Strain Localization in an Oscillating Maxwell Viscoelastic Cylinder

Abstract

The transient rotation responses of simple, axisymmetric, viscoelastic structures are of interest for interpretation of experiments designed to characterize materials and closed structures such as the brain using magnetic resonance techniques. Here, we studied the response of a Maxwell viscoelastic cylinder to small, sinusoidal displacement of its outer boundary. The transient strain field can be calculated in closed form using any of several conventional approaches. The solution is surprising: the strain field develops a singularity that appears when the wavefront leaves the center of the cylinder, and persists as the wavefront reflects to the outer boundary and back to the center of the cylinder. The singularity is alternately annihilated and reinitiated upon subsequent departures of the wavefront from the center of the cylinder until it disappears in the limit of steady state oscillations. We present the solution for this strain field, characterize the nature of this singularity, and discuss its potential role in the mechanical response and evolved morphology of the brain.

1 Introduction

Solutions for the responses of simple viscoelastic structures to small rotations are of value for magnetic resonance (MR) rheometry characterization of viscoelastic material response (Bayly et al., 2007; Bayly et al., 2012), for conventional rheometry (Oldroyd, 1951), for whole-planet models of the Earth (Peltier, 1974), and as first order models of the strain response of the brain to rapid skull rotation (Bycroft, 1973; Margulies and Thibault, 1989; Massouros and Genin, 2008). Our focus is the first area, in which noninvasive MR estimates of dynamic displacement fields can be combined with analytical models of mechanical response to estimate viscoelastic material properties through solution of an inverse problem. These measurements rely on the tracking of temporary (~0.5 second duration), sinusoidal MR “tag lines” superimposed on material points through the application of a gradient in the magnetization of proton spins (Axel and Dougherty, 1989). These methods have been used to track intersections of these tag lines to estimate displacement and strain fields in physical models (Bayly et al. 2004), material specimens (Bayly et al. 2007), animal models of myocardial infarction and brain injury (Bayly et al. 2006), and the brains of human volunteers (Abney et al., 2012; Bayly et al., 2005; Clayton et al. 2012; Feng et al., 2010; Ji et al., 2007; Ji and Margulies, 2007;Sabet et al. 2007). We have previously shown that strain measurements are possible using this approach to within a few percent error for measurements made on a gelatin cylinder that is well modeled by a three-parameter viscoelastic solid (Bayly et al., 2007).

In each of these cases, comparison to simplified models was central to interpreting material and structural responses. Material models such as the three parameter model, its generalizations, and the Maxwell model that allow for description of stress waves are of significantly greater utility in these experiments than those which cannot, such as the Kelvin viscoelastic material (e.g., Flügge, 1967). For these reasons, we studied the closed form solution for strain waves resulting from sinusoidal perturbation of the boundary of a Maxwell cylinder. We found that, for displacement boundary conditions such as these that impart a sudden change in angular velocity to the cylinder’s outer boundary, a singularity appears that alternately appears and annihilates as the wavefront departs from the center of the cylinder.

We note that many viscoelastic solutions exist that are of direct relevance to MR rheology measurements. From the traumatic brain injury community, model solutions are of interest to study the brain’s response to rapid rotation of the head, believed to be the most common source of mild traumatic brain injury (Bailey and Gudeman 1989; Holbourn 1943; Ommaya and Hirsch 1970). Here, skull acceleration may lead to propagation of shear waves through the brain, and the resulting strains may injure cells or tissue when a critical strain threshold is exceeded (Bain and Meaney 2000; Cohen et al. 2008; Geddes et al. 2003; Ommaya et al. 1967; Morrison et al. 2000). The dynamic strain field in the brain depends on the geometry and constitution of the brain, and on the details of the inertial loading.

Axisymmetric wave motion in spheres of Kelvin/Voigt viscoelastic material was modeled analytically for this purpose by Bycroft (1973), Ljung (1975) and Firoozbakhsh and DeSilva (1975), and numerically by Liu et al. (1975) and Misra and Chakravarty (1984). Margulies and Thibault (1989) considered approximately the periodic angular acceleration in a Kelvin/Voigt cylinder; Ljung (1975) studied a step angular loading. Lee and Advani (1970) solved the elastic case of sinusoidal loading of a sphere correctly, but made an error in the Maxwell viscoelastic solution that has never been corrected (cf. dimensional errors in their equation 30.) Misra and Chakraborty (2005) present a numerical solution to analogous problems, and review the efforts of others to do so.

Two other fields of study of which we are aware involve solutions of relevance. The first is simplified models of the Earth (e.g. Peltier, 1974) involving spherical geometries with radially varying material properties; these can yield singularities of a character different from that studied here (e.g. Fang and Hager, 1995). The second encompasses efforts to derive viscoelastic storage and loss moduli from measurements on Goldberg and Sandvik (1947) type coaxial oscillatory rheometers. Solutions to this problem (Bird et al. 1987; Markovitz 1952; Oka 1960; Oldroyd 1951) involve analytical steady state expressions for the oscillation of an annulus of viscoelastic material between two coaxially vibrating cylinders; however, these solutions cannot accommodate a zero radius of the internal cylinder, so a useful comparison with the present study cannot be made.

