Ghostscript wrapper for D:\Digitalizacja\MTS85_t23z1_4_PDF_artyku³y\mts85_t23z3_4.pdf M ECH AN I KA TEORETYCZNA I  STOSOWANA 3 - 4,  23( 1985) D ETERM IN ATION  OF  S TRES S  IN TEN S ITY  FACTORS   BY  OPTICAL  M ETH OD S J.  BALAS,  M .  D R Ź IK  ( BR AT I SLAVA) Institute of Construction and Architecture of the Slovak Academy of Sciences Introduction In  the  development  of  structures  the  fracture  mechanics  principles  and  criteria  are used  in  an  ever  greater  n um ber  of  applications.  The  stress  intensity  factor  (SIF )  provides a  quantitative  criterion  which  combines  critical  fracture  stresses  with  the  length  of  crack and  the  specimen  geometry.  Th e  determination  of  the  influence  of  geometry  and  loading on  stress  concentrations in  th e vicinity  of  the crack  tip  has  been  included  to  the problems which  are  solvable  by  optical  m ethods  of  stress  analysis.  I n  the Institute  of  Construction and  Architecture  of  the  Slovak  Academy  of  Sciences  several  methods in  the area  of holo- graphic  interferometry,  speckle  interferometry  and  some  other  optical  techniques  were developed  and  realized  for  this  purpose. 1.  St r ess  intensity  factors  in  tjie  generalized  plane  stress  state  cracked  specimens As  one  of  the  first  application s  we  used  th e  image- plane  holographic  technique  on models  of  P M M A  tran sparen t  material  to  th e  examination  of  isopachic  fringes.  The advantage  of  this  procedure is  in th e fact,  th at th e SI F  may  be determined from  isopachics data  without  requiring  troublesom e  separation  of  principal  stress  components.  A  holo- graphic  interferometer  was  developed  in  the  optical  arrangement  of  which  the  diffuser screen  is  n ot  included.  F o r  th is  reason  the  recorded  interference  pattern  can  be  photo- graphically  greatly  magnified  and  by  this  the  required  resolved  power  of  image  details can  be  secured. Stress  intensity  factors K t   and K n   were  determined  in  cracked  beams  with  inclined and  curved  edge  cracks.  Example  of  the  pattern  of  isopachics  is  shown  in  F ig.  1. In  order  to  determ ine  th e  quantitative  values  of K t   as  well  as K u   several  evaluating procedures  have  been  developed  based  on  the  description  of  the  stress  state  around  th e crack  tip  by  asym ptotic  series.  I t has  been  shown  th at  these  procedures  allow  reliably  t o calculate  SIF - s  generally  in  arbitrary  geometrical  and  loading  configurations  of  cracked specimens  in  the  plan e  stress  state  with  an  accuracy  of  better  than  ±  5  per  cent  [1], An  analogous  technique  of  the  holographic  record  an d  SI F   evaluating  we  used  also in  the  investigation  of  th e  dyn am ic  crack  propagation .  The  interferometer  functioned 556 J.  BALAS,  M.  D KŹ IK :  F ig. 1. Interference  pattern of  isopachics  in the vicinity of  the angled  crack  in a  bend  specimen Fig.  2. Isopachic  fringes  of  dynamic  crack  propagation  in PM M A  three- point  bend  specimen  at crack velocity  of  400  m/ s with  a  ruby  impulse  laser.  Isopachics  aroun d  the  crack  run n in g  with  the  speed  of  about 400  m/ s  in  the bend  P M M A  specimen  are  shown  in  F ig.  2.  F o r th e  quan titative  evaluating of KID  values  a  dynamic  correction  of  interference  con stan t  m ust  be  performed.  To  do this,  we  can  get  the  known  dynamic  dependencies  of  material  con stan ts  an d  by  using  the graphical  F ourier analysis  and iterative  procedure  th e  evaluating  interference  fringe  values may  be  calculated.  I t  has  been  shown,  th at  th e  differences  between  static  and  dynamic values  in  polymer  materials  (such  as  P M M A)  may  be  50  per  cent  or  even  m ore.  By  this procedure  the  relationship  between  the  dynamic  SI F   and  th e  crack  tip  velocity a,  which is  the  im portan t  fracture  characterization,  was  obtain ed  (F ig.  3). F or  th e  measurement  of  the time  functions  of  th e  dynamic  values  of  SIF - s  also  a new photo- electric  method  was  developed.  