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1 RESIDUAL STRESSES ANF Métallurgie Fondamentale Vincent Klosek (CEA / DSM / IRAMIS / LLB) 23/10/ NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 1

2 INTRODUCTION Residual Stresses? Static multiaxial stresses within an isolated solid in mechanical equilibrium, neither subjected to external force nor to external moment Result from the thermo-mechanical history of the considered material 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 2

3 REMINDERS (?) A da M n B T (M, n) =σ n 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 3

4 REMINDERS (?) 1 t ε( X ) = ( ξ ( X ) + ξ ( X )) 2 1D ε = lim x 0 u x = du dx M 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 4

5 REMINDERS (?) Constitutive Laws Local relations between σ, ε and T Thermoelasticity in the framework of infinitesimal strain theory ( linearized relation): σ = A : ε k ( T T 0 ) Isotropic elastic material, isothermal transformation: ij = λ ( ε ε ε ) δij + µε ij σ 2 Young modulus: Poisson ratio: µ (3λ + 2µ ) E = λ + µ ν = λ 2 ( λ + µ ) 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 5

6 RESIDUAL STRESSES? g rot (rot d ε) = 0 total stress free ε = ε + ε elastic accommodation 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 6

7 RESIDUAL STRESSES? Strain Incompatibility Application of a compatible strain field to a heterogeneous material: σ B A B A A+B A A σ res 0 B σ res ε RESIDUAL STRESSES 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 7

8 RESIDUAL STRESSES? Main sources various scales Atomic scale: point defects (very local influence) Single crystal scale: dislocations, precipitates, etc Polycrystal scale: Plastic strain incompatibilities phase transformations thermal strains Badulescu et al. (2011) 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 8

9 RESIDUAL STRESSES? Various scales res I σ ( X) = σ ( X) + σ ( X) + σ ( X) II III 1st order stresses: in equilibrium at the whole sample scale ( scale considered by engineers for structures calculations) I 1 res σ ( X ) = σ ( X ) V 2 nd order stresses: in equilibrium at the scale of a group of crystallites σ II 3 rd order stresses: in equilibrium at the scale of a crystallite Defects of crystallographic lattice (dislocations, precipitates, vacancies, grain boundaries, etc ) III V 1 res ( X) = v σ ( X) res σ ( X) = σ ( X) ( σ ( X) + σ ( X)) v I II 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 9

10 RESIDUAL STRESSES? Why evaluating them is so important?... Strongly influence the mechanical behaviour of a component - harmful: premature fracture (fatigue, cracking, etc ) - favourable: example of prestress treatments (shot peening, etc ) Significant effects on performance, safety, reliabillity of a component, a structure. Improvement of the processes (thermal or mechanical treatment required?...) At a more fundamental point of view: Understanding of the physical deformation mechanisms, and how they interact 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 10

11 RESIDUAL STRESSES? Examples of significant manifestations Drying of wood Silver Bridge (WV, USA), 1967 Stress-induced corrosion 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 11

12 DETERMINATION METHODS Experimental determination of elastic strains Destructive or semi-destructive methods Non destructive methods 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 12

13 DETERMINATION METHODS (semi-) destructive methods Principle: some matter is removed mechanical equilibrium is affected system tends to reach a new equilibrium it changes its shape Hole drilling method (incremental or not) or ring core method plane stress approximation, isotropic elasticity law, empirical coefs to determine Contour method 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 13

14 DETERMINATION METHODS Non destructive methods Ultrasonic technique dependence of the propagation velocity of ultrasonic waves on stress state (non linear elasticity, use of acoustoelastic coefficients) Barkhausen noise (ferromagnetic materials) Discontinuous motion of Bloch walls stress inverse magnetostrictive effect technique very sensitive to microstructural defects ( tricky to isolate contributions) Raman spectrometry dependence of optic phonon frequencies on stress state Diffraction Crystal lattice is used as a strain gauge 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 14

15 DIFFRACTION METHODS Theory of diffraction by distorted crystals (Krivoglaz, 1969) Diffraction vector K = g + s Bragg: I 0 s = 0 (non distorted perfect crystal) Scattered intensity (normalised): I ( K ) 2πin s 2πi u K = e e dxdn deformed 8 NOVEMBRE initial atom 2012 positions CEA 23 OCTOBRE 2012 PAGE 15 u( x, n) = u( x) u( x n)

