RESIDUAL STRESS MEASUREMENT USING INDENTATION AND A RADIAL IN-PLANE ESPI INTERFEROMETER

Transcription

1 RESIDUAL STRESS MEASUREMENT USING INDENTATION AND A RADIAL IN-PLANE ESPI INTERFEROMETER Ricado Sutéio, Amando Albetazzi G. J, Felipe K. Amaal, Andeson Pacheco. Univesidade Fedeal de Santa Cataina, Cx. Postal 5053, CEP , Floianópolis, SC, Bazil, albetazzi@labmeto.ufsc.b ABSTRACT A adial in-plane electonic speckle patten intefeomete (ESPI) has been developed by the authos goup. This intefeomete is used in this pape to measue esidual stesses in combination with the indentation method. A semiempiical mathematical model is developed to quantify the esidual stesses. Seveal tests wee made in a specimen with diffeent levels of esidual stesses imposed by a mechanical loading. Empiical constants wee computed fom those tests and ae used in combination with the developed model to pedict the esidual stesses levels. The adial displacement field aound a contolled indentation pint is measued, pocessed and fitted to a mathematical model to pedict esidual stesses. Seies of tests wee designed and executed. Diffeent indentation tip geomety and diffeent loading conditions wee involved. This pape pesents the measuement pinciple, implementation details and esults of the pefomance evaluation. The tests pesented hee ae not complete since they ae esticted to only one mateial, oneaxis stess state, two indentation tip geomety and only one indentation foce, but they ae sufficient to encouage futhe development. Keywods: esidual stess, indentation, ESPI; adial intefeomete. INTRODUCTION A new kind of electonic speckle patten intefeomete (ESPI) has been developed by the authos goup using conical mios to achieve adial in-plane sensitivity []. This device has been used fo seveal applications on the field of expeimental mechanics [6],[7],[8]. In this pape this adial in-plane (RIP) is used in combination with the indentation method to measue esidual stesses. To measue esidual stesses a contolled indentation pint is applied to the suface of the specimen by a conical diamond and spheical tip. As a consequence, a local yielding is developed and the mateial on the specimen suface moves away fom the indentation pint. The amount of the adial displacement component aound the indentation pint is influenced by the level and diection of esidual stesses state acting in the specimen. In many applications, it is vey impotant to know the magnitude and diection of esidual stesses. Since esidual stesses ae vey difficult to be pedicted using analytical o numeical models, thei values ae vey often detemined by diect measuement by expeimental methods. One of the most widely used expeimental methods is the hole-dilling technique. This technique involves monitoing the stains poduced when a small hole is dilled into a stessed mateial. The dilled hole poduces a local elease of esidual stesses that ae measued by a special type of stain gauges. Those values ae used into an appopiate mathematical method to quantify the esidual stesses level [4]. Altenatively, it is possible to obtain qualitative infomation about the esidual stesses acting in a mateial by the indentation method [8],[9]. The main idea is to poduce a local indentation pint using a tool with a conical o spheical tip and a contolled level of loading o total enegy. In opposition to the hole dilling method, the indentation does not elease stesses but add moe stesses ceating a local plastic zone. The local plastic defomation is a function of the geomety of the indentation tip, mateial popeties, and is also stongly dependent of the magnitude and diection of the Eighth Intenational Symposium on Lase Metology, edited by R. Rodiguez-Vea, F. Mendoza-Santoyo, Poc. of SPIE Vol (SPIE, Bellingham, WA, 2005) X/05/$5 doi: 0.7/

