sueralloy at 87 C. III. Anisotroic Cree Damage Model Modern cree continuum damage models are based off two fundamental equations, cree rate and damage evolution. Early work in the field consisted solely of the isotroic cree rate, such as the Norton ower law for secondary cree. These models are limited because they do not include the non-linear enhanced cree deformation during the tertiary cree stage. More advanced isotroic models included damage evolution which relates the net reduction in area of a material as it deforms to the alied stress. 6 These models are owerful but assume that damage is scalar and isotroic when it is not. Literature has shown that damage is fundamentally anisotroic. 7,8 Due to inconsistencies during material rocessing, secimen do not have comletely uniform, homogenous, grain structures and therefore uon loading exerience anisotroic damage. This damage causes the inelastic deformation of the material to differ deendent uon local orientation. Tensor based, anisotroic tertiary cree damage models are therefore requirement. The anisotroic cree damage model used to fulfill this requirement is fundamentally based off the isotroic Kachanov 9 -Rabotnov 0 cree damage model with the alication of the Hayhurst (triaxial) stress. The anisotroic tensor based formulation starts with the effective stress tensor, σ%. Murakami and Murakami and Ohno 3 roosed the tensor based, anisotroic extension of the effective stress and damage alied as σ % = σφ Φσ ( + ) ( ) Φ = I Ω 07MPa (30ksi) 4MPa (35ksi) 89MPa (4ksi) 379MPa (55ksi) 07MPa (30ksi) 4MPa (35ksi) 89MPa (4ksi) () where σ is the alied stress tensor and Ω is the secondary rank damage tensor. Damage is thus considered the three-dimensional degradation of the material due to voids and fissure growth in the material over time. Hayhurst 4 determined from cree ruture tests that for a number of cree resistant alloys, void and fissure growth on grain boundaries occurs at lanes 90 erendicular to the rincial stress direction, where a majority of damage is concentrated on the lane erendicular (0 ) to the first rincial stress, σ%. Using the terms of the effective stress tensor, the Hayhurst (triaxial) stress is alied relating the first rincial stress, σ%, hydrostatic (mean) stress, σ% H, and von Mises stress, σ% VM. The Hayhurst stress is given as 4

5 ( ) eq σ = ασ% Ω + 3βσ% H + α β % σvm () where α and β are weighing factors valued between zero and one and are determined from multiaxial cree exeriments. The damage evolution equation is given as l eq k l ( Ω ) tr ( ) ( ) Ω& = B σ Φ n n γ I + γn n (3) where n is the first rincile direction vector and B, k, l, and γ are tertiary cree constants. An attractive roerty of this anisotroic tertiary cree damage model is the inherit ability to revert to the isotroic Kachanov-Rabotnov tertiary cree damage model. When γ is equal to zero, the damage evolution equation will revert to a scalar form (i.e. an isotroic form). Conversely, increasing γ results in enhanced material degradation on lanes erendicular to the rincial direction n (γ =.0 is considered fully anisotroic). Taking the revious terms and alying them in the Norton ower law results in the following cree rate equation, cr 3 ε & = A % σ ( ) where s% is the deviatoric stress tensor and A and n are secondary cree material constants. Using Eqs. (3) and (4), the cree deformation resonse of an anisotroic material can be modeled. IV. Numerical Methodology The anisotroic tertiary cree damage model has been imlemented into the finite element analysis (FEA) software ANSYS. This was achieved by coding a user-rogrammable feature (UPF) material model subroutine in FORTRAN. This subroutine incororates the backward Euler imlicit-integration algorithm which allows long time simulations to be conducted with limited error. The material model was written into the USERMAT UPF. The USERMAT UPF allows the user to establish a unique total (reduced) stiffness tensor. This was done as follows 5 VM n s% % σ VM ε = SEL σ + SINEL σ Q = S + S ( ) ( ) TOT EL INEL where ε is the total strain tensor, σ is the stress tensor, S EL is the elastic comliance tensor, S INEL is the cree (inelastic) comliance tensor, and Q TOT is the total (reduced) stiffness tensor. Considering that GTD- is a transversely isotroic material, a total of six material roerties are necessary to determine the elastic comliance tensor, S EL. The elastic comliance tensor then takes the following form. ν ν z E E E z ν ν z E E E z ν z ν z E E E S z EL = (6) G z G z + ν E (4) (5)

