Myriad Radial Cavitating Equilibria in Nonlinear Elasticity

SIAM J. APPL. MATH. Vol. 63, No. 4, pp. 1461–1473 c 2003 Society for Industrial and Applied Mathematics MYRIAD RADIAL CAVITATING EQUILIBRIA IN NONLINEAR ELASTICITY∗ JEYABAL SIVALOGANATHAN† AND SCOTT J. SPECTOR‡ For Donald E. Carlson on the occasion of his 65th birthday Abstract. It is shown that every bounded strictly increasing smooth positive function of sufficiently slow growth is the Jacobian of a radial hole creating equilibrium deformation for an appropriately constructed compressible nonlinearly elastic energy. Key words. cavitation, elastic, equilibrium, singular minimizers AMS subject classifications. 74B20, 74G70, 35J55 PII. S0036139901397005 1. Introduction. Explicit solutions of model equations can be useful in gaining insight concerning the qualitative behavior of solutions to more general problems. Unfortunately, the explicit construction of radial equilibrium deformations that create new holes in a compressible nonlinearly elastic body has proven to be unexpectedly complicated. Consequently, although cavitation is a common occurrence in rubbery polymers, explicit solutions that exhibit this phenomenon are rare. The only such solutions (modulo a radial null-Lagrangian; see Horgan [3] and Steigmann [15]) that appear in the literature are for an elastic fluid (see, e.g., [3, 6]); for the Blatz–Ko constitutive relation for foam rubbers, which was obtained, in two dimensions, by Horgan and Abeyaratne [4] and, in three dimensions, by Tian-hu [18]; for a compressible neo-Hookean material, which was obtained in [11] (see also [1, section 7.6]); and for the generalized Carroll material, which was obtained by Murphy and Biwa [7] (see also Shang and Cheng [12]). The usual method of obtaining an explicit solution is to solve the differential equation for a postulated model problem. In this paper we take a different approach; we first posit deformations that, based upon prior results, have desired properties, and then construct differential equations that have these deformations as solutions. We show, in particular, that every function in a certain class of radial cavitating deformations will satisfy an equilibrium equation that is appropriately chosen for that particular function. The radial deformations we use are those for which the Jacobian is an increasing radial function. The appropriate stored energy is then constructed as the sum of two terms. The first is a homogeneous isotropic strongly elliptic storedenergy function, while the second is a function of the Jacobian of the chosen radial deformation. This second function is constructed so that the chosen deformation will automatically satisfy the radial equilibrium equation. Further information on radial cavitation is contained in the survey article by Horgan and Polignone [5]. ∗ Received by the editors October 24, 2001; accepted for publication (in revised form) January 7, 2003; published electronically May 29, 2003. This work was supported in part by the National Science Foundation under grant 0072414. http://www.siam.org/journals/siap/63-4/39700.html † Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, England (js@maths. bath.ac.uk). ‡ Department of Mathematics, Southern Illinois University, Carbondale, IL 62901–4408 (sspector@ math.siu.edu). 