2 esidual stesses initially pesent in the mateial. Pevious woks had epoted diffeent attempts to quantify esidual stesses using the indentation method fom: (a) hadness vaiation measuement; (b) elation between foce and indentation depth; (c) the final geometic shape of the indentation pint and (d) defomation o displacement measued aound the indentation pint [2],[3],[8] [9]. This behavio is vey difficult to be analytically and numeically modeled and it is not usually used fo quantitative measuement in a vey accuate way. Results of a quantitative evaluation of esidual stesses measuement using indentation is epoted in this pape. An indentation pint is poduced in the egion whee esidual stesses have to be measued. The in-plane adial displacement is measued aound it. Thee is a clea coelation between the esidual stesses level and the finge patten developed aound the indentation pint. A semi-empiical mathematical model was developed using the hole dilling model as a stating point. Additional paametes whee added to this model to best epesent the actual measued data. The possibilities to use those additional paametes to quantify the esidual stesses ae hee investigated. THE RADIAL INTERFEROMETER A double illumination adial in-plane intefeomete (RIP) using ESPI is used in this pape []. The basic pinciple is shown in Figue. This intefeomete was ecently incopoated in a potable and modula device fo esidual stesses measuement. The main optical element of the intefeomete is a conical mio, which is placed nea the specimen suface. Figue (a) shows a coss-section of the conical mio containing the mio axis, which displays two paticulaly chosen light ays fom a collimated illumination souce. Each light ay is eflected by the conical mio suface towads a point P ove the specimen suface, eaching this point symmetically. The illumination diections ae indicated by the unitay vectos na and nb and they have the same angle with espect to the suface nomal. The sensitivity diection is given by the vecto k obtained fom the subtaction of the two unitay vectos. Theefoe, in-plane sensitivity is eached in point P as well as in any othe point on the cicula double illuminated aea. The only exception is the cental point, thee is a singula point. CCD Camea Lase 45ºmio Collimated lase beam Conical mios Convegent lens PZT Specimen suface n A k P n B Conical mios Specimen suface (a) (b) Figue : Double illumination though the adial intefeomete. A pactical configuation of the adial in-plane intefeomete is shown in Figue (b). The lase light is expanded and collimated. The collimated beam is eflected towads the conical mio using a 45º tilted mio. The cental cicula window located at this mio has two main functions: (a) to avoid that the lase light to each diectly the measued 736 Poc. of SPIE Vol. 5776

3 suface to pevent tiple illumination, and (b) to povide a viewing window fo the camea. The conical mio is fomed by two pats with a small gap between them that is contolled by a PZT actuato do poduce phase shifting. Figue 2 shows two examples of phase diffeences pattens. Figue 2: Typical phase difeence pattens fo two diffeent esidual stesses levels. This adial displacement field is accuately measued by the RIP intefeomete. The phase patten is pocessed and few paametes ae extacted. An appopiate mathematical model was developed and is used to quantify the esidual stesses levels. RESIDUAL STRESS MEASUREMENT In ode to measue esidual stesses usually a small hole is dilled in the cente of the measued aea. A mathematical model developed by Kisch [4] based on the elastic solution fo an infinite plate, subjected to a unifom state of stesses, with a cylindical hole dilled all way though the plate thickness, is used. Accoding to that, the adial component of the displacement field ( ), developed by the hole dilling in pola coodinates is [],[5] : ( ρ)( σ + σ ) + ( ρ )( σ σ ) cos( 2θ β ) ( 2 2 ρ, θ ) = A B 2 ( ) whee the functions A(ρ) and B(ρ) ae given by: ( + υ) = ρ B ρ ) = 4ρ ( + υ) A( ) 2 E ρ 0 3 [ ] ( 0 ρ 2 E ( 2) and,, θ ae pola coodinates; o is the dilled hole adius; ρ is the nomalized adius (ρ = 0 / ); σ, σ 2 ae the pincipal esidual stesses (maximum and minimum espectively); E is the mateial s Young modulus; υ is the mateial s Poisson atio; β is the pincipal diection angle. Figue 3(a) shows a typical adial phase diffeence patten fo a one-axis esidual stess state due to hole dilling method combined with ESPI. The hole diamete is 0.8 mm and the esidual stess is about 200 MPa. Poc. of SPIE Vol

