ABSOLUTE STRESS DETERMINATION FROM ANGULAR DEPENDENCE

Transcription

1 ABSOLUTE STRESS DETERMINATION FROM ANGULAR DEPENDENCE OF ULTRASONIC VELOCITIES IN ORTHOTROPIC MATERIALS A. D. Degtyar and S. I. Rkhlin The Ohi State University Nndestructive Evaluatin Prgram Clumbus, Ohi 4321 INTRODUCTION The characteristic dependence f ultrasnic velcit:t n stress has fr a lng time been thught a prmising fr making residual stress measurements in materials. Fr a review f thery and experimental methds the reader is referred t [1-3]. Different wave mdes were prpsed fr stress determinatin. Thrugh-thickness average stress can be btained using lngitudinal wave data at nrmal incidence [4, 5] r transverse waves with different plarizatins [6-8]. Near-surface stresses can be determined using Rayleigh waves [9-11] and surface-skimming lngitudinal [12] and SH [12, 13] waves. It is nw well understd that there are cmplicatins in the practical utilizatin f the ultrasnic stress measurement technique. First f all velcity changes due t stress are small (typically belw.1 %) and very precise measurements f the time delay and travelling distance are required. Hwever the majr difficulty is the necessity f separating the effects f texture (anistrpy) and stress n the measured ultrasnic velcity. In mst cases materials under cnsideratin have unknwn anistrpy and even if the anistrpy is small its effect cannt be neglected in stress measurements. Several methds have been prpsed t vercme this difficulty. Thmpsn et a1. [12] cnsider the difference f tw SH-waves prpagating in the plane f the material in rthgnal directins. They shwed that fr this difference the effect f anistrpy is reduced by an rder f magnitude and fr small anistrpy it can be neglected. Als they demnstrate that the stress and anistrpy terms have different angular dependences which can be used fr their separatin. King and Frtunk [6] cnsidered bliquely incident SH-waves. Again a frmula was derived with separate anistrpic and stress terms. Man and Lu [14] generalized bth techniques and demnstrated the applicability f the ultrasnic methd fr stress measurements in materials which had undergne a cmplicated (pssibly plastic) histry f lading and unlading. Generalizing Thmpsn's results, they btained relatins between certain cmbinatins f stresses and velcities which d nt include elastic cnstants and thus are independent f initial texture and change f micrstructure due t plastic defrmatins. In this paper we will shw that stresses in a material can be fund simultaneusly with stress-dependent elastic cnstants frm the inversin f the Christffel equatin, using as input measured angular dependencies f ultrasnic velcities. The methd is applicable fr determinatin f bth applied and residual stresses fr materials f the Review f Prgress in Quantitative Nndestructive Evaluatin, Vl. 14 Edited by D.O. Thmpsn and D.E. Chimenti, Plenum Press, New Yrk,

2 mst general lading histries, in line with the discussin f Man and Lu [14J. Simulatin results are presented fr rthtrpic media t validate this technique. We will shw that the abslute errr in stress determinatin using this technique is independent f the degree f anistrpy and the stress level is defined nly by the accuracy f wave velcity measurement. CHRISTOFFEL EQUATION IN STRESSED MEDIA T describe the wave prpagatin in a prestressed medium we use the apprach prpsed by Man and Lu [14J. The prestressed cnfiguratin is the nly reference cnfiguratin in this apprach and the initial stress is included in the cnstitutive equatin: S = :E + C : E + H:E (1) where S is the first Pila-Kirchhff stress, :E is the initial stress, E is the elastic strain due t wave prpagatin, H is the displacement gradient and C is the furth rank tensr f stress dependent elastic cnstants. The equatin fr small elastic mtin due t wave prpagatin superimpsed n the prestressed state is: (2) where u is the displacement vectr. Making use f Eq. (1), Eq. (2) can be written in cmpnent frm: (3) The stress (J"ij can be bth applied and residual since there is n restrictin that the resulting defrmatin be elastic. Nw assuming that the material and lcal (ver the size f the transducer) stress are hmgeneus and using a plane wave slutin fr u u = peik(ll.x-vl) (4) where p is the unit vectr in the directin f particle mtin, k is the wave number and n is the unit vectr in the directin f wave prpagatin, ne has the Christffel equatin fr an anistrpic material under stress: (5) Eq. (5) was derived by Tkuka and Iwashimizu [15J and used by Thmpsn et al. [12J and King and Frtunk [6J. Man and Lu [14J reexamined cnstitutive equatins and extended the applicability f Eq. (5) t general types f lading histry, including plastic defrmatins. Eq. (5) has a cnvenient frm fr ur further use. The difference frm the Christffel equatin in an unstressed medium is the appearance f stress dependent elastic cnstants C ijkl instead f secnd rder elastic cnstants Cijkl and the additin f the stress 19

