Relaxation Spectrum Determination in Viscoelastic Materials

Viscoelastic materials exhibit time-dependent mechanical behaviour that combines both elastic spring-like response and viscous flow. Central to their characterisation is the relaxation spectrum, a distribution of discrete or continuous relaxation times (or frequencies) that encapsulates how stress decays under a step strain or how storage and loss moduli vary with frequency. Although the relaxation spectrum underpins constitutive models in polymer science, biomechanics, geophysics and advanced composites, it cannot be measured directly. Instead, it must be inferred from stress relaxation experiments or oscillatory shear tests, leading to an ill-posed inverse problem sensitive to experimental noise. To address this challenge, researchers have developed a range of approaches: discrete-element models such as the Maxwell or generalized Maxwell networks; stretched-exponential (Kohlrausch–Williams–Watts) formulations; and fully continuous representations expanded in orthonormal or power-exponential basis functions. Regularisation methods—most notably Tikhonov regularisation—are routinely applied to stabilise the inversion and suppress artefacts. Recent progress includes hierarchical schemes that optimise both model parameters and time-scale factors, as well as direct spectral identification frameworks that minimise an error measure between the unknown true spectrum and its finite-basis approximation. Collectively, these advances have improved the robustness of relaxation spectrum recovery, enabling more accurate prediction of long-term performance in applications spanning polymer processing, tissue mechanics and energy-dissipative materials.

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Relaxation Spectrum Determination in Viscoelastic Materials publication trend

The graph below shows the total number of articles in relaxation spectrum determination in viscoelastic materials across all publications each year (not limited to Nature Index journals).

Technical terms

Viscoelasticity: Material behaviour that exhibits both viscous flow and elastic deformation under load.

Relaxation spectrum: Distribution of characteristic times or frequencies governing the decay of stress following a deformation.

Relaxation modulus: Time-dependent stress response resulting from a sudden, unitary strain input.

Inverse problem: Mathematical procedure to infer the relaxation spectrum from observed mechanical data.

Tikhonov regularisation: Stabilisation technique that adds a penalty term to the inversion to mitigate ill-posedness.

Basis functions: Preselected functions (e.g. orthonormal polynomials or exponentials) used to represent the unknown spectrum in a finite series.

References

1. A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions. Polymers (2023).
2. Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor. Materials (2023).
3. Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials. Materials (2024).