6 The Young s modulus and Poisson s ratio on the <00> and <00> oriented x-y symmetry lane are defined as E and ν resectively. The <00> z-oriented Young s modulus, Poisson s ratios, and Shear modulus are defined as E z, ν z, ν z, and G z, resectively. The cree (inelastic) comliance tensor, S INEL, was derived using the cree strain rate Eq. (4) and effective stress tensor Eq. () into the following triclinic symmetric form. S INEL cr d SINEL = ε dσ% S S S3 S4 S5 S6 S S3 S4 S5 S 6 S33 S34 S35 S 36 = S44 S45 S46 S55 S 56 S66 Notice the use of effective stress instead of alied stress. This is accetable due to that fact the comonents of the alied stress terms within a comonent of effective stress are first order and thus terms of different indicial locations equate to zero uon differentiation. For comonents in the uer right symmetric zone (multilied by two), such as S 4, symmetry takes the following form, S 4 = S 4. An inut deck was coded in the ANSYS Parametric Design Language (APDL) and included all the commands necessary to erform a simulation. To establish the geometry, a single, three-dimensional, 8-noded block element was used. Constant force loading on the to four z-direction nodes was alied. Isothermal heating across the element was used. Aroriate dislacement controls were set to allow deformation consistent with cree tests. The damage tensor, cree strain rate, and cree strain were initialized at zero. The elastic material roerties E, E z, ν, ν z, ν z, and G z were related to temerature-deendence in third order olynomial form. Previous work by the authors using the isotroic Kachanov-Rabotnov model led to the develoment of cree material constants for GTD-. 5 Since the anisotroic model reverts back to the isotroic model when γ is equal to zero, the earlier develoed isotroic cree constants were used as initial values for the anisotroic model. The cree deformation resonse of the anisotroic model comared with the isotroic model is shown in Fig. 4. The model is shown to erform well with the given material. As shown in the figure, in the direction arallel to the uniaxial tensile load (loaded direction), the cree deformation remains the same between isotroic and the anisotroic model. Conversely, on directions erendicular we see a reduction in the tertiary cree resonse of the material using the anisotroic model (when γ =.0 tertiary cree is comletely removed leaving only secondary cree). This reduction is desired. Orientations where stress is not directly alied will not roduce the same cree resonse as uon those where load is directly alied. The anisotroic model is thus a more accurate measure of the cree resonse of the material and will roduce realistic and longer estimates of cree life. A limitation of the model is that the cree material constants are scalar and are based solely on the material orientation that interfaces with the uniaxial tensile load. The tertiary and secondary cree material constants remain the same and cree rate and damage evolution equations are simly scaled by γ and n. A more robust method would include tensor based cree material constants where the local cree rate is conveyed through a global transformation tensor. (7) 6

7 Cree Strain, ε/ε ref Cree Strain, ε/ε ref (a) (b) Uniaxial Tensile Load Parallel to L Direction (z) Uniaxial Tensile Load Parallel to T Direction (x) MPa 89MPa Loaded Direction ISO (Off Axis) ANI (Off Axis) Loaded Direction ISO (Off Axis) ANI (Off Axis) 4MPa 89MPa Loaded Direction ISO (Off Axis) ANI (Off Axis) Loaded Direction ISO (Off Axis) ANI (Off Axis) Figure 4. Comarison of exerimental and FEM simulated cree deformation of 87 C (a) L- oriented: unfilled symbols (b) T-oriented: filled symbols 7

8 Modulus, E/E ref (GPa) Shear Modulus, G/G ref Poisson's Ratio, ν/ν ref T-Oriented DS L-Oriented DS Temerature, T ( o C) T-Oriented DS L-Oriented DS Temerature, T ( o C) ZX XZ XY Temerature, T ( o C) Cree Coeff., A/A ref (MPa -n hr - ) Cree Exonent, n/n ref Tertiary Cree Coeff., B/B ref (MPa -χ hr - ) Temerature, T ( o C) V. Temerature Deendence All the cree curves can be generated through varying the cree constants, A, n, B, l, and k. After demonstrating that the anisotroic tertiary cree damage model can successfully model the cree deformation of GTD-, constants were develoed for temeratures range from 649 to 98 C. Regression analysis was utilized to determine temerature-deendence functions for each constant. The cree constants were considered stress-indeendent. The secondary cree constant, A, took on the standard Arrhenian temerature-deendent form, e.g T-Oriented DS L-Oriented DS T-Oriented DS L-Oriented DS Temerature, T ( o C) T-Oriented DS L-Oriented DS Temerature, T ( o C) Figure 5. Plots of the temerature-deendence of elasticity and cree material constants A Q RT ex cr = A0 (8) where Q cr is the aarent activation energy, A 0 is the temerature indeendent term, and T is temerature in Kelvin. R is the Boltzmann/universal gas constant. The secondary cree constant, n, was found to fit well as a third order olynomial, e.g. 8

