1. Introduction
Soft materials exhibit both viscous (damping) and elastic (stiffness) characteristics [
1,
2,
3,
4]. Quantification of the viscoelastic properties of soft materials is essential in numerous science and engineering applications [
5,
6,
7,
8,
9,
10,
11,
12]. Furthermore, next to elasticity, damping (or viscosity) could be an additional, relevant, diagnostic biomarker, and viscosity could enhance the current diagnosis in quantitative elastography [
13,
14,
15,
16,
17,
18,
19,
20,
21,
22]. Briefly, damping is the removal of energy from a system, and the energy can be either dissipated within the system or transmitted away by radiation [
23]. It should be remembered that material damping is the energy dissipation due to deformation in a medium, and radiation damping is the energy transfer to a surrounding medium [
23,
24]. In addition, the energy in a system can be dissipated, for example, via the friction between different parts in the system and air resistance [
25]. The properties of a structure such as mass and stiffness can be determined by preforming some static tests. However, identifying the damping of a structure or system requires a dynamic test [
26]. In general, both theoretical modelling and experimental identification of damping is quite difficult [
24,
27,
28,
29]. There are many research papers on determining the damping of materials, including biomaterials (e.g., [
30,
31,
32,
33,
34,
35,
36,
37]). The literature survey shows that there are different techniques for the identification of damping (e.g., dynamic indentation method, logarithmic decrement method, rheometry), and each method gives a different damping parameter, such as loss factor, loss modulus and viscous damping ratio [
23,
26,
38,
39,
40,
41,
42,
43]. The identification of the damping of conventional materials (such as ceramics and metals) is quite straightforward, and loss factor or viscous damping ratio is commonly used to quantify their damping [
44]. On the other hand, the identification of the damping of soft materials (e.g., agar, gelatine and collagen phantoms and tissue) is challenging, and different damping parameters such as loss modulus, loss angle, viscous damping ratio, or viscosity are used to describe their damping [
30,
34,
36,
45,
46].
Regarding the identification of the damping of materials, Nayar et al. [
30] used the dynamic indentation method to determine the storage and loss moduli of some samples of agar which is a representative material for biological tissues. Similarly, using the dynamic indentation method, Vriend et al. [
47] determined the viscoelastic properties of some elastomeric materials and Boyer et al. [
48] assessed the stiffness and damping of skin. Dakhil et al. [
31] identified the storage and loss moduli of cells using a rheometer. Peng et al. [
32] determined the dilute solution viscosities of some cellulose nanocrystal dispersions using a capillary viscometer. Wang et al. [
33] identified the viscous damping ratios of some beam-like hydrogel samples via resonant vibration tests. Esmaeel et al. [
36] determined the viscous damping coefficient of soft tissue by calculating the dissipated energy per cycle of harmonic motion by the material and the maximum stored energy in the sample using the displacement-force curve. Rosicka et al. [
49] identified the biomechanical and viscoelastic properties of skin, including the logarithmic decrement values. Based on the mathematical models for the dynamic response of a microbubble located at the soft medium interface [
50,
51,
52], Bezer et al. [
34] determined the shear modulus and viscosity of a tissue-mimicking gelatine phantom by matching the experimentally measured and predicted responses of the microbubble located at the soft medium interface exposed to ultrasound. Similarly, using the mathematical models for the dynamic response of a sphere located at the soft medium interface [
53,
54,
55], the shear modulus and viscous damping ratio of tissue-mimicking gelatine phantoms were identified by matching the experimentally measured and predicted responses of the sphere located at the soft medium interface [
37,
56,
57].
Li et al. [
58] presented the viscoelasticity imaging of biological tissues and single cells using shear wave propagation, including examples of ultrasound shear wave viscoelasticity imaging applications. Beuve et al. [
59] investigated the diffuse shear wave spectroscopy for the characterisation of the viscoelastic properties (shear modulus and viscosity) of soft tissue. Tecse et al. [
60] developed and validated a method for the characterisation of the viscoelastic properties of soft tissue using ultrasound elastography. Wang et al. [
61] investigated the effect of damping on ultrasound elastography algorithms. Koruk and Pouliopoulos [
62] presented the elasticity and viscoelasticity imaging based on the use of small particles located within the tissue and at the tissue interface exposed to static and dynamic external forces. Hirsch et al. [
45] measured the shear modulus and loss angle of liver and spleen using magnetic resonance elastography. Wang et al. [
63] derived the shear wave speed and loss angle for depicting hepatic fibrosis and inflammation in chronic viral hepatitis using magnetic resonance elastography.