Based  on  optical  observation  of  t h e  deformation of  a mirror like flat  close  to the crack  tip  and  on optical filtration  principle,  changes  of the light  intensity  are  produced.  These  are  transformed  to  electric  quantities  by  means  of  a photosensor.  In  this  simple  way  without  emploing  a  high  speed  cam era  a  continuous record  directly  of  the  stress  intensity  factor  can  be  obtained  immediately  in  tim e  on  the a im/s] DETERMINATION  OF  STRESS... K I O / K X C 1.0 2.0 _ 557 400 K,0lMN/m 3/2] Fig.  3.  The KtD  — a  relation  for  PMMA  Acrylon  obtained  from  the  holojraphic  interferomstry  data oscilloscope  screen.  Consequently,  very  fast non-stationary dynamic processes such as impact load may easily be solved. Moreover, the technique takes notice of the stress waves space distribution due to the fact that the deformation is watched close to the crack tip. Fig. 4 illustrates the method by the KID versus time dependence in the typical impact load process as was observed on the oscilloscope screen. The possibility of the displacement measurements on the specimen surfaces ranks among the most important characteristics of the holographic-speckle interferometry principle. For the study of the SIF determination from displacement data we can tried on application of image optics free speckle technique when the holographic plate is fixed immediately on the measured object. The main advantage of this technique is the self compensation of large non-controlled displacements of the rigid body motion. We employed Fig. 4. Stress intensity factor in the strip with- an edge stationar crack subjected to thres-point impact load bending as a function of time (time calibration 200 (xsldiv and Kla-10 s MNnTMdiv) 558 J.  BALAS,  M .  D R Ź IK th e  method by  means, when  one  of  the tran sparen t model  surfaces  is  roughed  (on  the side of  holographic  plate  emulsion). Collimated  laser  beam  passed  th rough  the  model  and  the groud  screen  surface  scatter  it  so  that  the  specklegram  may  be  recorded.  I n  the  step of reconstruction we  used  the F ourier filtration  procedure  and  by  this  in- plane orthogonal displacement  components are  optically  obtained.  As  a  test  specimen  we  chosen  the crack- emanating  from  a  hole  in  a  complicated  piercing  bending  specimen. In  evaluating  of  the  SI F   the  relation  displacements  versus  th e  distance  from  crack  tip is  graphically  drawn.  A  precise  interpolation  of  t h at  curve  to  the  zero  distance  from  the tip  can be  done on the basis  of  linear  fracture  mechanics  condition  of  zero  value  displace- ments  at  the  tip  (see  F ig.  5).  So  the  first  term  of  asymptotic  solution  for  displacements vitftmm] 8.0 Ki =  5.4- iO5MN rrf3/ 2 t - 6^90°- 0.2  0.5  1.0 3.0 4.0 5.0 6.0 r l m m ) F ig. 5. Displacement data as a function of square root of radial distance r from  the crack tip for the piercing cracked  specimen field  around  the  crack  may  be  successfully  used.  We  believe  t h at  th e  displacement  data with  respect  to  the  stress  intensity  factor  determination give  th e m ost  precise  results  of  all the  optical  methods.  \. 2.  Application  of  non- destructive  optical  methods  to  study  3- D  cracked  bodies One  of  the  up- to- date task  in th e field  of  optical m ethods of  experimental  stress analysis is  to  develop  reliable  and  effective  methods  for  th e  measurement, of  three- dimensional stress  state.  We  used  holographic  interferometry  as  a  non- destructive  m ethod for  the SIF determination  in  three- dimensional  cracked  specimens.  Th e  basic  principle  is  t o  record a  double- exposure  image  hologram  with  perpendicular  illumination  of  th e  crack  surface and  filtration  of  space  frequencies  for  th e  separation  of  the  out- of- plane  displacement component.  Arrangement  of  the  optical  system  with  th e  model  is  shown  schematically in  F ig.  6.  Surface  of  th e crack  is  illuminated  by  a  beam  passing th rough  t h e polished  side surface  of  the tran sparen t model. The image  of  the crack  is  observed  from  th e sam e  direc- tion  and  is  projected  by  an  objective  on  a  holographic  plate  where  the  hologram  with DETERMINATION  OF  STRESS... 