Viscoelasticity Semisolid Media

In Chapter 1, and thus far in the present chapter, the viscous nature of liquids has been emphasised. In the classical case of a perfectly viscous (Newtonian) liquid the applied stress is always proportional to the rate of change in the resulting strain, but is independent of the strain itself, which is not maintained. The correspondingly ideal, perfectly elastic, solid follows Hooke's law in which stress is always proportional to strain and is independent of the rate of strain. All real materials, however, exhibit some combination of these properties, i.e. the stress depends on the strain and the rate of strain (and higher time derivatives of strain) together, and are therefore viscoelastic in behaviour. Note also that real materials do not respond linearly to stress. The implications of this are discussed in Section 4.3 but, for the present, we continue to assume linearity.

A molecular picture of viscoelasticity may, for example, be gained by considering a liquid between two plates; one fixed and one oscillating to provide a sinusoidal shear stress. At low frequencies of oscillation all the driving energy is dissipated in viscous flow of the different layers of liquid over each other. Such flow occurs as a small directional drift superimposed on the random, thermal, molecular motion. If the frequency of the oscillating stress is increased until it is too fast for any molecular diffusion to occur during the period of shear strain, then the liquid will appear to possess shear rigidity; no energy is dissipated in viscous flow, instead it is stored elastically. The change from viscous to elastic behaviour with increasing frequency, intermediate between these extremes, is called the 'viscoelastic relaxation'. In this case the viscoelastic relaxation time is the diffusional jump-time of the liquid molecules.

With an alternating compression of low frequency the volume fluctuations remain in phase with the applied pressure by flow of molecules between positions of high and low density. At high frequencies the liquid structure is not able to respond to the pressure variations sufficiently quickly; volume or structural relaxation has occurred as the frequency was raised, the relaxation time being the time required for the liquid to adjust itself to its new equilibrium volume following a rapid change in the applied pressure.

An ultrasonic longitudinal wave contains both shear and compressional components, and its propagation may, in general, be discussed in terms of both shear and compressional elastic moduli and relaxation times.

The theory of viscoelasticity is essentially phenomenological and seeks to describe the mechanical behaviour of all macroscopically homogeneous solid and liquid media. Molecular or other mechanisms are not explicitly stated but may include the shear and volume relaxation mechanisms referred to in Section 4.3.1. Viscoelastic theory (Christensen 1971) has probably found its greatest application in describing the mechanical behaviour of polymers (Ferry 1961; Matheson 1971), both in solid form and in solution, which often display the transition from liquid to solid-like behaviour in a spectacular manner. The mainstream medical and biological ultrasonic literature makes little mention of viscoelasticity. However, since the earliest measurements of ultrasonic absorption by biological tissues it has been thought that similar theoretical descriptions might also be applied to tissue (e.g. Hueter 1958). The following is a glimpse of some aspects of viscoelastic theory, which should permit the reading of papers on this subject. The derivations provided are not mathematically rigorous and serve only to illustrate the relationships of results quoted to those in the medical ultrasound literature.

We begin by noting that equation (4.1) can be written in the form u(x, t) = U exp < ion t — x

c ioo which can be written back in the form of equation (4.1), u(x, t) = Ue'o(t x/c), as c is now a complex quantity given by:

c c' o where c' is the value that the phase speed took when there was no attenuation. In the presence of attenuation, therefore, the phase speed is frequency-dependent (dispersive) and complex. The one-dimensional wave equation for sound in solids is:

d2u M d2u

where M = K + 4/3G is the longitudinal elastic modulus, K is the bulk modulus and G is the shear modulus. Similar equations exist, where M is replaced by K or G, for the propagation of pure compressional or pure shear waves. From the form of the wave equation the propagation speed of sound waves in a solid is given by:

We see, therefore, that M must also be complex and frequency-dependent. This is also true of K and G. Generally this is written as:

The real parts are the elastic or storage moduli, each in phase with a sinusoidally varying strain, and the imaginary parts are those loss moduli, 90° out of phase with the strain.

The components of M may be written in terms of a and c' by combining equations (4.16) and (4.14). If the losses are small enough for (ac'/o)2 ^ 1, then the results reduce to:

In the literature on viscoelasticity most experimental results are presented in terms of M' and M", or the other moduli for non-longitudinal deformations. Equations (4.18) and (4.19) provide a useful means of quickly interpreting such data, in terms of the speed and attenuation coefficient which are usually quoted in biomedical ultrasonics; that is, c' / M0 and al / M"/M' = tan d (the 'loss tangent').

Using s and g to represent generalised stress and strain quantities respectively, the elastic moduli for perfectly elastic solids are defined by sv = Kgv and ss = Ggs, and the coefficients of viscosity for a Newtonian liquid are defined by sv = zv(dgv/dt) and ss = zs(dgs/dt), where the subscripts v and s denote volume and shear deformations respectively.