Overall, the literature review shows that the dynamic indentation method [
30], rheometry and viscometry [
31,
32,
64], atomic force microscopy [
65], hysteresis loop [
36], resonant vibration tests or experimental modal analysis [
33,
66], and logarithmic decrement [
49,
67] are commonly used to identify the damping of materials, including soft materials. In addition, a bubble or sphere placed inside the soft medium or located at the soft medium interface exposed to an external excitation such as acoustic radiation force or magnetic force has been recently used to identify the viscoelastic properties of soft materials [
34,
37,
68,
69,
70,
71,
72]. The ultrasound elastography [
17,
73,
74,
75], and magnetic resonance elastography [
76,
77,
78,
79] for determining tissue mechanical properties are quite common for the preclinical and clinical applications. It is seen that there are many parameters for the description and quantification of damping. The viscous damping ratio, loss factor, complex modulus (or storage and loss moduli), and viscosity are quite common to describe the damping of materials. In addition, some other parameters such as the specific damping capacity, phase lag or loss angle, half-power bandwidth, logarithmic decrement, and inverse quality factor are used to describe the damping of various materials. Often one of these parameters (e.g., loss factor) is measured in practical applications, and for comparison purposes the measured damping parameter needs to be converted into some other damping parameters (e.g., to viscosity). However, there is a limited number of studies that evaluated only few different damping parameters and presented their relationships [
38,
80]. Therefore, there is a need for a comprehensive study that presents the theoretical derivations of different damping parameters and their relationships.
This paper presents theoretical derivations of different parameters for the description and quantification of damping and their relationships, as well as the methods for damping identification. In this paper, the expressions for both high damping (i.e., accurate formulas) and low damping (i.e., approximate formulas) are presented and these approaches are evaluated. The structure of this paper is as follows. First, the elastic, viscous and viscoelastic materials are defined, and then the responses of single-degree-of-freedom (SDOF) systems with a viscous damper and a complex stiffness are presented in
Section 2. By exploiting the theoretical background presented in
Section 2, the theoretical derivations of different damping parameters and their relationships are presented in
Section 3. It should be noted that the MATLAB software (MathWorks, Natick, MA, USA) was used to present the relationship between different parameters whenever needed. The damping parameters investigated in this paper include the specific damping capacity, loss factor, viscous damping coefficient, viscous damping ratio, phase lag (or loss angle), logarithmic decrement, half-power bandwidth, complex modulus (or loss and storage moduli), inverse quality factor, viscosity, decay ratio in the step response, and structural reverberation time. The relationships between different damping parameters are summarised in
Section 4, and some sample damping identification applications of biomaterials using different sensing technologies are presented in
Section 5. It is anticipated that many researchers conducting research on damping, from very soft materials to very stiff conventional engineering materials used in different fields, will refer to this study. In addition, the material presented in this study can be exploited for teaching damping or viscoelasticity in various branches. Before the theoretical derivations of different parameters for the description and quantification of damping and their relationships are presented, the definitions of common damping parameters are listed in
Table 1 so that the reader can refer to these parameters as needed.
4. Summary of The Relationships Between Common Damping Parameters
In practical applications, often one of the damping parameters (e.g., loss factor) is measured, and for comparison purposes, it is needed to convert the measured damping parameter into some other damping parameters (e.g., viscosity). The measured parameter can be converted into the desired parameter using the expressions presented in
Section 3. Using the derived expressions in
Section 3, an important equation relating the loss factor (
) to the ratio of the dissipated energy per cycle (
) and maximum stored energy (
), the specific damping capacity (
), the loss angle (
), the ratio of the loss modulus (
) and storage modulus (
), and the viscous damping ratio (
) can be written as follows:
Again using the derived expressions in
Section 3, for small damping, another important equation relating the viscous damping ratio (
) to the ratio of the dissipated energy per cycle (
) and maximum stored energy (
), the specific damping capacity (
), the loss angle (
), the ratio of the loss modulus (
) and storage modulus (
), the loss factor (
), the logarithmic decrement (
), the ratio of the half-power bandwidth (
) and natural frequency (
), the quality factor (
) and the inverse quality factor (
) at
can be written as follows:
Overall, the important damping parameters measured in practical applications and their relations to other important damping parameters are summarized in
Table 2.
