photoelasticity

name	photoelasticity
description	Photoelasticity — fringe patterns, stress-optic law, isochromatic/isoclinic fringes, fringe order N, material fringe value f_sigma, digital photoelasticity, residual stress in glass.
metadata	{"priority":7,"promptSignals":{"phrases":["photoelasticity","photoelastic stress","fringe pattern","isochromatic fringes","stress-optic law","fringe order","birefringence stress"],"minScore":3}}

Photoelasticity — Complete Skill

Stress-Optic Law

When a birefringent material is stressed, the relative retardation of polarized light: δ = C × (σ₁ - σ₂) × t

δ = optical path difference [nm] C = stress-optic coefficient [nm/MPa/mm] (material constant) σ₁, σ₂ = principal stresses [MPa] t = specimen thickness [mm]

Fringe order N: N = δ/λ (integer at dark fringes in monochromatic light)

Stress difference from fringe order: (σ₁ - σ₂) = N × f_σ / t

f_σ = material fringe value [N/mm per fringe] = λ/(C)

Common Photoelastic Materials

Material	f_σ [N/mm/fringe]	E [GPa]	ν
Epoxy (Araldite B)	7–15	3.0–4.0	0.36
PSM-1 (polycarbonate)	7.0	2.5	0.38
CR-39	12	2.7	0.35
Glass (tempered)	f_σ varies	70	0.25
Gelatin	~0.5	0.001	0.50

Lower f_σ → more fringes for same stress → higher sensitivity but more crowded patterns

Fringe Types

Isochromatic Fringes

• Loci of constant (σ₁ - σ₂)
• Dark fringes (extinction) at N = 0, 1, 2, ...
• Bright fringes at N = 0.5, 1.5, 2.5, ...
• In white light: colored fringes (different λ extinct at different stresses)

Isoclinic Fringes

• Loci where principal stress directions align with polarizer axes
• Appear as dark lines in plane polariscope (not circular polariscope)
• Used to determine principal stress directions
• Disappear in circular polariscope (quarter-wave plates eliminate isoclinics)

Isostatic Lines

• Principal stress trajectories (orthogonal families)
• Constructed from isoclinic angle data

Polariscope Configurations

Plane Polariscope

Components: Light → Polarizer → Specimen → Analyzer (crossed at 90°) Shows both isochromatic and isoclinic fringes Field equation: I = I₀ × sin²(2α) × sin²(πNδ/λ)

Circular Polariscope

Components: Light → Polarizer → λ/4 plate → Specimen → λ/4 plate → Analyzer Eliminates isoclinics; shows only isochromatic fringes (better for analysis)

Bright-field circular: I = I₀ × sin²(π N) Dark-field circular: I = I₀ × cos²(π N)

Determining Stress from Fringe Order

At Free Boundary

Along a traction-free boundary, σ_n = 0 Therefore: σ_tangential = N × f_σ / t (Only one non-zero principal stress on free surface)

At Interior Points

Both σ₁ and σ₂ unknown; need additional information:

• Use stress equilibrium equations + fringe data
• Apply numerical (Shear difference) method
• Or use reflection photoelasticity + strain gauge

Shear Difference Method

Integrates equilibrium equations along a line: ∂σ_x/∂x + ∂τ_xy/∂y = 0

Step-by-step sum along horizontal line: Δσ_x = -Σ (Δτ_xy / Δy) × Δx τ_xy = (σ₁ - σ₂)/2 × sin(2α) [requires isoclinic angle α]

Digital Photoelasticity

Phase-Shifting Method

Capture 4 or 6 images at different analyzer/waveplate angles Compute fringe order continuously (not just at integers): N(x,y) = (1/π) × arctan[numerator/denominator from image combinations]

Provides full-field continuous N map; resolves fringe order ambiguity Software: PSIF, FRINGESOLVER

RGB Photoelasticity

White light fringe order from color calibration curve (τ_max vs. RGB triplet) Rapid but limited range (N < 3 before color repeats)

Thermoelastic Stress Analysis (TSA/Thermoelasticity)

Not strictly photoelasticity, but related full-field method ΔT ∝ (σ₁ + σ₂) for adiabatic conditions → complements photoelasticity (gives sum vs. difference)

Residual Stress in Glass (Tempered)

Photoelastic measurement of residual stress: Transmitted light method for thin glass plates σ_residual = N × f_σ / t

Tempered glass: compressive surface ~100 MPa; tensile core ~50 MPa Fringe pattern: concentric; surface in compression = low/zero fringes at surface

Scattered light photoelasticity: 3D stress within thick glass blocks (point-by-point slicing not needed)

3D Photoelasticity (Stress Freezing)

1. Load model at elevated temperature (stress freezing temperature ~80°C for epoxy)
2. Cool under load → strains locked in
3. Remove load; slice specimen
4. Analyze 2D slices in polariscope

Limitation: destructive; each slice gives 2D information from complex 3D state Application: turbine blade roots, complex castings, biomedical implants

Reflective Photoelasticity

Thin photoelastic sheet bonded to actual structure Measures surface strains on real component (any material) Fringe order related to actual part surface strain: N = f_σ_coating × ε_principal_difference / (2 × t_coating)

Advantage: can test real parts without model making Sheet materials: PC (PSM-1); adhesive: PC-1, PC-2 (Vishay)

Calibration (Disk-Under-Diametral-Compression)

Theoretical stress at center: σ₁ - σ₂ = 8P / (π × D × t)

Measure fringe order N at center → f_σ = (σ₁ - σ₂) × t / N

Output

Provide: principal stress difference (σ₁ - σ₂) [MPa] at all key locations, principal stress direction α [°] from isoclinics, σ_tangential along free boundaries, fringe value f_σ confirmation, comparison of fringe order N map to FEA (σ₁ - σ₂) plot, and uncertainty estimate (±N/2 fringe order = ±f_σ/(2t) MPa).