PVP2006-ICPVT

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1 Proceedings of PVP2006-ICPVT ASME Pressure Vessels and Piping Division Conference July 23-27, 2006, Vancoucer, BC, Canada PVP2006-ICPVT DAMAGE-2005: A POST-PROCESSOR FOR HIGH CYCLE FATIGUE UNDEX COMPLEX THERMOMECHANICAL LOADING Rodrigue Desmorat LMT Cachan ENS-Cachan/Univ. Paris 6/CNRS 61, av. du président Wilson F Cachan Cedex, France desmorat@lmt.ens-cachan.fr Frédéric Pauget First Coauthor LMT Cachan ENS-Cachan/Univ. Paris 6/CNRS 61, av. du président Wilson F Cachan Cedex, France Jean-Philippe Sermage Second Coauthor EDF/R&D 1 avenue du Général de Gaulle F Clamart Cedex, France sermage@edf.fr ABSTRACT On the idea that fatigue damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), a three-dimensional two scale damage model has been proposed in the past decade at LMT-Cachan for High Cycle Fatigue applications. It consists in the micromechanics analysis of a weak micro-inclusion subjected to plasticity and damage embedded in an elastic meso-element (the RVE of continuum mechanics). The consideration of plasticity coupled with damage equations at microscale, altogether with Eshelby-Kröner localization law, allows to compute the value of microscopic damage up to failure for any kind of loading, 1D or 3D, cyclic or random, isothermal or anisothermal, mechanical, thermal or thermomechanical. A robust numerical scheme makes the computations fast and the new programming of a graphical user interface gives a software simple to use with facilities for material parameters identification: DAMAGE Examples of thermal and thermomechanical fatigue as well as applications on E.D.F. FATHER and INTHERPOL structures subjected to complex thermo-mechanical fatigue are detailed. INTRODUCTION Various components in nuclear power plants are subjected to thermomechanical loading during service. Thermal fatigue microcracking is observed in the mixing zones of the Reactor Heat Address all correspondence to this author. Removal system of nuclear power plants [1, 2]. The evaluation of crack nucleation and their subsequent propagation in a pipe undergoing thermomechanical loading is very important to determine investigation periods and maintenance programs. On the remark that High Cycle Fatigue (HCF), either thermally or mechanically activated, occurs for an elastic regime at the Representative Volume Element (RVE) scale, the mesoscale of continuum mechanics, a two scale damage model has been built [3 7]. It accounts for micro-plasticity and micro-damage at the defects scale or microscale. The model is phenomenological, describing micro-plasticity with classical 3D von Mises plasticity equations, describing micro-damage by Lemaitre damage evolution law of damage governed by plasticity and enhanced by elastic energy [7, 8]. The scale transition is made by use of Eshelby-Kröner localization law, modified in order to take into account the coupling with micro-plasticity, micro thermal expansion and damage. Compared to classical approaches for fatigue [9 20] the present approach allows for the quantification of damage, even in 3D, even for non cyclic thermomechanical loadings. It works as a post-processor of a reference thermo-elastic (or thermo-elastoplastic) computation as detailed thereafter. TWO SCALE DAMAGE MODEL The two scales under consideration are represented in Fig. 1. The mesoscopic scale or mesoscale is the scale of the RVE of 1 Copyright c 2006 by ASME

