Dynamic modulus

Dynamic modulus (sometimes complex modulus) is the ratio of stress to strain under "vibratory conditions" (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property of viscoelastic materials.
Viscoelastic stress–strain phase-lag.
Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.
Stress and strain in a viscoelastic material can be represented using the following expressions:
where 
The stress relaxation modulus formula_8 is the ratio of the stress remaining at time formula_6 after a step strain formula_10 was applied at time formula_11:
formula_12,
which is the time-dependent generalization of Hooke's law.
For visco-elastic solids, formula_8 converges to the equilibrium shear modulusformula_14:
The fourier transform of the shear relaxation modulus formula_16 is formula_17 (see below).
Storage and loss modulus.
The storage and loss modulus in viscoelastic materials measure the stored energy, representing the elastic portion, and the energy dissipated as heat, representing the viscous portion. The tensile storage and loss moduli are defined as follows:
Similarly we also define shear storage and shear loss moduli, formula_20 and formula_21.
Complex variables can be used to express the moduli formula_22 and formula_23 as follows:
where formula_26 is the imaginary unit.
Ratio between loss and storage modulus.
The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the formula_27, (cf. loss tangent), which provides a measure of damping in the material. formula_27 can also be visualized as the tangent of the phase angle (formula_7) between the storage and loss modulus. 
Tensile: formula_30
Shear: formula_31
For a material with a formula_27 greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.