1461 1462 J. SIVALOGANATHAN AND S. J. SPECTOR 2. The constitutive relation. Let Ψ ∈ C 2 ((0, ∞)n ) be a symmetric function. We assume that the stored-energy function for the material is given by (2.1) W (F) = Φ(ν1 , ν2 , . . . , νn ) := Ψ(ν1 , ν2 , . . . , νn ) + h(ν1 ν2 . . . νn ) for all n×n matrices F with positive determinant, where ν1 , ν2 , . . . , νn are the principal stretches, i.e., the eigenvalues of the square root of FFT , and h ∈ C 2 ((0, ∞)) is a function to be determined. The problem of interest is to determine stationary points of the energy Z E(w) = (2.2) W (∇w(x)) dx Bo among orientation-preserving injective w that satisfy the boundary condition w(x) = λx for x ∈ ∂Bo , where Bo := B(0, Ro ) ⊂ Rn is the ball of radius Ro centered at the origin. For a radial deformation w(x) = r(R) x, R R := |x|, r : [0, Ro ] → [0, ∞), the principal stretches at any point x ∈ Bo are given by (see, e.g., [1]) ν1 (x) = r′ (R) and νi (x) = r(R)/R for i = 2, 3, . . . , n. Thus (2.2) reduces to1 (2.3) E(r) = Z 0 Ro  r(R) r(R) r(R) Φ r (R), , ,..., R R R ′  Rn−1 dR among those r : [0, Ro ] → [0, ∞) that satisfy r′ > 0 a.e. and r(Ro ) = λRo . A stationary point w of E corresponds to a solution r of the radial equilibrium equation  d  n−1 R Φ,1 = (n − 1)Rn−2 Φ,2 , dR (2.4) where   r(R) r(R) r(R) , ,..., Φ,i = Φ,i r′ (R), R R R and Φ,i (v1 , v2 , . . . , vn ) denotes differentiation of Φ with respect to its ith argument (see [1, Theorem 7.3]). Also, if r(0) > 0, then the deformed ball contains a spherical cavity, and r must satisfy the natural boundary condition (2.5)  R T (R) := r(R) n−1 Φ,1  r(R) r(R) r(R) r (R), , ,..., R R R ′  → 0 as R → 0+ , which corresponds to the radial component of the Cauchy stress vanishing on the cavity surface. For energies of the form (2.1) the radial equilibrium equation (2.4) becomes  n−1 !  r(R) d d  n−1 (2.6) , R Ψ,1 − (n − 1)Rn−2 Ψ,2 = −r(R)n−1 h′ r′ (R) dR dR R 1 The energy E is equal to E multiplied by the volume of the unit ball in Rn . MYRIAD RADIAL CAVITATING EQUILIBRIA 1463 and the natural boundary condition (2.5) reduces to "  n−1 !  n−1 #  r(R) R r(R) r(R) ′ ′ ′ + Ψ,1 r (R), (2.7) lim+ h r (R) = 0. ,..., R r(R) R R R→0 The main idea in this paper is that, given Ψ and r, (2.6) can be used to define the function h. In order to accomplish this, we will need the following hypotheses on the energy: (En1) for all q > 0 and t > 0 Ψ,11 (q, t, t, . . . , t) > 0; (En2) there exists λ∗ > 0 such that for every α ≥ λn∗ lim t1−n Ψ,1 (αt1−n , t, t, . . . , t) = 0; t→+∞ (En3) for every L > λn∗ there are