559 7/'' model 7% ' mirror objective load space filter Fig. 6. Scheme  of  optical  arrangement  of the  holographic interferometer  for  crack  opening  displacements measurement the  reference  beam  is  formed.  In  the  focal  plane  of  the  lens the  space  filter  — diaphragm with circle aperture — is positioned for  the separation  of the near-to-zero space frequencies. Moreover, in this manner, not only the separation of the out-of-plane displacements is performed but an improvement of the correlation between the first and second exposure records may be reached, too. The radius of the correlation which is characterized by the speckle size may be varied by a suitable choice of the aperture diameter in the space filter, consequently contrast of the fringes may be increased. Light rays immediatelly behind the reflection from the crack surface spread through the thickness of the transparent model material. This is the reason, why the value of material index of refraction n will be included in the expression for the quantitative evaluation of the out-of-plane component w. Using the diffraction theory this expression may be derived for the interpretation of interference fringes [2] 4 where N is the order number and X is the wave length. Illumination and observation of the inside crack was experimentally achieved with the aid of small mirror which turns the light beam to the polished side of the model. The mirror was placed in the sight-hole of jaw on one margin of the model By these groove steel chuck jaws the tension loaded beam including a surface flaw was fixed on both ends where the loading force was applied. Application of the „classic" speckle method to the measurement of the displacements inside a transparent body was another optical method for the non-destructive SIF evalua- tion on the same specimen as in the previous case of holographic interferometry. Laser beam with a diameter of 0.5 mm from the 60 mW power He-Ne laser was penetrating across the PMMA model near the crack front (see Fig. 7). Intensity of the light radiation in the beam was sufficient to the observable light scattering along the beam trace which arose from the Tyndal effect. Scattered coherent light was concentrated on a holographic plate where the image of the light line was focused by the photographic objective. Micro- structure of that image represents a speckle record and when the double-exposure procedure is used we can evaluate the displacements of the points on the light line. Evaluating 560 J.  BALAS,  M.  D R Ź IK leser beam photeplate F ig. 7. Schematic drawing  of  the speckle  interferometry  arrangement  in the measurement  of  inside  body displacements of  the  specklegrams  has  been  carried  out  by  the  usual  Youn g's  fringes  poin t  by  point method. The objective  used  for  the measurement was  wide open  H elios  lens  with  aperture number  1.5  and focal  distance 85 mm. M agnification  of  th e im age  was  4 times  an d corres- ponding  exposure  times  were  several  tens  of  seconds  on  Agfa- G evaert  holographic  plates. Both,  the  holographic  method  of  crack  opening  displacements  m easurem en t  and  the speckle  method  using  inside  scattered  light  were  experimentally  tested  on  th e  same  three tensile  loading  beam  specimens  including  a  semi  elliptical  surface  cracks.  T h e  surface flaws  as  an  imitation  of  n atural  cracks  were  machined with  t h e  aid  of  a  circular  milling cutter. By  holography  a  field  of  interference  fringes  on  a  crack  plan e  was  reached  several times  on  each  sample  for  different  loading  levels.  One  exam ple  of  the  measured  fringe patterns of crack opening displacements is shown in F ig. 8. I n addition to the direct  focused F ig.  8.  Interference  pattern  of  crack  opening  displacements  on the semi  elliptical  crack image  of  the crack  surface  a  total  reflex  on  an  inside  m irror  sm ooth  surface  of  th e model is  imaged,  too  (see  bottom  part  of  F ig.  8). Being  based  on  values  of  crack  opening  displacements  th e  stress  intensity  factor  can be  calculated  by  means  of  asymptotic  relations  for  t h e  displacements  aroun d  two- dimen- sional  crack  in  the  state  of  plane  strain  [3]. F o r  the  poin ts  on  a  perpendicular  line  to the DETERMINATION  OF  STRESS... 