9 n( T ) = n T + n T + n T + n, (9) where T is measured in degrees Celsius. The tertiary cree constant B, was found to fit well as a secondary order exonential equation of the form, e.g. ( ) B = B ex B T (0) 0 where B 0 and B are constants and T is measured in degrees Celsius. The additional tertiary cree constants l and k were found to be temerature-indeendent. These temerature-deendent functions were created for both L-oriented and T-oriented cree tests using temeratures ranging from 649 and 98 C. The lowest square of the Pearson roduct-moment correlation coefficient, R, found within all the temerature-deendent functions was The next lowest was This demonstrates that using these functions the temerature-deendence of the model will closely mirror that of the cree deformation data. VI. Orientation Deendence As reviously stated, GTD- is a transversely isotroic material. Both the elastic and cree resonse of this material deend on material orientation. To account for this henomenon, the elasticity comliance tensor is used and orientation-deendent functions are needed to adjust the cree resonse. To that end, a cree test was erformed at orientations of 45, L (0 ), and T (90 ). Cree Strain, ε/ε ref (T Oriented) 0.0 (L Oriented) Cree Strain, ε/ε ref Figure 6a shows the resulting cree deformation. Since cree material constants for L & T-orientations are already known, numerical simulations of the 45 orientation was necessary. The fit of the numerical simulation is found in Fig. 6b, and shows that the constants relicate the exerimental data well. Taking the cree constants for L, T, and 45 orientation-deendent constants were formulated. The secondary cree constant, A, was found to fit well in second order olynomial form, e.g Strain ISO ANI Figure 6. Cree deformation data with stress of 89MPa (a) L-T and 45 oriented secimen (b) comarison between exerimental 45 oriented secimen and FEM simulated cree deformation A( θ) = A θ + Aθ + A + A, () 0 0 where θ is measured in degrees. The McCauley brackets and the A 0 term are used to demonstrate that the minimum value of A(θ) can only be A 0. The secondary cree constant, n was found to fit well in the second order olynomial form, e.g. n( θ) = n θ + nθ + n, () 0 9

10 where θ is measured in degrees. The tertiary cree constant, B, was found to best fit within a secondary order exonential equation of the form, e.g. ( θ) B( θ) = B ex B (3) 0 where B 0 and B are constants and θ is measured in degrees. All together, regression analysis roduced functions with a square of the Pearson roduct-moment correlation coefficient, R, of or greater. Figure 7 shows the fit of these orientation-deendent functions. Cree Coeff., A/A ref (MPa -n hr - ) e+0 e-5 e-30 e-45 e-60 Using the develoed orientation-deendent functions, numerically the cree deformation of the model is arametrically exercised for various orientations. Figure 8 is the resulting cree deformation. It shows that the angle which roduces the weakest cree resistance is between 45 and 90. These orientations are weakest due to the ratio of textured and long grain boundaries. Literature has shown that grain boundary fluxes cause diffusion-induced grain-boundary migration and grain-boundary sliding activation. 6 The large volume fraction of long boundaries from the L-orientation couled with the textural boundaries from the T-orientation influence the grain boundary state and subsequently, the stability of the material. This causes the cree resistance of GTD- at angles between 45 and 90 to be reduced. Further develoment of these orientation-deendent functions using additional cree tests at varying angles is necessary to verify that the numerically simulated deformation mirrors that of exeriments. Cree Strain, ε/ε ref Angle from L Orientation, θ Cree Exonent, n/n ref Angle from L Orientation, θ Figure 7. Cree material constants related to secimen orientation Tertiary Cree Coeff., M/M ref (MPa -χ hr - ) e+0 e-5 e Angle from L Orientation, θ Figure 8. Numerically simulated cree deformation at 89MPa and 87 C for varies secimen orientations. 0

