Question 6 of 9: Textural Profile Analysis, the Maxwell Stress-Relaxation Model, and Brittle Fracture in Foods

Question 6: Textural Profile Analysis, the Maxwell Stress-Relaxation Model, and Brittle Fracture in Foods (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Effect of sample size and compression rate. Hardness (the peak force of the first compression, or the corresponding stress if expressed per unit area) rises with the sample's diameter-to-height ratio at a FIXED strain, because a squatter, wider cylinder deforms with proportionally more barrelling and more frictional restraint at the platens for the same fractional height reduction, so more force is needed to achieve the same strain; if the same test is instead read as a stress (force/original cross-sectional area) the effect is smaller but not eliminated, since the friction/barrelling contribution does not scale away with area alone. Cohesiveness (the ratio of the second-cycle to first-cycle positive work) is comparatively insensitive to sample size within a normal testing range, because it is a ratio of two areas measured on the SAME sample geometry and largely cancels the geometric effect — but it is not perfectly size-independent if the diameter-to-height ratio is pushed far enough that friction changes the FAILURE mode between the two cycles differently. Elasticity (springiness, the height the sample recovers between the two cycles, in cm) scales directly with the sample's own height, since it is reported as an absolute recovered distance rather than a ratio — a taller sample of the identical material will show more elastic recovery in cm at the same strain purely because there is more material to recover, so elasticity comparisons across differently sized samples must be normalized (recovered height / original height) to be meaningful. Rate of compression does affect all three parameters for a viscoelastic food, because a faster compression probes a shorter relaxation-time window of the material's viscoelastic spectrum: hardness typically rises with increasing compression rate (the material has less time to relax stress within the cycle, so peak force is higher), while cohesiveness and elasticity can fall at high rates if the faster test drives the sample into a different (more fracture-dominated) failure mode than the slower test.

(b) Maxwell model for stress relaxation. The Maxwell model represents a viscoelastic food as a spring (elastic modulus E) and a dashpot (viscosity η) in SERIES, so the same stress acts on both elements while the strains add. Under a step strain held constant, the model predicts an exponential stress decay, $$\begin{aligned} \sigma(t) &= \sigma_0\, e^{-t/\lambda}, \\ \lambda &= \frac{\eta}{E}, \end{aligned}$$ where \(\lambda\) is the relaxation time (the time for stress to fall to \(1/e\) of its initial value) and \(\sigma_0\) is the instantaneous stress at \(t=0^+\). To analyse real stress-relaxation data, the food's actual decay curve (which is rarely a single clean exponential) is fitted as a sum of several Maxwell elements in parallel (a generalized Maxwell model), $$\sigma(t) = \sum_i \sigma_{0,i}\,e^{-t/\lambda_i} + \sigma_\infty,$$ where the residual equilibrium stress \(\sigma_\infty\) captures any permanent, non-relaxing elastic component; each \(\lambda_i\) and its weight are extracted by successive-residual fitting (subtracting the slowest-decaying exponential's tail first, then fitting the residual, and so on) — a small number of relaxation times (2–3) is usually enough to fit most food stress-relaxation curves well.

(c) Non-fracture causes of brittle behaviour. A food can present as "brittle" — a sudden, large stress drop under compression — without a true propagating fracture crack in at least three other ways: (i) buckling of a thin-walled or cellular structure (a wafer, a puffed extrudate cell wall) collapsing suddenly under compressive load, which reads as a sharp force drop on the texture curve but is a structural instability, not a crack; (ii) internal air-cell collapse in an aerated/porous product (an extruded snack, a meringue), where the sudden escape or crushing of entrapped air pockets produces the same audible/force-curve "snap" signature as fracture; and (iii) a sudden slip or adhesive failure at the sample–platen interface (the sample slipping rather than continuing to compress) can be mistaken for brittle fracture if the test record is read from the force trace alone without also checking the deformation record for a genuine discontinuity in the sample's own geometry.