4 (a) (b) (c) (d) Figue 3: Typical phase pattens of the adial displacement fields component: (a) fo Holing Dilling Method (compute simulated), (b) in a stess-fee mateial afte indentation, (c) with 200 MPa stess afte indentation, and (d) phase diffeence: (c - b) In opposition to the hole dilling method, the indentation pint does not elease the stesses, but add moe stesses, poducing a local yielding. As a consequence, a pemanent displacement field is poduced aound it. In a stess-fee mateial, this pemanent displacement field is axi-symmetical and epeatable if the indentation tip geomety, indentation loading and mateial popeties ae constant. If mechanical o esidual stesses ae pesent in the mateial pio the indentation, the pemanent displacement field is affected in a way that depends on the esidual stesses levels. Figue 3(b) to (d) show some expeimental esults. Those phase diffeence pattens ae elated to the adial displacement component, measued by the adial in-plane ESPI intefeomete. The indentation was made using a 20 o conic tip. Figue 3(b) shows the adial displacement in a stess-fee mateial. Figue 3(c) shows the adial displacement in a mateial with a one-axis 200 MPa stesses field aligned about 30 with the hoizontal axis, and Figue 3(d) shows the esulting phase diffeence between the two pevious phase pattens. Compaing Figue 3(a) and Figue 3(d) it is possible to see that, afte emoving the pemanent displacement field elated to an indented stess-fee mateial, thee is a stong similaity between them. MATHEMATICAL MODEL In ode to find a model to measue esidual stesses the authos appoach uses this similaity. A mathematical model was deived stating fom equation ( ). It is noted that the tem that depend on cos(2θ) is elated to the pincipal stesses diffeence (σ - σ 2 ) and the tem independent of θ is elated to the pincipal stesses sum (σ + σ 2 ). The adial component of the displacement field ( ), is hee modeled in pola coodinates by the following equation: K K2 K3, θ ) = + + cos 2 ( 3 whee K, K2 and K3 ae unknown functions given by: ( 2θ β ) = f [( σ σ ),, υ] K = g[ ( σ σ ),, υ] K = h[ ( σ σ ),, υ] K + 2 E 2 2 E 3 2 E ( 3) ( 4) It was expeimentally veified that the adial displacement on a stess-fee mateial can be easonably descibed using the equation poposed by Giannakopoulos & Suesh [2] : υ ( ) = C E ( υ ) υ ( 5) 738 Poc. of SPIE Vol. 5776

5 Whee, C is a constant that depends upon the indentation foce, tip geomety and mateials popeties. This constant can be detemined by fitting this model to expeimental data fom the adial displacement field measued afte indenting a stess-fee specimen of the same mateial in identical indentation conditions. So, ou model to elate esidual stess with the adial displacement field is given by: (, θ ) = K K K cos( 2 2 ) + 3 θ β K4 υ ( 6) Whee K 4 depends of C, E and υ, as it is clea in equation ( 5), and can be detemined a pioi in a stess-fee specimen. In a ecent investigation [6] the authos vefied that the tem K 3 / 3 in equations ( 3) and ( 6) is negligible if compaed with K 2 /. So, equations ( 3) and ( 6) can be simplified and ewitten as: K K, θ ) = + cos 2 2 ( ( 2θ β ) ( 7) K (, θ ) = K ( 2θ β ) 2 + cos 2 + K 4 υ ( 8) Equation ( 7) is applied when the adial displacement field fom a stess-fee specimen is subtacted fom the adial displacement field measued fo a specimen with esidual stesses. By the othe hand, the authos wish to investigate if equation ( 8) can be successfully used to quantify esidual stesses without emoving the adial displacement field fom a stess-fee specimen. So, the constant K 4 must be fitted using the actual total adial displacement field. In ode to ty to keep those equations mateial independent, they wee again ewitten as equations below. This independence must be expeimentally veified in a futue wok. ( + υ) 2, θ ) = ψ + ψ cos 2 2 E E ( 2 ( + υ) ( 2θ β ) 2 υ (, θ ) = ψ + ψ 2 cos( 2θ 2β ) + ψ 2 E E E υ 3 ( υ ) Whee ψ, ψ 2 and ψ 3 ae constants that depend upon the indentation foce and tip geomety. Additionally, ψ is also a function of σ + σ 2 and ψ 2 is a function of σ - σ 2. ( 9) ( 0) Poc. of SPIE Vol

