Ansys Mooney Rivlin Tissue

Finite–Amplitude Inhomogeneous Plane Waves

in a deformed Mooney–Rivlin material.

Abstract

The propagation of finite–amplitude linearly–polarized inhomoge- neous transverse plane waves is considered for a Mooney-Rivlin ma- terial maintained in a state of finite static homogeneous deformation.

It is shown that such waves are possible provided that the directions of the normal to the planes of constant phase and of the normal to the planes of constant amplitude are orthogonal and conjugate with respect to the B–ellipsoid, where B is the left Cauchy-Green strain tensor corresponding to the initial deformation. For these waves, it is found that even though the system is non-linear, results on energy flux are nevertheless identical with corresponding results in the clas- sical linearized elasticity theory. Byproducts of the results are new exact static solutions for the Mooney-Rivlin material.

Introduction

An isotropic homogeneous elastic material maintained in a state of arbitrary finite static homogeneous deformation exhibits three privileged orthogonal directions, namely those of the principal axes of strain. For a general mate- rial held in such a state, Green showed that transverse finite–amplitude homogeneous plane waves can travel in a principal direction and Carroll showed that homogeneous transverse plane waves of finite amplitude, linearly–polarized in a principal direction, may propagate in a principal plane.

John considered the possibility of having finite–amplitude plane waves propagating in any direction in a prestressed material; he showed that the so- called Hadamard material was the most general compressible one in which it may happen. Then Currie and Hayes extended his result to incompressible bodies and the corresponding material was found to be of the Mooney–Rivlin type, a model used to describe the mechanical behaviour of rubber . Later, Boulanger and Hayes [6, 7] gave a detailed study of finite–amplitude waves propagating in a deformed Mooney–Rivlin material. For these waves, the displacement is of the form g(n · x −vt)a, where g is a function of arbitrary magnitude, n is a unit vector in the direction of propagation (which may be any direction), v is the speed at which the wave travels and a is a unit vector in the direction of polarization.

In contrast to this type of wave, inhomogeneous plane waves have distinct planes of constant amplitude and of constant phase. The purpose of this paper is to consider the superposition of an inhomogeneous motion upon an arbitrary static homogeneous deformation in a Mooney–Rivlin material. We show that waves with displacement of the form f(b · x)g(n · x −vt)a, where f and g are certain functions of finite magnitude, and the vectors n and b satisfy a geometric relationship, may propagate in the deformed Mooney– Rivlin material, for any orientation of the plane of b and n.

The plan of the paper is as follows. In Section 2, we recall the basic equations describing the Mooney–Rivlin incompressible material.

In Section 3, we assume that this material is subjected to a finite homoge- neous static deformation, upon which a finite–amplitude linearly–polarized inhomogeneous plane wave is superposed.

The equations of motion are derived in Section 4 and solved for the func- tions f and g, the speed v and the pressure. Two types of solutions arise. When the directions of a, b, and n are along principal axes of the initial strain ellipsoid, the solutions are called ‘special principal motions’ (Section 5); when the orientation of the plane containing b and n is arbitrary, we show that these directions must be conjugate with respect to the elliptical

2

section of the strain ellipsoid corresponding to the initial static deformation by the plane orthogonal to the direction of polarization (Section 6). In Section 7, we study a class of solutions which are of interest in some physical contexts (gravity waves, surface waves, waves in layered media, in- terfacial waves, etc. . . ). For these solutions, the amplitude decays exponen- tionally in one direction while it varies sinusoidally with time in another. We show how the possible directions of polarization, propagation and attenua- tion may be constructed geometrically, and give bounds for the values of the phase speed. For these waves, the propagation of energy is also considered.

Although the theory is non-linear, well-known aspects of the linearized the- ory of wave propagation in conservative media are recovered. In particular, it is shown that the direction of the mean energy flux vector is parallel to the planes of constant amplitude, and that the component of this vector in the direction of propagation is equal to the phase speed times the total energy density.