Here, we present a simple derivation of the transient strain field inside a Maxwell cylinder enclosed by a rigid shell that is perturbed sinusoidally from rest. We initially analyzed the problem using simple finite difference techniques, but found that the solution diverged as the wave departed from the center of the cylinder. After a straightforward derivation of the transient strain field, the nature of a singularity that arises or annihilates as waves leave the center of the cylinder is investigated by exploring the behavior of an infinite series that arises in the solution. The singularity is multiplied by a term that decays exponentially with time, allowing the solution to be applied to the steady state case. We conclude with a discussion of how interpretation of our MR observations of the mechanical response of the brain are informed by this singular behavior, and of implications for MR rheometry experiments.

2 Analytical solution

The system considered was a rigid, infinitely long cylindrical shell of radius a filled with an incompressible, homogeneous Maxwell material of density ρ, shear modulus μ and viscosity η. No slip is allowed at the shell boundary. The entire system is at rest before the shell is subjected to a uniform, sinusoidal, axisymmetric rotation of small amplitude but arbitrary frequency. Displacements in the radial and axial directions (r,z) are neglected; the only displacement of the Maxwell material is the tangential displacement, u_θ, which is a function of radial position and time only. This assumption is reasonable for the experiments of interest, in which a compliant, nominally incompressible gel is encased in a relatively rigid cylinder (Bayly et al., 2007). Consequently, the only non-zero strain component is the tensorial shear strain ε_rθ, and the only stress component of interest is the shear stress σ_rθ.

In the following, the partial differential equation for the displacement u_θ is constructed from the fundamental governing relations. This partial differential equation is solved in the appendix. While a host of other, more general approaches to the solution of this problem exist (e.g. Hunter 1967), direct solution of the governing equations suffices for the purposes of this article. The closed form expression for ε_rθ that follows directly involves an infinite series that is shown to vanish at large times.

Governing equation

The only non-trivial kinetic equation is:

$$\frac{\partial {\sigma }_{r\theta }}{\partial r}+2\frac{{\sigma }_{r\theta }}{r}=\rho \frac{{\partial }^2{u}_{\theta }}{\partial t^2},$$ (1)

and the only non-trivial strain-displacement equation is:

$${\epsilon }_{r\theta }=\frac{1}{2}\left(\frac{\partial u_{\theta }}{\partial r}-\frac{u_{\theta }}{r}\right).$$ (2)

The Maxwell viscoelastic stress-strain relation is:

$${\sigma }_{r\theta }+\tau \frac{\partial {\sigma }_{r\theta }}{\partial t}=\eta \frac{\partial {\epsilon }_{r\theta }}{\partial t}.$$ (3)

Combining these governing equations in a single differential relation for the displacement u_θ we obtain:

$$\tau \rho \frac{{\partial }^3{u}_{\theta }}{\partial t^3}-\frac{\eta }{2}\frac{{\partial }^3{u}_{\theta }}{\partial t\partial r^2}-\frac{\eta }{2r}\frac{{\partial }^2{u}_{\theta }}{\partial r\partial t}+\rho \frac{{\partial }^2{u}_{\theta }}{\partial t^2}+\frac{\eta }{2r^2}\frac{\partial u_{\theta }}{\partial t}=0.$$ (4)

Using the non-dimensional parameters x =r / a, t̂ = t /τ, û_θ = uθ / a, and $v_m=\sqrt{\frac{\tau \eta }{2\rho a^2}}$ yields:

$$\frac{1}{v_M^2}\left[\frac{{\partial }^3{\widehat{u}}_{\theta }}{\partial {\widehat{t}}^3}+\frac{{\partial }^2{\widehat{u}}_{\theta }}{\partial {\widehat{t}}^2}\right]-\frac{{\partial }^3{\widehat{u}}_{\theta }}{\partial \widehat{t}\partial x^2}-\frac{1}{x}\frac{{\partial }^2{\widehat{u}}_{\theta }}{\partial x\partial \widehat{t}}+\frac{1}{x^2}\frac{\partial {\widehat{u}}_{\theta }}{\partial \widehat{t}}=0.$$ (5)

We next define the non-dimensional speed $\mathrm{\Phi }=\frac{\partial {\widehat{u}}_{\theta }}{\partial \widehat{t}}$. Equation 5 becomes:

$$\frac{1}{v_M^2}\left[\frac{{\partial }^2\mathrm{\Phi }}{\partial {\widehat{t}}^2}+\frac{\partial \mathrm{\Phi }}{\partial \widehat{t}}\right]-\frac{{\partial }^2\mathrm{\Phi }}{\partial x^2}-\frac{1}{x}\frac{\partial \mathrm{\Phi }}{\partial x}+\frac{1}{x^2}\mathrm{\Phi }=0.$$ (6)

Analytical solution

As described in the appendix, Equation (6) can be solved by straightforward separation of variables. After some manipulation, the solution can be written in the following compact form:

$$\begin{array}{l}{u}_{\theta }=Re\{-Ui[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}+\\ +\sum _{k=1}^{\infty }\left\{\frac{{\lambda }_k{v}_M^2}{\sqrt{1-4{\lambda }_k^2{v}_M^2}}\frac{J_1\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}\right[\frac{1+i2\mathrm{\Omega }+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}-\\ \frac{1+i2\mathrm{\Omega }-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\left]\right\}\left]\right\}.\end{array}$$ (7)

The tensorial shear strain ε_rθ follows from Equations 2 and 7:

$$\begin{array}{l}{\epsilon }_{r\theta }=Re\left\{\frac{U}{2a}i\right[\frac{\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}{J}_2\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}+\\ +\sum _{k=1}^{\infty }\left\{\frac{{\lambda }_k^2{v}_M^2}{\sqrt{1-4{\lambda }_k^2{v}_M^2}}\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}\right[\frac{1+i2\mathrm{\Omega }+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\\ -\frac{1+i2\mathrm{\Omega }-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\left]\right\}\left]\right\}.\end{array}$$ (8)

Interpretation of Non-Dimensional Parameters

The significance of the two basic parameters, Ω and v_M, lies in their interpretation as a Deborah number and a dimensionless wave speed, respectively.

