Stress-optic coefficient ($C(\lambda )$)

The stress-optic coefficient $C$ measures the birefringence induced per unit principal-stress difference in the linear stress-optic law:Here $\Delta n$ is a difference of refractive indices, so $C$ has SI units ${\mathrm{Pa}}^{-1}$. The wavelength $\lambda$ matters: a coefficient determined with one illumination wavelength should not simply be reused with another. The signs of $C$ and $\Delta n$ also require a consistent ordering of the optical axes and stresses.^[ref]

For a uniform specimen of thickness $t$ viewed along a principal optical direction, the measured phase retardation isA known uniaxial load supplies $\Delta \sigma =F/A$ away from grips and other stress concentrations. Fitting the slope $s=d\delta /dF$ givesSubtracting the unloaded retardation removes a constant background, but a load-dependent background must be investigated rather than absorbed into the coefficient.

For an original illustrative measurement, let $\lambda =500\mathrm{nm}$, $t=2\mathrm{mm}$ and $\Delta \sigma =20\mathrm{MPa}$ produce $\delta =0.80\mathrm{rad}$. ThenDoubling the stress difference doubles the retardation only within the same linear regime. This numerical value is illustrative, not a material specification.

An equivalent magnitude convention uses the photoelastic fringe value $f_{\sigma }=\lambda /|C|$. One additional full retardation cycle corresponds to $|\Delta \sigma |=f_{\sigma }/t$ for the uniform model. A fringe magnitude alone does not determine the sign of $C$ or the separate principal stresses.

The Illinois primary apparatus description makes the measurement chain concrete: a leadscrew loads the specimen, a load cell measures force, monochromatic light passes through the specimen between polarizing elements, and a camera records fringes. Establishing $C$ connects those optical observations to a mechanical stress scale. Wavelength, thickness, unloaded birefringence and loading conditions belong with the reported coefficient.^[ref]

The photoelastic fringe value $f_{\sigma }$ is the material-and-wavelength scale connecting photoelastic fringe order to principal-stress difference times specimen thickness in the linear stress-optic law. With vacuum wavelength $\lambda$ and stress-optic coefficient $C$, use the magnitude conventionHere $N=|\delta |/(2\pi )$ counts full relative-phase cycles and $t$ is specimen thickness. Since $C$ has units $\mathrm{P}{\mathrm{a}}^{-1}$, $f_{\sigma }$ has units $\mathrm{Pa}\mathrm{m}=N/m$. A value expressed in $\mathrm{MPa}\mathrm{mm}$ equals the same numerical value in $kN/m$. The wavelength dependence of $C$ must be retained.^[ref]

For an original example, take $\lambda =600\mathrm{nm}$ and $|C|=50\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$. Then $f_{\sigma }=12000N/m$. A3mm specimen at fringe order2.5 has stress-difference magnitude $2.5(12000)/0.003=10\mathrm{MPa}$. One additional complete phase cycle changes the inferred magnitude by4MPa for that thickness.

A uniaxial calibration supplies a known stress difference away from grips and other stress concentrations. For a rectangular strip of width10mm and thickness3mm, load300N gives stress10MPa. If the corresponding referenced fringe order is2.5, the measured fringe value is $10(3)/2.5=12\mathrm{MPa}\mathrm{mm}$. Repeating at several loads tests the assumed linear regime; counting from a referenced unloaded state helps separate the applied-load change from pre-existing retardation.

A primary instructional apparatus uses a manual leadscrew, load-cell readout, monochromatic illumination, polarizing elements and a camera. Those components connect a known mechanical load to the observed optical pattern.^[ref] A single intensity value does not uniquely identify an absolute phase cycle, so fringe order requires continuity, reference information or another phase-resolving method.

The magnitude convention loses the sign of $C$ and does not separately determine the two principal stresses. It also assumes the stated through-thickness optical/stress model. A varying stress field along the light path requires the appropriate retardation model rather than inserting one arbitrary local stress into the uniform-specimen formula.

The photoelastic fringe order $N$ expresses the magnitude of the full relative phase retardation between two principal optical components in cycles:For a uniform specimen obeying the linear stress-optic law, $N=t|{\sigma }_1-{\sigma }_2|/f_{\sigma }$, where $t$ is thickness and $f_{\sigma }$ is the photoelastic fringe value. The order is dimensionless and need not be an integer. One unit means one additional complete relative-phase cycle, not a separate stress component.

The connection to a visible fringe depends on the optical arrangement. For ideal crossed linear polarizers and a uniform retarder with optical axes at angle $\alpha$ to the incident polarization, the transmitted intensity satisfiesHere $I_0$ is the intensity entering the specimen after the first polarizer, and $\delta$ denotes the full relative phase. At $\alpha =45^{\circ }$, integer orders are dark and half-integer orders are bright. At axis alignment $\alpha =0$, the image is dark for every order. The crossed-polarizer derivation therefore separates retardation-related extinction from axis-alignment extinction.^[ref]

For an original ambiguity example at45°, orders0.25,0.75 and1.25 all give $I/I_0=1/2$. With $f_{\sigma }=12000N/m$ and $t=3\mathrm{mm}$, they correspond to stress-difference magnitudes1,3 and5MPa. Brightness alone does not choose among them. Referencing an unloaded state and tracking changes continuously during loading can supply information for photoelastic phase unwrapping.