Finally, in Section 8, we investigate the case where the time dependency of the superposed deformation is removed. The corresponding solutions provide examples of inhomogeneous finite static deformations possible in a Mooney– Rivlin material, in the absence of body forces.

Basic Equations

Here we recall the basic equations governing the deformation of Mooney– Rivlin rubberlike materials. The constitutive equation for a Mooney–Rivlin material is T = −p1 + (C + DI)B −DB2 = −(p −DII)1 + CB −DB−1.

(2.1)

Here T is the Cauchy stress, C and D are constants, B is the left Cauchy- Green strain tensor defined by B = FFT, where F = ∂x/∂X is the defor- mation gradient, and the deformation is x = x(X, t). Also, I and II are

Invariants Of B, Given By I = Tr B,

II = tr B−1. In the constitutive equation (2.1), p(x, t) corresponds to an arbitrary pressure. Because the material is incompressible, we must have det F = 1.

(2.2)

For this material, the strain energy density Σ, measured per unit volume

(2.4)

in order that the strong ellipticity condition be satisfied (see for a short proof). If D = 0, the material is said to be neo–Hookean which case is being examined elsewhere. Here we exclude this possibility, that is we assume that (2.4)1 holds.

The equations of motion in the absence of body forces are

(2.5)

where ρ is the mass density of the material, measured per unit volume in any configuration (because of the incompressibility constraint).

(2.6)

where ˙x is the particle velocity. Rα gives the rate at which the mechanical energy crosses, at time t, a material element which is normal to the xα–axis at time t, in the final state of deformation, measured per unit area of surface in this configuration.

Also, the kinetic energy density K measured per unit volume is given by K = ρ( ˙x · ˙x)/2.

Deformation

Here we consider the propagation of a linearly–polarized inhomogeneous plane wave of finite amplitude in the material, when it is held in a state of finite static homogeneous deformation. We determine the corresponding stresses and energy flux for the motion.

We assume that the material is held in the state of finite static homoge-

(3.1)

in which the particle initially at position X is displaced to x. The constants λ1, λ2, λ3 are such that λ1 > λ2 > λ3 > 0 and satisfy λ1λ2λ3 = 1, due to (2.2).

4

In this case, the principal stresses tα, necessary to support the deformation

2 + Λ2

3 and p0 is a constant. Let a linearly–polarized inhomogeneous plane wave of finite amplitude propagate in the deformed body, so that the final position of the particle, which is initially at X, and at x in the state of finite static homogeneous

Deformation, Is At X Where

x = x + f(b · x)g(n · x −vt)a.

(3.3)

Here, (n, a, b) form an orthonormal triad. The planes defined by n · x = const. are the planes of constant phase and those defined by b · x = const. are the planes of constant amplitude. Also, v is the real speed of propagation and (f, g) are two real functions to be determined.

We use bars, for example W, to denote quantities in the final state of deformation. The deformation gradient F associated with the deformation

(3.4)

where the prime denotes the derivative of a function with respect to its argument. The corresponding left Cauchy-Green tensor B = F F

−1 Are

B = [1 + f ′g a ⊗b + fg′ a ⊗n]B[1 + f ′g b ⊗a + fg′ n ⊗a],

B

−1 = [1 −f ′g b ⊗a −fg′ n ⊗a]B−1[1 −f ′g a ⊗b −fg′ a ⊗n].

= I + 2[F ′G(A · Bb) + Fg′(A · Bn)]

+ (f ′g)2(b · Bb) + 2ff ′gg′(n · Bb) + (fg′)2(n · Bn),

−1

= II −2[f ′g(a · B−1b) + fg′(a · B−1n)] + [(f ′g)2 + (fg′)2](a · B−1a).

Ξ = A · X = Ξ + F(Ζ)G(Η −Vt),

ζ = b · x = ζ.