The Deborah number is the ratio between a characteristic timescale for the material to the characteristic time for the system. The parameter Ω has been defined as the ratio of the relaxation time of the material τ=η/μ to the characteristic time for the sinusoidal acceleration, 1/ω. Consequently, Ω =ωτ is a Deborah number for the problem.

The wave speed for a Maxwell-type material is $v_c=\sqrt{\frac{(\mu /2)}{\rho }}$. The characteristic velocity v* by which to normalize this wave speed is the ratio of the characteristic length scale a to the characteristic time scale τ: $v^{\ast }=\frac{a}{\tau }$. Normalizing by this characteristic wave speed yields the second dimensionless parameter, v_M:

$$\frac{v_c}{v^{\ast }}=\sqrt{\frac{\mu {\tau }^2}{2\rho a^2}}\equiv v_M.$$ (9)

Since the interplay of two different time scales governs mechanical response in the problem, a second Deborah number is also important. The first time scale is the period of the sinusoidal boundary condition, 2π/Ω; the second is the time needed for each wave that is initiated at the boundary to travel through the material and return to the boundary. This period is a function of the material properties. Since v_M is the normalized wave speed, this period is 2/v_M. We thus define the ratio ζ of the two time scales as a second type of Deborah number:

$$\zeta =\frac{2\pi /\mathrm{\Omega }}{2/v_M}=\frac{v_M\pi }{\mathrm{\Omega }}$$ (10)

ζ also serves as a wave number. In the extreme cases of ζ very large, a single wave initiated at the boundary will bounce back and forth many times before a next wave is initiated. On the other hand, if ζ is very small a number of waves will be initiated before the first wave has time to bounce back. For ζ=1 each wave returns to the boundary at the same moment as the next wave is initiated. More generally, if ζ is an integer, waves initiated at the boundary exactly meet waves returning from the center.

3 The singular nature of the transient solution

The analytical solution includes an infinite series that vanishes in steady state. However, in the transition from rest to steady state it is found that the solution is singular along specific characteristic space-time lines. In this section, this series is studied and regions on which this series diverges are found. Approximations are made for the dominant transient terms in the series in Equation 8; the approximations are valid for all x > 0, but not at x = 0 itself. From these approximate expressions, the nature of the singularity, the conditions under which it occurs, and the conditions under which it self-annihilates are studied.

Steady state expression

The steady state wave patterns are given by the first terms in Equation 8. The term in the exponential will be either a complex number with negative real part or a negative real number depending on the sign of the term $1-4{v_M}^2{\lambda }_k^2$. For those λ_k where 4v_M²λ_k²>1, the exponent will be a complex number with a negative real part, while for any λ₁ where 4v_M²λ_i²<1, the exponent will be a negative real number, since $1-4{v_M}^2{\lambda }_i^2$ will be a positive number less than unity. It follows that the terms $exp\left(\frac{-1\pm \sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}\right)$ represent exponential decay and they approach zero as the steady state is reached.

Thus, the steady state tensorial shear strain is given by the first term in Equation 8:

$${\epsilon }_{r\theta }=Re\left\{\frac{U}{2a}i\left[\frac{\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}{J}_2\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}\right]\right\}$$ (11)

Asymptotic form for the terms of the series with increasing k

The second term in Equation 8 incorporates an infinite sum whose terms decrease exponentially in time. Its importance lies in the transient state, from initial rest to the eventual formation of the steady state. Here, we analyze the nature of this term and identify the regions on which divergence occurs.

The form of the transient term is the following:

$$S=\frac{U}{a}Re\sum _{k=1}^{\infty }{T}_k,$$ (12)

where:

$$\begin{gathered} T_k=\frac{i}{2}\frac{{\lambda }_k^2{v}_M^2}{\sqrt{1-4{\lambda }_k^2{v}_M^2}}\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)} \\ \left[\frac{1+i2\mathrm{\Omega }+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}-\frac{1+i2\mathrm{\Omega }-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\right]. \end{gathered}$$ (13)

Divergence will be shown to initiate upon departure of the wave from the center of the cylinder and persist as the wavefront travels to the outer boundary. Thus, the focus is a form that is valid for 0 < x ≤ 1; even close to x = 0, the focus is on the dominant terms T_k as k → infin;. Since λ_k → ∞ as k → ∞,

$$T_k~\frac{e^{-\frac{\widehat{t}}{2}}}{4}\frac{{\lambda }_k{v}_M}{{\lambda }_k^2{v}_M^2+i\mathrm{\Omega }}\left[-4\mathrm{\Omega }sin\left({\lambda }_k{v}_M\widehat{t}\right)+4i{\lambda }_k{v}_{M}cos\left({\lambda }_k{v}_M\widehat{t}\right)\right]\frac{J_2\left(x{\lambda }_k\right)}{J_0({\lambda }_k)},$$ (14)

where much smaller terms as k → ∞ have been eliminated. Removing imaginary parts from the denominator and again eliminating much smaller terms as k → ∞:

$$T_k~e^{-\frac{\widehat{t}}{2}}\frac{{\lambda }_k^3{v}_M^3-i{\lambda }_k{v}_M\mathrm{\Omega }}{{\lambda }_k^4{v}_M^4}\left[-\mathrm{\Omega }sin\left({\lambda }_k{v}_M\widehat{t}\right)+i{\lambda }_k{v}_{M}cos\left({\lambda }_k{v}_M\widehat{t}\right)\right]\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}.$$ (15)

Taking the real part of the above expression and eliminating much smaller terms we have:

$$Re[T_k]~-\frac{e^{-\frac{\widehat{t}}{2}}\mathrm{\Omega }}{{\lambda }_k{v}_M}sin\left({\lambda }_k{v}_M\widehat{t}\right)\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}$$ (16)

as k → ∞.

