STRESS ERRORS ASSOCIATED WITH MINIATURIZATION OF Ψ ASSEMBLY X-RAY STRESS ANALYZERS

Transcription

1 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN STRESS ERRRS ASSCIATED WITH MINIATURIZATIN F Ψ ASSEMBLY X-RAY STRESS ANALYZERS ABSTRACT Toru GT Department of Mechanical Engineering, Fukui Universit of Technolog, Gakuen, Fukui , Japan Using a model and a simulation method, the problems caused b miniaturization of goniometers are discussed from the viewpoint of the stress error caused b sstematic errors including the mis-setting of the specimen and the misalignments of the collimator and detector of the Ψ assembl X-ra stress analzer with a position sensitive detector. As a concluion, it was found that stress errors increase with decreasing goniometer size, and it is believed that the goniometer size could be optimized b the approach adopted in this stud. INTRDUCTIN There are man mechanical parts whose un-reliabilit depends heavil on their residual stresses. It is also true that the residual stress of most of these parts cannot be measured because of size limitations. Therefore, the miniaturization of an X-ra stress analzer is one important subject in this field. Although various efforts have been made, the current minimum size of an analzer seems to be determined mainl b the miniaturization of the X-ra tube and the detector. Little discussion has been held on the size of the X-ra path from the X-ra focus to detector through the collimator and specimen. In this paper, the problems caused b miniaturization of the goniometer are discussed from the view point of the stress error caused b the mis-setting of the specimen and the misalignment of the collimator and detector for the Ψ assembl X-ra stress analzer using a position sensitive detector (PSD). The stress errors are calculated using a model and a simulation method. A stud on the Ω assembl analzer was reported in a previous paper []. MDELING AND SIMULATIN F X-RAY STRESS MEASUREMENT X-Ra Stress Analzer The strain measurement in the Ψ direction using a Ψ assembl X-ra stress analzer is sche- maticall illustrated in Figure. The scanning plane of the detector (Ψ plane) is inclined b Ψ against the normal of the specimen surface. The X-ra beam hits the specimen from η direction on the Ψ plane through a collimator, where η=π/ (θ 0 )/ and θ 0 is the diffraction angle of a specimen with stress σ = 0. The distance from the goniometer center to PSD is the goniometer radius R 0. α is the detectable angle range of the PSD. Specimen mis- X-ra tube Collimator Cl z' 0 R0 PSD ' (a) Ψ plane (b) η plane Figure. Ψ assembl X-ra stress analzer z L z' PSD x x'

2 ISSN This document was presented at the Denver X-ra Conference (DXC) on Applications of X-ra Analsis. Sponsored b the International Centre for Diffraction Data (ICDD). This document is provided b ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific communit. All coprights for the document are retained b ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN setting is given b the off set L. Modeling Let us consider an arbitrar incident X-ra element UW through a collimator having an arbitrar cross section. At the entrance of the collimator, the point U is defined b the distance η from the center of incident beam z 0 in the direction on the Ψ plane and x ψ from z 0 in the direction on the plane normal to the Ψ plane. The point W is defined b η and x ψ at where UW hit the specimen while mis-set. Figure shows the element UW, the center of the incident beam z 0 and PSD projected on Ψ plane with the specimen surface for illustration. The goniometer center is the irradiated point of the normall set specimen b z 0 and the beam is diffracted to C (θ 0 ) on PSD if the specimen has zero stress. For the mis-set specimen with zero stress, the irradiated point is and the diffracted point on PSD is C. For UW, the irradiated point is defined b and the diffracted point on PSD is defined b G. The points,, C, C and G are on the Ψ plane. Let δ be the distance from C to G on PSD, then the diffraction angle θ at the point G is given b the following equation: δ θ = θ 0 + tan R. () 0 The distance δ can be obtained as follows. The angles z 0 C and z 0 C are η. Let η be the angle U G, then the diffraction angle θ of the crstal plane diffracting the incident beam UW to G is written as θ = π η. () Giving the normal of the crstal plane as F, the point F is defined as the point bisecting the angle U G on the line UG as shown in Figure. As is illustrated in Figure3, let us make the coordinate axes at the origin, which are the x axis in the direction of specimen stress σ, axis perpendicular to the x axis on the specimen surface and the z axis normal to the surface. These axes are parallel to those given in Figure, respectivel. The vector F has the angles φ, φ and φ 3 against the axes x, and z, respectivel. Rotate the axes z and x on the plane z- -x b Ψ and rotate the inclined z axis around the inclined x axis b η to coincide with the final z axis to the vector F. Then the angles Ψ and η are given b the following equations. z' z z U PSD C F C' G' ' 3 " F " W ' ' ' " Specimen surface ' " x, Figure. X-ra element and diffraction to the PSD Figure 3. Vector F