2 STRUCTURE CALCULATION p σ ij (t), ε ij (t), T(t), (ε ij (t)) Thermo-elastic E, ν, α (Plastic σ y, C ) Figure 1. MESO RVE Localization law Plasticity and damage at microscale µ Thermo-elastic E, ν, α =α µ Plastic σ y = σ f, C Damage S, s, Dc µ µ σ ij (t) e µ εij (t) p εij (t) micro T(t) D(t) MICRO-ELEMENT EMBEDDED IN A THERMO-ELASTIC REPRESENTATIVE VOLUME ELEMENT continuum mechanics, the microscopic scale or microscale is the scale of the microdefects. At the mesoscale, the stresses are denoted σ i j, the total, elastic and plastic strains ε i j, ε e i j, εp i j. They are known from a thermo-elastic Finite Element (FE) computation as for High Cycle Fatigue ε p i j 0. The values at the microscale have an upper-script µ. For HCF, plasticity and damage are assumed to occur at the microscale only, ε pµ i j 0, D 0, where for simplicity the damage variable at the microscale has no upper-script (D = D µ ). The thermoelastic law for the RVE reads: ε i j = 1+ν E σ i j ν E σ kkδ i j + α(t T re f )δ i j (1) with E the Young modulus, ν the Poisson ratio, α the thermal expansion coefficient and T re f the reference temperature. The temperature field in the structure T(x) is usually determined from an initial heat transfer computation. The mechanical properties E, ν, α may depend on the temperature. A law of thermo-elasto-plasticity coupled with damage is considered at microscale. The elasticity law reads then (recall that µ-upper-script stands for variable at microscale ): ε µe σ µ i j i j = 1+ν E 1 D ν σ µ kk E 1 D δ i j + α µ (T T re f )δ i j (2) where the thermal expansion coefficient α µ is taken next equal to the meso coefficient α. In the yield criterion, the hardening X µ is kinematic, linear, and the yield stress is the asymptotic fatigue limit of the material, denoted σ f, is then: ε µ i j = εµe i j + εµp i j ε µe i j = 1+ν E σµ i j ν E σµ kk δ i j + α(t T re f )δ i j σ µ i j = σµ i j 1 D ε µp i j = 3 σ µd i j X µ i j 2 ( σ µ X µ ṗ µ ( ) eq µ ) d X i j = 2 dt C y 3 εµp i j (1 D) ( ) Y µ s Ḋ = ṗ µ if p µ > p D S D = D c crack initiation with the plastic modulus C y, the damage strength S, the damage exponent s and the critical damage D c as material parameters. The damage evolution (last equation of previous set) is smaller in tension than in compression due to the consideration of the micro-defects closure parameter h within Y µ as usually for metals h 0.2 [8] and: Y µ = 1+ν [ σ µ + : σ µ + 2E ν 2E (4) (1 D) 2 + h σµ : σ µ ] (1 hd) 2 [ tr σ µ 2 + (1 D) 2 + h tr σµ 2 ] (5) (1 hd) 2 p µ is the accumulated plastic strain at micro-scale and p D the damage threshold taken equal to zero in the following (for loading dependent threshold refer to [7]). The plastic multiplier λ = ṗ µ (1 D) is determined from the consistency condition f µ = 0, ḟ µ = 0. The scale transition meso micro is governed by modified Eshelby-Kröner localization law [21 23]: ε µd i j = 1 ( )] [ε D i j 1 bd + b (1 D)ε µp i j ε p i j ε µ H = 1 1 ad [ε H + a((1 D)α µ α)(t T re f )] rewritten by considering ε = ε D + ε H 1, ε µ = ε µd + ε µ H 1: (6) f µ = ( σ µ X µ ) eq σ f (3) where σ µ = σ µ /(1 D) is the effective stress and where f µ < 0 elasticity. The set of constitutive equations at microscale ε µ = 1 [ ] (a b)d ε+ 1 bd 3(1 ad) tr ε 1+b((1 D)εµp ε p ) + a((1 D)αµ α) (T T re f )1 1 ad (7) 2 Copyright c 2006 by ASME

3 with a et b the Eshelby parameters for a spherical inclusion, a = 1+ν 3(1 ν), b = 2 4 5ν 15 1 ν If S E is Eshelby fourth order tensor, one has S E : 1 = a 1, S E : x D = b x D, with 1 the unit tensor and x D any deviatoric tensor. NUMERICAL SCHEME FOR ANISOTHERMAL TWO SCALE DAMAGE MODEL The strain and temperature histories at mesoscale are known from the reference FE thermoelastic computation. The numerical scheme is classically strain driven: at each time step t and for known strain increment at mesoscale ε = ε ε n and temperature increment T = T T n, the numerical scheme must calculate, by time integration of the constitutive equations at microscale altogether with the consideration of the localization law, the strain ε µ, stress σµ, plastic strain εµp, accumulated plastic strain p µ and damage D at microscale. The 3 stages for the numerical resolutions of the model equations classically are [3]: 1) an elastic prediction at