We now make use of the asymptotic forms of Bessel’s functions for large arguments (e.g. Abramowits and Stegun, 1972):

$$J_v(z)~\sqrt{\frac{2}{\pi z}}cos\left(z-\frac{1}{2}v\pi -\frac{1}{4}\pi \right),z\to \infty .$$ (17)

Applying this, we can write as k → ∞:

$$\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}~-\frac{1}{\sqrt{x}}\frac{cos\left(x{\lambda }_{\kappa }-\frac{\pi }{4}\right)}{cos\left({\lambda }_{\kappa }-\frac{\pi }{4}\right)}.$$ (18)

Furthermore as k → ∞:

$${\lambda }_k~k\pi +\frac{\pi }{4}.$$ (19)

Equation 18 then becomes, as k → ∞:

$$\frac{J_2\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}~\frac{{\left(-1\right)}^{k+1}}{\sqrt{x}}cos\left(k\pi x+\frac{\pi }{4}x-\frac{\pi }{4}\right).$$ (20)

Combining Equation 20 as well as the property in Equation 19 with Equation 16 and eliminating very small terms, we obtain the asymptotic form of the real part of the terms in the examined series as k → ∞:

$$Re[T_k]~\frac{{\left(-1\right)}^k{e}^{-\frac{\widehat{t}}{2}}\mathrm{\Omega }}{k\pi v_M\sqrt{x}}cos\left(k\pi x+\frac{\pi }{4}x-\frac{\pi }{4}\right)sin\left(k\pi v_M\widehat{t}+\frac{\pi }{4}{v}_M\widehat{t}\right).$$ (21)

The above terms can be written in the following more interesting and convenient form:

$$\begin{gathered} Re[T_k]~\frac{{\left(-1\right)}^{k+1}{e}^{-\frac{\widehat{t}}{2}}\mathrm{\Omega }}{2k\pi v_M\sqrt{x}}\left[sin\left(k\pi \left(x-v_M\widehat{t}\right)+\frac{\pi }{4}\left(x-v_M\widehat{t}-1\right)\right) \\ -sin\left(k\pi \left(x+v_M\widehat{t}\right)+\frac{\pi }{4}\left(x+v_M\widehat{t}-1\right)\right)\right] \end{gathered}$$ (22)

as k → ∞. For any x > 0, Equation 22 will be a valid approximation of Re[T_k] for k sufficiently large; this range of k is discussed below. Therefore, Equation 8 will diverge when the associated summation of Equation 22 diverges.

Region of divergence

A series of the form:

$$S=\sum _{k=1}^{\infty }\frac{{\left(-1\right)}^k}{k}sin(k\pi u+\phi )$$ (23)

diverges for u = {±1, ±3, ±5, ±7} unless sin(φ)= 0.

The series obtained in Equation 22 is of the form:

$$S=\sum _{k=1}^{\infty }\frac{{\left(-1\right)}^k}{k}sin\left(k\pi u+\frac{\pi }{4}(u-1)\right).$$ (24)

To find the values of u for which this series diverges, we have to exclude the values of u for which $sin\left(\frac{\pi }{4}(u-1)\right)=0$ from those values for which Equation 23 diverges. We thus find that the values of u for which Equation 24 diverges are:

$$u_D=-1,+3,-5,+7,\dots$$ (25)

Applying this result we can see that the region of divergence in the series is:

$$x-v_M\widehat{t}=-1,+3,-5,+7,\dots$$ (26a)

and

$$x+v_M\widehat{t}=-1,+3,-5,+7,\dots$$ (26b)

Since the series in the solution (Equation 8) is asymptotic to the examined series, it also becomes singular in the region described by Equations 26a,b.

Divergence occurs along some characteristic directions

The partial differential equation through which the problem is formulated is linear hyperbolic; therefore, there exist two families of characteristic directions. These are given by:

$$x-v_M\widehat{t}=K_{-}$$ (27a)

and

$$x+v_M\widehat{t}=K_{+}$$ (27b)

where K₊ and K₋ are constants.

The slopes of these characteristic lines are ± v_M (the dimensionless wave speed). Waves initiated at the boundary travel along these characteristics. From Equations 26a and 26b it follows that the region of divergence is located along some of these lines. The characteristics on which singular behavior occurs and that are within the domain of the problem are those with:

$$K_{-}=-1,-5,-9,-13,\dots$$ (28a)

and

$$K_{+}=+3,+7,+11,+15,\dots$$ (28b)

Furthermore it can be observed that the entire region of divergence lies exclusively on the path of the first wave.