In modelling the macroscopic aspects of viscoelasticity there are many possible ways of combining the elastic and viscous components (often referred to in this context as 'springs' and 'dashpots' respectively). Two such ways, which have often been used, are known commonly as the Maxwell model and the Voigt model. Elements of these models may be simulated using combinations of 'springs' and 'dashpots' as shown in Figure 4.2a and b. Consideration of these models, or their equivalent electric networks3 leads to the observation that in the Maxwell element the entire applied stress (voltage) is felt equally by the elastic (capacitance) and viscous (resistance) components, but their strains, or rate of change of strain (current), are additive.

For the Voigt element each component experiences the same strain, but they share the stress (potential division). By inspection of the figures we can write, for the Maxwell model, dg. ss 1 dss dT = +7^ (4.20)

dt zs Gm dt and for the Voigt model

where shear deformations only have been considered. The subscript infinity is used to indicate the value of the elastic modulus under conditions of totally solid-like behaviour, i.e. at a limiting high frequency.

Solutions to equations (4.20) and (4.21) may be found in texts discussing electrical networks. For example, the transient response of the system described by equation (4.21), when an applied stress is suddenly removed, is given by gs = gsoe_t/ts' (4.22)

where gso represents the value of the shear strain at the instant that ss is set to zero, and tsv = zs/Gx is the shear response or relaxation time for the Voigt element. The transient response describes the phenomenon of creep.

Under a sinusoidally varying stress the strain is also sinusoidal, and the frequency-dependent complex elastic moduli corresponding to the two models can be obtained by substituting tsGw for zs and replacing the time derivatives in equations (4.20) and (4.21) by io. Thus, for the Maxwell element,

3 A description of equivalent electrical and mechanical quantities may be found, for example, in Braddick (1965) p. 42.

Maxwell Element
Figure 4.2. The Maxwell element and its electrical equivalent (a), the Voigt element and its electrical equivalent (b) and their respective generalised models (c and d), which have been used to represent the macroscopic behaviour of viscoelastic materials (adapted from Ferry 1961)

and for the Voigt element

Compressional deformations are dealt with using a model like that of Figure 4.2b, in which the spring represents the zero frequency elasticity (modulus K0) but the dashpot is replaced by a Maxwell model described by a compressional equivalent of equation (4.20). The combined stress-strain relationship amounts to dgv Zv dsv s = Kogv+- Kzm

By defining a bulk relaxation time tb = Zv/KR, and assuming the volume to vary sinusoidally, the bulk modulus may be expressed, using equation (4.25), as k=s = K (4-26)

gv 1 + W2t2 1 + a2t2b where KR = Km — K0 is the relaxational part of the bulk modulus.

By combining equation (4.26) with either (4.23) or (4.24), according to (4.16), Maxwell or Voigt expressions for the longitudinal modulus, M, can be obtained. Equations (4.18) and (4.19) can then be used to derive expressions for the propagation speed and attenuation coefficient. This is done in an excellent comparison of the two models provided by Raichel (1971). Usually the two relaxation times ts and tb are assumed equal.

Both Maxwell and Voigt models yield dispersion relations and relaxation expressions similar to equation (4.11). The essential differences are that greater speed dispersion is predicted by the Maxwell model, which also predicts that the attenuation coefficient should plateau at a maximum high frequency value similar to the curve shown in Figure 4.1a. Attenuation according to the Voigt model, however, has a factor constant with a/f2 included in the relationship and therefore continues to increase with frequency. An alternative view of this is that a/f2 for the Maxwell model decreases to zero for infinite frequency, whereas for the Voigt model it decreases to assume a nearly constant high frequency value. This latter behaviour is similar to the curve shown in Figure 4.1c, for a single two-state equilibrium process added to classical processes. The differences as regards speed dispersion can be appreciated by examining equations (4.23), (4.24) and (4.26). When o approaches infinity, the effective longitudinal modulus, M», can be seen in both models to approach K0 + KR+ 4G1/3. However, if o is zero, then M0 = K0 for the Maxwell model, and K0 + 4G1/3 for the Voigt model.

It should be noted that, as is often necessary to describe experimental results, equations (4.24), (4.25) and (4.26) can be generalised to represent distributions of relaxation times (Figure 4.2c and d). Whilst the Maxwell theory has been found adequate for describing sound propagation in liquids, the additional existence of a static shear modulus in the Voigt model seems to provide a better description for tissues. Indeed, Ahuja (1979), using published data on ultrasonic attenuation in tissues, has successfully modelled tissue as a Voigt body using only one relaxation time to describe the data over the medical range of frequencies.

One final point is worthy of note with regard to viscoelasticity. From equation (4.19)

a 2n2M"

Expressions for M", obtained from equations (4.23), (4.24) and (4.26), reduce at low frequencies, for both models, to

M" = ffl^ KrTJ + 4 Git where ts is either tsm or tsv.

At low frequencies equation (4.27) becomes, therefore, a 2P2 / 4 \

which can be expressed as

We have now come full circle, back to the classical expression for absorption in liquids [equation (4.12)].

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