microscale, taking into account the localization law, 2) a test over the criterion function f µ, and 3) if f µ is found positive, a plasticdamage correction (still at microscale). Elastic Prediction The elastic prediction assumes a behavior (thermo-)elastic with constant plastic strain ε µp = εµp n, constant kinematic hardening X µ = X µ n and constant damage D = D n. The elastic prediction gives a first estimate for the total strain, the elastic strain and the effective stress at microscale at time t, ε µ = 1 1 bd n [ε + (a b)d n 3(1 ad n ) tr ε 1 + b ( (1 D n )ε µp n εp ) ] + a((1 D n)α µ α) 1 ad n (T T re f )1 ε µe =ε µ ε µp n αµ (T T re f )1 σ µ =E : ε µe σ µ =(1 D n ) σ µ where E is Hooke s tensor. Plastic-Damage Correction The previous elastic prediction gives the estimate σ of the effective stress σ at time t, with unchanged kinematic hardening X = X n, and allows for the calculation of the yield criterion. If the condition f µ 0 is fulfilled, the calculation is over (8) (9) and ε µp = εµp n, X = X n, D = D n is set. If not, this elastic solution is corrected by ensuring the consistency condition f µ = 0 (resolution by Euler implicit scheme). For non monotonic application, it is judicious to assume damage constant over a time increment [7]. In High Cycle Fatigue, D could even be considered as constant over a full cycle as if N R cycles are necessary to break the material, the maximum damage increment per cycle is of the order of magnitude of D c /N R < 1/N R, small value indeed! Assuming a constant damage D = D n, the incremental form of the nonlinear equations to be solved reads: ε µe + 1 b ε µe + α µ 1 b T 1 ε+ ε p 1 bd n 1 bd n 1 bd n (a b)d n (1 bd n )(1 ad n ) tr ε1 a((1 D n)α µ α) T 1 = 0 1 ad n f µ = ( σ µ Xµ ) eq σ f = 0 (10) These equations can of course be solved by use of Newton iterative method, but to write them in a form of 2 residuals R p and R s [23], R s = sµ + Γ s µd E (s µ ) p µ + Q s = 0 eq R p = (s µ ) eq σ f = 0 (11) function of the unknown accumulated plastic strain and of the unknown variable s µ = σ µ Xµ allows to explicitly gain a closed-form solution. One has set: Q s = C X µ n C n E 1 1 bd n σ σ σµ n E [ E : ε+k (a b)d n 1 ad n tr ε1 2G b ε p + 3K (1 ad n ) ((1 a)αµ + aα) T 1 Γ = 1 ( 3G 1 b ) +C (1 D n ) E 1 bd n ] (12) with E = E(T ), C n = C y (T n ), C = C y (T ) and E = E(T )/E, G = G(T )/E, K = K(T )/E, G and K being respectively the shear modulus and the compressibility modulus of the material. 3 Copyright c 2006 by ASME

4 The exact solution of (10) rewritten as (11) is: s µ H = E Q sh p µ = 1 ( ) Γ Q seq σ f E s µd = E Q D s 1+ Γ E σ p µ f (13) with the exponent D for the deviatoric part, Q seq for von Mises norm of Q s, Q sh for its hydrostatic part and with s µ = sµd + s µ H 1. Variables Updating Once the previous correction made, all the micro values are updated as follows, normal to the yield surface: m µ = 3 s µd 2 σ f plastic strain : ε µp = εµp n + mµ p µ kinematic hardening: X µ = 2 3 C (1 D n ) ε µp + C X µ n effective stress: σ µ = sµ + Xµ elastic strain: ε µe = E 1 : σ µ + αµ (T T re f )1 damage: ( ) µ s Y D = D n + p µ if p > p D with S C n Y µ = 1+ν 2E [ σ µ + : σ µ + ( ) 1 2 Dn + h σ µ 1 hd : σ µ ] n ν 2E [ tr σ µ 2 ( ) 1 2 Dn + h tr σ µ 1 hd 2 ] n stress tensor : σ µ = (1 D ) σ µ and one can then start the calculation at time t n+2. (14) DAMAGE-EDF-2004 AND DAMAGE-2005 POST-PROCESSORS A FORTRAN program solves previous constitutive equations and allows to launch batch calculations. For a given material parameters file (extension.mate) and for a given loading sequence (made of the repetition by blocks of complex cycles, file Figure 2. MATERIAL PARAMETERS of extension.load), the program calculates the time to crack initiation, i.e. the time to reach the critical damage D c. The inputs (mesostrains and temperature) come from a FE reference computation and