Towards Tomographic Photoelasticity

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1 Towards Tomographic Photoelasticity Dr Rachel Tomlinson Department of Mechanical Engineering,

2 Outline What is photoelasticity? 3D Photoelasticity Methods Advances in data collection and processing Future plans

3 What is Photoelasticity? Principal Stresses Polarised light in transparent component θ σ1 direction Light source Polariser with vertical axis σ direction Stressed photoelastic model with σ 1 and σ axes inclined θº to vertical Polariser with horizontal axis

4 Photoelastic Fringe Pattern

5 Stress freezing photoelasticity Load model in thermal cycle Thinly slice model D fringe pattern *Technique expensive *Destroys the model

6 Integrated Photoelasticity

7 Tomographic Photoelasticity Photoelasticity gives a tensor quantity Data collection is tedious Overall aim consider possibilities of integrated photoelasticity and its combination with tomography

8 Work carried out Determine the parameters to describe a 3D Stress Field using phase stepping Use reconstruction algorithms to obtain 3D stresses tomographically

9 General 3 D photoelastic model True model y θ χ θ+χ x y Rotator x Retarder Optically equivalent model

10 Characteristic Parameters δ Characteristic retardation θ Primary characteristic direction θ + χ Secondary characteristic direction

11 Phase stepping P β R χ Q φ φ β y P 0 Q ρ ρ θ M χ θ ( δ ) Retarder Rotator Analyser Output quarter wave plate o x Input quarter wave plate Polariser Sout = P β Q φ Pk R χ M θ ( δ) Q ρ P0S

12 Experimental Procedure L PQ Model Transmission Polariscope I CCD camera QP PC L = Light source; P = Polariser; Q = Quarter wave plate; I = Immersion Tank 6 images captured Digital intensity maps 56 x 56 pixels Periodic maps of δ, θ and (θ + χ) Continuous maps of δ, θ and (θ + χ)

13 Data Processing

14 Advantages of the method Resolution * Full field analysis * Copes with dense fringe patterns Speed * 10 minutes collection and processing

15 Photoelastic Instruments

16 Conclusions on phase stepping Obtain three full field parameters Tomographic photoelasticity is now a real possibility Measure rotation term indication of through thickness stress field

17 CT photoelasticity Developed filtered back projection method to reconstruct axi symmetric stress fields Experimental example

18 Projections through a photoelastic model projection Photoelastic model (stress field) Projection (Integrated Retardation and Primary Characteristic Direction)

19 Grooved cylinder under tension

20 Projection data through grooved cylinder degrees isoclinic angle fringes retardation

21 Reconstructed data F/(P/A)

22 Reconstructed data F = (σ z σ θ )+1/(σ θ σ r ) Separate the stress differences to obtain (σ z σ θ ) (σ θ σ r ) Issues: Case specific Complex procedure Further study required

23 Separated stress differences ( σ )/( P A) ( σ θ σ )/( P A) z σ θ / r / Reconstructed Theory

24 Conclusions Full field 3 D stress field characteristic parameters Tomographically reconstruct axi symmetric stress field data from projections Next step Generalised 3 D stresses Future Integrated automated instrument

25 Six intensity maps Image Q ρ Q φ P β No. 1 π/4 π/4 π/4 π/ π/4 0 π/ 4 π/4 0 π/4 5 π/ 0 π/4 6 π/ 0 π/ i i i i i i a = k x1( 1+ sin δ cos( θ + χ ) a = k x( 1 sin δ sin ( θ + χ ) a = k x( 1+ sin δ sin ( θ + χ ) a = k x1( 1+ cosδ) a = k x1( 1+ sin θ sin δ) a 1 cos θ cos θ + χ = k x sin θ sin θ + χ cosδ ( ) ( )

26 Solution of Characteristic Parameters ( i ) Rk x 1 Rk x1 i1 ( θ + χ) = tan δ ( ) = tan 1 i1 cos ( θ + χ)( Rk x1 i 4 ) Rk θ = 1 tan 1 sin θ = x1 sin θ( cos ( θ + χ ) ( ) 1 i 6 sin ( θ + χ) sin θ cos δ Rk ( i ) 5 Rk sin δ x where 1 and + R = i i k x1 3 x

Phase-Shifting Interferometry and Combined Methods

38 Chapter 3 Phase-Shifting Interferometry and Combined Methods 3.1 Introduction The chapter describes the hybrid phase-shifting full-field experimental CGS-photoelasticity method for in-plane tensorial