In Figure 2, characteristic directions of the two families are plotted, for K₊ and K₋ integers, inside the domain of the problem. Waves begin at x=1 and v_M t̂ = 0, and progress towards the center of the cylinder (x=0), reaching this point at v_M t̂ = {1,3,5,…}. Lines with positive slope are of the form of Equation 27a, while lines with negative slope are of the form of Equation 27b. The values of K for the lines shown are from left to right: K₋= 0, −1, −2, −3 and −4 for those with positive slope and K₊= 1, 2, 3, 4 and 5 for those of negative slope. Dotted lines represent characteristics with K even, while the solid line is the path of the leading wave, its location being on characteristics with K odd.

Figure 2.

Characteristic directions along which the strain solution converges (blue) and diverges (red). The wave begins at the outer extremity of the cylinder (x=1, t̂=0), and the solution converges along the wavefront (solid line) until the wave reaches the center of the cylinder (x=0, t̂=1/v_M). The solution diverges on the wavefront as the wave reflects outwards and returns back to the center of the cylinder. This singularity alternately vanishes and reappears upon the departure of the wave from the center of the cylinder.

The singularity initiates as the wave reflects from x=0, and persists as the wave returns to the outer boundary and back to the center. The singularity then annihilates upon returning to the center. Singular behavior then occurs along every other pair of these characteristics; lines along the path of the first wave in Figure 2 on which the solution is divergent are presented in red, while those on which the solution converges are denoted are presented in blue.

4 Discussion

Application of a sinusoidal displacement to the boundary of an initially quiescent cylinder inserts a discontinuity in the first derivative of the displacement, and thus a discontinuity in strain rate. This discontinuity travels inside the material at the first wavefront with a speed of v_M, along a characteristic direction. While this first strain wave can be sharp, it is of finite magnitude. When the discontinuity reaches the center of the cylinder, the linear viscoelastic mathematical model is unable to accommodate the strain rate discontinuity, presumably at the central pole, and the series in the solution becomes divergent upon departure of the wavefront from the center; note that our approximations are valid for x>0, but not for x=0. The singularity that is formed travels back along a different characteristic and reflects from the outer boundary until it reaches the center again. The same axisymmetric process that created the singularity then eliminates upon departure from the center, and the series becomes convergent. The wave continues its path along the characteristics, decreasing in magnitude due to the viscous aspect of the material and alternately becoming singular and convergent each time it departs from the center. The alternately appearing and disappearing singularity will propagate in steady state too, but it can be ignored because of a prefactor that tends to zero in steady state.

The accuracy of Equation 22, which is the asymptotic form of Equation 13, depends upon the position x (Figure 3, with A–D corresponding to the points noted in Figure 2). For x = 0.95 (Figure 3A and D), the accuracy is within a few percent for k > 5; for x = 0.05 (Figure 3B and C), the approximation is valid only for k greater than approximately 25. Divergence of the series is evident in panels C and D, which correspond to a characteristic upon which the singularity exists: at high k, the terms of the series do not oscillate around zero (right column).

Figure 3.

Assessment of the asymptotic form of Equation 13 for the case of v_M=1 and ζ = 0.5. Plotted are values of T_k for low (left column) and high (right column) values of k. A–D correspond to points shown in Figure 2; C and D are on a characteristic where the singularity persists on the wavefront.

Subsequent waves resulting from the sinusoidal oscillation of the boundary do not yield singular behavior like that described above, since subsequent waves do not originate from a discontinuity in strain rate. As a consequence, the steady state solution is free of these singularities (Massouros and Genin, 2008).

Finite difference simulations of the problem studied in this article to predict maximum strains on the wavefront do not converge upon mesh refinement on the characteristics identified in Equation 28 (Massouros and Genin, 2013). However, convergence can be reached artificially by imposing an additional boundary condition at the node in the center of the cylinder that requires the strain rate to be zero at that point, which eliminates the singularity by eliminating the discontinuity in strain rate over a region surrounding the central pole. Although this is not rigorous, we consider it to be a reasonable approach for the reasons discussed in the below.

First, representative results (Figure 4 for v_M=0.5 and 1, and ζ=0.5 and 1.5) generated using this additional boundary condition possess all features needed to make qualitative sense. Using a standard implicit finite difference scheme (e.g., Press et al., 2002) plots generated in this way of normalized shear strain vs. position and non-dimensional time (Figure 3) show a clear transition from initial rest to the steady state. The sinusoidal boundary perturbation at x=1 sends a wave towards the center (x=0). Wave amplitude tends to increase as energy approaches the center, but this increase can be outweighed by viscous dissipation. In the absence of the singularity the wave reflects back towards the outer boundary. If wave energy persists upon the return of the wave to the outer boundary, the boundary motion can interfere constructively or destructively with the returning wave. Second, simulations performed in this way converge towards the analytical strain field at space-time regions in which the singularity is known to vanish. We apply this method in subsequent work to analyze conditions under which strain amplification can occur in the mechanical response of an idealized model of the brain (Massouros et al., 2013).

Figure 4.

Normalized shear strain in the interior of a Maxwell viscoelastic cylinder in the absence of singularities at x=0. Singularities were suppressed in these strain fields by forcing the strain rate to equal zero at x=0; strains were estimated using standard finite difference methods.