are then at a user chosen Gauss point. The maximum number of points used to describe a cycle being large (actually 5000), the program allows for random fatigue calculations. The first version of the damage and fatigue post-processor was LMT DAMAGE-90 program [3, 8], followed by DAMAGE- 98 [4], DAMAGE-2000 [5]. The mean feature of fatigue such as the mean stress effect, the non-linear accumulation of damage, initial hardening effect and the non proportional loading effect in bi-axial fatigue are recovered. Note that only DAMAGE-EDF-05 and the LMT version DAMAGE-2005 allows for anisothermal conditions, as illustrated next. DAMAGE-2005 also has a graphical interface with facilities for material parameters identification and for results plotting. The outputs of any calculation are a standard results file (damage history, file of extension.out), and optional files for complete results at mesoscale and microscale (files of extension.meso and.micro). DAMAGE-2005 allows to enter quite simply the material parameters distinguished as (Fig. 2 and 3), the temperature independent parameters: the Poisson ratio ν (nu), the critical damage D c for mesocrack initiation, the 4 Copyright c 2006 by ASME

5 Figure 3. INPUT OF TEMPERATURE DEPENDENT PARAMETERS Figure 4. LOADING DEFINITION Figure 5. ENTERING A COMPLEX LOADING microdefects closure parameter h, the temperature dependent parameters: the Young modulus E, the thermal expansion coefficient α (ALPHA), the plastic modulus C y, the asymptotic fatigue limit σ f (SigFinf), the yield stress σ y (SigY), the damage strength S, the damage exponent s. The damage threshold in pure tension ε pd (epd) is used to describe the loading dependency of the accumulated plastic strain threshold p D. As plasticity and damage laws are differential equations, non vanishing initial values may be entered (initial plastic strain p 0, initial damage D 0 ). Note that essai or example are the name of the projects used here for the illustrations of DAMAGE-2005 features. A complex loading is defined as repetitions of blocks made of cycles (Fig. 4). Both applied temperature and stresses (or strains) may vary. A maximum of 5000 points per cycle may be used, practically enough to address most thermo-mechanical fatigue loadings. In the simple example of Fig. 5, the block simple40 made of 6 cycles is followed by the block simple80 made of 8 cycles. For the material parameters identification, one proceeds as follows: 1. The mesoscale parameters (E, ν, α, σ y, C y ) are identified at Figure 6. LOADING AN EXPERIMENTAL WOHLER CURVE each temperature on the monotonic tensile curve. 2. The asymptotic fatigue limit is guessed from an experimental Wöhler curve as the asymptote at very high numbers of cycles. 3. Take for the parameters h and D c the default constant values for metals, h = 0.2, D c = 0.3 [8]. 4. The damage parameters S and s are pre-identified from a nonlinear fitting. The experimental Wöhler curve entered as 5 Copyright c 2006 by ASME

6 Figure 9. LAUNCHING THE COMPUTATION y x Figure 7. FAST IDENTIFICATION PROCEDURE B C E A D Figure 10. THERMAL FATIGUE OF A BEAM Last, the computations for the complex loadings under consideration are performed (Fig. 9) by saving or not all the results at mesoscale (.meso file) and/or at microscale (.micro file) for later curves plotting (microstresses or damage history over a complex cycle for instance). Figure 8. FINAL IDENTIFICATION OF S THERMAL AND THERMOMECHANICAL FATIGUE Thermal Fatigue from Damage Analysis A simple academic case to illustrate the anisothermal facilities of the model and the corresponding post-processors is the example of a bar blocked at its two extremities and uniformly heated and cooled (Fig. 10). The corresponding thermal fatigue loading is: a text file (reference curve of Fig. 6), an approximate closedform solution for the number of cycles to rupture [6, 24] is used to obtain automatically a first set of parameters S, s (Fig. 7). 