3 principal axis Os axis (a) plane 1-2 plane / ~~ 2 3 Fig. 1. (a) Principal stresses alng symmetry axes in the 1-2 plane and wave prpagatin in the 1-3 and 2-3 symmetry planes; (b) Principal stresses ff symmetry axes in the 1-2 plane and wave prpagatin in the plane f stresses. (b) term O"ilninl in the diagnal elements f (5). The effect f the stress term n the wave velcity can be separated ff. T validate ur apprach by cmputer simulatin and t create a synthetic set f experimental data we need t be able t cmpute stress dependent elastic cnstants Gijkl as a functin f stress. This can be dne fr a finitely stressed hyperelastic bdy using secnd and third rder elastic cnstants. While the velcity data set thus generated is fr hypere!astic material, ur recnstructin prcess, using Eq. (5), des nt have this limitatin and has the same applicability as Eq. (5) itself. DEPENDENCE OF ULTRASONIC VELOCITY ON STRESS. DETERMINATION OF STRESS We will cnsider tw cases f principal stress rientatin with respect t material axes f symmetry. When principal axes cincide with symmetry axes f the material, we analyze the prpagatin f quasilngitudinal (QL) and quasi shear (QT) waves in symmetry planes 1-3 and 2-3 perpendicular t the symmetry plane f the material 1-2 (Fig. 1a). In the general plane stressed state, i.e. when principal directins deviate frm symmetry axes, we utilize waves prpagating in the plane f stress (1-2 plane) and plarized in the same plane. Principal Stresses alng Symmetry Axes When principal axes cincide-with symmetry axes f an rthtrpic material, the stresses d nt change the symmetry and the number f independent Gij in the stressed slid is the same as the number f secnd rder elastic cnstants C ij. Let us cnsider plane stress in the 1-2 plane with the shear stress cmpnent "12 being zer and wave prpagatin in the 1-:3 r 2-3 symmetry planes (see Fig. 1a). Fr wave prpagatin in the 1-3 plane the Christffel equatin (5) has the fllwing frm: 191

4 - (2 2) Gll - pv -O"l1S 22 - (pv 2 - "11 S2 ) G (2 pv - O"llS 2) = (6) where On = 11S2 + 55C2 j 13 = (13 + (55)SCj S = sin()j 33 = 55S2 + 33C2 j 22 = 66S2 + 44C2 j C = cs (). (7) The angle () is between the directin f prpagatin and the 3-axis (Fig. 1a). The equatin (6) can be decupled and simple clsed-frm slutins fr QL and QT wave velcities are: (8) As we can see the QL and QT wave velcities in this symmetry plane depend nly n five parameters, namely 11,33,13,55 and O"n. Similarly, QL and QT wave velcities in the 2-3 plane depend n fur stress dependent elastic cnstants 22,33,23,44 and the stress cmpnent "22. Frm the angular dependence f the measured velcities we can determine these five unknwn parameters fr each plane separately. We emply the least squares methd fr the minimizatin f the sum f squared deviatins between experimental and calculated, using Eq. (8), velcities cnsidering effective elastic cnstants and stress cmpnent as variables in multidimensinal space: (9) where n is the number f parameters t be defined, m is the number f velcity measurements fr different directins, and ve and VC are the experimental and calculated phase velcities, respectively. This apprach was used by Chu and Rkhlin [17] and Chu et al. [18] t find secnd rder elastic cnstants frm velcity data in symmetry and nn symmetry planes fr rthtrpic materials. The flw chart f the inversin prcedure fr the 1-3 plane is presented in Fig. 2a. Fur effective elastic cnstants (11,33,13,55) and ne stress cmpnent (O"n) frm a 5-dimensinal space f unknwn parameters which determine the velcities f the QL and QT waves. The functin t be minimized in this space is defined by the sum f squared differences between experimental and calculated velcities at different prpagatin angles. Thus in the inversin prcedure Gij and stress are cnsidered t be 192

