Spectrum Representation
Viscoelastic materials exhibit a distribution of characteristic times over which they dissipate stress after a deformation. This distribution, known as the relaxation spectrum, is a fundamental material function that relates the molecular dynamics of a polymer to its macroscopic viscoelastic properties. It cannot be measured directly but must be calculated from experimental dynamic mechanical data such as the storage and loss moduli.
Mathematical Inverse
Calculation of the spectrum from experimental moduli is a classic ill-posed inverse problem that requires regularization to obtain a stable solution. The moduli are related to the spectrum through integral equations that act as a smoothing filter, which means that small amounts of experimental noise can produce large, spurious oscillations in the calculated spectrum. Solvers use methods like Tikhonov regularization or non-negative least squares to ensure that the computed spectrum is smooth and physically realistic.
The resulting spectrum is represented either as a discrete set of relaxation strengths and times or as a continuous function. This selection depends on the requirements of the downstream simulation or design task.
Calibration Protocol
To ensure that the input moduli are accurate enough for this mathematical reconstruction, the dynamic mechanical analyzer must be calibrated carefully. Mechanical compliance of the instrument frame must be compensated, as frame deformation can alter the measured phase angle and moduli, particularly for high-stiffness polymer specimens. Traceable reference standards are measured to verify the phase angle accuracy of the instrument before testing begins.
Operational Constraint
The calculated spectrum is only valid within the time or frequency window defined by the experimental data. Extrapolating the spectrum beyond this window is unreliable and can lead to incorrect predictions of long-term material behavior. For this reason, the experimental sweeps should cover as wide a frequency range as possible.