constants β ∈ [0, n − 1) and K > 0 such that Ψ,2 (κt1−n , t, t, . . . , t) ≤ Ktβ for all λ∗ < t < ∞ and λn∗ ≤ κ ≤ L; (En4) for q 6= t define (2.8) R(q, t) := qΨ,1 (q, t, t, . . . , t) − tΨ,2 (q, t, t, . . . , t) . q−t Then for every µ > λ∗ we assume that there exists a Bµ > 0 such that |R(q, t)| ≤ Bµ for all λ∗ < t < µ and 0 < q < t. Remark 2.1. Hypothesis (En1) is a consequence of the strong-ellipticity of the energy Ψ. Hypotheses (En2)–(En4) are satisfied by many examples of stored energies (see [1, 16, 17]). In particular, Stuart [16, 17] requires a more stringent growth hypothesis than (En4): 0 ≤ R(q, t) ≤ A + Btβ for 0 < q < t. 3. The construction. Let λcrit > λ0 > 0, and suppose that J ∈ C 1 ([0, ∞)) is a strictly monotone increasing function that satisfies Z ∞ n n J(0) = λ0 , lim J(R) = λcrit , (3.1) J ′ (t)tn dt ≤ 1. R→+∞ 0 Define ρ : [0, ∞) → [1, ∞) by n ρ(R) := 1 + n (3.2) Z R J(t)tn−1 dt 0 so that (3.3) ρ′ (R)  ρ(R) R n−1 = J(R) for 0 < R < ∞. For future reference we note the following properties of ρ. 1464 J. SIVALOGANATHAN AND S. J. SPECTOR Lemma 3.1. The function ρ given by (3.1) and (3.2) satisfies (i) 0 < ρ′ (R), (ii) d ρ(R) dR [ R ] < 0, ρ′ (R) < ρ(R) R , ′′ (iii) (iv) 0 < ρ (R), ′ (v) limR→+∞ ρ(R) R = limR→+∞ ρ (R) = λcrit . Remark 3.2. Properties (i)–(v) are standard properties of radial minimizers and radial equilibrium deformations (see, e.g., [1, 10, 16, 17]). Since our proof shows that (3.1)3 is necessary and sufficient for (iii), it is now clear that (3.1)3 is also a standard property of such deformations. Note also that by (3.2), ρ(0) = 1. Theorem 3.3. Let Ψ satisfy (En1)–(En4) and let ρ be given by (3.1) and (3.2), where λ0 > λ∗ . Suppose that h ∈ C 2 ((0, ∞); (0, ∞)) and satisfies   Z R ρ(s) (n − 1)sn−2 ′ ds Ψ̂ ρ (s), h′ (J(R)) = 2 ρ(s)n−1 s 0   n−1 ρ(R) 1 n−1 ′ −R Ψ̂1 ρ (R), (3.4) R ρ(R)     Z R ρ(s) d 1 + ds sn−1 Ψ̂1 ρ′ (s), s ds ρ(s)n−1 0 ρ(s) ρ(s) ρ(s) ′ for R ∈ (0, ∞), where Ψ̂i (ρ′ (s), ρ(s) s ) := Ψ,i (ρ (s), s , s , . . . , s ). Then ρ is a solution of the radial equilibrium equation (2.6) on (0, ∞) and satisfies the natural boundary condition (2.7). Remark 3.4. It follows from the proof of the above result (see (4.4)) that h′ (λn0 ) = 0. Finally, we use ρ to construct a family of equilibrium deformations of Bo . For any Ro > 0 and δ > 0 (3.5) uδ (x) := rδ (|x|) x, |x| rδ (R) := ρ(δR) δ is an orientation-preserving injective radial deformation of Bo , and, in view of (3.3), (4.1), and [1, Lemma 4.1], J(δ|x|) is the Jacobian of uδ at any point x 6= 0. Clearly, each uδ is a cavitating deformation that creates a new hole of radius 1/δ at the center of the ball and satisfies the boundary condition R δR [1 + n 0 o J(t)tn−1 dt] n n (3.6) uδ (x) = λx, λ = λ(δ, Ro ) := (δRo )n for x ∈ ∂Bo . Moreover, results in [1] show that each of these deformations is contained in the Sobolev space W 1,p (Bo ; Rn ) for every p ∈ [1, n), while Theorem 3.3 shows that uδ is a stationary