residual stresses residual stresses analysis of residual stresses

residual stresses

residual stresses analysis of residual stresses in crystalline materials by use of X-rays - determination of the stress/strain tensor - possibilities to investigate strain gradients - line positions are converted into strains and those into stresses (stress measurements yield a couple of extra information) - used for bulk and layered materials

residual stresses a body is under stress when its atoms are shifted away from their eqilibrium positions DEF: a force, which increases/decreases atomic distances os positive/negative F F theoretical strength Fth a 0 r + - a 0 r ~ linear (= elastic behaviour) 3

residual stresses analysis of residual stresses in crystalline materials by means of X-ray diffraction - kinds of residual stress The 3 kinds of residual stress are additive!

residual stresses origins of residual stresses - lattice misfit at boundaries different coefficients of thermal expansion of phases material processing (e. g. surface modifications) welding coatings deformation (in situ) 5

residual stresses stress and strain are tensorial properties http: //www. maschinenbau-wissen. de/skript 3/mechanik/festigkeitslehre/116 -spannungstensor 6

residual stresses concept of residual stress analysis by XRD - let a sample be under stress, parallel to its surface - change of interplanar distances in a crystallite depending on its orientation relative to the (field of) stress in the sample coordinate system - residual stress analysis by XRD determines elastic deformation of the lattice as a function of the direction Welzel et al. , J. Appl. Cryst. 38 (2005) 1 7

residual stresses concept of residual stress analysis by XRD - change of interplanar distance is a function of the external (field of) stress - it varies also as a function of the mutual orientation of lattice plane normal and stress - measurement of this directional dependence Welzel et al. , J. Appl. Cryst. 38 (2005) 1 8

residual stresses concept of residual stress analysis by XRD - change of interplanar distance is a function of the external (field of) stress - it varies also as a function of the mutual orientation of lattice plane normal and stress - measurement of this directional dependence Welzel et al. , J. Appl. Cryst. 38 (2005) 1 9

residual stresses concept of residual stress analysis by XRD - measurement of the position of a single diffraction line - definition of residual stresses of 1 st kind - interplanar spacings are measured only along the diffraction vector (lattice planes are normal to them) - only a certain fraction of crystallites diffracts - -> residual stresses determined by mechanical methods likely to differ from XRD values Welzel et al. , J. Appl. Cryst. 38 (2005) 1 10

residual stresses concept of residual stress analysis by XRD j…rotation around the surface normal of the sample y…angle between surface normal and diffraction vector measured: all diffracting crystallites along L 3 ODF rotation around diffraction vector angeles j and y describe the direction of the diffraction vector relative to the strain field Welzel et al. , J. Appl. Cryst. 38 (2005) 1 11

residual stresses basic equations – isotropic crystallites - assumption: polycrystal, assembled from elastically isotropic crystallites - elastic properties of the polycystal = elastic properties of an isotropic crystallite elongation of crystallites along the diffraction vector rotation of sample coordinate system into the laboratory coordinate system - relation between strain and stress for polycrystals composed of elastically isotropic crystallites 12

residual stresses basic equations – isotropic crystallites the sin 2 y method - holds for strain from diffraction measurements as well as for mechanical measurements - for homogeneous stress fields and elastically isotropic crystals also valid upon presence of textures 13

residual stresses basic equations – anisotropic crystallites - realistic case: polycrystals are composed of elastically anisotropic crystallites - stress and strain vary from crystallite to crystallite, given the mutually different orientations of crystallites and stress field - resultant stress/strain is described by grain interaction models -> quasi-isotropic samples: sample appears elastically isotropic although each crystallite is elastically anisotropic - no texture !!!! - grain interaction is direction-dependent - for quasi-isotropic samples: use of X-ray elastic consants (XEC = S 1, ½S 2) - for makroscopically anisotropic samples (e. g. textured), the X-ray stress factors (XSF) have to be used 14

residual stresses basic equations – anisotropic crystallites the sin 2 y equation, differs from the isotropic case only in form of the XEC - XEC‘s depend only on the diffracting lattice plane family and the chosen grain interaction model - obtained stress and strain differ from the mechanical values - sin 2 y method is applicable for all stress states in quasi-isotropic polycrystals 15

residual stresses grain interaction models - calculation of XEC or XSF - can be measured (strain measurement with known external applied stress) - usually calculated from the 2 nd order elastic constants of single crystals - Hooke‘s law Einstein summation - 6 independent equations - 12 independen components - solving a linear system of equations, where 6 of the independent components are known - assumption: 6 of the components are equivalent to the mechanical values averaged for all crystallites 16

residual stresses grain interaction models – Voigt model - assumption: homogeneous strain - causes stress jumps at grain boundaries, which violate the mechanical eqilibrium - incompatible with realistic cases - individual (lengthy) expressions for all crystal systems possible - e. g. cubic systems: - Voigt XECs have no hkl dependence 17

residual stresses grain interaction models – Reuss model - assumption: homogeneous stress - strain inhomogeneities at grain boundaries (crystal would break up) - incompatible with realistic cases - individual (lengthy) expressions for all crystal systems possible - e. g. cubic: - Reuss XECs depend on hkl 18

residual stresses grain interaction models – Eshelby-Kröner model - assumption: an anisotropic crystallite is embedded in an isotropic matrix with the properties of the full polycrystal - expresses the mismatch of the elastic properties between particle and matrix (!!!: averaging of the deviations for all crystallites = 0 interative solution) - calculate stress/strain for an embedded spherical particle - analytical solution only for isotropic cases - extendable to textured samples (no analytical solution) - currently the most realistic model, but hard to calculate 19