4 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN ψ η cos = tan cos π φ φ 3 = φ. (4) Therefore, the diffraction angle θ of the crstal plane for the case of specimen with stress is θ = σ cos η sin ψ ( σ σ ) sin η sin φ sinψ K K ν, 5 + ( σ sin φ + σ cos φ ) sin η ( σ + σ ) + θ 0 K K + ν where K is the X-ra stress constant, ν is Poisson s ratio, σ and σ are principal stresses and φ is the angle between σ and σ. Finall, the distance δ can be defined satisfing the following equation: θ = θ. (6) Figure 4 shows a point G on PSD of an arbitrar angle θ. The incident angle β of G to PSD can be defined b the angle between G and the vector C G which is the vector having the shortest distance from to PSD. Let β as β for θ. Then, the diffraction intensit of UW at θ can be described b the following equation []. R0 θ = I0 sec β0 cos β exp b R [ ( β β) ] I, (7) where β0 = θ θ 0 and R is the distance between and G. The X-ra diffraction distribution is assumed to be a Gaussian distribution of β (θ) with β as its center. The parameter b is controlling the half-value breadth (HVB) of the X-ra diffraction profile of the specimen. Simulation of X-ra stress measurement A simulation of stress measurement of ferrite steels using a Cr target with four incident angles of 0, 30, 45 and 60 degrees was conducted. The values used for θ 0 and K were 56.4 deg and 38 MPa/deg, respectivel. The detectable angle α of the PSD used was 8 deg. (3) z' z x' z U PSD G C C G G d:collimator diameter ' " ' ' 37 points Specimen surface Figure 4. Diffraction angle and incident angle Figure 5. Elements for the circular collimator

5 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN (a) Normal (b)translation (c)rotation (d) (b)&(c) Figure 6. Collimator misalignment Figure 7. Translation error vs.its direction The incident beam through a circular collimator was represented b elements, as shown in Figure 5. The intensit contributed b all elements was summed at θ in the necessar range of diffraction angle to produce the diffraction peak for the incident angle Ψ. The collimator misalignment was investigated for translation and rotation as shown in Figure 6. The amount of translation is defined b t and that of rotation is defined b κ. The stress error caused b the collimator misalignment was found to depend on its direction. Figure 7 shows an example plotting the stress error against the angle φ c starting from the direction normal to the Ψ plane. It is seen that the effect of misalignment has its maximum at φ c = 0 and 80 deg. Therefore, the stress errors caused b collimator misalignment in the direction normal to the Ψ plane are used in the following paragraphs. The stress errors associated with miniaturization of the goniometer were investigated under the condition of Cl=R 0 using a specimen of HVB = deg. RESULTS Intrinsic Stress It was found that a zero stress specimen without mis-setting shows an apparent stress if we use a collimator with a large diameter and/or short effective length. We call this apparent stress the intrinsic stress σ c []. In Figure 8, σ c was calculated for two cases of collimator diameter d under the conditions of Cl = R 0. It is seen in the figure that larger diameters cause larger σ c values and that the σ c increases with the decrease in goniometer radius. It is also recognized that σ c increases rapidl as the R 0 Figure 8. Intrinsic stress becomes less than about 50mm. Stress Error caused b Mis-Setting of Specimen It was found that the stress errors σ caused b sstematic errors were well explained using the following equation: σ = σ x σ σ c (8) where σ x is the X-ra stress or stress obtained. Figure 9 shows the changes in stress error σ L caused b specimen mis-setting with the change in the ratio

6 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN Figure 9. Stress error σl vs. ratio L/R 0 Figure 0. Stress error σl vs. goniometer radius at L=0.mm L/R 0 of off set L to goniometer radius R 0 for d=mm. Though some variation is seen, a linear approximation seems possible as a relation between σ L and L/R 0. Figure0 is a plotting of σ L against R 0 at L = 0.mm as a possible off set. The absolute value of σ L tends to increase rapidl as R 0 decreases below about 50mm. Stress Error caused b Collimator Misalignment Figure shows the changes in the stress error σ t caused b collimator translation with the change in the ratio t/r 0 for d=mm. It is seen that a linear approximation seems possible as a relation between σ t and t/r 0. Figure is a plotting of σ t against R 0 at t = 0.05mm as a possible misalignment. The absolute value of σ t tends to increase rapidl once R 0 decreases below about 50mm. Figure 3 shows the changes in the stress error σ κ caused b collimator rotation with the change in κ for d=mm. It is seen that a linear approximation seems possible as a relation between σ κ and κ. Figure 4 is a plotting of σ κ against R 0 at κ = 0.05deg.as a possible misalignment. σ κ is almost constant over a wide range of R 0. Stress Error caused b Misalignment of Detector It was found that the PSD misalignment has little effect on the stress error for Ω assembl stress measurement []. From this, it is believed that the misalignment of the PSD should have little effect on the stress error also for Ψ assembl stress measurement. Figure. Stress error σ t vs. ratio t/r 0 Figure. Stress error σ t vs. goniometer radius at =0.05mm

7 Copright JCPDS - International Centre for Diffraction Data 004, Advances in X-ra Analsis, Volume ISSN Figure 3. Stress error σ κ vs. κ Figure 4. Stress error σ κ vs. goniometer radius at κ=0.05deg Figure 5. Total error and component errors Figure 6. Total error vs. goniometer radius SUMMARY It was confirmed that the total error is the sum of the absolute values of the intrinsic stress, the stress error caused b specimen mis-setting and those b the misalignments of the collimator and detector []. Figure 5 shows the plotting of the total error against goniometer radius with its component errors for d=mm. Figure 6 is the plotting of the total error against goniometer radius for various collimator diameters. It is seen from the figures that about 50mm is the minimum size of the goniometer radius if we allow a stress error of 50MPa under the conditions of minimum commitment of 0.mm for specimen mis-setting, 0.05mm for collimator translation and 0.05deg for collimator rotation. The sstematic error is composed of the experimental error of the specimen mis-setting and the assembl errors of the analzer components, that is, collimator and detector. From Figure5, one of the main errors seems to be the collimator misalignment translation, therefore, it ma be possible to further reduce the size of the goniometer if we can appl more advanced technolog in the assembl of analzer components. REFERENCE.Goto and Y.Gong, Advances in X-Ra 45(00) Analsis, 57