The primary Illinois apparatus makes these controls concrete: a leadscrew loads the specimen, a load cell records force, monochromatic light and polarizing elements reveal the pattern, and a camera records it. Its magnitude-observation setting uses quarter-wave plates to remove the direction-related fringes.^[ref] That changes the observation geometry but does not remove the need to establish absolute order.

An order map under a known photoelastic fringe value gives principal-stress difference in the stated specimen model. It does not by itself give the two stresses individually or the sign of the stress-optic coefficient. Wavelength, initial retardation and the through-thickness stress model remain part of the interpretation.

Photoelastic phase unwrapping assigns consistent cycle counts to retardation observations that determine phase only modulo a full cycle. If a phase-resolved method reports ${\varphi }_k\in (-\pi ,\pi ]$, the full relative phase retardation has the formThe integers carry the fringe-order information that the wrapped values omit. A known starting order and sufficiently small changes can make the branch choices recoverable.

For an original one-dimensional example, take wrapped phases $2.80,3.05,-2.95,-2.70$ radians. Choosing the first value as the known starting branch, the nearby increasing continuation isor approximately $2.80,3.05,3.3332,3.5832$ radians. The apparent large negative jump is a change of representation at $\pi$, not a demonstrated physical reversal. The final photoelastic fringe order on this branch is about0.5703.

A nearest-branch rule is justified in a noiseless sequence when successive true phase changes have magnitude strictly below $\pi$: among increments differing by $2\pi$, exactly one lies in that interval. With phase errors bounded by $ϵ$, the stronger condition $|{\delta }_{k+1}-{\delta }_k|+2ϵ<\pi$ is sufficient for that local choice. Larger steps, discontinuities or inadequate signal can cause a photoelastic branch-slip error. One wrong cycle changes inferred stress-difference magnitude by $f_{\sigma }/t$ in the uniform-specimen model.

The algorithm assumes phase or equivalent branch information is available. A single crossed-polarizer intensity $I/I_0={\sin }^2(\pi N)$ does not itself supply a unique signed wrapped phase. Its ambiguity includes reflection within a cycle as well as the integer count. Additional polarization measurements, a known load path or other reference information must justify phase recovery before applying the simple continuation rule. The primary stress-analysis derivation explicitly shows how intensity depends on retardation and axis orientation.^[ref]

The Illinois apparatus supplies controlled loading, a load-cell record and camera observations, making a load-indexed sequence physically meaningful.^[ref] Continuity alone still leaves a common $2\pi$ offset unless the absolute starting order is known. An unloaded image can reveal residual retardation rather than automatically establish zero stress or zero order.

A photoelastic branch-slip error is an incorrect integer-cycle choice during photoelastic phase unwrapping. If the measured wrapped phase retardation is ${\varphi }_k$ and the true unwrapped value is ${\delta }_k={\varphi }_k+2\pi m_k$, replacing the correct integer $m_k$ by $m_k+j$ changes the recovered phase by $2\pi j$. A small error in one branch decision can therefore become a full-cycle offset in many later observations.

For an original noiseless counterexample, suppose a known starting phase is $2.2\pi$ and the next true phase is $3.4\pi$. Their wrapped values in $(-\pi ,\pi ]$ are $0.2\pi$ and $-0.6\pi$. The nearest-increment rule chooses $-0.8\pi$ instead of the true $1.2\pi$, producing $1.4\pi$ as the second recovered phase. The recovered photoelastic fringe order is $0.7$ rather than $1.7$. Every later small increment can be locally consistent while retaining that wrong cycle.

In the uniform-thickness stress-optical model, $N=t|{\sigma }_1-{\sigma }_2|/f_{\sigma }$, where $f_{\sigma }$ is the material fringe value. While staying on a known nonnegative stress-difference branch, one cycle changes the inferred stress difference by $f_{\sigma }/t$. Stipulated $f_{\sigma }=10\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$ and thickness $t=5\mathrm{mm}$ make the example's one-cycle error $2\mathrm{MPa}$. The material coefficient depends on the optical conditions, including wavelength; the primary photoelastic derivation explains that dependence and the analyzer's periodic intensity response.^[ref]

A sufficient local noise margin iswhen each phase error is bounded in magnitude by $ϵ$. The triangle inequality bounds the error in the phase increment by $2ϵ$, leaving it inside the nearest-branch interval. Equality gives no strict guarantee. Reducing the loading increment helps only if the assumed phase recovery and error bound are valid.

The Illinois polariscope provides controlled loading, a load readout and camera observation, which support a proposed check using smaller repeated load increments.^[ref] Agreement between a coarse sequence and a finer sequence tests branch stability over those measurements. It cannot establish an unknown common starting order. A lone crossed-polarizer intensity also does not supply a unique signed phase: the additional phase-recovery information assumed by the algorithm must be measured or independently justified.