residual stresses grain interaction models – Vook-Witt model - special model for thin films assumption: crystallite is in 3 dimensions sourrounded by other crystals 2 extremal cases for the grain interaction models (Voigt, Reuss) layers are low dimensional structures - Vook-Witt model: makroskopische transverse Isotropie (in der Schichtebene) - strain is rotational symmetric in layer plane - strain is equal for all crystallites (V) - stresses normal to the layer plane vanish (R) - inverse Vook-Witt model: - stress is rotational symmetric in layer plane - stress is equal for all crystallites (R) - strain normal to the layer plane is equal for all crystallites (V) 20

residual stresses grain interaction models – Neerfeld-Hill model (effective grain interaction) - XECs of limiting grain interaction models do not corespond to reality - averagingn the extreme cases: parts of the crystals follow one extreme case, the other part the other extreme case - Neerfeld-Hill: averaging of Voigt and Reuss models for compact polycrystals - empitical: arithmetic or geometric average from V and R - weighting: ½ - weighting can also be refinement parameter 21

residual stresses U grain interaction models – examples Sn 22

residual stresses Kornwechselwirkungmodelle – Beispiele Kröner 1/2 S 2 [mm 2/N] -S 1 [mm 2/N] 23

residual stresses measurement strategies - angle y sets direction of measurement relative to the sample surface (normal) - y, j represent the relative orientation of {hkl} lattice plane normals in the sample coordinate system (sin 2 y equation) - f, w, c (instrument angles) stand for the sample orientation in the laboratory coordinate system - 2 q is used to select the requested {hkl} lattice plane

residual stresses measurement strategies w mode (c = 0) - variation of w (q = qhkl) causes variation in y according to y = w-q angle of incidence = w, exit angle = q-y limited range of y (beam parallel to surface |y|<q)

residual stresses measurement strategies w mode (c = 0) - variation of w (q = qhkl) causes variation in y according to y = w-q angle of incidence = w, exit angle = q-y limited range of y (beam parallel to surface |y|<q) c mode (w = q) - c = y (‚y mode‘) variation of c causes variation of y angle of incidence, exit angle sin w·cos c j rotated by 90° with respect to w mode y limit (theoretically) at 90°

residual stresses measurement strategies w mode (c = 0) - variation of w (q = qhkl) causes variation in y according to y = w-q angle of incidence = w, exit angle = q-y limited range of y (beam parallel to surface |y|<q) c mode (w = q) - c = y (‚y mode‘) variation of c causes variation of y angle of incidence, exit angle sin w·cos c j rotated by 90° with respect to w mode y limit (theoretically) at 90°

residual stresses measurement strategies - defocussing caused by inclination of the sample parallel beam geometries recommended

residual stresses measurement strategies - defocussing caused by inclination of the sample parallel beam geometries recommended

residual stresses measurement strategies glancing angle method - measurement of residual stress in thin films/surface layers - method has constant depth of penetration (stress-depth-profiling) - angle of incidence fixed (penetration depth!) limited freedom to vary y - multiple c: angle of incidence w small, variation of c causes variation in y (combination of w- and c-mode) - Multiple {hkl}: angle of incidence w small, c = 0, variation in 2 q causes variation in y according to y = qhkl-w (y, qhkl nicht unabhängig voneinander) - multiple l: change of wavelength changes qhkl and accordingly y, angle of incidence needs to be corrected for the wavelength

residual stresses measurements - categories

residual stresses data treatment: macroscopic elastically isotropic samples - determination of the lattice strain along a certain direction measurement (y, j) with respect to the components of the strain tensor in the sample coordinate system - theoretically: 6 independent components of the stress tensor -> strains have to be measured along 6 independent directions (solving a linear system of equations) - traditional: reshaping of the sin 2 y law in order to obtain the components of the strain/stress tensor, from linear regressions of functions in j, y, hkl - single {hkl} or multiple {hkl} - least squares methods

residual stresses data treatment: macroscopic elastically isotropic samples single {hkl}: sin 2 y-sin(2 y) + triaxial stress state

residual stresses data treatment: macroscopic elastically isotropic samples single {hkl}: sin 2 y-sin(2 y) + triaxial stress state - for general stress states, the functions in sin 2 y / sin(2 y) are non-linear - linearisation: vs. sin 2 y vs. sin(2 y) j = (0°, 45°, 90°)

residual stresses data treatment: macroscopic elastically isotropic samples single {hkl}: sin 2 y-sin(2 y) + tiaxial stress state - for general stress states, the functions in sin 2 y / sin(2 y) are non-linear - from 3 j-angles (0°, 45°, 90°) 3 slopes (Ai) for a+j are obtained - from 2 j-angles (0°, 90°) 2 slopes (Ai) for a-j are obtained - additional condition for the 4 th stress component in a+j from ej 0 hkl

residual stresses no j-dependence mainly used in combination with glancing angle diffraction methods, measurement of multiple {hkl}

residual stresses biaxial stress state http: //paulino. ce. gatech. edu/courses/cee 570/2013_prior/FAQs/Plane_stress. png 37

residual stresses glancing angle diffraction (GAXRD) for thin films S S 0 Diffracting crystallites 38

residual stresses glancing angle diffraction (GAXRD) for thin films S S 0 Diffracting crystallites 39

residual stresses sin 2 y method PLANE STRESS in principle reference frame: Welzel et al. , J. Appl. Cryst. 38 (2005) 1 40

residual stresses sin 2 y method (data analysis) for rotational symmetric stres state - no texture - homogeneous sample - isotropic elastic properties - … djy d 0 sin 2 y Welzel et al. , J. Appl. Cryst. 38 (2005) 1 41

residual stresses biaxial, rotational symmetric stress state

residual stresses

residual stresses choice of {hkl}