5. Some Damping Identification Applications of Biomaterials
The dynamic indentation test is widely used to identify the viscoelastic properties of biomaterials. For example, the dynamic indentation method was used to determine the storage and loss moduli of some agar samples [
30]. The average storage modulus (
) and loss modulus (
) for a 5% agar sample obtained with the frequency sweep load function with a 1500 μN static load and 2 μN dynamic amplitude was found to be between 2 and 2.3 MPa and 0.013 and 0.02 MPa, respectively, in the frequency range of 100-200 Hz [
30]. Using Equation (54), i.e.,
, the loss factor of the 5% agar sample can be calculated to be around 0.07 and 0.09 at 100 and 200 Hz, respectively.
It is quite common to measure the storage and loss shear moduli of soft materials using an oscillatory rheometer, and then calculate the loss factor or viscosity from the measured storage and loss shear moduli. For instance, the storage shear modulus (
) and loss shear modulus (
) of a hydrogel were measured using an oscillatory rheometer test [
124]. Using the relationship between the loss factor and the storage and loss shear moduli given before (i.e.,
), the average loss factor of the hydrogel for the given frequency range (i.e., 1-10 Hz) can be calculated to be
. Similarly, using the relationship between the viscosity and the loss shear modulus (i.e.,
) and the given frequency, the viscosity of the hydrogel at
Hz can be calculated to be
Pa·s.
The logarithmic decrement method is effective for determining the damping of a structure when a single mode of vibration can be isolated from the others. Furthermore, this time-domain method does not require input measurement, it requires only response measurements. For example, the vibration damping characteristics of some spider silk threads were determined through the nanoindentation and the time decay waveform obtained from a laser vibrometer [
125]. Using the measured time decay waveform and Equation (56), the logarithmic decrement of the so-called spiral thread was calculated. Then, the viscous damping ratio of the spiral thread was calculated using Equation (57). It should be noted that, although the measured time decay waveform given in the reference [
125] is not pure harmonic, it is still dominated by a frequency component, and the logarithmic decrement can be used to identify the damping of the structure. Overall, the viscous damping ratio for the spiral thread was found to be
[
125]. Using Equation (71), i.e.,
, the loss factor of the spiral thread can be calculated to be
.
The resonant vibration test or experimental modal analysis is quite commonly used to identify the damping of a structure. The viscous damping ratios of some hydrogel beam-shaped samples were identified via resonant vibration tests for the first bending mode [
33]. For this purpose, the frequency response functions using an accelerometer and a laser Doppler vibrometer were measured. The modal viscous damping ratio was determined by fitting the Euler-Bernoulli beam model to the experimental data. Using Equation (72), i.e.,
, the loss factor of the hydrogel sample was determined, and using the simplified relation between the loss factor and the viscous damping ratio (i.e.,
), the viscous damping ratio of the hydrogel sample was calculated. For example, the viscous damping ratio for the hydrogel 0.8% Bis sample was found to be
[
33].
As mentioned before, although the half-power bandwidth concept for the identification of the loss factor was presented in
Section 3, more sophisticated methods such as the circle-fit and line-fit methods are commonly used to identify the modal loss factors of a structure using the measured frequency response functions [
92]. For instance, the circle-fit is based on fitting a circle to the measured frequency response function data around the vicinity of a natural frequency. Although the viscous damping ratio can be identified using Equations (59-61) based on the half-power bandwidth method, the modal loss factor for the r
th mode (
) of a structure is determined using
in the circle-fit method where
is the natural frequency of the r
th mode, and
and
correspond to the angles
and
around
when the frequency response function is plotted using the Nyquist diagram. For example, the loss factor of a biofibre based plate for the first mode using the circle-fit method was determined to be
[
102]. Using the simplified relation between the loss factor and viscous damping ratio (i.e.,
), the viscous damping ratio of the biofiber based plate can be determined to be
.