561 crack  front  the  factor Kx  may  be  given  by  the well-known relationship described here for the polar angle 6 = 180° where E, v are material constants and r is the polar distance from the crack tip. To calculate the factor K{ by Eq. (2), points in the close vicinity of the crack tip are taken into account. The stress intensity factors are obtained along the whole crack front. 0./. K 0 =K, 0 / / E(k) / ft. hi / T5. e o A r \" c { c /=40 t c c i 0 10 ^ / }/ —2c a/t=0-5 j a/c=0.267 a/t=0.33;a/c =0.217- xa/t=0.167;a/c=0.152 o - holography x - speckle — BEM 1 O 7 1.4 1.2 — 1.0 o.e - 0.6 5- 0 10 20 30 40 50 60 70 80 90 [ ° ] c Fig. 9. Variations of the stress intensity factor along the crack front as were measured by holographic and speckle interferometry and calculated by boundary element method [4] To calculate the factor Kt from the data obtained by the speckle interferometry the same asymptotic expression but for the polar angle 0 = 90° may be used (3) 2(1+*) Results for the test specimens are summarized in the form of Kj relationships on the radial angle of the given crack front point. Values of the factor Kj plotted in Fig. 9 are normalized by the maximum value of the factor Kt for an elliptical crack in an infinite space. The evaluated data obtained by holography and speckle methods are compared with those calculated by the numerical boundary element method [4]. 3. Conclusions Results and experience acquired during the last years show that the exploitation of optical methods in the field of stress state investigation in the bodies including cracks are adequately exact and effective in the solution of both engineering tasks and problems of fundamental research. Problems connected with a material evaluation and testing with 14  Mech.  Teoret.  i  Stos. 3-4/85 562  J-   BALSA,  M.  D R Ź IK regard  to  the  influence  of  cracks  measurement  of  fracture  toughness  param eters  as  well as  a research  of  non- linear effects  in th e cracked  bodies,  all  these  questions  call  for  experi- mental  measurements.  M oreover,  methods  of  experimental  mechanics,  an  important part  of  what  are  optical  methods,  allow  successfully  work  out  also  very  complicated problems  in  static  and  dynamic  conditions.  I n  comparison  with  the  numerical  computa- tional  method  an  experiment  has  the  advantage  where  th e  specified  boun dary  conditions are  inaccurate  or  complicated  (thermal,  dynamical  problems). This  is  why  we  can  say  that  application  of  optical  m ethods  in  fracture  mechanics  has a  bright  perspective. References 1.  M. D R Ż I K, Stress analysis in the vicinity of angled cracks by holographic interferometry  (in Slovak), Staveb.  cas. 30  (12), 913- 927,  1982. 2.  J. BALAS, M.  D RZ IK, Measurement ofKi on three- dimensional cracked bodies by holographic interferometry, 5th  Int. Congress  Exp. Mech.,  M ontreal,  June  10- 15,  1984  (to  be  published). 3.  M. K. KASSIR, G . C.  SIH ,  T hree- dimensional stress distribution around an elliptical crack under arbitrary loadings,  J.  Appl.  Mech.  33(4),  602- 611, 1966. 4.  J.  SLADEK,  V.  SLADEK,  Boundary element  method in fracture  mechanics, Acta  Techn. CSAV,  27(6), 718- 732,  1982. 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Streszczenie WYZN ACZAN IE  WSPÓŁCZYN N IKÓW IN TEN SYWN OŚ CI N AP RĘ Ż EŃ   M ETOD AM I OPTYCZN YMI D o  wyznaczenia  współ czynników  intensywnoś ci  naprę ż eń w przypadku  pł askiego i  trójwymiarowego stanu  naprę ż enia, w warunkach  statycznych jak i  dynamicznych, zastosowano  szereg  metod optycznych, a  mianowicie  interferometrii  holograficznej,  interferometrii  plamkowej  i  optycznej  filtracji  połą czonej z zapisem fotoelektrycznym.  Wyniki  i doś wiadczenia  ostatnich lat pokazują ,  że zastosowanie metod optycz- nych w dziedzinie  badania stanu naprę ż eń spę kanych  ciał  są   odpowiednio  dokł adne i efektywne  zarówno w  rozwią zaniach  zagadnień  inż ynierskich  jak  i  badaniach  podstawowych. Praca został a zł oż ona w  Redakcji  dnia 20 kwietnia  1985 roku