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 By definition, the strain components ε$_{r}$, ε$_{θ}$, γ$_{rθ}$ are given by the matrix form and u (0) N . 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. ( ∂ 2 r + r − $^{1}$∂$_{r}$ + r − $^{2}$∂$_{θ}$ ) 2 φ ( r, θ ) = 0 , (A1) σ$_{r}$ = ( r − $^{1}$∂$_{r}$ + r − $^{2}$∂2 θ ) φ, σ$_{θ}$ = ∂ 2 $_{r}$φ, (A2) − τ$_{rθ}$ = ∂$_{r}$ ( r $^{1}$∂$_{θ}$ ) φ. (A3)   ε$_{r}$ ε$_{θ}$ γ$_{rθ}$   = 1 E$_{M}$   1 − ν 2 M − ν$_{M}$ (1 + ν$_{M}$ ) 0 − ν$_{M}$ (1 + ν$_{M}$ ) 1 − ν 2 M 0 0 0 2(1 + ν$_{M}$ )     σ$_{r}$ σ$_{θ}$ τ$_{rθ}$   =   ∂$_{r}$ 0 r − 1 r − $^{1}$∂$_{θ}$ r − $^{1}$∂$_{θ}$ ∂$_{r}$ − r − 1   [ u v ] , (A4) σ ( n ) r = { − n ( n +1) a$_{n}$r − $^{2}$− ( n − 1)( n +2) b$_{n}$ } r − $^{n}$cos nθ, (A8) σ ( n ) θ = { n ( n + 1) a$_{n}$r − $^{2}$+( n − 1)( n − 2) b$_{n}$ } r − $^{n}$cos nθ, (A9) τ ( n ) rθ = { − n ( n + 1) a$_{n}$r − $^{2}$− n ( n − 1) b$_{n}$ } r − $^{n}$sin nθ, (A10) u ( n $^{)}$( r, θ ) = (1 + ν$_{M}$ ) E$_{M}$ [ na$_{n}$r − 2 + { n + 2(1 − 2 ν$_{M}$ ) } b$_{n}$ ] r 1 − $^{n}$cos nθ, (A11) v ( n $^{)}$( r, θ ) = (1 + ν$_{M}$ ) E$_{M}$ [ na$_{n}$r − 2 + { n − 4(1 − ν$_{M}$ ) } b$_{n}$ ] r 1 − $^{n}$sin nθ. (A12) leading to the results 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 τ ( n ) rθ ∣ r = r$_{N}$ = ( k$_{21}$δ ¯ µ$_{N}$ + k$_{22}$δ ¯ ν$_{N}$ ) sin nθ, (A14) σ ( n ) r ∣ r = r$_{N}$ = ( k$_{11}$δ ¯ µ$_{N}$ + k$_{12}$δ ¯ ν$_{N}$ ) cos nθ, (A13) k$_{11}$ = 2( n + 1)(1 − ν$_{M}$ ) − 1 (1 + ν$_{M}$ )(3 − 4 ν$_{M}$ ) E$_{M}$ , (A15) k$_{12}$ = 2( n + 1)(1 − ν$_{M}$ ) − n (1 + ν$_{M}$ )(3 − 4 ν$_{M}$ ) E$_{M}$ , (A16) 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 We now evaluate the explicit forms of U (0) M and Δ U ( n ) 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$_{M}$ 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 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 f$_{0}$ = a$_{0}$ log r + b$_{0}$r $^{2}$log r + c$_{0}$r $^{2}$+ d$_{0}$ + α$_{0}$θ + β$_{0}$r $^{2}$θ, (A5) f$_{1}$ = a$_{1}$r − $^{1}$+ b$_{1}$r log r + c$_{1}$r 3 + d$_{1}$r + α$_{1}$θ, (A6) f$_{n}$ = a$_{n}$r − $^{n}$+ b$_{n}$r 2 − $^{n}$+ c$_{n}$r 2+ $^{n}$+ d$_{n}$r $^{n}$, ( n ≥ 2) (A7) 7