point for the energy. Theorem 3.5. Let W be given by (2.1) and satisfy (En1)–(En4). Let λ > λcrit > λ0 > λ∗ . Then there exists a unique δ = δ(λ) such that uδ , given by (3.1), (3.2), (3.5), and (3.6), is a stationary point of the energy (2.2) and satisfies the boundary condition uδ (x) = λx for x ∈ ∂Bo . Proof. Let λ > λcrit . Then since ρ(0) = 1, it is clear from Lemma 3.1(ii) and (v) that there exists a unique Rλ > 0 such that ρ(Rλ ) = λRλ . Define δ = Rλ /Ro . Then MYRIAD RADIAL CAVITATING EQUILIBRIA 1465 by (3.5) uδ (x) := ρ(δRo ) ρ(Rλ ) x= x = λx for |x| = Ro . δRo Rλ 4. Proofs for the construction. Proof of Lemma 3.1. We first note that (i) is clear from (3.2), (3.3), and the nonnegativity of J. Next, if we divide (3.2) by Rn and use the quotient rule to differentiate the result with respect to R, we find that
RR  n−1   nJ(R)Rn−1 Rn − nRn−1 (1 + n 0 J(t)tn−1 dt) ρ(R) d ρ(R) = R dR R nR2n R R J(R)Rn − (1 + n 0 J(t)tn−1 dt) = (4.1) Rn+1 RR ′ −1 + 0 J (t)tn dt , = Rn+1 where an integration by parts has been used to deduce (4.1)3 from (4.1)2 . It is now clear from (4.1) that (3.1)3 is necessary and sufficient for (ii). Next, if we differentiate ρ(R)/R with respect to R, we see that     1 ′ ρ(R) d ρ(R) (4.2) = ρ (R) − , dR R R R and consequently (iii) is equivalent to (ii). Similarly, if we differentiate (3.3) with respect to R, we discover that n−1 n−2     d ρ(R) ρ(R) ρ(R) ′ ′ ′′ (4.3) , = J (R) − ρ (R)(n − 1) ρ (R) R R dR R and thus (iv) follows from (i), (ii), and the fact that J is increasing. Finally, to obtain (v) we first note that (ii)–(iv) imply that both limits exist and are finite. Thus, if we divide (3.2) by Rn and take the limit as R → +∞, we find, using L’Hôpital’s rule and (3.1)2 , that n  ρ(R) = lim J(R) = λncrit , lim R→+∞ R→+∞ R which together with (3.3) yields (v). Proof of Theorem 3.3. Assume for the moment that s 7→ sn−2 Ψ̂2 (ρ′ (s), ρ(s) s ) and ρ(s) n−1 ′ s 7→ s Ψ̂1 (ρ (s), s ) are integrable on (0, R), so that the right-hand side of (3.4) is well defined on (0, ∞). Then, if we differentiate (3.4) with respect to R, it is clear that ρ satisfies the radial equilibrium equation (2.6) on (0, ∞). In order to show that s 7→ sn−2 Ψ̂2 (ρ′ (s), ρ(s) s ) is integrable on (0, R) we use (3.3), (En3), ρ′ ≥ 0, and the fact that J(s) ∈ [λn0 , λncrit ] for each s to conclude that !  n−2 1−n    ρ(s) ρ(s) s ρ(s) , sn−2 Ψ̂2 ρ′ (s), Ψ̂2 J(s) = ρ(s)n−2 s ρ(s) s s n−2 ≤ Kρ(s)  ρ(s) s β−n+2 ≤ Kρ(R)β sn−2−β , 1466 J. SIVALOGANATHAN AND S. J. SPECTOR which is clearly integrable on (0, R) since β < n − 1. In order to prove that s 7→ sn−1 Ψ̂1 (ρ′ (s), ρ(s) s ) is integrable on (0, R), we will show that   ρ(s) n−1 ′ lim s Ψ̂1 ρ (s), (4.4) = 0. s s→0+ Now, by (3.3), n−1 (4.5) s Ψ̂1  ρ(s) ρ (s), s ′  n−1 = ρ(s)  s ρ(s) n−1 Ψ̂1  ρ(s) J(s) s 1−n ρ(s) , s t := ρ(s) . s ! . Moreover, since λn0 ≤ J(s) ≤ λncrit , hypothesis (En1) implies (4.6)


