Mechanical Properties of Hair 238 amplitude between one cycle and the next. The log decrement, Δ, is defined as: where a 1 and a n are the amplitudes of the first and the nth peak, respectively. The exponential decay of peak amplitude due to frictional damping is illustrated in Figure 7B. Δ is related to the shear loss modulus G’’ by: As with dynamic extensional measurements, the shear loss tangent can be defined and is given by: The log decrement and thus the shear loss tangent, tan(δ), will be increased by treatments or conditions that either increase frictional loss or decrease the storage of elastic potential energy as the fiber twists. Wolfram and Albrecht51 carried out torsional measurements on hair using a torsion pendulum. They concluded that the cuticle is very stiff in the dry state and may make a significant contribution to the torsional rigidity, especially for fine hairs. However, in the wet state the cuticle was found to be so plasticized as to make no contribution to torsional properties. Pesuad and Kamath52 of the Textile Research Institute have recently described a torsion pendulum for the study of hair. A photograph of their current device is shown in Figure 8. Torsional measurements of the shear modulus are further discussed below in the section on models for mechanical behavior. Bending is also a key component of hair style, but bending measurements are generally not simple to perform. Bending is a combination of compression and extension as illustrated in Figure 9. Because the extent of extension and compression at any given bending angle increases with the diameter of the object, bending strength depends on the fourth power of dimension. The bending strength of an elliptical hair fiber is given by E*I b where E is bending modulus and I b is the bending moment of inertia, given by (π/4)ab3,