6 EXPERIMENTAL SETUP Figue 4(a) shows the potable ESPI adial in-plane intefeomete used in this wok. The system is composed by a diode lase souce with 658 nm wavelength, a 30 conical mio and a CCD camea. The illuminated egion is about 0 mm in diamete (Figue 4b). The indentation module always applied the same level of constant impact enegy. Two diffeent tip geometies wee used: a 20 o conical diamond tip and a 2.5 mm diamete spheical tungsten cabide tip (see Figue 5). It is vey difficult to evaluate the measuement pefomance of a esidual stesses measuement device. Since thee is no available efeence standad, o even a efeence measuement system, the authos applied a known mechanical stesses level in a peviously annealed specimen. That appoach mechanically simulates a known esidual stesses state and it was sufficiently good fo a fist measuement pefomance evaluation. In ode to ovecome this difficulty, known esidual stesses fields wee mechanically simulated by the device shown in Figue 4(a). Basically, a long esidualstesses fee specimen is loaded by taction though 6 bolts connected to a U shaped stuctue. The specimen is a ectangula AISI 020 steel ba with appoximate dimensions of 3 x 50 x 3000 mm (Figue 6b). The long specimen was instumented with ten stain gages to monitoing the actual tension level applied to the specimen in ode to guaantee unifom loading and to povide a efeence value fo it. The specimen size is long enough to accommodate ove 300 measuement points. The potable adial in-plane intefeomete was clamped to the loading device in such a way that its measuement axis was aligned in about 30 with the loading diection. (a) Hole dilling module, univesal base, and adial in-plane intefeomete. (b) Measuement aea with 0 mm in diamete Figue 4: Potable ESPI adial in-plane intefeomete used in this wok. Figue 5: Indentation module with a 20 conical diamond o a 2.5 mm diamete spheical tungsten cabide tips. 740 Poc. of SPIE Vol. 5776

7 stain gage Top View Bottom View Figue 6: Loading system, specimen, and stain gages configuation. EXPERIMENTAL RESULTS A set of measuement tests was designed and executed with nine diffeent stesses levels. Fo each stess level the same long specimen was indented thee times. Both 20 conic tip and 2.5 mm spheical indentation tips wee used applying about the same amount of enegy given by a mechanical impact of a sphee diven by a sping at constant compession ate. So, a total of six images wee acquied fo each loading level. Figue 7(a) to Figue 7(d) show diffeent expeimentally obtained images. A 229 MPa one-axis stesses filed was applied in the long specimen. The two kinds of indentation tips wee used. Test at 229 MPa, 20 conical tip Test at 229 MPa, 2.5 mm spheical tip (a) total phase diffeence afte indentation in a loaded mateial. (b) phase diffeence: loaded mateial minus a stess-fee mateial. (c) total phase diffeence afte indentation in a loaded mateial. Figue 7: Typical test esults fo two indentation tips. (d) phase diffeence: loaded mateial minus a stess-fee mateial. Poc. of SPIE Vol

