Ghostscript wrapper for D:\Digitalizacja\MTS85_t23z1_4_PDF_artyku³y\mts85_t23z3_4.pdf M E C H A N I K A TEORETYCZNA 1 STOSOWANA 3 - 4, 13 (1985) THEORETICAL AND EXPERIMENTAL STUDY OF MICRO-CRACKING INDUCED BY THE RAYLEIGH WAVE J. F. CARDENAS-GARCIA and D. C. HOIXOWAY Departme nt of Mechanical Engineering Univetsit y of Maryland College Park, MD 20742 Abstract This paper examines the extension of surface micro-cracks induced by a surface or Rayleigh wave (R-wave). This problem is examined both theoretically and experimentally. The theoretical approach involves a full-field reappraisal of the Lamb solution for a surface wave propagating in a homogeneous, isotropic, elastic, two-dimensional material, for the cases of plane strain and plane stress. Using the Griffith-Irwin energy release rate fracture criterion for cracks under combined Mode I and Mode II loading, a prediction is made of the path and final length of the surface micro-crack extension produced by the R-wave. Predictions of the crack extension direction are also obtained using the maximum norma! stress fracture criterion. The experimental approach uses dynamic photoelasticity to observe the isochromatic patterns associated with a R-wave propagating along the narrow edge of a transparent, birefringent plate, examining in detail the process of crack extension. When the theoretically and experimentally obtained results are compared, reasonable agreement is obtained. Introduction A previous paper [1] dealt with a detailed appraisal of the solution by Rayleigh for the surface wave phenomena that bears his name. This theoretical appraisal led to a hypot- hesis for the initiation and propagation of cracks by the passage of a Rayleigh wave (R- wave). The hypothesis was verified experimentally using large plates of glass and Homalite 100. This paper delas with a parallel appraisal of surface waves, using the Lamb solution [2], but following the treatment of Dally and Thau [3, 4], which considers the entire field of the R-wave. Both plane strain and plane stress solutions of the R-wave generated by a line load applied to a semi-infinite body have been examined. The expressions for the characteristic or Rayleigh equation, the displacements, and the stresses over the full-field have been derived. Detailed numerical calculations have been IS Mech. Teoret. i Stos. 3-4/85 578 J. F . CARDENAS-GARCIA, D . C. HOLLOW AY carried out and computer-generated isochromatic plots have been obtained which compare quite favorably to those observed experimentally. In addition, calculations of the cumulative strain energy contained in the R-wave field have been made, which confirm that a large portion of the energy contained in a propagating R-wave lies within one wavelength depth into the medium. Finally, the maximum normal stress fracture criterion [5] is used to obtain estimates of the crack extension direction, and crack extension is predicted using the Griffith-Irwin strain energy release rate fracture criterion [6, 7]. , The model The geometry considered is shown in Figure 1. Examination of the boundary conditions yields the Rayleigh equation: (A:?)3-8(A:?)2 + (24-16a?)(Je?)+(16a?-16) = 0 (i) line load plane strain b-óo plane stress h-h 0 /' boundary conditions on the X-2 plane: Fig. J. Geometry of the semi-infinite plate used for the Lamb solution where for plane strain I _ 2(1 and -2v_ IkA* 7C3 (2) (3) (4) where k,, kp, k,, kR and kE are the wave numbers associated with the various waves. CE represents the R-wave velocity in plane stress. The Rayleigh equation shows that the K > _ ( k t \ 2 I co Y ! c R \ 2 and for plane stress (where a t and ATr change t o for plane strain, is given by: 2E For plane stress, it is given by: I Uo m --— (Oxx + 0-y) — — 0XX + - (20) (21) which implies that the strain energy can be expressed as: / b (22)UmJJf Uodzdxdy = h.f j Updxdy o o o o o where h is the thickness of the material in which the plane R-wave travels. Figure 