In the recent years, a bubble or sphere placed inside the soft medium [
68,
69,
70] or located at the soft medium interface [
34,
56,
57] exposed to an external excitation such as acoustic radiation force or magnetic force has been widely used to identify the viscoelastic properties of soft materials. For instance, using the deformation curve for a microbubble administered into a wall-less hydrogel channel exposed to an acoustic pulse obtained by the high-speed microscopy, and the curve fitted to the measured deformation curve exploiting a mathematical model, the viscosity of the gel was estimated [
34]. Overall, the maximum displacement of the bubble was determined to be around 2.2 μm, and the viscosity of the hydrogel was estimated to be 0.12 Pa·s [
34]. Using a novel approach based on the dynamic response of a spherical object placed at the sample interface, the shear modulus and viscosity of a gelatine sample with a density of 1105 kg/m
3 were determined to be 3000 Pa and 1.5 Pa⋅s, respectively [
56].
An ultrasound elastography for the characterisation of the viscoelastic properties of soft tissue was developed and validated [
60]. The reverberant shear wave ultrasound elastography was used to scan plantar soft tissue and gelatine phantom at 400–600 Hz. The shear wave speed was determined using the ultrasound particle velocity data. The viscoelastic parameters were extracted by fitting the Young’s modulus as a function of frequency derived using different rheological models to the shear wave dispersion data. For example, the Young’s modulus and viscosity of plantar soft tissue were determined to be 13628 Pa and 3.3 Pa⋅s, respectively, using the Kelvin-Voight model [
60]. It should be noted that there have been many attempts to exploit the damping (or viscosity) in quantitative ultrasound [
58,
60,
126,
127,
128,
129]. For example, the reconstructions of viscosity maps in different tissues (e.g., ex vivo normal porcine liver, fatty duck liver and fatty goose liver) with inclusions were presented in [
60]. In addition, modifications have been made to existing magnetic resonance elastography via using a damping parameter (e.g., loss angle) to improve its accuracy [
45,
63,
76,
130,
131].
6. Conclusions
The literature review shows that the dynamic indentation method, rheometry and viscometry, atomic force microscopy, hysteresis loop or power input method, resonant vibration tests or experimental modal analysis, and logarithmic decrement are commonly used to identify the damping of materials, including soft materials. In addition, a bubble or sphere placed inside the soft medium or located at the soft medium interface exposed to an external excitation such as acoustic radiation force or magnetic force is nowadays used to identify the viscoelastic properties of soft materials. The ultrasound elastography and magnetic resonance elastography for determining tissue mechanical properties are quite common for the preclinical and clinical applications. The viscous damping ratio, loss factor, complex modulus (or storage and loss moduli), and viscosity are quite common to describe and quantify damping in practical applications. In addition, the specific damping capacity, loss angle, half-power bandwidth, logarithmic decrement, and inverse quality factor are used to describe and quantify damping in many applications. In practice, usually one of the damping parameters (e.g., loss factor) is measured, and for comparison purposes the measured damping parameter needs to be converted into some other damping parameters (e.g., viscosity).
The theoretical derivation of different damping parameters and their relationships have not been presented in the literature so far. Therefore, the theoretical derivations of different parameters for the description and quantification of damping and their relationships, as well as the methods for damping identification are covered in this comprehensive review. Both accurate formulas (i.e., for systems with any amount of damping) and approximate formulas (i.e., for systems with low damping) are presented and compared. The damping parameters investigated in this paper include the specific damping capacity, loss factor, viscous damping coefficient, viscous damping ratio, loss angle or phase lag, logarithmic decrement, half-power bandwidth, complex modulus (or loss and storage moduli), inverse quality factor, viscosity, decay ratio in the step response, and structural reverberation time. It is believed that the material presented in this paper will be a primary resource for damping or viscoelasticity research and teaching in the future.