## 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) = σ0 sin(ω t)

where σ0 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:

dσ/dt = ω σ0 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) = ε0 sin(ω t)

where ε0 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:

dε/dt = ω ε0 cos(ω t) = ε0 sin(ω t + π/2),

and when multiplied by the material's viscosity:

σ = η dε/dt = ω η ε0 cos(ω t) = σ0 sin(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) = ε0 sin(ω t + δ)

This equation can be rewritten as

ε(t) = ε0 sin(ω t) cos(δ) + ε0 cos(ω t) sin(δ) = ε' sin(ω t) + ε'' cos(ω t)

or

σ(t) = ε0 E* [sin(ω t) cos(δ) + cos(ω t) sin(δ)] = ε0 sin(ω t) E' + ε0 cos(ω t) E''

ε0 cos(δ) and ε0 sin(δ) are the in- and out-of-phase strain (ε', ε''), E* is the ratio of maximum stress σ0 to maximum strain ε0, and σ0 cos(δ)/ε0 and σ0 sin(δ)/ε0 are the storage and loss modulus (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' + iε''

E* = E' + iE''

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) = γ0 sin(ω t) G' + γ0 cos(ω t) G''

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

σ(t) = ω γ0 sin(ω t) η'' + ω γ0 cos(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
1. J. H. Poynting, Proc. Royal Soc., Series A, 82, 546 (1909).
• ## Summary

### Dynamical Mechanical Analysis

The stress in a sample is measured as function of an oscillatory deformation (strain).

• The tensile or shear strain is a sinusoidal function of time.

• DMA instruments are mainly used to measure moduli and glass transition temperatures of amorphous or semi-amorphous materials (polymers).

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