2 shows the normalized strain energy plotted against normalized depth for plane strain and plane stress. The normalized strain energy represents that portion, up U T - total strain energy - wavelength 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 .1.80 2.00 normalized depth , ( y , l ) Fig. 2. The Lamb solution — normalized cumulative strain energy as a function of normalized depth, for plane strain and plane stress, v = 0.35 to a depth v0 (0 < y0 ^ 2e), of the total strain energy per unit width contained in a R-wave which is one wavelength (e) long and two wavelengths deep. More of the strain energy associated with the plane stress solution lies close to the surface than that of the plane strain solution. Or, at a given normalized depth, the plane stress solution shows a greater relative amount of cumulative strain energy than the plane strain solution. Even so, more than 80% of the strain energy is contained within one wavelength from the surface for both cases. Figure 3 shows a representation of the magnitudes and directions, on and below the surface, of the principal stresses associated with a R-wave. The R-wave is moving to the right and the principal stresses have been normalized with respect to the largest principal stress occurring at the surface. The larger principal stress, ot, at any. particular point is represented by solid lines and the smaller principal stress, a,, is represented by dotted MlCU O- CRACKING   BY  R A Y L B I G H   WAVE 581 0.00 0.10 n?n 0.30 run 0.50 0.60 0.00 — I — - .V - _  v - -   V I normalised  distance along  the surface , x/ l 0.25  0.50  0.75 1   1 ——  — i — •   — . . . * -   ,.,'• • •   - 4 -  '• , ,   i  +0.25 •»...,   x x  X k y k X - 0.2S - 025 x x /   /  \  x y y / i \ \ x  x  •/.   •/•  * \ (a| )mox= 37.3  MN/ rn 2(plane  strain) (02)max=32.3MN/ m 2(pIane  stress) 1 = wavGlength  , 1.00 —i—i N  _ - - i Fig.  3.  The  Lamb  solution  —  normalized  principal  stresses  and  directions  on  and  below  the  surface for  plane  strain  and  plane  stress, v  =   0.35 lines. If  a  stress  is  positive  a  small  bar  is  attached at  each end  of  the respective  representa- tion  of  stress  and if  it is  negative,  n o bar is  added. The first  half- wavelength  of  the  R- wave, at  the surface,  exhibits  a  tensile  com ponent while  the second  half- wavelength  of  the  R- wave exhibits a compressive  com pon en t tan gen t to the surface.  Considering only the first  quarter- wavelength  of  the  R- waye  an d  applying  the  maximum  normal  stress  fracture  criterion, where  the  crack  propagation  takes  place  perpendicular  to  the  direction  of  the maximum principal  stress,  it  is  clear  t h at  the  crack  would  grow  into  the  material,  in  the  direction opposite to t h at of  R- wave propagation , at  some  angle  from  the vertical.  But consideration of  the  second and th ird quarter- wavelengths  of th e R- wave,  which  seem to be the dominant portions, shows  t h at th e crack  would  grow into th e material, away  from  the direction where the  R- wave originated,  at  som e  angle  from  the vertical.  The last  quarter- wavelength  could again cause th e crack to chan ge direction, but th e magnitudes of th e  stresses  would  probably not  support  con tin ued  crack  propagation .  If  a  prediction  were  to  be  made  of  the  final overall  R- wave  crack  propagation  length,  it  would  be  of  the  order  of  one- quarter  of  a wavelength.  This  assumes  a  crack  to  R- wave velocity  ratio  of  one- third and  that the crack would  start  propagatin g  at  a  value  of  normalized  distance  along  the surface, x/ e,  of  0.75. Experimental  results  and  con ,  • '.  i  i » Ł ^ a  : -   •   ?• The  experimental  m odel  is  shown  in  F igure  4.  