5 ~ INITIAL GUESS Gl~), G~g), Gig), G~g), q\~) D COMPUTE VQL AND VQT ( V;C ) FROM CHRISTOFFEL EQUATION MINIMIZE ~ EXPERIMENTAL VELOCITY DATA IN 1-3 PLANE ( v;e ) ~ ~ we _ v;c)2 2i=1 t l D RECONSTRUCTED VALUES (a) , EXACT SOLUTION... RESULT OF THE INVERSION CALCULATED THEORETICALLY FROM THE VALUE OF STRESS FOUND ,---~ NUMBER OF ITERATIONS (b) Fig. 2. (a) Scheme f the inversin prcedure in 1-3 plane; (b) Results f the inversin fr Gll after each iteratin. independent parameters althugh actually Gij depend n stress. T validate the prcedure we checked numerically that the recnstructed values fr G ij have the crrect dependence n the recnstructed value f stress (i.e. actual value f stress). Fig. 2b shws with triangles the results after each iteratin during determinatin f Gn using nnlinear least squares minimizatin. The circles shw the values f Gu calculated exactly using the stress (ltn) determined at this iteratin step and the secnd and third rder elastic cnstants used fr creatin f the synthetic set f velcity data. We see that starting with the third iteratin bth results fr Cn are very clse t each ther and t the riginal value (shwn by the straight slid line). This shws that the crrect dependence between Gn and ltn hlds fr values determined frm inversin althugh their relatin is nt specified in the inversin algrithm. When the recnstructin frm velcity data is perfrmed fr bth the 1-3 and 2-3 planes, bth stress cmpnents ltn and lt22 and seven f nine effective elastic cnstants (all except G12 and ( 66) can be fund. If lt12 =I then the slutin f Christffel equatin in the 1-3 plane cannt be presented in the frm f Eq. (8) because this shear stress cmpnent alters the material symmetry and the 1-3 plane becmes the nnsymmetry plane. As has been shwn in [18] the recnstructin frm a nnsymmetry plane is unstable. Our calculatins shw that the effect f even very small scatter in the velcity data destabilizes the inversin prcess fr stress. Mre effrt will needed t develp stable algrithms fr this case. Principal Stresses Off Symmetry Axes Nw let us cnsider an arbitrary plane stress state in the 1-2 plane (Fig. 1b), ltu, lt22 and lt12 are the nly nnzer stress cmpnents. In this case the symmetry reduces t mnclinic. G 16, G 26, G 36, G 45 are stress-induced effective elastic cnstants. They depend nly n the shear stress cmpnent lt12. Instead f cnsidering wave prpagatin in planes rthgnal t the stress plane, we cnsider the wave prpagatin in the plane f the stresses (Fig. 1b). This plane still remains the plane f symmetry and the Christffel equatin (5) can again be decupled. The slutins fr quasilngitudinal and quasishear (plarized in the stress plane) waves have the fllwing frm: 193

