Even if an object is completely isolated from the outside world, its mechanical vibration will gradually attenuate. This phenomenon, where mechanical energy is dissipated into heat energy, is called internal friction. It refers to the energy consumption caused by internal reasons during the solid vibration.

In English literature, “internal friction” is a common term used to describe this phenomenon. In engineering, the term “damping capacity” is also used. For high-frequency vibrations, it is referred to as “ultrasonic attenuation.” However, these terms represent the same physical process as internal friction.

Internal friction (damping) in solids is caused by their structural characteristics and defects. Measuring internal friction (damping) can sensitively reflect the internal structure and changes in movement and interactions of various structural defects.

This makes internal friction a useful tool for studying grain boundaries without damaging the material. By combining internal friction measurements with static observations, a deeper understanding of grain boundary properties and dynamic behavior can be obtained.

Relaxation and aftereffects are characteristics of inelasticity in the static process, while damping and internal friction are characteristics of inelasticity in the dynamic process. Inelasticity has a greater influence on the vibration process, so researchers often use internal friction (damping) measurements to study inelasticity rather than conducting experimental studies.

**1. Relationship between damping and stress-strain**

According to Hooke’s Law in elastic theory, the relationship between stress and strain of materials during elastic deformation is as follows:

Where M represents the elastic modulus E or shear modulus G.

The above formula must satisfy three conditions:

- The response of strain to stress is linear.
- The stress and strain phases are the same.
- Strain is a single-value function of stress.

However, in actual loading processes, it is not always possible to meet all three conditions simultaneously, resulting in non-ideal elasticity.

In such cases, damping occurs. Inelasticity is typically represented by anelasticity and viscoelasticity, with anelasticity being classified as either linear or nonlinear based on whether the stress-strain relationship is linear.

As a result, damping can also be categorized as linear and nonlinear anelastic damping and viscoelastic damping, as depicted in Figure 1.

Fig. 1 Stress Strain Loop

Fig. 2 Stress Strain Relationship under Periodic Stress

When the material is subjected to cyclic load, the actual relationship between stress and strain is as follows:

Where σ_{0} and ε_{0} are the amplitudes of stress and strain; t is the time; τ is the time when the strain waveform lags behind the stress waveform; ω is the angular frequency of vibration, φ is the phase angle difference of strain lag stress; T vibration period, as shown in Fig. 2.

According to the definition of complex modulus:

Among them, η is the loss factor of viscoelastic damping materials (also known as loss tangent or damping coefficient), which is one of the main indicators to measure the vibration energy dissipated by damping materials.

It is in direct proportion to the ratio of energy lost by weekly vibration to stored energy.

Expressed as:

Where, E * is the complex tensile modulus; E ‘is the real part of the complex tensile modulus, also known as the energy storage tensile modulus, which can be expressed as:

E “is the imaginary part of the complex tensile modulus, which determines the energy loss when the damping material is deformed under tension and compression, so it can be expressed as:

**2. Common parameters used to characterize the damping properties of materials and their relationships**

**(1) Loss factor η, loss tangent tan φ and loss angle φ**

The loss coefficient is the ratio of loss modulus to storage modulus, and its relationship with loss tangent and loss angle is as follows:

The greater the damping capacity of the material, the higher the phase difference angle. Thus, the phase difference angle φ can be utilized to characterize the material’s damping capacity.

However, in practical applications, if the internal friction is very low, measuring the phase difference angle can be challenging. Hence, this method is suitable for cases with significant internal friction.

**(2) Specific damping (S.D.C. or ψ)**

The material is subject to cyclic load, and the strain lags behind the stress, forming a hysteresis loop on the stress and strain curve, as shown in Fig. 1.

During one cycle of vibration, the energy loss ΔW is:

The maximum stored energy W is:

When researching high damping alloys, it is common practice to use “specific damping property” (SDC) to measure internal friction, which is typically expressed as ΔW/W.

In physics, to represent damping in correspondence with the damping electromagnetic circuit, Q⁻¹ is often used, where Q is the quality factor of the vibration system.

Similar to the definition of quality factor in electromagnetic circuit:

**(3) Logarithmic decrement δ**

Fig. 3 Free attenuation curve of vibration

In the process of free vibration, the vibration amplitude of the material will gradually attenuate, as shown in Fig. 3.

The faster the attenuation is, the higher the damping capacity of the material is.

The relationship between the damping performance of the material and the two adjacent amplitudes is as follows:

Further derivation shows that:

This applies to very small internal friction, i.e

The internal friction value is:

Where, δ is the logarithmic decay rate;

An and An+1 represent the amplitudes of the nth and the (n+1)th vibrations, respectively. Here, n represents the number of vibrations, which can take values 1, 2, 3, and so on.

It is apparent that the logarithmic decrement rate indicates the degree of amplitude attenuation. The higher the value of the logarithmic decrement rate, the greater the amplitude attenuation and, therefore, the better the damping performance.

This method belongs to the resonance method and is suitable for testing damping in the audio frequency range.

**(4) Reciprocal of quality factor Q ⁻ ¹**

Fig. 4 Resonance peak in forced vibration

The external force with different frequencies is used to excite the sample.

When the frequency of the external stress is equal to the resonance frequency of the sample, the vibration amplitude of the sample is the largest, as shown in Fig. 4.

In the same case, the higher the damping property of the material, the smaller the resonance amplitude and the wider the resonance peak.

Therefore, the sharpness of the resonance peak can be used to characterize the damping capacity of the material, that is, the damping of the material has the following relationship with the corresponding frequency difference and common frequency when the vibration amplitude is half of the resonance amplitude:

Where, Q ⁻ ¹ is the reciprocal of the quality factor;

Δ F is the frequency difference f2-f1 (Hz) at half of the resonance amplitude;

Fr is the resonant frequency value (Hz).

When the internal friction is small and the resonance peak is sharp, the width of the resonance peak is difficult to test;

The larger the internal friction, the wider the resonance peak, and the more accurate the measurement.

This method, like logarithmic decay rate, is suitable for testing audio frequency damping.

**(5) Ultrasonic attenuation**

The pulse method is commonly used to excite vibration in the megafrequency range.

The internal friction is measured by the attenuation of the pulse sound wave passing through the material. The attenuation coefficient β is defined as

Therefore, δ can be expressed as follows:

λ is the acoustic wave length, then:

**3. Conversion of internal friction (damping) measurement value and selection of measurement method**

In the case of small attenuation, tanφ<0.1, tanφ, Q⁻¹ or η are usually used to characterize the damping properties of materials, and the relationship between them is as follows (approximate formula):

However, when the damping is large (Q^{-1} ≥ 10 ^{-²}), there are two views: one is the exact expression:

According to the above formula, when Q ⁻ ¹= 10⁻ ² , then the error of the approximate formula is about 0.5%;

When Q ⁻ ¹= 10⁻ ¹, the error of the approximate formula is about 5%.

Another view was put forward by Zhu Xianfang and Shui Jiapeng:

According to the above formula, when Q ⁻ ¹= 5× 10⁻ ³, the error of the approximate formula reaches 1%;

When Q ⁻ ¹= 10⁻ ¹, the error of the approximate formula is more than 50%.

There is such a big deviation between the two arguments on this basic issue that they need to be clarified urgently.