The solution presented here is useful both as a comparison problem for analyzing the structural response of the brain to skull acceleration, and as a guide for design of MR rheology experiments. The brain is a structure that is most certainly not an isotropic cylinder (e.g. Namani et al., 2012), and is most certainly not a Maxwell fluid (e.g. Bayly et al., 2012). However, even in cases in which linear viscoelasticity is an inadequate representation of tissue mechanics over a broad range of timescales and strain levels, most materials can be well represented incrementally by linear viscoelasticity (e.g. Nekouzadeh et al., 2007; Pryse et al., 2003), and it is reasonable to expect amplification of strains analogous to those we observe in a rate-dependent material that is loaded suddenly.

Why is strain localization of the character we describe not observed in MR movies of displacement fields and associated strain fields within the brains of living humans during oscillation of the skull (Abney et al., 2011; Sabet et al., 2007)? While, again, it is not surprising that the brain is not a solid, isotropic, homogeneous viscoelastic cylinder, comparison of observations in the literature to an idealized solution suggest mechanical roles for three features of brain anatomy. First, the hemispheres of the brain are largely separated by a relatively stiff membrane, the falx cerebelli, that appears to transmit little shear stress (e.g. Clayton et al., 2012), and by one of the ventricles, which is filled with fluid. Principal component analysis of the mechanical response of the brain shows a decoupling of motion on either side of the falx cerebelli (Abney et al., 2011). This serves to interrupt strain amplification of the type we report. Second, a major crossing of the hemispheres of the brain, the corpus callosum, presents some of the most dense and aligned fibers in the brain, creating a stiff and possibly more elastic region that reduces rate dependence. Principal component analysis of brain motion of the deformation of a horizontal slice of the brain imaged near the corpus callosum suggests that this stiffening reduces distortion of a region of tissue near the corpus callosum (Abney et al., 2011). Finally, the cerebellum and lower brain stem has been shown by us and others to be insulated from rotation of the rest of the brain through internal structures including the tentorium cerebelli (Clayton et al., 2012; Ji et al., 2007; Ji and Margulies, 2007). While all of these adaptations serve a number of roles, they additionally serve to reduce strain levels associated with propagating shear waves within the brain.

The specific guidance that the solution here provides for MR rheology experiments is in the choice of boundary conditions. Regardless of whether the viscoelastic object being rotated is a sphere or a cylinder, strains associated with a sudden change in velocity of the outer boundary will lead to elevated strain near the axis of rotation. For MR rheology experiments performed by imaging internal displacement fields of a viscoelastic cylinder, conditions that lead to a singularity are undesirable, as interpretation of results relies upon comparison to solutions for idealized model problems. For MR studies of the human brain, elevated strains are undesirable. Because these techniques require the assembly of partial images acquired in frequency space from multiple, repeated loadings of the head, localization and factors that make strain fields less repeatable, even by only a few percent, can complicate analysis and interpretation of data.

These difficulties can be avoided by careful selection of experimental conditions. In our studies of strain fields within the brains of living humans, we do not apply sinusoidal pulses to the periphery of the skull, but rather perform imaging during a cosine-like stopping pulse applied to the boundary. We have, in other work eliminated the discontinuity in angular velocity by rotating a cylinder into a stopper that provided a sinusoidal deceleration pulse, rather than by applying a sinusoidal displacement to an initially quiescent cylinder (Bayly et al. 2007). The singularity disappears if the discontinuity in angular velocity, and hence the discontinuity in strain rate, is eliminated. Spreading of wavefronts due to nonlinearity (e.g. Nekouzadeh et al., 2005) may ameliorate the singularity in some situations as well.

5 Conclusions

For a specific set of boundary conditions that led to a discontinuity in the strain rate field, the transient shear strain field in a Maxwell cylinder contains a singularity at the wavefront that initiates after the first shear wave reflects from the center of the cylinder. The singularity propagates along a characteristic direction to the outer boundary, then returns to the center and annihilates itself upon departure from the center. The singularity re-initiates and annihilates upon subsequent departures from the center, and disappears in the limit of steady state oscillations.

The solution shows that wave motion and strain severity are governed by a pair of dimensionless parameters: a dimensionless frequency that can be understood as a Deborah number, and a second dimensionless parameter that can be understood as a dimensionless wave speed. A third parameter, a ratio of timescales that is effectively a wave number, is also valuable for gaining a parametric understanding of wave behavior.

The derived equations are valid only in the case of small strain. The resulting strains scale with the ratio U/a, where U is the amplitude of oscillation at the boundary. Since this is in fact the amplitude of the angle of oscillation, it follows that the derived equations are valid only for small rotations. The small strain formulation can be modified in a straightforward way for large rigid body rotations superimposed upon these small angular rotations (Bayly et al., 2007). Modification for such rotations will not affect the nature of the singularity.

Finite difference simulations in which the central point is over-constrained to require that the strain rate be zero suppress the singularity; while these produce reasonable results, these are not accurate along the characteristic lines upon which the singularity propagates. MR rheometry experiments can be designed to eliminate the discontinuity in angular velocity, and thereby eliminate the discontinuity in strain rate and interpretation problems associated with the singularity.

Figure 1.

The response of a Maxwell viscoelastic cylinder to sinusoidal perturbation of its boundary was studied.

