Relationship Between Static and Dynamic Properties

Still another relationship between experimental parameters is a direct consequence of the Boltzmann superposition principle. We will derive the equations relating the shear stress relaxation modulus G(t) to the in-phase and out-of-phase dynamic shear moduli G oi) and G (co) starting from equation (2-46) 

Consider the application of a sinusoidal strain, which may be represented by 

It is apparent that equation (2-56) and equation (2-57) embody a Fourier sine and cosine transformation of G(t) thus normal Fourier transform8 methods 

APPENDIX 1 CONNECTING CREEP COMPLIANCE AND STRESS RELAXATION MODULUS USING LAPLACE TRANSFORMS 

At sufficiently low strains, the Boltzmann superposition principle allows us to express moduli and compliances in terms of one another even in the time-dependent case. We will derive these relationships directly from equations (2-45) and (2-46) with the aid of Laplace transforms. A short introduction or review of Laplace transform techniques will be presented first. 

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