Use of Zarka s Method at FHL

Transcription

1 Use of Zarka s Method at FHL Hartwig Hübel Andreas Kretzschmar (FH Lausitz) Theory Implementation in ANSYS xperience Cracow March 11, 2003

2 Background, History 3 in nuclear industry: elastic-plastic response required for life assessment under cyclic loading: strain range ε cyclic accumulated (ratchet-) strain ε acc displacements at end of life 20 years ago: only rough estimates available as simplified elastic-plastic methods (knock-down factors), without capturing effects of: individual geometry of structure kind of loading (thermal, displacement-, force-controlled,...) hardening of material

3 Background, History (ctd.) 4 by end of seventies Zarka s method came up, claiming it was able to account for these effects publication of some examples obtained by Zarka with Zarka s method revealed: superior quality of results (approximations almost exact) obtained with little numerical effort (few lin. elastic analyses) however: method remained largely obscure: description of theory appeared to be not complete some assumptions seemed to be heuristically motivated some applications outside Zarka s team showed poor results question: why that ambiguous? when advantageous?

4 Outline of Zarka s method 5 material model: - yield surface must exist - hardening required:» purely kinematic» linear or multilinear - temperature dependence: only size of yield surface, i.e. y (not, ν, t ) infinite ratcheting ruled out: limit state = S or PS, associated with finite ratcheting simplified method: - space t y ε only partial information: - no evolution of, ε with cycles - only post-shakedown quantities - loss in accuracy (Vp,, ε) only reduced effort: - direct method - few elastic analyses sufficient + some local calculations

5 Specialisation, Modification 6 Zarkamethod specialisation, modifications simplified theory of plastic zones (VFZT) - moderate levels of strain (e.g. no deep drawing) - isotropic material - no strength differential effects (tensile = compressive) - linear and bilinear kinematic hardening (bilin., trilin. ε) - Mises yield surface - temperature-dependent yield stress - mainly cyclic loading between 2 extreme states of loading - logic of iterative improvement: identifying Vp, estimating TIV Y

6 here: kind of analysis exact (el-pl) fictitious elastic (f.el) difference Reformulation of xact Plastic Theory - uniaxial stress state - bilinear ε-diagram (kinem. hardening) - monotonic loading modif. elast. (mea) in plastic zone Vp ε el-pl = el-pl / + ξ/c ε f.el = f.el / ε el-pl -ε f.el =( el-pl - f.el )/ + ξ/c ε * def ρ ξ = ρ+y => ε * = ρ (1/ * ) + ε 0 with 1/ * =1/+1/C and ε 0 =Y/C in el. zone Ve (=V-Vp) ε el-pl = el-pl / ε f.el = f.el / ε el-pl -ε f.el =( el-pl - f.el )/ ε * ρ => ε * = ρ (1/) load applied residual state can be obtained by 1 mea with modif. elastic material data (*) and modif. loading (ε 0 ), if Vp and Y are known idea of Zarka-Method: estimation of Vp and Y y t ε yes yes no initial strain ε 0 in Vp C 7 ε pl

7 Mon. Loading: stimation of Y 8 here: bilinear ε-diagram (lin. kinem. hardening) - space: ξ=c ε pl.*.. def Y = ξ ( f.el ) ρ Mises yield condition: f(,ξ, y ) f(y, f.el, y ) Y space:... * f.el Y projection is unknown; estimation not possible, since center ξ (i.e. ε pl ) also unknown Y is unknown; but estimation possible, since center f.el is known uniaxial stress: Y exactly known by projection; only Vp unknown multiaxial stress: Vp and Y unknown; local estimation of Y sufficient, since field effects of Y are relieved compared to (=heuristic)

8 Mon. Loading: Algorithm 9 fictitious elastic analysis (f.el) of max. loading estimate TIV Y estimate extension of plastic zone Vp f.el Vp Y modify loading: real loading initial strain ε 0 modify elastic parameters: Ε, ν *, ν* modified elastic analysis (mea) ρ elastic-plastic solution by superposition, e.g. el-pl = f.el + ρ el-pl, iterative improvement of Vp and Y numerical effort required: few linear analyses (fictitious elastic and modified elastic) local calculations ε el-pl, displace. etc.

