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Phase Shifting Full-Field Interferometric Methods for Determination of In-Plane Tensorial Stress

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Abstract

A new method that combines phase shifting photoelasticity and transmission Coherent Gradient Sensing (CGS) is developed to determine the tensorial stress field in thin plates of photoelastic materials. A six step phase shifting photoelasticity method determines principal stress directions and the difference of principal stresses. The transmission CGS method utilizes a standard four step phase shifting method to measure the x and y first derivatives of the sum of principal stresses. These stress derivatives are numerically integrated using a weighted preconditioned conjugate gradient (PCG) algorithm, which is also used for the phase unwrapping of the photoelastic and CGS phases. With full-field measurement of the sum and difference of principal stresses, the principal stresses may be separated, followed by the Cartesian and polar coordinate stresses using the principal stress directions. The method is demonstrated for a compressed polycarbonate plate with a side V-shaped notch. The experimental stress fields compare well with theoretical stress fields derived from Williams solution for a thin plate with an angular corner.

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Abbreviations

A x,y :

magnitude of \(\boldsymbol{E}\) along x and y axes

c i :

constant of integration

C :

relative photoelastic constant

C f :

fitting parameter for V-notch stress solution

d shear :

shearing distance in CGS

D 1,2 :

absolute photoelastic constants

D 3,4 :

photoelastic constants related to D 1,2, ν, E, and n

E :

Young’s modulus

\(\boldsymbol{E}\) :

electric field vector

I :

intensity (irradiance)

h :

specimen thickness

h d :

phase discontinuity height

k :

wave number

n o :

isotropic index of refraction

n i :

principal index of refraction (\(i=\lbrace 1,2,3\rbrace\))

N :

photoelastic fringe order

p :

pitch of Ronchi gratings

R x,y :

reflection coefficients along x and y directions for the non-polarizing beamsplitter

T x,y :

transmission coefficients along x and y directions for the non-polarizing beamsplitter

α :

isoclinic angle

β :

angle of ouput polarizer

γ :

small angle of first order diffraction in CGS

δ :

isochromatic phase

\(\tilde{{\mathit\Delta}}\) :

Ronchi grating separation

θ :

polar angle

λ :

wavelength

ν :

Possion’s ratio

ξ :

fast axis angle of input λ/4 plate

ρ :

angle of input polarizer

σ 1,2 :

principal stresses

σ ab :

in-plane stresses (Cartesian for \((a,b)=\lbrace1,2\rbrace\) and polar for \((a,b)=\lbrace r,\theta\rbrace\))

ϕ :

fast axis angle of output λ/4 plate

ϕ x,y :

arbitrary phase constant in initial electric field

φ diff :

phase related to σ 1 − σ 2

φ sum :

phase related to σ 1 + σ 2

References

  1. Patterson EA, Ji W, Wang ZF (1997) On image anlaysis for birefringence measurements in photoelasticity. Opt Lasers Eng 28:17–36

    Article  Google Scholar 

  2. Siegmann P, Backman D, Patterson EA (2005) A robust approach to demodulating and unwrapping phase-stepped photoelastic data. Exp Mech 45(3):278–289

    Article  Google Scholar 

  3. Tippur HV, Krishnaswamy S, Rosakis AJ (1991) A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results. Int J Fract 48:193–204

    Article  Google Scholar 

  4. Rosakis AJ (1993) Experimental techniques in fracture. In: Optical techniques sensitve to gradients of optical path difference: the method of caustics and the coherent gradient sensor (CGS) (Chapter two). Society of Experimental Mechanics, Bethel, pp 327–425

    Google Scholar 

  5. Haake SJ, Patterson EA, Wang ZF (1996) 2d and 3d separation of stresses using automated photoelasticity. Exp Mech 36(3):269–276

    Article  Google Scholar 

  6. Greene RJ, Clarke AB, Turner S, Patterson EA (2007) Some applications of combined thermoelastic-photoelastic stress analysis. J Strain Anal Eng 42:173–182

    Article  Google Scholar 

  7. Berghaus D (1991) Combining photoelasticity and finite-element methods for stress analysis using least squares. Exp Mech 31(1):36–41