6 On + 22 J(Gn - G22 )2 + 4G~ (<1n - (122)S + < <112 SC On + G22 J( Gn - G 22 )2 + 4G~2 ( ) <1n - <122 S + < <112 SC (1) where On = Cn s2 + C66C2 + 2CI6 SC; 22 = C66S2 + C22C2 + 2C26 SC; S = sino; C = cso. (11) We use the inversin prcedure in the space f unknwns C11, C22, C12, C66, C16, C26, <1n, <122, <112. Frm (1) we can see that the difference f nrmal stresses <1n - <122 and shear stress <112 have different angular dependences which we can determine using the inversin prcedure described abve. SIMULATION RESULTS T validate the technique described cmputer simulatin tests were perfrmed fr materials with different degrees f anistrpy. T btain the synthetic set f data pints we assume that the secnd and third rder elastic cnstants are knwn. Fr a given stressed state, Cij can be fund assuming that the material is hyperelastic and the velcities can be determined frm the Christffel equatin (5). Different levels f randm scattering are intrduced int this velcity data and this data set is used fr the recnstructin f effective elastic cnstants and stresses. The values f stress and Cij fund are cmpared with thse riginally selected in the simulatins t determine the precisin f the prcedure. First, cmputatins were made fr principal stresses alng symmetry axes in the 1-2 plane and bliquely incident quasilngitudinal and quasishear waves. It was bserved that the stress cmpnent <122 des nt affect the results f the recnstructin f <1n frm velcity data in the 1-3 plane. This happens because <122 affects the velcity data in the 1-3 plane nly thrugh the effective elastic cnstants and in the recnstructin prcess we cnsider <111 as an independent unknwn. We present here the results fr the case f uniaxial stress alng the I-axis and velcity data in the 1-3 plane. The angular range fr we take t be clse t that experimentally available using the duble-thrugh transmissin technique [19] fr metals. Fr a quasilngitudinal wave it is apprximately - 6 and fr a quasishear wave 3-7. The ttal number f synthetic data pints is 45. As an initial guess fr the effective elastic cnstants we take the secnd rder elastic cnstants fund assuming n stress [17]. The initial guess fr stress was set t be zer. The first material cnsidered was textured aluminum (anistrpy was 1 %). The third rder elastic cnstants were taken frm' [16] fr istrpic aluminum. Results f the recnstructin are presented in Table 1 a. Different stress levels and different scattering are cnsidered. The same cmputatins were als made fr a graphite/epxy cmpsite which exhibits strngly anistrpic prperties. It was assumed that this material is rthtrpic. The third rder elastic cnstants were chsen arbitrarily assuming nly that they are apprximately ne rder f magnitude higher than the secne! rder elastic cnstants. It was bserved that a different chice f third rder elastic cnstants did nt affect the accuracy f the recnsructin prcess. The recnstructin results are presented in Table 1 b. Cmparing these results we cnclude that the precisin f the recnstructed stress values des nt depend n the degree f anistrpy. Fr each scattering level we have 194

7 Table 1. Results f stress recnstructin fr (a) textured aluminum and (b) graphite/epxy cmpsite samples fr different stress levels and scattering. Original value Standard deviatin f the recnstructed (Tn, MPa value frm the riginal, MPa n scatter.1 % scatter.5 % scatter.1 % scatter a b a b a b a b apprximately the same abslute errr in stress determinatin regardless f the stress level. T cnsider the effect f errr in the velcity data we intrduced different levels f scattering in synthetic velcity data using a randm functin generatr. We changed the scattering frm zer t.1 % and recnstructed the knwn value f stress fr anistrpic aluminum. The results are presented in Fig. 3a fr the case f applied tensile stress equal t 1 MPa. Fr each scattering level we made 5 runs f the recnstructin prgram with different synthetic data. We see that upn increasing the scattering frm.1 % t.1 % the accuracy drps by a factr f rughly ten. Fig. 3b represents the distributin f recnstructed values f stress fr the case f applied stress 1 MPa and scattering.2% (ttal number f runs = 1). The distributin btained can be apprximated by a nrmal distributin with deviatin (T = S the results f the thery f prbability fr the nrmal distributin are applicable in ur case. SUMMARY An apprach fr abslute stress determinatin frm angular dependence f ultrasnic velcities has been described. It is based n inversin f the Christffel equatin in a multidimensinal space frmed by effective elastic cnstants and stress cmpnents. The technique is applicable fr determinatin f bth applied and residual stresses. In the case when principal plane stress directins cincide with symmetry axes f the rthtrpic material, ne can recnstruct principal stresses frm the angular dependence f quasilngitudinal and qua.sishear waves measured in these planes. When the principal axes d nt cincide with symmetry axes the angular dependence f quasilngitudinal and quasishear waves prpagating and plarized in the plane f stresses can be used t determine the difference between nrmal stress and shear stress cmpnents. Numerical simulatin has been perfrmed using a set f synthetic velcity data frm all three symmetry planes. The recnstructed values are nt affected by the selectin f the initial guesses. The results shw that the abslute errr in stress determinatin des nt depend n the stress level and the degree f anistrpy f the material. 195

