Dynamic modulus


Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.

Viscoelastic stress–strain phase-lag

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force is applied to a material and the resulting displacement is measured.
Stress and strain in a viscoelastic material can be represented using the following expressions:
where
The stress relaxation modulus is the ratio of the stress remaining at time after a step strain was applied at time :
which is the time-dependent generalization of Hooke's law.
For visco-elastic solids, converges to the equilibrium shear modulus:
The fourier transform of the shear relaxation modulus is .

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, and.
Complex variables can be used to express the moduli and as follows:
where is the imaginary unit.

Ratio between loss and storage modulus

The ratio between the loss and storage modulus in a viscoelastic material is defined as the, which provides a measure of damping in the material. can also be visualized in vector space as the tangent of the phase angle between the storage and loss modulus.
Tensile:
Shear:
For example, a material with a greater than one will exhibit more damping than a material with a less than one. The reason that a material with a greater than one shows more damping is because the loss modulus of the material is greater than the storage modulus, which means the energy dissipating, viscous component of the complex modulus prevails as the defining factor of the material's characteristics.