11 VII. Conclusion The anisotroic tertiary cree damage model successfully models the exerimental data for GTD- in L- oriented, T-oriented, and 45 secimen. It has been roven that the anisotroic model erformance better than revious isotroic models due to the fact that it reduces the tertiary cree exerienced by direction erendicular to the uniaxial tensile load similar to what is observed exerimentally. Cree material constants were determined over a range of temeratures using cree deformation data from exerimental cree tests. Using these constants, temerature-deendent functions were created. These functions allow the use of non-isothermal boundary conditions on comonents. Taking the cree deformation data, constants were determined for the three orientations L-oriented, T-oriented, and 45. Orientation-deendent functions were develoed. This results in the model being able to adjust the cree resonse of a finite element deendent on the direction vector uon which stress load is alied. Further develoment of this model can significantly enhance the ability of the aerosace and ower generation industries to accurately redict the cree resonse of gas turbine comonents. Acknowledgments The authors would like to thank Richard W. Neu of Georgia Institute of Technology for 45 -oriented cree data. Calvin Stewart is thankful for the suort of a McKnight Doctoral Fellowshi through the Florida Education Fund. References Daleo, J.A. and Wilson, J.R., GTD alloy material study Journal of Engineering for Gas Turbines and Power, Transactions of the ASME, Vol. 0, No., 998, Li, L., Reair of directionally solidified sueralloy GTD- by laser-engineered net shaing, Journal of Materials Science, Vol. 4, No. 3, 006, Ibanez, A. R., Srinivasan, V. S., and Saxena, A., Cree deformation and ruture behaviour of directionally solidified GTD sueralloy, Fatigue & Fracture of Engineering Materials & Structures, Vol. 9, No., 006, Sajjadi, S. A., and Nategh, S., A high temerature deformation mechanism ma for the high erformance Ni-base sueralloy GTD-, Materials Science and Engineering A, Vol. 307, No. -, 00, Hale, J. M., Procedure develoment for the reair of GTD- gas turbine bucket material, Eighth Congress &Exosition on Gas Turbines in Cogeneration and Utility, Portland, OR, Stewart, C. M., and Gordon, A. P., Modeling the Temerature-deendence of Tertiary Cree Damage of a Ni-Base Alloy, ASME Early Career Technical Journal, Vol. 7, No., 008, Murakami, S. and Sanomura, Y. Cree and cree damage of coer under multiaxial states of stress, In Plasticity Today, edited by A. Sawczuk and G. Bianci, 985, Altenbach H., Huang C., and Naumenko K., Cree-damage redictions in thin-walled structures by use of isotroic and anisotroic damage models, Journal of strain analysis for engineering design, Vol. 37, No. 3, 00, Kachanov, L. M., "Time to Ruture Process Under Cree Conditions," Izv. Akad. Nank., Vol. 8, 958, Rabotnov, Y. N., Cree Problems in Structural Members, North-Holland Publ. Co, North Holland, Amsterdam, 969. Hayhurst, D. R., On the Role of Continuum Damage on Structural Mechanics, Engineering Aroaches to High Temerature Design, edited by B. Wilshire and D. R. Owen, Pineridge Press, Swansea, 983, Murakami, S. A continuum mechanics theory of anisotroic damage, In Yielding, Damage and Failure of Anisotroic Solids, edited by J. P. Boehler, Mechanical Engineering Publications, London, 990, Murakami, S. and Ohno, N. A continuum theory of cree and cree damage., In Cree in Structures, edited by A. R. S. Ponter and D. R. Hayhurst, 98, Hayhurst, D. R., Cree Ruture Under Multi-Axial States Of Stress, Journal of the Mechanics and Physics of Solids, Vol. 0, No. 6, 97, Gordon, A. P., Khan, S., and Nicholson D. W., Temerature and Orientation Deendence of Cree Damage of Two Ni-Base Sueralloys, Proceedings of the 007 ASME Pressure Vessels and Piing/CREEP8 Conference, San Antonio, Texas, Kolobov, Yu.R., Grabovetskaya, G.P., Ivanov, K.V. and Ivanov, M.B., Grain Boundary Diffusion and Mechanisms of Cree of Nanostructured Metals, Interface Science, Vol. 0, 00,