8 15!i 14 vi 13 Vl ~ 12 t- Vl 11 ~ 1 U ~ 9 t-!q 8 results fr different experimental data sets = standard deviatin I i I 8 8 u 7 w a:: 6 I SeA HERING (percent) (a) r t=.7 Vl z w.5 '.2 - nrmal distributin STRESS, MPa Fig. 3. (a) Results f stress recnstructin fr anistrpic aluminum under applied tensile stress f 1 MPa frm velcity data with different scattering; (b )Distributin f recnstructed values f stress fr different synthetic data. Applied tensile stress is 1 MPa and scattering is.2%. (b) REFERENCES 1. Y. H. Pa, W. Sachse and H. Fukuka, in Physical Acustics, Vl. 17, eds. W. P. Masn and R. N. Thurstn (Academic, New Yrk, 1984), Chap J. H. Cantrell and K. Salama, Internatinal Materials Reviews 36, 125 (1991). 3. R. B. Thmpsn, W.-Y. Lu and A. V. Clark, Jr., in SEM Mngraph f Techniques fr Residual Stress Measurement, (in preparatin), Chap J. J. Dike and G. C. Jhnsn, J. Appl. Mech. 57, 12 (199). 5. G. S. Kin, J. B. Hunter, G. C. Jhnsn, A. R. Selfridge, D. M. Barnett, G. Hermann and C. R. Steele, J. Appl. Phys. 5, 267 (1979). 6. R. B. King and C. M. Frtunk,.I. Appl. Phys. 54, 1339 (1983). 7. A. V. Clark,.I. C. Mulder, R. B. Migngna and P. P. Delsant, in Review f Prgress in Quantitative NDE, Vl. 5B, eds. D.O. Thmpsn and D.E. Chimenti, (Plenum, New Yrk, 1986) p C. M. Sayers and D. R. Allen,.I. Phys. D 17, 1399 (1984). 9. Y. Iwashimizu and O. Kbri, J. Acust. Sc. Am. 64, 91 (1978). 1. P. P. Delsant and A. V. Clark, Jr.,.I. Acust. Sc. Am. 81,952 (1987). 11. G. T. Mase and G. C. Jhnsn,.I. Appl. Mech. 54, 126 (1987). 12. R. B. Thmpsn, S. S. Lee and J. F. Smith,.I. Acust. Sc. Am. 8,921 (1986). 13. R. B. Thmpsn, S. S. Lee and J. F. Smith, Appl. Phys. Lett. 44,296 (1984). 14. C.-S. Man and W. Y. Lu, J. Elasticity 17, 159 (1987). 15. T. Tkuka and Y. Iwashimizu, Ir.t. J. Slids Structures 4, 383 (1968). 16. Landlt-Brnstein, in Numerical Data and Functinal Relatinships in Science and Technlgy, Vl. III/ll, (Springer, New Yrk,1979) 17. Y. C. Chu and S. 1. Rkhlin, J. Acust. Sc. Am. 95, 213 (1994). 18. Y. C. Chu, A. D. Degtyar and S. I. Rkhlin, J. Acust. Sc. Am. 95,3191 (1994). 19. S. 1. Rkhlin and W. Wang,.I. Acust. Sc. Am. 92, 333 (1992). 196