(3.8)

Now we write the Cauchy stress tensor T in (n, a, b). For a Mooney– Rivlin material, the stress-strain relation is given by equation (2.1) or, in the

(3.9)

where the arbitrary pressure p may be decomposed into the pressure p cor- responding to the primary homogeneous deformation and an incremental pressure p∗(say) corresponding to the superposed dynamic deformation: p = p + p∗. We assume p∗to be of a form similar to the superposed motion (3.3), that is p∗= p∗(n · x, b · x, t) = p∗(η, ζ, t).

Using (3.5), (3.6), (3.9), and the notation: Tηη = n · Tn, Tηξ = n · Ta,

Etc., The Components Of T Are Found As

Tηη = Tηη −p∗−2Df ′g(a · B−1b) + D(f ′g)2(a · B−1a), Tηξ = Tηξ + C[fg′(n · Bn) + f ′g(n · Bb)] + D(fg′)2(a · B−1a), Tηζ = Tηζ + D[f ′g(a · B−1n) + fg′(a · B−1b)] −Dff ′gg′(a · B−1a),

Tξξ = Tξξ −P∗+ 2C[F ′G(A · Bb) + Fg′(A · Bn)]

+C[(f ′g)2(b · Bb) + 2ff ′gg′(n · Bb) + (fg′)2(n · Bn)], Tξζ = Tξζ + C[f ′g(b · Bb) + fg′(n · Bb)] + Df ′g(a · B−1a), Tζζ = Tζζ −p∗−2Dfg′(a · B−1n) + D(fg′)2(a · B−1a).

(3.10)

In passing, we introduce some quantities associated with the energy car- ried by the disturbance (3.3). First, from (2.7) and (3.3), the kinetic energy

Density K Per Unit Volume Is Given By

K = ρv2(fg′)2/2.

(3.11)

Next, the stored–energy density W per unit volume associated with the

(3.12)

where I and II are given by equations (3.6) in our context. Finally, we introduce the energy flux vector R associated with the motion (3.3). From (2.6) it is given here by R = −T· ˙x. It is related to the energy flux vector R measured in the intermediate static state of deformation through

(3.13)

Using (3.3) we have ∂x/∂x = 1 + f ′ga ⊗b + fg′a ⊗n, so that R = v(fg′)[1 −f ′ga ⊗b −fg′a ⊗n]T · a.

(4.1)

Here div represents the divergence operator with respect to position x, that is with respect to the coordinates (η, ξ, ζ) = (η, ξ, ζ). However, by inspection of equations (3.10), we see that T depends on η, ζ and t only, so that div T computed with respect to x is equal to div T computed with respect to x.

(4.3)

where commas denote differentiation with respect to the coordinates (η, ξ, ζ); thus, for example, Tζξ,ξ = ∂Tζξ/∂ξ.

[C(B · Bb) + D(A · B−1A)]F ′′G + 2C(N · Bb)F ′G′

+ [C(n · Bn) + D(a · B−1a) −ρv2]fg′′ = 0.

(4.4)

Also, the second derivatives of p∗must be compatible, that is p∗

Or, From (4.3)1,3,

f ′′′g(a · B−1n) −f ′′g′(a · B−1b) + (f ′f ′′ −ff ′′′)gg′(a · B−1a) = fg′′′(a · B−1b) −f ′g′′(a · B−1n) + (g′g′′ −gg′′′)ff ′(a · B−1a).

(4.5)

Equations (4.4) and (4.5) are the two equations to be solved for f and g in order that the inhomogeneous motion (3.3) may propagate in the deformed Mooney–Rivlin material. The solutions are established in Appendix A, and a distinction needs be made, according as to whether or not the orthogonal vectors n, a, and b are along the principal axes of the B–ellipsoid.

Special Principal Motions

Here we present solutions valid when n, a, and b are in the principal direc- tions of the initial strain ellipsoid. For small-amplitude homogeneous plane waves propagating in a homo- geneously deformed body, the term ‘principal wave’ is used to describe a wave travelling along a principal axis of the strain ellipsoid corresponding to the initial static deformation. This terminology can readily be extended to the case of inhomogeneous motions for which the planes of constant phase are orthogonal to such an axis.

