Viscosity and Oscillatory Rheology - Practical Food Rheology - page 19

The phase lag, δ , and the amplitude ratio depend on the material

is 0 ◦ for purely elastic materials (no phase lag) and 90 ◦
for purely viscous materials (out of phase due to viscous losses).
Equation 2.12 can be written as follows:

being tested; δ

σ = σ 0 cos δ sin ω t + σ 0 sin δ cos ω t (2.13)

This allows for the following quantities to be defined:

G = σ 0 cos

δ in-phase stress amplitude

strain amplitude

= (2.14)

γ 0

G = σ 0 sin

δ out-of-phase stress amplitude

strain amplitude

= (2.15)

γ 0

and

G
G

tan δ = (2.16)

and therefore the stress response can be written as follows:

G γ G γ σ = 0 sin ω t + 0 cos ω t (2.17)

where G is the storage modulus and G is the loss modulus. The storage
modulus is an indicator of the degree of elasticity of the material, and
G is a measure of the degree of viscous behaviour. A large value of G ,
in comparison to that for G , indicates that the product being analysed
has predominantly elastic properties. The tan

is directly related to the
energy lost, per cycle, divided by the energy stored per cycle; because
this can vary from zero to infinity, 0 ◦ ≤ δ ≤

δ

90 ◦ .
Fig. 2.6 shows a typical viscoelastic behaviour as identified by a
frequency sweep within the LVR. Frequencies are an indicator of the
time scale of process, i.e. the higher the frequency, the shorter the time
scale of the process and vice versa.
The crossover frequency is the frequency where the G and G curves
intersect, i.e. the frequency at which the elastic and viscous responses are
equal. The crossover frequency is an important rheological parameter
and is inversely proportional to the relaxation time (the time at rest
required for the sample to relax a stress received from an external body),
i.e.:

1
relaxation time

f crossover ∝ (2.18)