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1.4  Determining the complex modulus

A number of methods and a few standards [], exist for the experimental determination of the complex modulus. Only the main test categories will be listed here

• traction/compression tests under sinusoidal excitation are used to measure the properties of materials that are sufficiently stiff to allow testing without combination of the material sample with metallic components. With significant experimental precautions, this technique has also been applied to films. Depending on the experimental setup, one will determine the complex modulus directly on isofrequencies or isotherms.
  
• Oberst [] and sandwich beams are used to determine the properties of a viscoelastic layer glued onto a metallic beam. Stiff materials work in traction compression in a free layer, while soft ones work in shear in a metal/visco/metal sandwich. Vibration analysis of the beam gives resonance frequencies and associated damping ratio. By inverse analysis of analytical [] or numerical solutions, one determines the complex modulus at the resonances under the modal damping assumption damp12.bib(see [])(see section ??). Using changes in the beam/plate dimensions and temperatures, one can obtain a large number of points on the nomogram.
  
• shear tests allow the direct determination of the complex modulus of films between two rigid surfaces. This test is more representative of material solicitation for sandwich structures. It is the only one applicable to test the effect of pre-stress in sandwich structures.

Building a test rig with no perturbing modes being quite difficult, modulus characterization is always performed on fairly narrow frequency bands. The frequency-temperature superposition hypothesis is thus made to create master curves over a wide frequency band. The next possible step is the determination of the coefficients of an analytical representation. Identification tools developed in control theory (ARMA models, ...) are suited for rational fraction models. It is however difficult to enforce a good reproduction of quantities that are typically judged as important : low and high frequency moduli, maximum loss factor, ... In other case a non linear optimization is readily implemented using optimization tools available in MATLAB.