Appendix

The governing Equation (6) can be solved analytically by straightforward separation of variables. We express Φ as:

$$\mathrm{\Phi }(x,\widehat{t})=A(x)B(\widehat{t}).$$ (A1)

Substituting this into Equation 6, multiplying by $\frac{1}{AB}$ and rearranging, we obtain an equation in which the left side is a function of time only and the right side a function of space only and thus should be constant:

$$\frac{1}{v_M^2}\left[\frac{\ddot{B}}{B}+\frac{\stackrel{.}{B}}{B}\right]=\left[\frac{A^{″}}{A}+\frac{1}{x}\frac{A^{\prime }}{A}-\frac{1}{x^2}\right]=-p^2.$$ (A2)

The dot denotes differentiation with respect to t̂ and the prime denotes differentiation with respect to x. p is the above-mentioned constant to be determined. Thus, from Equation A2, we get a space equation and a time equation. The space equation reads:

$$A^{″}+\frac{1}{x}{A}^{\prime }+\left(p^2-\frac{1}{x^2}\right)A=0.$$ (A3)

This is Bessel’s equation and the solution is:

$$A(x)=a_1{J}_1(px)+a_2{Y}_1(px).$$ (A4)

At this point we can use the boundary condition u_θ|x=0 = 0. Since the displacement u_θ is zero at x=0, Φ is also zero at x=0. It follows that for a nontrivial solution A(x)=0 at x=0. Since J₁(0)=0, it follows that a₂=0. So we have:

$$A(x)=a_1{J}_1(px).$$ (A5)

The space equation reads:

$$\ddot{B}+\stackrel{.}{B}+p^2{v}_M^{2}B=0.$$ (A6)

The solution to Equation A6 is:

$$B(\widehat{t})=b_1{e}^{\left(\frac{-1+\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}+b_2{e}^{\left(\frac{-1-\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}.$$ (A7)

Then,

$$\mathrm{\Phi }(x,\widehat{t})=\frac{\partial u_{\theta }}{\partial \widehat{t}}=A(x)B(\widehat{t})\Rightarrow u_{\theta }=A(x){\int }_0^{\widehat{t}}B(n)dn+C(x),$$ (A8)

where C(x) is a function of space that results from the indefinite integral.

Before using the second boundary condition we substitute the values of the functions A and B to Equation A8 and evaluate the integral. Then,

$$u_{\theta }=a_1{J}_1(px)\left[\frac{2b_1}{-1+\sqrt{1-4p^2{v}_M^2}}{e}^{\left(\frac{-1+\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}+\frac{2b_2}{-1-\sqrt{1-4p^2{v}_M^2}}{e}^{\left(\frac{-1-\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}\right]+C(x)$$ (A9)

Combining the various constants we arrive at:

$$u_{\theta }=J_1(px)\left[c_1{e}^{\left(\frac{-1+\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}+c_2{e}^{\left(\frac{-1-\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}\right]+C(x)$$ (A10)

We now use the non-dimensional parameter Ω = ωτ and satisfy the boundary condition at x=1:

$$u_{\theta }{\mid }_{x=1}=Usin\mathrm{\Omega }\widehat{t}=\frac{U}{2i}\left(e^{i\mathrm{\Omega }\widehat{t}}-e^{-i\mathrm{\Omega }\widehat{t}}\right)$$ (A11)

Equation A10 gives:

$$u_{\theta }{\mid }_{x=1}=J_1(p)\left[c_1{e}^{\left(\frac{-1+\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}+c_2{e}^{\left(\frac{-1-\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}\right)}\right]+C(1)$$ (A12)

To match Equations A11 and A12 we need to match the frequencies of the two expressions and set C(1)=0, then determine the constants c₁ and c₂. This matching gives two values for p:

$$p_1=\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)},\text{and}$$ (A13a)

$$p_2=\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}.$$ (A13b)

Taking a linear combination for the two values of p we have:

$$J_1(p_1)\left[c_{11}{e}^{i\mathrm{\Omega }\widehat{t}}+c_{12}{e}^{(-i\mathrm{\Omega }-1)\widehat{t}}\right]+J_1(p_2)\left[c_{21}{e}^{(i\mathrm{\Omega }-1)\widehat{t}}+c_{22}{e}^{-i\mathrm{\Omega }\widehat{t}}\right]=\frac{U}{2i}\left(e^{i\mathrm{\Omega }\widehat{t}}-e^{-i\mathrm{\Omega }\widehat{t}}\right).$$ (A14)

From the above expression:

$$c_{11}=\frac{U}{i2J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)},\text{and}$$ (A15a)

$$c_{22}=-\frac{U}{i2J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)},$$ (A15b)

while c₁₂ and c₂₁ are equal to zero.

We can now write the first part of the solution for u_θ:

$$u_{\theta 1}=\frac{U}{i2}\left[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}-\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{e}^{-i\mathrm{\Omega }\widehat{t}}\right].$$ (A16)

For the second part, the solutions for which u_θ|x=1 = 0 are added. From Equation A16 we have:

$$J_1(p)\left[c_1{e}^{\frac{-1+\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}}+c_2{e}^{\frac{-1-\sqrt{1-4p^2{v}_M^2}}{2}\widehat{t}}\right]=0.$$ (A17)

Since the second term cannot be zero for a non-trivial solution, we are left with J₁(p) = 0. The values of p that satisfy this equation are the roots λ_k of J₁(s):

$$p_k=\pm {\lambda }_k.$$ (A18)

Thus u_θ becomes:

$$u_{\theta }(x,\widehat{t})=\frac{U}{i2}\left[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}-\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}\right]+\sum _{k=1}^{\infty }{J}_1\left(x{\lambda }_k\right)\left[c_{1k}{e}^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}+c_{2k}{e}^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\right].$$ (A19)

Since J₁(−s)=J₁(s) we have combined the ±λ_k.

