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Finite-shell bias in boundary-density calibration is the difference between a density averaged through a layer of nonzero thickness and the local density at an erosion boundary. Making a layer thinner reduces spatial averaging, but can increase relative uncertainty in the small removed mass and volume. Boundary-density calibration for erosion fronts must manage both effects.

For an original spherical model with outer radius $R$, sampled thickness $\delta$ and smooth radial density $\rho (r)$,Write $s=R-r$. Expanding $\rho (R-s)=\rho (R)-s{\rho }^{\prime }(R)+O(s^2)$ and the volume weight $(R-s{)}^2$ givesThe leading bias is set by the local density gradient. Curvature changes higher-order terms; a uniform density has no such averaging bias, even for a thick shell.

A first-order polymer erosion front speed conversion uses $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R)$. Replacing ${\rho }_s$ by ${\rho }_{\delta }$ givesA density increasing toward the surface therefore leads to overestimated speed when the sampled layer has lower mean density. This calculation assumes a genuine moving front and negligible independent interior density change.

An original local planning model makes the thickness tradeoff explicit. Approximate the sampled area by $A$, the density-gradient magnitude by $|g|$, and the standard uncertainty of removed mass by a thickness-independent $u_M$. Neglect other uncertainties. Squared bias plus variance is approximatelyMinimization givesFor stipulated $A=1\mathrm{c}{\mathrm{m}}^2$, $|g|=0.2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-4}$ and $u_M=0.0001$ g, ${\delta }_{\ast }\approx 0.0316$ cm, or 0.316 mm. At that thickness, bias magnitude and measurement standard deviation are each about $0.00316\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. This is a conditional planning result, not a universal required slice thickness.

Layer-thickness tradeoff in erosion-density sampling also depends on uncertain layer thickness, geometry, conditioning and the accuracy of the gradient model. Shared balance corrections can correlate successive mass differences; the GUM requires those covariance contributions to be included.^[ref]

A proposed check compares several layer thicknesses and looks for a stable extrapolation toward zero thickness, with matched drying and dimensional protocols. A changing gradient or damage caused by sectioning can invalidate the extrapolation. Kureha's polyglycolic acid experiment measures filtered, dried retained solids after water exposure.^[ref] A local shell-density calibration would be an additional measurement, not a quantity already determined by that residue curve.

Nonuniform-density erosion inference estimates a polymer erosion front speed when local density varies through a specimen. Total mass divided by total volume gives an average density; the material being removed at the boundary can have a different density. Substituting the average into a surface-removal formula can therefore misestimate the speed.

For an original spherically symmetric model, let $R(t)$ be the surviving radius and $\rho (r,t)$ the local mass density. ThenDifferentiating the integral and defining positive inward speed $v=-R^{\prime }$ givesThis radial form of the Reynolds transport theorem separates loss at a moving boundary from density change inside the surviving volume. All quantities must refer to the same material and conditioning state; a water-filled wet volume and a subsequently dried polymer mass do not automatically define the local density used here.

If the interior density profile is stationary in spatial coordinates over the measurement interval, the integral term vanishes. The appropriate boundary conversion is then $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R,t)$. Writing average density as $\overline{\rho }=3M/(4\pi R^3)$, a uniform-density substitution instead givesThe ratio is measurable information about the profile, not a shape correction already supplied by knowing the outer radius.

An illustrative fixed profile is $\rho (r)={\rho }_0[1+(r/R_0{)}^2]$, with $R_0$ a fixed initial radius, not the changing current radius. At $R=R_0$, the boundary density is $2{\rho }_0$ and the mean is ${\rho }_0(1+3/5)=1.6{\rho }_0$. The uniform-density inference overestimates the instantaneous front speed by 25%. For $R_0=1$ cm, ${\rho }_0=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ and true $v=0.001\mathrm{cm}\mathrm{da}{\mathrm{y}}^{-1}$, mass leaves at $0.008\pi \approx 0.02513\mathrm{g}\mathrm{da}{\mathrm{y}}^{-1}$; dividing by the mean-density surface factor gives $0.00125\mathrm{cm}\mathrm{da}{\mathrm{y}}^{-1}$. These are stipulated profile values, not measured properties of a named polymer.

Time-dependent hydration, porosity or chemical change can make the interior integral nonzero. One mass slope cannot independently identify both that contribution and boundary speed. Boundary-density calibration for erosion fronts therefore needs spatial or compositional evidence in addition to a bulk mass measurement. NIST's mass-density definition is mass divided by volume; it does not equate a whole-specimen average with an unknown boundary value.^[ref]

A discriminating experiment measures outer dimensions, conditioned mass and density or composition at several depths, using matched specimens or accounting for destructive sampling. Kureha's polyglycolic acid protocol filters and dries the remaining solids after water exposure.^[ref] That observation supplies retained dry mass, not a radial density field. Establish an inward-moving degradation boundary before interpreting the inferred speed as polymer erosion, rather than another cause of material loss.