Residual Stress analysis

Transcription

1 Residual Stress analysis

2 Informations: strains Macro elastic strain tensor (I kind) Crystal anisotropic strains (II kind) Fe Cu C Macro and micro stresses Applied macro stresses

3 Residual Stress/Strain definition!!' : Macrostress!'' : Microstress!''' : r.m.s. Microstress!''!' x!'''

4 Experimental setting Measurement of a high 2theta peak position for different tilting of the sample The sample can be tilted in omega or psi (chi) Changing phi we will scan different direction for the stress in the sample The simple behavior is a linear relationship between the measured d- spacing and sin 2 psi. The slope is proportional to the macrostrain.

5 Strain measurement The full formula: e L 33 / e R N ' d RN / d 0 & 1 e R N ' e 33 % [e 11 cos 2 N % e 12 sin2n % e 22 sin 2 N & e 33 ] sin 2 R % [e 13 cosn % e 23 sinn]sin2r Isotropic planar stresses: " ' ˆ d hkl '; d 0 Š=d 0 ˆ 1 2 shkl 2 ' sin 2 2s hkl 1 ' ;

6 (22) eviation, ae, find example of the procedures. Since the averages will be taken over single-crystal compliances s'33u, the X-ray elastic constants will result in the Reuss limit. It will be assumed that theshear orientation distribution of the stresses crystallites can be idealized by some combination of 1 (23a) (23b) cept of al vs ult, note that components ). For q~=0, (e22-/333) is an then be 45. From e23 when ~0 nto account, d from tp=45? 0.5./ t-- to= 90~" [ o/ -~0. -.5' L..S" o. I.5 ol.5 o..5 sin2qu Fig. 4. Lattice strain vs sin 2 ~Omeasured at the 211 reflection of ground steel (D611e & Cohen, 1979). Measured values: ~b>0; ~O<0. ( - - ) Calculated from stress tensor (25). (---) Average strain at (23a).

7 Non linear behavior In many cases oscillation of d vs. sin 2 psi are observed; some possible causes: Textured sample -> the elastic tensor is anisotropic. Plastic deformation: anisotropy of the plasticity behaviour and elastic tensor results in anisotropy of the residual stresses/strains Thermal expansion anisotropy Shear stresses normal to the surface Coherent and semicoherent interfaces (in thin film.) Dolle in 1979 (J. Appl. Cryst., 12, 489) analyzed the problem in general and was followed by other authors: Noyan and Nguyen for the plastic deformation, Barral et al. for the texture connection.

8 Procedure: Pro: Cons: Texture-Stress Measurement of the texture ODF by traditional pole figures Measurement of the d-spacing vs. sin 2 psi for high angle reflections Computation of the effective macro-elastic tensor using single crystal elastic constants and the ODF Different theories can be used to average the elastic tensor over the ODF: Voigt (stress compatibility) Reuss (strain compatibility) Hill (mean value between Reuss and Voigt) Self Consistent, FEA... (costly) Geometrical mean Analysis of the d-spacing vs. sin 2 psi using the averaged elastic tensor You control the entire process Lengthly procedure, two measurements, two analyses Does not work (very difficult) for highly stressed or strongly textured materials

9 Traditional methods for the ZrO 2 films d 113 [Å] Voigt model (no texture) Reuss model (texture) sin 2 (!) Elastic modulus [GPa] <C> vs. psi ZS1 ZS2 ZS ! [degrees]

10 ZrO2 thin films Intensity [counts] ! 65

11 Macro residual stress on the ZrO 2 serie 0-1 Whole pattern analysis sin 2! method Residual stress [GPa] -2-3 sin 2! with texture Voigt model F&L model Reuss model Thickness [µm]

12 Measuring the stress also by the curvature! " = K # d s 2 # a 11 d f data fit 10 [µm] ! 2 y/!x 2 = e-7 µm scan length [µm]

13 Comparison of results method: XRD: sin 2! curvature method XRD (220) plane (200) plane(113) plane Stoney's formula modified formula Ferrari and Lutterotti " 11 = " [GPa] * " c, i = j * " c, i # j [GPa]

14 Stress-texture for a zirconia film (WIMV)

15 Voigt model + WIMV Residual Stresses/Texture analysis

16 ZrO 2 film: results Reconstructed pole figures Very high in plane residual stresses (compression): Reuss model: 3.6 GPa Bulk Path GEO: 3.47(5) GPa Curvature method: > 10 Gpa!? Thickness: 320 Nanometer

17 Experimental errors Example: the CPT film shows big shift of the peaks increasing!. The shift is not smaller at low 2theta angle. In the fitting was perfectly reproduced by a beam 0.59 mm higher than the goniometer center. Using the Rietveld method peak shifts from low angle positions are also used normally -> good sample positioning required, perfect alignment of the instrument also.