We introduce the term ‘special principal motion’ to describe an inhomo- geneous motion for which the normal to the planes of constant phase, the normal to the planes of constant amplitude, and the linear polarization are in the directions of the principal axes of strain. With our notation, a special principal motion is of the form f(b · x)g(n · x −vt)a where n, a, and b are the vectors i, j, and k.

The most general functions f and g, solutions to the equations of motion (4.4) and (4.5), where, without loss of generality, n = i, a = j, and b = k, are such that one is of exponential type while the other is of sinusoidal type (see Proof in Appendix A). For these motions, the quantity v may be arbitrarily prescribed (within an interval).

Explicitly, the possible inhomogeneous special principal motions of the

F(Λ3Z) = F(Z) = A1Ekσz + A2E−Kσz,

g(λ1X −vt) = g(x −vt) = d1 cos k(x −vt) + d2 sin k(x −vt),

F(Λ3Z) = F(Z) = A1 Cos Kσz + A2 Sin Kσz,

g(λ1X −vt) = g(x −vt) = d1ek(x−vt) + d2e−k(x−vt).

(5.3)

Here, a1, a2, d1, d2 are constants, σ is defined by

3 + Dλ−2

2 ).

General Solutions

Here, we seek the most general solutions f and g of equation (4.4) satisfying (4.5) such that f(b · x)g(n · x −vt)a is an inhomogeneous motion in the deformed Mooney–Rivlin material, (a, n, b) form an orthonormal triad, and the plane of n and b is arbitrary. It is shown in Appendix A that f and g

F(Ζ) = A1 Cos Kζ + A2 Sin Kζ,

g(η −vt) = d1ek(η−vt) + d2e−k(η−vt).

(6.2)

Here a1, a2, d1, d2 and k are arbitrary constants, v is such that

(6.4)

must be satisfied. The condition (6.4) means that the unit vectors n and b are conjugate with respect to the B–ellipsoid, defined by x · Bx = 1. Because they are orthogonal, they must lie along the principal axes of the elliptical section of the B–ellipsoid by the plane orthogonal to a. We note that the speed v given by (6.3) is real when ˆm and ˆn are in the directions of the minor and major axes of the elliptical section, respectively.

(6.5)

so that the speed may be written in terms of the basic strain B and the corresponding basic stress T. Finally, using the compatibility equations, the incremental pressure p∗is determined. To within an arbitrary constant, it is found to be either

(6.7)

when f and g are given by (6.2). Thus, a finite displacement of the form f(b · x)g(n · x −vt)a, where (a, b, n) is an orthonormal basis, f and g are given by (6.1) or (6.2), v is given by (6.3), and b, n satisfy (6.4), is an exact solution to the equations of motion in a homogeneously deformed Mooney– Rivlin material, for any orientation of the plane of b and n.

Sinusoidal Evanescent Waves

In this Section, we restrict our attention to the propagation of a linearly– polarized finite–amplitude inhomogeneous plane wave in a homogeneously deformed Mooney–Rivlin material. The phase of the wave fluctuates sinu- soidally in the direction of a unit vector n, its amplitude decreases exponen- tionally in the direction of b, orthogonal to n, and its polarization is in the direction of a, orthogonal to both n and b. These waves are a subclass of the general solutions to the equations of motion found in the previous Section, and arise in a variety of contexts such as Raleigh waves, Love waves, Stoneley waves, etc . . .

We give a geometrical construction for the triads (n, a, b) such that the wave may propagate, and present expressions for the phase speed and the pressure. Then we examine the propagation of the energy carried by the wave.

(7.1)

where α and k are real arbitrary constants, and v is assumed to be real. Recall that in equation (7.1), x corresponds to the static homogeneous predeforma- tion x = diag(λ1, λ2, λ3)X. As proved in Section 6, this motion is possible as long as n and b are along the minor and major axes of the elliptical section of the B–ellipsoid by the plane orthogonal to a, respectively.

