content_image/381619.jpg content_image/381620.jpg content_image/381621.jpg content_image/381622.jpg content_image/381639.jpg content_image/381640.jpg content_image/381641.jpg content_image/381623.jpg content_image/381624.jpg content_image/381625.jpg content_image/381626.jpg content_image/381627.jpg content_image/381632.jpg content_image/381631.jpg content_image/381633.jpg content_image/381634.jpg
stress function φ that satisfies the so-called compatibility equation 54 stress τ$_{rθ}$ as follows. 

where ∂$_{r}$ = ∂/∂r and ∂$_{θ}$ = ∂/∂θ . Once φ is obtained, we can deduce the radial and circumferential components of the normal stress, σ$_{r}$ and σ$_{θ}$ , respectively, and the shear 

<img src='content_image/381619.jpg'>

By definition, the strain components ε$_{r}$, ε$_{θ}$, γ$_{rθ}$ are given by the matrix form 

<img src='content_image/381620.jpg'>

<img src='content_image/381621.jpg'>

<img src='content_image/381622.jpg'>

where u = u ( r, θ ) and v = v ( r, θ ) are respectively the radial and circumferential displacements of a volume el- ement in the host medium. 

The general solution of Eq. (A1) is given by φ ( r, θ ) = ∑ ∞ n =0 φ$_{n}$ ( r, θ ), where φ$_{n}$ ( r, θ ) = f$_{n}$ ( r ) cos nθ + g$_{n}$ ( r ) sin nθ and 55 

<img src='content_image/381639.jpg'>

<img src='content_image/381640.jpg'>

<img src='content_image/381641.jpg'>

with similar definitions of g$_{n}$ . The zeroth component φ$_{0}$ represents a uniform contraction of the circular cross sec- tion, thus corresponding to the energy U (0) M that we have introduced in Eq. (4). The first one φ$_{1}$ implies a rigid body translation that is irrelevant to our consideration. Other components φ$_{n}$ for n ≥ 2 describe radial corruga- tions with mode index n , thus providing the energy Δ U$_{M}$ given by Eq. (5). In the following, we set f$_{0}$ = a$_{0}$ log r and f$_{n}$ = a$_{n}$r − $^{n}$+ b$_{n}$r 2 − n instead of Eqs. (A5) and (A7), respectively, in order to obtain physically relevant solu- tions of σ$_{r}$, σ$_{θ}$, τ$_{rθ}$ that decay with increasing r . 55 (0) ( n ) 

We now evaluate the explicit forms of U M and Δ U M . To derive U (0) M , we consider a specific solution of (A1) that has the form of φ = φ$_{0}$ , and then substitute it in Eqs. (A2) and (A3) to obtain σ (0) r = a$_{0}$r − $^{2}$, σ (0) θ = − a$_{0}$r − $^{2}$, and τ (0) rθ = 0. Hence, it follows from Eq. (A4) that u $^{(0)}$( r ) = (1 + ν$_{M}$ ) a$_{0}$/ ( E$_{M}$ r ), v (0) ≡ 0. Com- plete contact between the medium and the outermost wall implies u $^{(0)}$( r$_{N}$ ) = u (0) N , and the elastic nature of the medium implies σ (0) r |$_{r}$$_{=}$$_{r}$$_{N}$ = κ$_{0}$u (0) N with the stiff- ness coefficient κ$_{0}$ . Obviously, κ$_{0}$ is identified with σ (0) r |$_{r}$$_{=}$$_{r}$$_{N}$ associated with the unit radial displacement u (0) N ≡ 1. Hence, imposing an appropriate value of a$_{0}$ , we obtain κ$_{0}$ = − E$_{M}$ / [(1 + ν$_{M}$ ) r$_{N}$ ]. Moreover, the uni- form contraction energy U (0) M should be represented as U (0) M = ( κ$_{0}$/ 2) ∫ 2 π 0 { u (0) N } 2 r$_{N}$ dθ . Consequently, we ob- tain the explicit form of U (0) M that depends on E$_{M}$ , ν$_{M}$ 

and u (0) N . 

Next, we consider the energy Δ U ( n ) M ( n ≥ 2) that cor- responds to the radial corrugation of the n th order. We only consider the cosine term in φ$_{N}$ without loss of gen- erality, which is based on our assumption of cosine radial displacement δu$_{i}$ ( θ ) (see Eq. (6)). A similar procedure to the case of n = 0 yields 

<img src='content_image/381623.jpg'>

<img src='content_image/381624.jpg'>

<img src='content_image/381625.jpg'>

leading to the results 

<img src='content_image/381626.jpg'>

<img src='content_image/381627.jpg'>

Parallel discussions to the n = 0 case together with the formula 

lead us to attain the stiffness coefficients k$_{11}$ = − σ ( n ) r |$_{r}$$_{=}$$_{r}$$_{N}$$_{,δ}$$_{¯}$ µ$_{N}$ =1 ,δ ¯ ν$_{N}$ $_{=0}$/ cos nθ and k$_{12}$ = − σ ( n ) r |$_{r}$$_{=}$$_{r}$$_{N}$$_{,δ}$$_{¯}$ µ$_{N}$ =0 ,δ ¯ ν$_{N}$ $_{=1}$/ cos nθ as 

<img src='content_image/381632.jpg'>

<img src='content_image/381631.jpg'>

<img src='content_image/381633.jpg'>

<img src='content_image/381634.jpg'>

