Anelasticity and internal friction

5.5 Anelasticity and internal friction

For an elastic solid it is generally assumed that stress and strain are directly proportional to one another, but in practice the elastic strain is usually dependent on time as well as stress, so that the strain lags behind the stress; this is an anelastic effect. On applying a stress at a level below the

conventional elastic limit, a specimen will show an initial elastic strain ε e followed by a gradual increase in strain until it reaches an essentially constant value, ε e +ε an , as shown in Figure 5.8. When the stress is removed the strain will decrease, but a small amount remains, which decreases slowly with time. At any time t the decreasing anelastic strain is given by the relation ε =ε an exp( −t/τ), where τ is known as the relaxation time, and is the time taken for the anelastic strain to decrease to 1/e ∼ = 36.79% of its initial value. Clearly, if τ is large, the strain relaxes very slowly, while if small the strain relaxes quickly.

In materials under cyclic loading this anelastic effect leads to a decay in amplitude of vibration and therefore a dissipation of energy by internal friction. Internal friction is defined in several different but related ways. Perhaps the most common uses the logarithmic decrement δ = ln (A n / A n +1 ), the natural logarithm of successive amplitudes of vibration. In a forced vibration experiment near a resonance, the factor (ω 2 −ω 1 )/ω 0 is often used, where ω 1 and ω 2 are the frequencies on the two sides of √

the resonant frequency ω 0 at which the amplitude of oscillation is 1/

2 of the resonant amplitude.

252 Physical Metallurgy and Advanced Materials

␧ an ␧ e

Strain ␧ e ␧

an

Figure 5.8 Anelastic behavior.

vibrational energy E, i.e. the area contained in a stress–strain loop. Yet another method uses the phase angle α by which the strain lags behind the stress, and if the damping is small it can be shown that

By analogy with damping in electrical systems, tan α is often written equal to Q −1 . There are many causes of internal friction arising from the fact that the migration of atoms, lattice defects and thermal energy are all time-dependent processes. The latter gives rise to thermoelasticity and occurs when an elastic stress is applied to a specimen too fast for the specimen to exchange heat with its surroundings and so cools slightly. As the sample warms back to the surrounding temperature it expands thermally, and hence the dilatation strain continues to increase after the stress has become constant.

The diffusion of atoms can also give rise to anelastic effects in an analogous way to the diffusion of thermal energy giving thermoelastic effects. A particular example is the stress-induced diffusion of carbon or nitrogen in iron. A carbon atom occupies the interstitial site along one of the cell edges, slightly distorting the lattice tetragonally. Thus, when iron is stretched by a mechanical stress, the crystal axis oriented in the direction of the stress develops favored sites for the occupation of the interstitial atoms relative to the other two axes. Then if the stress is oscillated, such that first one axis and then another is stretched, the carbon atoms will want to jump from one favored site to the other. Mechanical work is therefore done repeatedly, dissipating the vibrational energy and damping out the mechanical oscillations. The maximum energy is dissipated when the time per cycle is of the same order as the time required for the diffusional jump of the carbon atom.

The simplest and most convenient way of studying this form of internal friction is by means of a K ˆe torsion pendulum, shown schematically in Figure 5.9. The specimen can be oscillated at a given

varies smoothly with the frequency according to the relation:

E E ωτ

E =2 E max 1 + (ωτ) 2

and has a maximum value when the angular frequency of the pendulum equals the relaxation time of the process; at low temperatures around room temperature this is interstitial diffusion. In practice, it is difficult to vary the angular frequency over a wide range and thus it is easier to keep ω constant and vary the relaxation time. Since the migration of atoms depends strongly on temperature according

to an Arrhenius-type equation, the relaxation time τ 1 = 1/ω 1 and the peak occurs at a temperature T 1 . For a different frequency value ω 2 the peak occurs at a different temperature T 2 , and so on

Physical properties 253

Pin chuck

Wire specimen

Weight

Torsion bar

Mirror

Figure 5.9 Schematic diagram of a K ˆe torsion pendulum.

Temperature, °C 1.00 100 80 60 40 20 0 ⫺10

Frequency 0.8 Hz

A 2.1 B 1.17

0.6 C 0.86 D 0.63 E 0.27

0.4 A

Internal friction, normalized

Figure 5.10 Internal friction as a function of temperature for Fe with C in solid solution at five different pendulum frequencies ( from Wert and Zener, 1949; by permission of the American

Physical Society).

producing the damping by plotting ln τ versus 1/T , or from the relation: ln(ω 2 /ω 1 )

H =R

1/T 1 − 1/T 2 In the case of iron the activation energy is found to coincide with that for the diffusion of carbon in

iron. Similar studies have been made for other metals. In addition, if the relaxation time is τ the mean

254 Physical Metallurgy and Advanced Materials time an atom stays in an interstitial position is ( 3 2 1 )τ, and from the relation D = 24 a 2 v for bcc lattices

derived previously the diffusion coefficient may be calculated directly from

Many other forms of internal friction exist in metals arising from different relaxation processes to those discussed above, and hence occurring in different frequency and temperature regions. One important source of internal friction is that due to stress relaxation across grain boundaries. The occurrence of a strong internal friction peak due to grain boundary relaxation was first demonstrated on polycrystalline aluminum at 300 ◦

C by K ˆe and has since been found in numerous other metals. It indicates that grain boundaries behave in a somewhat viscous manner at elevated temperatures and grain boundary sliding can be detected at very low stresses by internal friction studies. The grain boundary sliding velocity produced by a shear stress τ is given by ν = τd/η and its measurement gives values of the viscosity η which extrapolate to that of the liquid at the melting point, assuming the boundary thickness to be d ∼ = 0.5 nm.

Movement of low-energy twin boundaries in crystals, domain boundaries in ferromagnetic materials, and dislocation bowing and unpinning all give rise to internal friction and damping.