Ψ̂1 λn0 t1−n , t ≤ Ψ̂1 J(s)t1−n , t ≤ Ψ̂1 λncrit t1−n , t , Since ρ → 1 and t → +∞ as s → 0+ , (4.4) now follows from (En2), (4.5), and (4.6). In addition, the natural boundary condition (2.7) follows from (3.4), (4.4), together with n−1 the integrability of s 7→ sn−2 Ψ̂2 (ρ′ (s), ρ(s) Ψ̂1 (ρ′ (s), ρ(s) s ) and s 7→ s s ) on (0, R). Finally, we need to show that the right-hand side of (3.4) is bounded as R → ∞ in order that h can be extended smoothly as a real-valued function on (λncrit , ∞). Let R1 > 0. Then by (3.4) h′ (J(R)) − h′ (J(R1 )) = (4.7) R   ρ(s) (n − 1)sn−2 ′ ds Ψ̂ ρ (s), 2 ρ(s)n−1 s R1   ρ(R) ρ(R)1−n − Rn−1 Ψ̂1 ρ′ (R), R   ρ(R1 ) + R1 n−1 Ψ̂1 ρ′ (R1 ), ρ(R1 )1−n R1   Z R  ρ(s) d sn−1 Ψ̂1 ρ′ (s), + ρ(s)1−n ds s ds R1 Z for R ∈ (R1 , ∞). Now, it is clear from Lemma 3.1(v) that (4.7)2 is bounded as R → ∞. Next, the sum of the integrals on the right-hand side of (4.7)1 , (4.7)4 is equal to (4.8) (n − 1) Z R −1 s R1  s ρ(s) n      ρ(s) ρ(s) ρ(s) ′ ′ ′ − ρ (s)Ψ̂1 ρ (s), ds. Ψ̂2 ρ (s), s s s ρ(R1 ) However, by Lemma 3.1, λcrit < ρ(s) =: µ for R1 ≤ s < ∞ and hence, in s ≤ R1 view of (En4), the absolute value of (4.8) is bounded by a constant times Z R R1     Z R 1 ρ(s) d ρ(s) ds − ρ′ (s) ds = − s s ds s R1 = which is bounded as R → ∞. ρ(R1 ) ρ(R) − , R1 R MYRIAD RADIAL CAVITATING EQUILIBRIA 1467 5. The energy: Uniqueness. In this section we note that, whenever the function h is convex, the cavitating radial equilibrium solution we have constructed is the unique global minimizer of the energy among radial deformations. The following three results can be found in Sivaloganathan [13] (see also [14]). Proposition 5.1. Assume that (5.1) Φ̂11 (q, t) > 0 for all q > 0 and t > 0. Let rc ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) be a cavitating equilibrium solution; i.e., rc satisfies (2.4) on (0, ∞), (2.5), rc (0) > 0, and rc′ > 0 a.e. Suppose that Ro > 0, and define λ = rc (Ro )/Ro . Let r ∈ Aλ , (5.2) Aλ := {r ∈ W 1,1 ((0, Ro )) : r(Ro ) = λRo , r(0) ≥ 0, r′ > 0 a.e.}, satisfy  r(R) . 0 < lim sup R R→0+  (5.3) Then E(rc ) < E(r) unless r ≡ rc , where E is given by (2.3). Corollary 5.2. Let λ > 0 and Ro > 0. Then, under the hypotheses of the previous proposition, there exists at most one cavitating equilibrium solution rc ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) that satisfies rc (Ro ) = λRo . Remarks. 1. The statement of Theorem 6.8 in [13] actually requires that (5.3) be satisfied with lim sup replaced by lim inf. However, the remark after the proof of [13, Theorem 6.9] notes that the result remains valid under this weaker hypothesis. 