16 DIFFRACTION METHODS -Intensity distribution I(K) directly related to the projection of the relative displacement between atoms u along K Directional measurement! ε KK ( x) = 1 K ² K ε(x) K = u K lim n 0 nk - Homogeneous deformation diffraction peak shift (back to Bragg law!) - Diffraction technique only sensitive to elastic strains! 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 16

17 DIFFRACTION METHODS Selectivity of diffraction methods: Strain is measured for grains in diffraction condition only!!! K 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 17

18 DIFFRACTION METHODS Selectivity of diffraction methods: Strain is measured for grains in diffraction condition only!!! K 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 18

19 DIFFRACTION METHODS What do we measure?... 1st order moment 2 nd order (centered) moment (1) µ + = s I( s) ds (pos. of the «gravity centre» of the peak (2) µ + = s ² I( s) ds («peak width») Strain along K averaged over the diffracting volume Variance of elastic strains along K over the diffracting volume ε KK = d d hkl 0 hkl ε KK Ω = θ = tanθ 0 (1) µ K If homogeneous strain over Ω! ε = 2 KK Ω µ K ² (2) Ideal for coupling with homogenization techniques! 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 19

20 DIFFRACTION METHODS Interpretation of peak shifts Stress at a given point x: = with σ = 0 σ( x) B( x) : σ + σ ( x) res res σ σ = 0 + General Formulation : Strain: ε ( x) = S( x) : σ( x) K K K K K K < ε KK > Ω = : S : B : σ + : S : σ 2 Ω 2 res Ω 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 20

21 DIFFRACTION METHODS The «sin² ψ law» 1st approximation σ res = 0 < ε KK > Ω = K K : K S : B 2 Ω : σ 2 nd approximation: isotropic elasticity ε < KK > Ω = K K 2 K : S : σ 1+ ν ε KK = ( σ cos ² ϕ + σ sin 2ϕ + σ sin ² ϕ σ )sin ² ψ Ω E 1+ ν ν + σ 33 ( σ11 + σ 22 + σ 33) E E 1+ ν + ( σ13 cosϕ + σ 23 sinϕ)sin 2ψ E σ, σ σ 11 22, 33 Too often applied out of hypothesis validity 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 21

22 EXAMPLES Example 1: welding Process involving melting Temperature gradients Withdrawal during solidification Residual stresses If no cohesion (incompatibility strain field) 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 22

23 EXAMPLES Determination by means of neutron diffraction Tensile residual stress tangential stress depth from inner surface (mm) axial stress (MPa) distance from centre (mm) (coll. CEA/DEN) 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 23

24 EXAMPLES Example 2: Intergranular stresses in Zr SC estimates isotropic elastic behaviour T = -600 C, σ = {00.2} reflections 0 Isotropic texture αc 2α a 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 24 Coll. O. Castelnau (PIMM)

25 EXAMPLES Example 2: Intergranular stresses in Zr Experimental observations (neutron diffraction) Σ = 450 MPa (Coll. O. Castelnau (PIMM)) Importance to take microstructure into account for data analysis!!! 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 25

26 CONCLUSIONS Various scales various mechanisms : need to consider physical, chemical and mechanical phenomena (i.e. metallurgical phenomena!) Stress fields are most of the time complex and heterogeneous Characterization of residual stresses is of fundamental and applied importances (but relevant scales are not the same!) Various techniques to determine residual stresses Diffraction (RX & neutrons) allows to characterize intra- and intergranular heterogeneities ideal for coupling with micromechanical modelling BUT in every case: interpretating measurement results is far from being trivial RESIDUAL STRESS MEASUREMENT! Numerical prediction of residual stresses within components? optimization of geometries, processes, performances (ex: ) 8 NOVEMBRE 2012 CEA 23 OCTOBRE 2012 PAGE 26

27 PAGE 27 CEA 23 OCTOBRE 2012 Commissariat à l énergie atomique et aux énergies alternatives Centre de Saclay Gif-sur-Yvette Cedex T. +33 (0)1 XX XX XX XX F. +33 (0)1 XX XX XX XX DSM IRAMIS LLB 8 NOVEMBRE 2012 Etablissement public à caractère industriel et commercial RCS Paris B