8 The measuement sampling egion was delimited by two concentic cicles with adius.5 and 4.5 mm. A pola mesh of 2 x 360 = 4320 sampling points was used in all calculations. In each case the efeence values wee given by the stain gauges with an uncetainty anging fom 5 to 9% with 95% confidence level [7]. In this pape the emphasis was given to investigate if is it possible to successfully quantify the esidual stesses level without emoving the adial displacement field of stess-fee indentation. This goal is motivated by pactical aspects. Table and Table 3 pesent measuement esults. The fist column is the efeence stess level. Coolum two shows the elative stess elated to mateial s yield point. The emaining columns pesent the paametes ψ, ψ 2 and ψ 3, fitted by least squaes fom the adial displacement data fo each image. Figue 8 and Figue 9 plot the dependence between ψ, ψ 2 and the efeence stess level. Two quadatic polynomials, named Η(σ +σ 2 ) = ψ x (σ + σ 2 ) and Γ(σ σ 2 ) = ψ 2 x (σ - σ 2 ), wee fitted to the expeimentally computed values, as show equations ( ) to ( 4). In this paticula case both sum and diffeence of pincipal stesses ae equal since σ 2 = 0. ψ 3 did not show a coelation with the applied stesses. Its value is moe coelated to the indentation enegy that, ideally, should be kept constant. Table : Paametes ψ tough ψ 3 detemined using least squae fitting fo 20 o diamond cone tip. σ ef σ ef / σ y ψ ψ 2 ψ 3 [%] % % % % % % % % % σ ef ψ Figue 8: Indentation paamete vesus efeence esidual stess fo 20 o diamond cone tip. ψ2 Quadatic egession shows a quite good ageement with expeimental data. The coelations of H and Γ functions ae bette than 0.95 (R 2 ). Fom those pedicted values, σ and σ 2 can be computed by: σ = (Η + Γ) / 2 and σ 2 = (Η - Γ) / 2. The esulting computed values fo σ and σ 2 ae pesented in Table 2 and Table 4 espectively. Η(σ+σ2) = ψ ψ (R 2 = ) ( ) Γ(σ σ2) = ψ ψ 2 (R 2 = ) ( 2) 742 Poc. of SPIE Vol. 5776

9 Table 2: Pedict stesses values fo a 20 o diamond cone tip. Ref. Value Relative to Yield Point Pedict Values Diffeence Eo σ ef σ σ ef / σ σ 2 σ ef -σ y (σ ef -σ ) / σ ef % % % % % % % % % % % % % % % % % Table 3: Paametes ψ tough ψ 3 detemined using least squae fitting fo 2.5 mm diamete tungsten cabide spheical tip σ ef σ ef σ ef / σ y ψ ψ 2 ψ 3 [%] % % % % % % % % % ψ ψ2 Figue 9: Indentation paamete vesus efeence esidual stess fo 2.5 mm diamete tungsten cabide spheical tip. Η(σ+σ2) = ψ ψ (R2 = ) ( 3) Γ2(σ-σ2) = ψ ψ 2 (R2 = ) ( 4) Poc. of SPIE Vol

10 Table 4: Pedict stesses values fo a 2.5 mm diamete tungsten cabide spheical tip. Ref. Value Relative to Yield Point Pedict Values Diffeence Eo σ ef σ ef / σ y σ σ 2 σ ef -σ (σ ef -σ ) / σ ef % % % % % % % % % % % % % % % % % CONCLUSIONS This pape investigates the possibility to quantify esidual stesses combining the indentation method with a adial inplane ESPI intefeomete. The classical blind hole dilling mathematical model was modified by intoducing thee unknown paametes: ψ, ψ 2 and ψ 3. A set of well contolled expeiments was designed to expeimentally evaluate this measuement pinciple in only one mateial. Expeimental data wee fitted to this model and the esulting paametes wee analyzed. A quadatic elationship was found between the esidual stesses levels and the paametes ψ and ψ 2. This elationship was used to pedict esidual stesses fom the measued adial displacement field. As a peliminay evaluation the esults wee consideed vey encouaging. It is clea that ψ and ψ 2 in equations (9) and (0) ae vey good choices to be coelated with the esidual stesses sum and diffeences espectively. The tem ψ 3 seams to be moe coelated to the indentation enegy, and pehaps with mateial hadness, what can be vey useful to veify the epeatability of the indentation device o the mateial s hadness homogeneity. Both conic and spheical indentation tips shows about the same behavio. The values fo ψ and ψ 2 ae obviously diffeent fo each tip geomety. Diffeent conical angles o sphee adius must poduce diffeent values fo ψ and ψ 2. Fo the two tips investigated in this pape it was noted that the sensitivity fo the spheical tip is highe than fo the conical tip. The expeimental data dispesion was smalle fo the spheical tip. In addition, the amount of damage in the measued mateial s suface is smalle if a spheical tip is used. The numeical esults obtained in this investigation ae vey encouaging. It seams that is possible to quantify the esidual stesses level without subtacting the adial displacement field of an indented stess-fee mateial. Best esults wee found fo stesses levels highe than 20% of the mateial s yield stess. In this wok only stesses levels below 80% of the yield stess wee investigated. The maximum deviation between the pedicted and efeence stesses values ae of about 8% fo the spheical tip in the ange of 20% to 80% of the yielding stesses. Fo the conical tip the diffeence was about the same, except fo an outlie point the eached a diffeence of 7%. Since only a one-axis stesses state was applied to the specimen, it was not possible to individually veify the dependence of ψ and ψ 2 with the sum and diffeence between the pincipal stesses espectively. Although, since the absence of pincipal stesses diffeence poduces only an axis-symmetical phase diffeence patten, it is clea that the pincipal stesses diffeence is esponsible fo intoducing a dependence in tems of cos(2θ), what is only elated to ψ 2. So, ψ is not affected by the pincipal stesses diffeences. 744 Poc. of SPIE Vol. 5776