2. Theorems 6.8 and 6.9 in [13] also appear to require the weakened Baker– Ericksen inequality R(q, t) ≥ 0, where R is given by (2.8). However, an examination of the proofs in [13, 14] shows that this inequality is used only to extend a solution of the radial equilibrium equation from (0, Ro ) to (0, ∞), a step not needed in our presentation. Proof of Corollary 5.2. Let λ > 0 and Ro > 0. If rc is any cavitating equilibrium solution that satisfies rc (Ro ) = λRo , then rc ∈ Aλ and rc satisfies (5.3). Thus, by the previous proposition, two distinct cavitating equilibrium solutions rc1 and rc2 would satisfy E(rc2 ) < E(rc1 ) and E(rc1 ) < E(rc2 ), which is a contradiction. Corollary 5.3. Let λ > 0 and Ro > 0. Suppose that rc ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) is a cavitating equilibrium solution that satisfies rc (Ro ) = λRo . Then, under the hypotheses of the previous proposition, E(rc ) < E(rh ), where rh (R) := λR. Proof. For any λ > 0 and Ro > 0 the homogeneous deformation rh (R) := λR satisfies rh ∈ Aλ and (5.3). The result then follows from Proposition 5.1. In order to make use of Proposition 5.1 we will need the following additional hypothesis on the energy: (En5) there exist φ, ψ : (0, ∞) → R and Ψ∗ ∈ C((0, ∞)n ; R), with φ > 0, ψ ≥ 0, and Ψ ≥ 0, that satisfy Ψ(ν1 , ν2 , . . . , νn ) = n X i=1 φ(νi ) + X i6=j ψ(νi νj ) + Ψ∗ (ν1 , ν2 , . . . , νn ), 1468 J. SIVALOGANATHAN AND S. J. SPECTOR where t 7→ Ψ∗ (t, t, . . . , t) is bounded on (0, λ∗ ] and ψ ≡ 0 if n = 2. We now use Proposition 5.1 to show that if our equilibrium solutions are supersolutions for the energy Ψ, then they are energy minimizers among the radial deformations. Theorem 5.4. Let Ψ satisfy (En1)–(En5) and let ρ be given by (3.1) and (3.2), where λ0 > λ∗ and J ′ > 0 on [0, ∞). Suppose that      d ρ(R) ρ(R) (5.4) Rn−1 Ψ̂1 ρ′ (R), < (n − 1)Rn−2 Ψ̂2 ρ′ (R), . dR R R Finally, suppose in addition that h ∈ C 2 ((0, ∞); (0, ∞)) satisfies (3.4) and h′′ (s) ≥ 0 for s ∈ (0, λn0 ) ∪ [λncrit , ∞). Then the radial cavitating deformation rδ(λ) given by Theorem 3.5 satisfies E(rδ(λ) ) < E(r) for every r ∈ Aλ (see (5.2)). Proof. We first show that h′′ is nonnegative on its domain of definition. Let s ∈ (0, ∞). If s ∈ / [λn0 , λncrit ), then h′′ (s) ≥ 0, by hypothesis. If s ∈ [λn0 , λncrit ), then by (3.1) there exists an R ∈ [0, ∞) such that J(R) = s. Therefore, by (3.3) and the radial equilibrium equation (2.6), (5.5)  d  n−1 R Ψ,1 − (n − 1)Rn−2 Ψ,2 = −ρ(R)n−1 h′′ (J(R)) J ′ (R). dR The desired result now follows from (5.4), (5.5), and the assumed positivity of ρ and J ′. Now let λ > λcrit , r ∈ Aλ , and suppose that   r(R) 0 < lim sup (5.6) . R R→0+ Then, by Proposition 5.1, all we need show is that (5.1) is satisfied. If we differentiate (2.1) twice with respect to ν1 and set ν1 = q and ν2 = ν3 = · · · = νn = t, we find that Φ̂11 (q, t) = Ψ̂11 (q, t) + t2n−2 h′′ (qtn−1 ). Consequently, in view of (En1), a sufficient condition for (5.1) is that h′′ be nonnegative on its domain of definition, which has previously been shown. Before proceeding further we note that the convexity of h, together with h′ (λn0 ) = 0 (see the remark following Theorem 3.3), implies that h is bounded below by h(λn0 ) on (0, ∞). Next, suppose alternatively that (5.6) is not satisfied and therefore that   r(R) . 