Boulanger and Hayes considered the propagation of finite–amplitude homogeneous plane waves in a deformed Mooney–Rivlin material and found

10

that it was possible in any direction of propagation along n. Here, we deal with finite amplitude inhomogeneous plane waves and it is the plane of n and b (propagation and attenuation directions) that may be arbitrarily pre- scribed.

Thus we construct a sinusoidal evanescent inhomogeneous plane wave as follows. First, consider any plane passing through the origin. Let n and b be unit vectors along the respective major and minor axes of the elliptical section of the B–ellipsoid by the chosen plane. Then the motion (7.1) is possible in the deformed Mooney–Rivlin material, with a linear polarization in the direction of a = b ∧n, an amplitude exponentially attenuated by factor k in the direction of b, and a velocity vn, where ρv2 = C[(n · Bn) −(b · Bb)].

In order to write explicit expressions for the directions of n and b, and for the value of ρv2, we use a method developed by Boulanger and Hayes [9, Section 5.7]. First we prescribe a plane cutting the B–ellipsoid in a central elliptical section. We denote by a the unit vector normal to this plane. Then we write the Hamiltonian cyclic decomposition of the tensor B as [9, Section

3)(M+ ⊗M−+ M−⊗M+),

where the unit vectors m± are in the directions of the ‘optic axes’ of the

2 −Λ2

3 k.

(7.3)

We now seek n and b, unit vectors in the directions of the principal axes of the elliptical section of the B–ellipsoid by the plane a · x = 0. Hence, n and b are eigenvectors of the tensor ΠBΠ where Π, the orthogonal projection upon the plane a · x = 0, is defined by Π = 1 −a ⊗a. With the aid of (7.2),

1 −Λ2

3)(Πm+ ⊗Πm−+ Πm−⊗Πm+).

(7.4)

Calling ψ± the angles between the polarization direction a and the optic axes m± (0 ≤ψ± ≤π), and noting that the vectors Πm±/ sin ψ± are of unit length, we find from (7.4) that Πm+/ sin ψ+ ± Πm−/ sin ψ−are eigenvectors of ΠBΠ with eigenvalues γ± given by [9, Section 5.7]

1 −Λ2

3)(Πm+ · Πm−± sin ψ+ sin ψ−).

(7.5)

Recall that the phase speed v, which is given by equation (6.3),

11

was assumed to be real. Therefore, n is the eigenvector of ΠBΠ with the greater eigenvalue, which is γ+ according to (7.5), and b is the eigenvector of ΠBΠ with the lesser eigenvalue γ−. Hence, we obtain the propagation and

B = Πm+/ Sin Ψ+ −Πm−/ Sin Ψ−,

b · Bb = γ−.

Phase Speed And Pressure

We now write the phase speed in terms of the initial stretches and the angles

1 −Λ2

3) sin ψ+ sin ψ−.

(7.8)

In this connection, it may be noted that for finite–amplitude homogeneous plane waves propagating in a homogeneously deformed Mooney–Rivlin mate- rial, two linearly–polarized waves exist for each direction of propagation, and the difference between the two corresponding squared speeds is proportional

1 −Λ2

3) sin φ+ sin φ−, where φ± are the angles between the propagation direction and the optic axes of the B−1–ellipsoid (also called the ‘acoustic axes’) .

From equation (7.8), it follows that the minimum value of v is vmin given

(7.9)

and is attained only when a = m±. In this case, n · Bn = b · Bb = λ2

2,

and the plane orthogonal to a is a plane of central circular section of the B–ellipsoid. In other words, when the plane of n and b is prescribed to be orthogonal to an optic axis of the B–ellipsoid, the superposed deformation must be static. We treat this case in Section 8.

The maximum value for the phase speed is vmax given by

(7.10)

which is attained when a = j. In that case, n = i and b = k. In other words, the fastest wave occurs for propagation in the direction of the great- est initial stretch. A similar result was established by Ericksen for ac- celeration waves in a homogeneously deformed neo–Hookean material and also by Boulanger and Hayes for finite–amplitude homogeneous waves in a homogeneously deformed Mooney–Rivlin material.

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.