11 It has to be clea that these esults ae not definitive. Only one mateial was involved, only one-axis esidual stesses induced was analyzed, only one enegy level was used fo the indentation. Futhe wok will be focused on extending those conditions. Diffeent mateials and diffeent esidual stesses states ae cuently been investigated. The final goal of this eseach wok is to develop a potable esidual stesses measuement unit. The authos believe that the combination of indentation and this adial in-plane ESPI intefeomete can be the basis fo a vey pactical device. ACKNOWLEDGMENTS The authos would like to thank the institutions that have been suppot this wok: ANP-PRH34, CTPETRO, PADCT, CNPq/RHAE and CAPES. REFERENCES [] ALBERTAZZI JR, A.; KANDA, C; BORGES, M. R.; HREBABETZKY, F. - Potable Residual Stesses Measuement Device Using ESPI and a Radial In-Plane Intefeomete - Lase Metology fo Pecision Measuement and Inspection in Industy; Albetazzi J., A.; Eds., Poc. SPIE, v. 4420, p. 2-22, Sep [2] GIANNAKOPOULOS, A. E.; SURESH, S. - "Indentation of Solids with Gadients in Elastic Popeties. Pat I: Point Foce" - Jounal of Solids and Stuctues, v.34, n.9, p , 997. [3] GIANNAKOPOULOS, A. E.; SURESH, A. - "Theoy of Indentation of Piezoelectic Mateials" - Elsevie Science Ltd., Acta Mateialia, v.47 n.7 p , 997. [4] LU, JIAN - "Handbook on Measuement of Residual Stesses" - The Faimont Pess, SEM - Society fo Expeimental Mechanics, GA, USA, 996. [5] MAKINO, A.; NELSON D. - Residual-Stess Detemination by Single-Axis Hologaphic Intefeomety and Hole Dilling. Pat I: theoy - Exp. Mech. 34, (994). [6] SUTERIO, R; ALBERTAZZI JR., A. G.; AMARAL, F. K.; - Residual Stess Measuement using Indentation and a Radial ESPI Intefeomete Recent Pogess - ICEM2-2th Intenational Confeence on Expeimental Mechanics, Politecnico di Bai, Italy, 29 August - 2 Septembe, 2004 [7] SUTERIO, R.; ALBERTAZZI JR., A. G.; PACHECO, A.; FERREIRA, R. P. - "Metology Analysis of a Simulation Device fo Residual Stess" (in Potuguese) - METROLOGIA 2003 Bazilian Congess of Metology, SBM Bazilian Society of Metology, Recife, PB, Bazil, sep. -5, [8] SUTERIO, R; ALBERTAZZI JR., A. G.; CAVACO, M. A. M. - Peliminay Evaluation: The Indentation Method Combined with a Radial Intefeomete fo Residual Stess Measuement - SEM Annual Confeence and Exposition on Expeimental and Applied Mechanics, SEM - Society fo Expeimental Mechanics, Chalotte, Noth Caolina, USA, June 2-4, [9] UNDERWOOD, J. H. - "Residual Stess Measuement using Suface Displacements aound an Indentation" - Expeimental Mechanics, p , Sep., 973. Poc. of SPIE Vol