9 Mon. Loading: Iterative Improvement 10 iterative improvement of Vp: with el pl ij el pl > v y from previous iteration Vp Ve iterative improvement of Y: Y space: f.el. * Y projection Y*=-ρ ρ ij with from previous iteration

10 xample 1: Mon. Loading: 2 Bars in Series y hand calculation analytical solution e.g. for specific configuration (A 1 /A 2, l 1 /l 2 ; t /): u t ε 11 ε el-pl ε f.el without iteration iteration exact f.el / y iteration of Vp partly necessary exact results obtained after maximum 1 iteration

11 hand calculation analytical solution; for specific t /: ε el-pl ε f.el xample 2a: Mon. Loading: Beam of ideal I-Profile w 0 1 iteration exact y t without iteration ε after 1 iteration close to exact results f.el / y

12 xample 2b: Mon. Loading: Beam of rect. Cross Section w 0 y t =0 ε hand calculation possible for t =0 (via initial stress): ε el-pl 2.5 ε f.el exact f.el / y 1 iteration without iteration clamped exact 1 iteration plastic zone top surface without iteration middle surface mid length

13 Implementation in F-codes 13 implementation in any commercial F code (ANSYS, ABAQUS, ADINA,...) usually not easy: - modification of elastic material properties and of loading (initial strain) required during iterative solution - access to elastic material properties and application of initial strain usually inconsistent: element, ε 0 Gauss or nodal point - exact definition of Vp somewhat arbitrary (Vp, if v > y in > 50% of Gauss-points ) implementation at FHL in ANSYS : for general structures (1D, 2D, 3D): - bilinear ε - T-dependent y - mon. + cyclic loading (2 states of loading) extension for specific structures (trusses, beams): - trilinear ε - increm. projection in S, if > 2 states of loading y,1 * * * * Vp Ve t T 1 T 2 t y,2 ε

14 Ways of Implementation in ANSYS VFZOT: APDL-Macro (=ANSYS parameter language; slow) initial strain simulated by: * thermal strain at nodal points * orthotropic coefficient of thermal expansion ( deviatoric) * orthotropic directions changing from el. to el. sometimes poor qual. 2. Userpl.f: FORTRAN user subroutine (compiling and linking required) initial strain simulated by: * inelastic strain at Gauss points * formally equilibrium iterations performed, although not necessary 3. IS-Macros (APDL-Macro) or Ustress.f (FORTRAN-subroutine): initial strain simulated by initial stress at Gauss points 4. Particular implementation for trusses and bending beams: initial strain simulated by ts and t across section

15 xample 3: Mon. Loading: Hollowed Plate 15 y t ε displacement + Mises effective stress plastic zone exact plastic zone VFZT axial strain exact VFZT

16 xample 4: Mon. Loading: Crack in a Plate 16 crack monotonic loading t ε Mises effect. stress: fictitious elastic VFZT, 3. mea plastic zone evolutive

17 max min load histogram: const. load cyclic load Cyclic Loading: S: stimation of Y time time f.el min * Y space Ω * f.el max elastic shakedown (S): if Ω exists at each point of structure 17 Vp, if Y*=-ρ is outside Ω: Y*=-ρ f.el min *. Ω Y * f.el max f.el min * Y Ω. Y*=-ρ * f.el max f.el min* Y Ω. Y*=-ρ * f.el max

18 xample 5: S: 2 Bars Parallel 18 T F T y t F time ε y ε exact: ε el-pl /ε y no. of cycles ε el-pl /ε y numerical effort required to achieve S: exact: 1000 cy. * 10 loadsteps * 5 iterations ~ ca linear elast. anal. VFZT: 2 f.el. + 2 mea = 4 linear elast. anal. ε acc elastic shakedown no. of cycles identical results

19 xample 6: S: Thermal Stratification 19 p hot * cold T p T elastic shakedown time ε y t ε hoop ε p ε y shakedown fictitious elastic analysis hoop ε acc mea VFZT evolutive analysis 10 5 evolutive 0 VFZT: good approxim. with 1 mea (i.e. 3 linear elastic analyses) no. of cycles

20 max min load histogram: const. load cyclic load Cyclic Loading: PS: stimation of Y time time ε * ε mean ε acc ε f.el min * Y space 20 * f.el max range values: Vp, if v > 2 y : f.el min * Y * f.el max f.el min * Y * f.el max plastic shakedown (PS): if there is no intersection at least at 1 point of structure

21 Cyclic Loading: PS: stimation of Y mean 21 mean values: Vp, if v,min > y or v,max > y : in Ve, : f.el min * in Vp, : Y mean * f.el max projection of Y*± ½ Y on max (or on min, if closer): f.el min * Y*=-ρ mean ½ Y ½ Y Y mean * f.el max

22 xample 7: PS: Beam under N, w 0 22 w 0 N N w 0 time plastic shake down y t ε ε acc evolutive analysis ε fictitious elastic analysis mea VFZT

23 xample 7 (ctd.): PS: Beam: ε 23 fictitious elastic VFZT: w 0 N 1 mea N 2 mea w 0 plastic shakedown time 3 mea evolutive: 15 cycles

24 xample 7 (ctd.): PS: Beam: ε acc 24 fictitious elastic VFZT: 1 mea N w 0 N 2 mea 3 mea w 0 * plastic shakedown time 4 mea evolutive: 15 cycles 5 mea

25 xample 8: PS: Cylindrical Shell under Axial Force and Temperature 25 C L radius N T N T time N f.el / y =0.8 T f.el / y =2.2 ( PS) t t /=0.03 ν=0.3 ε axial ε acc evolutive analysis fictitious elastic analysis axial ε mea VFZT

