Dynamic modulus

Dynamic modulus (sometimes complex modulus^[1]) 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.

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.^[2]

• In purely elastic materials the stress and strain occur in phase, so that the response of one occurs simultaneously with the other.
• In purely viscous materials, there is a phase difference between stress and strain, where strain lags stress by a 90 degree ( π / 2 {\displaystyle \pi /2} radian) phase lag.
• Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.^[3]

Stress and strain in a viscoelastic material can be represented using the following expressions:

• Strain: ε = ε 0 sin ⁡ ( ω t ) {\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}
• Stress: σ = σ 0 sin ⁡ ( ω t + δ ) {\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta )\,} ^[3]

where

ω = 2 π f {\displaystyle \omega =2\pi f} where f {\displaystyle f} is frequency of strain oscillation,

t {\displaystyle t} is time,

δ {\displaystyle \delta } is phase lag between stress and strain.

The stress relaxation modulus G ( t ) {\displaystyle G\left(t\right)} is the ratio of the stress remaining at time t {\displaystyle t} after a step strain ε {\displaystyle \varepsilon } was applied at time t = 0 {\displaystyle t=0} : G ( t ) = σ ( t ) ε {\displaystyle G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}} ,

which is the time-dependent generalization of Hooke's law. For visco-elastic solids, G ( t ) {\displaystyle G\left(t\right)} converges to the equilibrium shear modulus^[4] G {\displaystyle G} :

G = lim t → ∞ G ( t ) {\displaystyle G=\lim _{t\to \infty }G(t)} .

The Fourier transform of the shear relaxation modulus G ( t ) {\displaystyle G(t)} is G ^ ( ω ) = G ^ ′ ( ω ) + i G ^ ″ ( ω ) {\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )} (see below).

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.^[3] The tensile storage and loss moduli are defined as follows:

• Storage: E ′ = σ 0 ε 0 cos ⁡ δ {\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }
• Loss: E ″ = σ 0 ε 0 sin ⁡ δ {\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta } ^[3]

Similarly we also define shear storage and shear loss moduli, G ′ {\displaystyle G'} and G ″ {\displaystyle G''} .

Complex variables can be used to express the moduli E ∗ {\displaystyle E^{*}} and G ∗ {\displaystyle G^{*}} as follows:

E ∗ = E ′ + i E ″ {\displaystyle E^{*}=E'+iE''\,}

G ∗ = G ′ + i G ″ {\displaystyle G^{*}=G'+iG''\,} ^[3]

where i {\displaystyle i} is the imaginary unit.

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the tan ⁡ δ {\displaystyle \tan \delta } , (cf. loss tangent), which provides a measure of damping in the material. tan ⁡ δ {\displaystyle \tan \delta } can also be visualized as the tangent of the phase angle ( δ {\displaystyle \delta } ) between the storage and loss modulus.

Tensile: tan ⁡ δ = E ″ E ′ {\displaystyle \tan \delta ={\frac {E''}{E'}}}

Shear: tan ⁡ δ = G ″ G ′ {\displaystyle \tan \delta ={\frac {G''}{G'}}}

For a material with a tan ⁡ δ {\displaystyle \tan \delta } greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.

• Dynamic mechanical analysis
• Elastic modulus
• Palierne equation