The two constants for each term will be evaluated through the initial condition:

$$u_{\theta }(x,0)=0.$$ (A20)

This gives the following relation:

$$\frac{U}{i2}\left[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}-\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}\right]+\sum _{k=1}^{\infty }{J}_1(x{\lambda }_k)(c_{1k}+c_{2k})=0.$$ (A21)

We now use the property (Bowman, 1958):

$$\frac{J_1(kx)}{J_1(k)}=2\sum _{k=1}^{\infty }\frac{{\lambda }_k{J}_1\left({\lambda }_{k}x\right)}{\left({\lambda }_k^2-k^2\right)J_2\left({\lambda }_k\right)}$$ (A22)

in combination with J₂(λ_k) = − J₀(λ_k) for all the roots λ_k of J₁(s) to arrive at:

$$\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }\pm i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }\pm i)}\right)}=-2\sum _{k=1}^{\infty }\frac{{\lambda }_k{J}_1\left({\lambda }_{k}x\right)}{\left({\lambda }_k^2-\frac{{\mathrm{\Omega }}^2\pm i\mathrm{\Omega }}{v_M^2}\right)J_0({\lambda }_k)}.$$ (A23)

Substituting the above result in Equation A21 and equating term by term we obtain:

$$c_{1k}+c_{2k}=\frac{U{\lambda }_k{v}_M^2}{i{J}_0\left({\lambda }_k\right)}\left(\frac{1}{\left({\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }\right)}-\frac{1}{\left({\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }\right)}\right).$$ (A24)

To obtain a second relation for the two unknown coefficients we examine u_θ as t̂ → 0:

$$\begin{array}{l}{u}_{\theta }(x,\widehat{t})\to \frac{U}{i2}\left[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}(1+i\mathrm{\Omega }\widehat{t})-\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}(1-i\mathrm{\Omega }\widehat{t})\right]\\ +\sum _{k=1}^{\infty }{J}_1(x{\lambda }_k)\left[c_{1k}\left(1+\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}\right)+c_{2k}\left(1+\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}\right)\right].\end{array}$$ (A25)

Using the property in Equation A22 once again and eliminating some terms through Equation A24 we arrive at:

$$c_{1k}-c_{2k}=\frac{U}{i{J}_0\left({\lambda }_k\right)}\frac{{\lambda }_k{v}_M^2}{\sqrt{1-4v_M^2{\lambda }_k^2}}\left(\frac{1+i2\mathrm{\Omega }}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}-\frac{1-i2\mathrm{\Omega }}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }}\right).$$ (A26)

We have now evaluated the two coefficients c_1k and c_2k:

$$c_{1k}=\frac{U{\lambda }_k{v}_M^2}{i{J}_0\left({\lambda }_k\right)}\frac{1}{\sqrt{1-4v_M^2{\lambda }_k^2}}\left(\frac{1+i2\mathrm{\Omega }+\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}-\frac{1-i2\mathrm{\Omega }+\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }}\right),\text{and}$$ (A27a)

$$c_{2k}=\frac{U{\lambda }_k{v}_M^2}{i{J}_0\left({\lambda }_k\right)}\frac{1}{\sqrt{1-4v_M^2{\lambda }_k^2}}\left(-\frac{1+i2\mathrm{\Omega }-\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}-\frac{1-i2\mathrm{\Omega }-\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }}\right).$$ (A27b)

Thus, the solution is:

$$\begin{array}{l}{u}_{\theta }(x,\widehat{t})=i\frac{U}{2}\{-\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}+\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }+i)}\right)}{e}^{-i\mathrm{\Omega }\widehat{t}}+\sum _{k=1}^{\infty }\\ \left[\frac{J_1\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}\frac{{\lambda }_k{v}_M^2}{\sqrt{1-4v_M^2{\lambda }_k^2}}\right[\left(-\frac{1+i2\mathrm{\Omega }+\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}+\frac{1-i2\mathrm{\Omega }+\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }}\right)e^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}}\\ +\left(\frac{1+i2\mathrm{\Omega }-\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}-\frac{1-i2\mathrm{\Omega }-\sqrt{1-4v_M^2{\lambda }_k^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2-i\mathrm{\Omega }}\right)e^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\left]\right]\}.\end{array}$$ (A28)

This final expression incorporates additions and subtractions of pairs of complex conjugates (for cases of the term $\sqrt{1-4v_M^2{\lambda }_k^2}$ being either real or imaginary) and by combining them appropriately we can express Equation A28 in a more compact form:

$$\begin{array}{l}{u}_{\theta }=Re\{-Ui[\frac{J_1\left(\frac{x}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{J_1\left(\frac{1}{v_M}\sqrt{\mathrm{\Omega }(\mathrm{\Omega }-i)}\right)}{e}^{i\mathrm{\Omega }\widehat{t}}\\ +\sum _{k=1}^{\infty }\left\{\frac{{\lambda }_k{v}_M^2}{\sqrt{1-4{\lambda }_k^2{v}_M^2}}\frac{J_1\left(x{\lambda }_k\right)}{J_0\left({\lambda }_k\right)}\right[\frac{1+i2\mathrm{\Omega }+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1+\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\\ -\frac{1+i2\mathrm{\Omega }-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{{\lambda }_k^2{v}_M^2-{\mathrm{\Omega }}^2+i\mathrm{\Omega }}{e}^{\frac{-1-\sqrt{1-4{\lambda }_k^2{v}_M^2}}{2}\widehat{t}}\left]\right\}\left]\right\}.\end{array}$$ (A29)