26 xample 8 (ctd.): PS: Cylindrical Shell: axial ε C L N fictitious elastic Zarka s method 1 mea 2 mea 3 mea evolutive (100 cycles) radius T

27 Application: Ratcheting- Interaction Diagrams 26 which combination of load parameters yields S, PS, R for specific configuration of structure and loading? by shakedown theorems no post-shakedown quantities load 2 PS S R load 1 Zarka s method: S, PS known a priori R not possible: subregions S 2, PS 2 post-shakedown quantities ( ε, ε acc ) load 2 PS 1 S 1 PS 2 S 2 load 1

28 T xample 9: Ratch-Inter: 2 Bars Parallel F F T time hand calculation for any configuration of loading (T, F) and hard. param. ( t /): y ε y t ε 27 t y 2 1 Sb Sa P Sc exact with 1 iteration exact without iteration 5 ε el-pl ε y exact after max. 1 iteration without iteration p / y = 0.5 p / y = p y exact after maximum 1 iteration t / y

29 xample 10: Ratch-Inter: 2 Bars Parallel (Trilinear) 28 t y1 2 g b e j k d S P f l h ε el-pl ε y y2 y1 ε 1,0 0,5 p y a c 0 0,5 1 p y t y1 0,2 0,05

30 p T xample 11: Ratch-Inter: Bree Problem p T time hand calculation for any configuration of loading (T, P) and hard. param. ( t /): T 29 P 4 t / y Pb 2 iterations ( exact) Sb Pc Sa 1 iteration 0 iteration p / y close to exact after 2 iterations Sc 1.5 ε el-pl ε y p y = 0.4 exact without iteration 0 iteration t / y 2 iterations exact 1 iteration

31 xample 12: Ratch-Inter: Beam under N, w 0 N 30 w 0 N w 0 * max. load level time 3,50 accumulated strain: f.el w0 y 3,00 2,50 2,00 1,50 1,00 0,50 ε acc /ε y : 10,0-12,0 8,0-10,0 6,0-8,0 4,0-6,0 2,0-4,0 0,0-2,0 0,00 0 0,5 1,0 f.el N / y

32 1.Benchmark CUT-FHL: Thick-walled Cylinder 31 thick walled cylinder under internal pressure p mon. loading, load level v f.el / y =1.665 p time material: lin. kin. hardening y ε y t ε combination ν t / A B C D

33 1.Benchmark CUT-FHL (ctd.): Thick-walled Cylinder 32 results: combinations A, B, D: all residual stress- and strain-components after 6 mea close to exact values (difference <1%); combination C ( t /=0.001, ν=0.3): hoop res. stress [Mpa] axial radial res. strain VFZT exact hoop axial radial inner radius outer inner radius outer no improvement with more mea loss in accuracy: why?

34 2.Benchmark CUT-FHL: Hertz Contact 33 q (parabolic, regularized) Hertz contact (plane stress) stationary cycl. loading (S); load level v f.el / y =1.388 q time lin. kin. hardening t / = 0.1 y t ε

35 2.Benchmark CUT-FHL (ctd.): Hertz Contact results: residual stresses along vertical section: 34 y x section top evolutive ρ x ρ y top 1 mea 7 mea 30 mea slow approximation to exact solution: why?

36 3.Benchmark CUT-FHL: Bree Problem 35 thin-walled cylinder under internal pressure and radial temperature gradient p T cycl. loading (S, PS) p T time combination Sa Sb Sc Pb Pc f.el p / y f.el t / y lin. kin. hardening: t / = 0.1, T-dependent yield stress y2 y1 ε y1 t ε

37 3.Benchmark CUT-FHL (ctd.): Bree Problem 36 y = 200 at inner and outer surface for min. load level (no temperature), y = at inner surface and at outer surface for max. load level (fully developed T-gradient) results for combination Pc: - range values = exact after 2 mea - accum. state at minimum load level: [%] 0.5 fictitious elastic analysis ε acc,hoop ε acc,axial mea VFZT evolutive analysis ε acc reasonable, but not superior: why?

38 Further Development: Non-radial Loading 37 3-Bar Problem: T 1 T 2 T 3 T 1 T 2 T 3 time Nozzle: coolant medium Ṫ - space path of fictitious elastic stress Moving Load:

39 Summary 38 Zarka s Method (and it s modifications by FHL) is a simplified method to provide post-shakedown quantities in S and PS by direct means ( ε for fatigue, ε acc for ratcheting-assessment, displacements) is implemented in commercial F-program ANSYS for monotonic and cyclic loading examples show in case of radial loading: - good approximation to exact solution - at small numerical effort (few linear elastic analyses) - in particular if directional redistribution is not strongly developed - range-values (PS) usually better than mean values benchmarks between CUT and FHL raised some questions