5. The value for s is kept when the parameter S is adjusted by comparison with the reference curve but using this time DAMAGE-2005 to compute the Wöhler curve instead of an approximate formula (Fig. 8). T = T(t), ε 1 = 0, ε 2 = ε 3 = α(1+ν)(t T re f ) (15) where the temperature varies between a minimum value T min and a maximum value T Max. Micro-plasticity (and damage) occurs from points A to B, C to D, E to next summit, with elastic stages at microscale in-between. The numbers of cycles to rupture N R = N(D = D c ) are plotted in Fig. 11. This is the computed thermal fatigue curve T Max vs N R (with here T min = T re f = 150 C). 6 Copyright c 2006 by ASME

7 T Max N R Figure 11. COMPUTED THERMAL FATIGUE CURVE e θ Figure 13. THERMOMECHANICAL FATIGUE IN PHASE (CASE 1), OUT OF PHASE (CASE 2) LOADINGS Figure 12. NAL PRESSURE Internal pressure P(t) Uniform temperature field T(t) THERMOMECHANICAL FATIGUE OF A PIPE WITH INTER- Anisothermal Structures Computations Complex thermomechanical loadings are encountered in real structures. As an illustration, one can calculate the time to rupture of a pipe subject to both temperature T(t) and internal pressure P(t) variations. The cylinder is assumed thin, the temperature uniform. The stresses due to the thermal loading and to the mechanical loading are of the same order of magnitude. Different pressure loadings are considered: constant : P = P Max, in phase with the temperature (case 1): P = 0 for T = T min, P = P Max for T = T Max, out of phase with the temperature (case 2) : P = 0 for T = T Max, P = P Max for T = T min. e r T The results are given in Fig. 13 where T = T Max T min. A strong effect of temperature-pressure out of phase loading on the number of cycles to rupture is obtained, comforting us in the idea that one must develop adequate tools for anisothermal thermomechanical cases. One has last performed successfully post-processing computations of thermo-hydraulic computations (giving the temperature fields) followed by thermo-elastic computations (giving the strains history) [25]. The structures studied were E.D.F. FA- THER and INTHERPOL [1] real size testing structures representative of pressure vessels subjected to complex temperature variation due to inside vessel chaotic mixture of hot and cold fluids. The loading history for FATHER case is similar to random thermomechanical fatigue, a cycle of a few seconds being made of 1000 points to describe both the strains and the temperature variations. The loading history for INTHERPOL case is similar to out of phase thermomechanical loading with only one (but complex!) increase and decrease of the temperature over a repeated cycle. In both cases the stress state is multi-axial (in fact mainly bi-axial). The location of the crack initiated is recovered as well as the order of magnitude for the time to crack initiation. Applications to FATHER and INTHERPOL must be developped and comparisons made with measurements. CONCLUSION The two scale damage model allows to compute the mesocrack initiation condition in mechanical fatigue, thermal fatigue and thermo-mechanical fatigue. For HCF applications, it is implemented in a post-processor from, post-processor of Finite Element 3D computations. The model being written in an incremental form (in opposition to a stress or strain amplitude forms), 7 Copyright c 2006 by ASME