I t  consists  of  a  18 x 24 x  1 / 2  inch  plate of  H omalite  100,  a  tran sparen t, birefringent  m aterial  which  is homogeneous  and isotropic. A  micro- crack  or  flaw  across  the  plate  thickness  was  induced  on  th e  edge,  by  tapping an  X- acto  knife  edge  softly  with  a  ham m er.  Th e  micro- crack  is  10  inches  away  from  th e 582 J. F. CARDENAS-GARCIA, D.C. HOLLOWAV detonator cord homalite 100 Fig. 4. The experimental model Homalite 100 cylindrical charge holder 6,35mm — diameter x /" long), which was hollowed out with a 3,17mm diameter drill and epoxied on to the edge of the model plate. The explosive used was 200 mg of PJETN. A 1 " x l " grid of one-inch squares was drawn with black ink on the side of the Homalite 100 plate facing the camera. This experimental model is used to simulate, under controlled conditions, the extension of a surface micro-crack as the R-wave moves past it along the edge of the plate. Care was taken in choosing the size of the plate so as to prevent reflected stress waves from impinging upon the crack which would alter the results. Using this experimental set-up several tests were run. A typical test is presented here and the results are compared to the theory. Figure 5 shows the isochromatic patterns associated with a R-wave moving from left to right, and the location and size of the initial crack. Notice that the leading fringes of the R-wave appear unaffected by the presence of the crack which has an initial length of 0,53 mm inches which corresponds to a length normalized with respect to the wavelength of 0,006. The initial normalized depth is less than that which Dally [8] found to have negligible effect on the R-wave characteristic : MICRO-CRACKING BY RAYLEIGH WAVE 583 stress distribution, which was 0.018. Figure 6 shows the final crack size. It also shows some effects of R-wave component reflections as is evident from the jagged look if some of the isochromatic fringes. The crack in Figure 5 is located, with respect to the R-wave, at a fringe of order of 2.5 on the leading portion of the R-wave. Figure 7 shows a comparison of the experimentally obtained isochromatics with the theoretical predictions using the Classical and Lamb solutions, for the plane stress case. The resemblance between the experimental and the Lamb solution result is very close. Therefore, the theoretical solution and the methods used to obtain it from the experimental result are adequate. This also implies that the description of the stresses in the entire R-wave field is being modelled correctly. Figure 8 shows the experimental result as a composite of the crack tip location in each of the frames. As such, it incorporates a greater amount of experimental error because CLASSICAL SOLUTION Fig. 7. A Comparison of the experimental and theoretical full-field isochromatic patterns: a) classical solution, b) experimental lesult, c) Lamb solution 584 J. F. CARDENAS-GARCIA, D . C . HOLLOW AY 0.0 0.020 normalized horizontal distance, xo/ 0.0 0.020 0,040 0.060 O.Oi.0 'a. cv "O 0.060 0.080 0.100 0.120 T T 1,l',l" normalized initial crack length 2 =0.006crack initially located, 3 with respect to thtt R-wave at: ,» x/l =0.90 experiment x/l =0.90classical solution x/l =089 lamb solution t=26.9psec Fig. S. A comparison in space and time of the experimental and theoretical results of crack extension induced by the Rayleigh wave of the changing perspective in the Cranz-Schardin camera, since each frame has a separate lens location. Also plotted in Figure 8 are the predictions obtained using the Classical and Lamb solutions. Lines have been drawn in Figure 8 to show equal absolute times. The large differences between the experimental composite result and the Classical solution result can be explained in terms of the isochromatics previously shown in Figure 7, i.e., the simulation or modelling of the full stress field is not at all similar. Therefore, even assuming that the fracture models the phenomena of R-wave cracking correctly, the results would be expected to differ. The small differences between the experimental