    Article  Google Scholar 

  8. Huang YM, Mohsen Abdel HH, Rowlands RE (1990) Determination of individual stresses thermoelastically. Exp Mech 30(1):88–94

    Article  Google Scholar 

  9. Huang YM, Rowlands RE, Lesniak JR (1990) Simultaneous stress separation, smoothing of measured thermoelastic isopachis information and enhanced boundary data. Exp Mech 30(4):398–403

    Article  Google Scholar 

  10. Barone S, Patterson EA (1996) Full-field separation of principal stresses by combined thermo- and photoelasticity. Exp Mech 36(4):318–324

    Article  Google Scholar 

  11. Sakagami T, Kubo S, Fujinami Y, Kojima Y (2004) Experimental stress separation technique using thermoelasticity and photoelasticity and its applciation to fracture mechanics. JSME Int J A 17(3):298–304

    Article  Google Scholar 

  12. Barone S, Patterson EA (1998) Polymer coatinf as a strain witnessin thermoelasticity. J Strain Anal Eng 33(3):223–232

    Article  Google Scholar 

  13. Yoneyama S, Morimoto Y, Kawamura M (2005) Two-dimernsional stress separation using phase-stepping interferometric photoelasticity. Meas Sci Technol 16:1329–1334

    Article  Google Scholar 

  14. Ghiglia DC, Romero LA (1994) Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J Opt Soc Am A 11(1): 107–117

    Article  Google Scholar 

  15. Baldi A, Bertolino F, Ginesu F (2002) On the performance of some unwrapping algorithms. Opt Lasers Eng 37:313–330

    Article  Google Scholar 

  16. Kobayashi AS (ed) (1993) Handbook of experimental mechanics. Wiley, New York

    Google Scholar 

  17. Narasimhamurty TS (1981) Photoelastic and electro-optic properties of crystals. Plenum, New York

    Google Scholar 

  18. Ekman MJ, Nurse AD (1998) Completely automated determination of two-dimensional photoelastic parameters using load stepping. Opt Eng 37(6):1845–1851

    Article  Google Scholar 

  19. Ramesh K, Tamrakar DK (2000) Improved determination of retardation in digital photoelasticity by load stepping. Opt Lasers Eng 33(6):387–400

    Article  Google Scholar 

  20. Ashokan K, Ramesh K (2006) A novel approach for ambiguity removal in isochromatic phasemap in digital photoelasticity. Meas Sci Technol 17:2891–2896

    Article  Google Scholar 

  21. Patterson EA, Wang ZF (1998) Simultaneous observation of phase-stepped images for automated photoelasticity. J Strain Anal Eng 33(1):1–15

    Article  Google Scholar 

  22. Kramer SLB, Ravichandran G, Bhattacharya K (2009) Transmission wavefront shearing interferometry for photoelastic materials. GALCIT Report 2009.001, California Institute of Technology

  23. Shimizu K, Suetsugu M, Nakamura T, Takahashi S (1998) Evaluation of concentrated load by caustics and its application in the measurement of optical constant. JSME Int J 41(1):134–141

    Google Scholar 

  24. Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19:526–528

    Google Scholar 

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Acknowledgements

We gratefully acknowledge the support of the National Science Foundation (DMR # 0520565) through the Center for Science and Engineering of Materials (CSEM) at the California Institute of Technology, of the American Society for Engineering Education National Defense Science and Engineering Graduate (NDSEG) Fellowship Program, and of the National Science Foundation Graduate Research Fellowship Program. We thank Prof. E.A. Patterson for valuable discussions during this project.

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  1. California Institute of Technology, 1200 E California Blvd., Pasadena, CA, 91125, USA

    S. L. B. Kramer, M. Mello (SEM Member), G. Ravichandran (SEM Member) & K. Bhattacharya

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  1. S. L. B. Kramer
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  2. M. Mello
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  3. G. Ravichandran
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  4. K. Bhattacharya
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Correspondence to S. L. B. Kramer.

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Kramer, S.L.B., Mello, M., Ravichandran, G. et al. Phase Shifting Full-Field Interferometric Methods for Determination of In-Plane Tensorial Stress. Exp Mech 49, 303–315 (2009). https://doi.org/10.1007/s11340-009-9230-0

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