## Dynamical Mechanical Analysis

The dynamical mechanical analysis (DMA) is one of the most important tools in the polymer laboratory. The idea to apply oscillatory deformations and to measure the mechanical response goes back to Poynting (1909), who probably conducted the first oscillatory experiment to measure the elasticity of a material. The dynamical mechanical analysis found little use until the late sixties when commercial instruments became more user-friendly. These early instruments were slow, and limited in their ability to process data. Today, several companies have developed very powerful DMA's. They are mainly used to measure the moduli and the glass transition temperature.

The DMA technique is based on a rather simple principle; when a
sample is subjected to a sinusoidal oscillating stress, its response
is a sinusoidal oscillation with similar frequency provided the
material stays within its elastic limits. When the material responds
to the applied oscillating stress perfectly elastically, the
responding strain wave is in-phase (*storage or elastic response*),
while a viscous material responds with an out-of-phase strain wave (*loss or a
viscous response*). For a Newtonian liquid the phase angle
will be 90 degrees and for Hookean solid it will be 0 degrees, whereas
the phase angle of viscoelastic material falls in between these two extremes.

### Dynamic Mechanical Analysis

For any point on the stress curve, the applied stress is described as

*σ*(*t*) = *σ _{0}* sin(

*ω t*)

where *σ _{0}* is the maximum stress, and

*ω*is the frequenzy of the oscillation. The rate of stress is the derivative of the above equation in terms of time:

d*σ*/d*t* = *ω σ _{0}* cos(

*ω t*)

The two extremes of a material are the pure elastic and
pure viscous response. The behavior can be understood by looking at each of the two extremes.
A material at the spring-like or *Hookean limit* responds elastically with the oscillating
stress. The strain at any time is:

*ε*(*t*) = *ε _{0}* sin(

*ω t*)

where *ε _{0}* is the maximum amplitude of the strain and

*ω*is the frequenzy of the oscillation.

In the case of the *viscous limit*, the stress is proportional to the strain
rate which is the first derivative of the strain:

d*ε*/d*t* = *ω ε _{0}* cos(

*ω t*) =

*ε*sin(

_{0}*ω t*+ π/2),

and when multiplied by the material's viscosity:

*σ* = *η* d*ε*/d*t* = *ω η ε _{0}* cos(

*ω t*) =

*σ*sin(

_{0}*ω t*+ π/2)

The behavior of viscoelastic materials lies between these two limits, that is, the difference between the applied stress and the resultant strain is an angle,
*δ*

*ε*(*t*) = *ε _{0}* sin(

*ω t*+

*δ*)

This equation can be rewritten as

*ε*(*t*) = *ε _{0}* sin(

*ω t*) cos(

*δ*) +

*ε*cos(

_{0}*ω t*) sin(

*δ*) =

*ε'*sin(

*ω t*) +

*ε''*cos(

*ω t*)

or

*σ*(*t*) = *ε _{0}*

*E*[sin(

^{*}*ω t*) cos(

*δ*) + cos(

*ω t*) sin(

*δ*)] =

*ε*sin(

_{0}*ω t*)

*E'*+

*ε*cos(

_{0}*ω t*)

*E''*

*ε _{0}* cos(

*δ*) and

*ε*sin(

_{0}*δ*) are the in- and out-of-phase strain (

*ε*',

*ε*''),

*E**is the ratio of maximum stress

*σ*to maximum strain

_{0}*ε*, and

_{0}*σ*cos(

_{0}*δ*)/

*ε*and

_{0}*σ*sin(

_{0}*δ*)/

*ε*are the storage and loss modulus (

_{0}*E*',

*E*''). The two moduli are also known as the

*elastic*and

*viscous modulus*. The vector sum of the two components gives the overall or complex strain on the sample (modulus):

*ε*^{*} = *ε*' + i*ε*''

*E*^{*} = *E*' + i*E*''

The ratio of the loss to the storage modulus is the phase angle and is called damping:

tan(*δ*) = *E*'' / *E*'

The same relations can be applied when the material is subjected to shear
forces. In this case, the Young's modulus *E* has to be substituted by the shear
modulus *G*, and the tensile strain *ε* by the shear strain
*γ*:

*γ*(*t*) = *γ'* sin(*ω t*) + *γ''* cos(*ω t*)

*σ*(*t*) = *γ _{0}* sin(

*ω t*)

*G'*+

*γ*cos(

_{0}*ω t*)

*G''*

The later equation can be also expressed in terms of the real and imaginary viscosity:

*σ*(*t*) = *ω γ _{0}* sin(

*ω t*)

*η''*+

*ω γ*cos(

_{0}*ω t*)

*η'*

and

*η** = (*η*''^{2} + *η*'^{2})^{1/2} =
[(*G*''/*ω*)^{2} + *G*'/*ω*)^{2}]^{1/2} = |*G**| / ω

where *η*' = *G*''/*ω* is the in-phase or real component of the viscosity which is often called the
*dynamic* or *absolute*
viscosity, and *η*'' = *G*'/*ω* is the
out-of-phase or *imaginary* component of the complex viscosity.

##### References

- J. H. Poynting,
*Proc. Royal Soc.*, Series A, 82, 546 (1909).