8 it easily allows to deal with random fatigue (no need of arbitrary equivalent cycles!). Damage maps can be drawn. A robust numerical scheme allows for the computation a million cycles in a few minutes on a PC. Note that a jump in cycle procedure can be used to reduce the calculation time [3, 7]. The abilities of the present Continuum Damage Mechanics approach to deal with complex thermo-mechanical loadings are illustrated. Further validation work is in progress. REFERENCES [1] Curtit, F., and Stephan, J.-M., Mechanical aspect concerning thermal fatigue initiation in the mixing zones piping. In 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18), Beijing, China, August [2] Taheri, S., High cycle thermal fatigue on a mixing zone of auxiliary cooling system. In ASME-PVP conference, San Diego. [3] Lemaitre, J., and Doghri, I., Damage 90 : a post processor for crack initiation. Comput. Methods Appl. Engng., 115, pp [4] Lemaitre, J., Sermage, J. P., and Desmorat, R., A two scale damage concept applied to fatigue. International Journal of Fracture, 97, pp [5] Sauzay, M., Effet de surface libre en fatigue policyclique. PhD thesis, Université Paris 6. [6] Desmorat, R., Modélisation et estimation rapide de la plasticité et de l endommagement, Habilitation à diriger des recherches. Tech. rep., Université Pierre et Marie Curie. [7] Lemaitre, J., and Desmorat, R., Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures. Springer. [8] Lemaitre, J., A Course on Damage Mechanics. Springer Verlag. [9] Miner, M. A., Cumulative damage in fatigue. J. Appl. Mech., 67, pp. A159 A164. [10] Crossland, B., Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. Proc. of the Inter. Conf. on fatigue metals Inst. Mech. Engr., pp [11] Dang-Van, K., Sur la résistance à la fatigue des métaux. Sciences et Tecniques de l Armement, 3, pp [12] Kujawski, D., and Ellying, F. A., A cumulative damage theory of fatigue crack initiation and propagation. International Journal of Fatigue, 6, pp [13] Manson, S. S., and Halford, G. R., Re-examination of cumulative fatigue damage analysis an engineering perspective. Engineering Fracture Mechanics, 25, pp [14] Leiholz, H. H. E., On the modified s-n curve for metal fatigue prediction and its experimental verification. Engineering Fracture Mechanics, 23, pp [15] Leis, B. N., A nonlinear history-dependent damage model for low cycle fatigue. In Low Cycle Fatigue, ASTM STP 942, American Society for Testing Materials, Philadelphia, P.A., USA, L. K. H. D. Salomon, G.R. Halford and B. Leis, eds., pp [16] Zuchowski, R., Specific strain work as both failure criterion and material damage measure. Res. Mechanica, 27, pp [17] Leis, B. N., An energy-based fatigue and creepfatigue damage parameter. Journal of of Pressure Vessels and Technology, ASME Transactions, 99, pp [18] Taheri, S., and Docquet, V., Evaluation of non conservatism of combined rainflow counting and miner s rule for damage cumulation in strain controlled fatigue. In 16th International Conference on Structural Mechanics in Reactor Technology (SMiRT 16), Whashington D.C. [19] Angles, J., Taheri, S., and Papaconstantinou, T., High cycle fatigue under multiaxial loading damage, accumulation models applied to an industrial structure. In 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18), Beijing, China, August [20] Fatemi, A., and Yang, L., Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. International Journal of Fatigue, 20, pp [21] Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, related problems. Proc. Roy. Soc., London, A241, p [22] Kröner, E., On the plastic deformation of polycrystals. Acta Metall., 9, pp [23] Seyedi, M., Desmorat, R., and Sermage, J.-P., A two scale model for thermo-mechanical high cycle fatigue failure. In European Conferrence on Fracture ECF 15, Advanced Fracture Mechanics for Life and Safety, Stockholm, Sweden. [24] Desmorat, R., and Lemaitre, J., Two scale damage model for quasi-brittle and fatigue damage, in Handbook of Materials Behavior Models, chapter Continuous Damage, section Academic Press. [25] Pasutto, T., Péniguel, C., and Sakiz, M., Chained computation using an unsteady 3d approach for the determination thermal fatigue in a t-junction of a pwr nuclear plant. In ICAPP Seoul. 8 Copyright c 2006 by ASME