compo- site result and the Lamb solution can also be looked at in terms of the isochromatics previously shown in Figure 7, but in this case the close correlation between the R-wavc cracking results would imply the validity of the fracture analysis which was performed. Conclusions Several observations can be made with regard to R-wave crack extension: (1) R-wave extension of micro-cracks is an experimentally verifiable phenomena. (2) Though both the Classical and Lamb solutions can be used to model the R-wave, the Lamb solution appears to best describe it. (3) Using the Lamb solution in combination with the Griffith-Irwin maximum strain energy release rate criterion it is possible to approximate R-wave micro-crack extension reasonably well. MICRO- CRACKING   BY RAVLEIGH   WAVE  585 Bibliography j_  j , F,  G ARDKN AS- G ARCIA,  and  D . C.  HOLLOW AY, T heoretical and Experimental Investigation of the Propagation of Surface Cracks by the Rayleigh W ave, Proceedings  of  the  Society  for  Experimental Stress  Analysis  Annual  Spring  Conference,  Cleveland,  Ohio,  pp.  937- 944,  May  1983. 2.  H . LAMB, On the Propagation of T remors over the Surface of an Elastic Solid,  Phil. Trans.  Royal  So c , Vol.  A2O3,  p p .  1 - 4 2, 1904. 3.  J. W.  D ALLY, an d S. A.  T H AU , Observations of Stress W ave Propagation in a  Half- Plane  with  Boundary L oading,  International  Journal  of  Solids  and  Structures,  Vol.  3,  pp.  293 -  308,  1967. 4.  S. A.  TH AU ,  and  J. W.  D ALLY,  Subsurface Characteristics  of  the  Rayhigh  W ave,  International Journal of  Engineering  Science,  Vol.  7,  pp.  37 -  52,  1969. 5.  E. H .  YOFFE,  T he Moving Griffith Crack, The Philosophical Magazine, Vol. 42, Seventh Series, N o .  330 pp.  739- 750,  1951. 6.  A. A.  G RIPF ITH ,  T he Phenomenon of  Rupture and Flow In Solids,  Philosophical Transactions of  the  Roya Society  of  London,  Vol.  A221,  pp.  163 - 198,  1920. 7.  G .R .  IRWIN ,  Fracture Dynamics, F racturing  of  M etals,  American  Society  for  Metals,  p .  152,  194S. S.  J. W.  D ALLY,  Dynamic  Photoelastic  Studies  of  Stress  W ave  Propagation, Modern  Problems  in Elastic Wave P ropagation,  J.  M iklowitz  and  J. D .  Achenbach  (Eds.),  John  Wiley  and  Sons, Inc., New  York  (1978), P  e  3 io  M  e TEOPETOTECKI- IE  H   3KCriEP M M EH TAJlBH BIE H CCJIEflOBAH H fl  MH KPOTPEIIIMH BH 3BAH H LI X  BOJIH AM H  P 3JIEH A B  paGoTe  o6cy>iKfleirHH   an epn iH   n.na i p e m n a  n o ^ - nepmeHHHx narpy3KoM   I  H  I I Tim oB, npe^ycMoTpeH O  flopory  ii win n y  MHKpoipeimiH  nbi3D anH bix  BOJI - B  3KcnepHMeHTaJii>H0M   noflxoAe  npH M eneiio  dpoToynpyrocTh  ^Jia  H a6jn ofl«aia  paen peflen em w n3oxpoM   CBji3aHHbrx  c  n po6eroM   BOJI H M  Pajiefta  B,CI;OJIŁ  6epera  ABynpejionoieH H oro  MaTepaajia.  Ilo.ny- yAoBJieTBopnTejiLHyio  cxofliiAiocTb  TeoperimecKiix  u  3KcnepHMeHTajiMaix  pe3yjn.TaT0B. S t r e s z c z e n i e TEORETYCZNE  I  D OŚ WIAD CZALNE BAD AN IA  M IKROPĘ KN IĘ Ć  POWODOWANYCH   FALĄ RAYLEIG H A W  pracy  omówiono  teoretyczne  i  doś wiadczalne  badania  rozwoju  mikropeknię ć  powierzchniowych powodowanych  falą   Rayleigha.  Podejś cie  teoretyczne  wykorzystuje  polowe  rozwią zanie  Lamba  dla pro- pagacji  fali  powierzchniowej  w  dwuwymiarowym  oś rodku  jednorodnym  izotropowym,  dla  przypadków płaskiego  pola  naprę ż eń  i  odkształ ceń. Stosują c  kryterium  G riffitha- Irwina  prę dkoś ci  zwolnienia  energii dla pę knięć poddanych I i I I typowi  obcią ż eń, przewidywano  drogę  i dł ugość mikropę knięć  wywoływanych przez  fale  Ra}'Jeigha.  W  podejś ciu  doś wiadczalnym  stosowano  dastooptykę   dla  obserwacji  rozkł adów izochrom  zwią zanych  z  przebiegiem  fali  Rayleigha  wzdł uż  brzegu  materiał u  dwójł omnego.  Otrzymano zadowalają cą   zgodność  wyników  badań  teoretycznych  i  doś wiadczalnych. Praca został a  zł oż ona  w  Redakcji  dnia  20  kwietnia  1985 roku