Question 9 of 9: Stress Relaxation Analysis by the Successive-Residual Method (3-Term Maxwell Model)

Question 9: Stress Relaxation Analysis by the Successive-Residual Method (3-Term Maxwell Model) (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.

Given. A food material is subjected to a stress-relaxation test — a step strain ε₀ applied instantaneously at t = 0 and held constant, while the resulting stress σ(t) (equivalently the relaxation modulus E(t) = σ(t)/ε ₀) is recorded as it decays with time — and the material's behaviour is to be represented by a generalized (3-element) Maxwell model, $$E(t)=E_1e^{-t/\lambda_1}+E_2e^{-t/\lambda_2}+E_3e^{-t/\lambda_3}$$ with $\lambda_1 < \lambda_2 < \lambda_3$, i.e. six unknown parameters (three moduli E_i, three relaxation times λ_i).

Find. A procedure to extract all six parameters from the single measured E(t) curve.

Approach. Exploit the fact that at sufficiently long times only the slowest-relaxing (largest λ) term still contributes measurably, and peel the terms off the data one at a time from slowest to fastest — the successive residual (or "peeling-off") method.

1. Fit the longest-time tail. Plot ln E(t) against t. At large t, the two faster terms (λ₁, λ₂) have decayed to negligible size, so the curve becomes a straight line governed by the slowest term alone: ln E(t) ≈ ln E₃ − t/λ₃. A least-squares line through the last several points gives the slope (−1/λ₃) and intercept (ln E₃), hence E₃ and λ₃.
2. Subtract the fitted term and repeat. Compute the first residual, $$R_1(t)=E(t)-E_3e^{-t/\lambda_3},$$ over the full data range. R₁(t) is now governed, at its own longest remaining times, by the second-slowest term alone: ln R₁(t) ≈ ln E₂ − t/λ₂. Fitting a straight line to the mid-time tail of R₁ gives E₂ and λ₂.
3. Subtract again for the fastest term. Compute the second residual, $$R_2(t)=R_1(t)-E_2e^{-t/\lambda_2}=E(t)-E_3e^{-t/\lambda_3}-E_2e^{-t/\lambda_2},$$ which over the short-time data is governed by the fastest term alone: ln R₂(t) ≈ ln E₁ − t/λ₁. A line fit through the earliest points gives E₁ and λ₁, completing all six parameters.
4. Check. Sum all three fitted terms and compare against the original E(t) data over its full range; a good 3-term fit leaves a residual that is small and structureless (noise-level) everywhere — a systematic remaining curvature indicates a fourth relaxation mode is needed and the model should be extended, not forced to fit with only three terms.