0 = lim+ R R→0 Then, in particular, r(0) = 0, and hence Z Ro Z Ro Z Ro d  n n n n ′ n−1 (5.7) λn Rn−1 dR. r dR = r(Ro ) = λ Ro = n nr r dR = dR 0 0 0 Now, in view of the convexity of h,    n    n r(R) r(R) n ′ n ′ − λ h′ (λn ), h r (R) ≥ h(λ ) + r (R) R R 1469 MYRIAD RADIAL CAVITATING EQUILIBRIA and consequently, by (5.7), Z (5.8) Ro 0  r(R) h r′ (R) R n−1 ! R n−1 dR ≥ Z Ro h(λn )Rn−1 dR. 0 If E(r) = +∞, we are done. If instead E(r) < ∞, then by (En5) and since h is bounded below,   r(R) (5.9) R 7→ Rn−1 φ ∈ L1 ((0, Ro )), R 2 !  r(R) n−1 (5.10) ∈ L1 ((0, Ro )). R 7→ R ψ R We claim that n 0 = lim inf R Ψ̂ (5.11) R→0+ Otherwise, Rn Ψ̂ r(R) r(R)
R , R  r(R) r(R) , R R  . ≥ K > 0 for small R, and thus   r(R) r(R) K n−1 R Ψ̂ ≥ . , R R R This inequality, together with (En5), implies nR n−1 φ  r(R) R  1 + n(n − 1)Rn−1 ψ 2  r(R) R 2 ! ≥ K − Ψ̂∗ R  r(R) r(R) , R R  , which, since r(R)/R and hence Ψ∗ are bounded, contradicts (5.9) or (5.10). Next, Ψ,11 > 0. Therefore Ψ̂(q, t) ≥ Ψ̂(t, t) + (q − t)Ψ̂,1 (t, t), and hence         r(R) r(R) r(R) r(R) r(R) r(R) ′ ′ Ψ̂ r (R), ≥ Ψ̂ + r (R) − , , Ψ̂,1 R R R R R R    r(R) r(R) 1 d = R1−n Rn Ψ̂ , , n dR R R which, when multiplied by nRn−1 and integrated over (Rk , Ro ), yields (5.12) Z Ro Rk     h i  r(R) r(Rk ) r(Rk ) nΨ̂ r′ (R), Rn−1 dR ≥ Ron Ψ̂ (λ, λ) − Rkn Ψ̂ , . R Rk Rk
k ) r(Rk ) In particular, choose a sequence Rk → 0+ as k → ∞ so that Rkn Ψ̂ r(R Rk , Rk converges to its lim inf, which is zero by (5.11). Then, if we let k → ∞ in (5.12) and apply the dominated convergence theorem, we find that  Z Ro Z Ro  r(R) n−1 ′ Ψ̂ (λ, λ) Rn−1 dR, R dR ≥ Ψ̂ r (R), R 0 0 which, together with (2.1) and (5.8), yields E(r) ≥ E(rh ). The desired result now follows from Corollary 5.3. 1470 J. SIVALOGANATHAN AND S. J. SPECTOR 6. Examples. 6.1. Ogden materials. In order to illustrate the form that hypotheses (En1)– (En5) take for a well-analyzed class of materials, we now restrict our attention to three dimensions and consider materials whose constitutive relation is of the form (6.1) Ψ(λ1 , λ2 , λ3 ) = φ(λ1 ) + φ(λ2 ) + φ(λ3 ) + ψ(λ1 λ2 ) + ψ(λ2 λ3 ) + ψ(λ1 λ3 ), where φ, ψ ∈ C 2 ((0, ∞)). (Such constitutive relations were used by Ogden [8] to match theory with experiments.) For such materials we make the following assumptions (cf. [1, 16, 13, 14, 9] and especially [17, 10]): (Og1) for all s > 0 φ′′ (s) > 0, ψ ′′ (s) ≥ 0; (Og2) there exist β ∈ [1, 2), γ ∈ [0, 1), and B > 0 such that for every s > 0   |φ′ (s)| ≤ B s−γ + sβ , |ψ ′ (s)| ≤ B[s−γ + s(β−1)/2 ]. We now show that (Og1) and (Og2) imply (En1)–(En5). First, it is clear that (En5) is satisfied with Ψ∗ ≡ 0. Next, we differentiate (6.1) with respect to λ1 and let λ1 = q and λ2 = λ3 = t to get (6.2) Ψ̂1 (q, t) = φ′ (q) + 2tψ ′ (qt), Ψ̂11 (q, t) = φ′′ (q) + 2t2 ψ ′′ (qt). Then (Og1) and (6.2)2 yield (En1). If we differentiate (6.1) with respect to λ2 and let λ1 = q and λ2 = λ3 = t, we get (6.3) Ψ̂2 (q, t) = φ′ (t) + qψ ′ (qt) + tψ ′ (t2 ), and hence, when q = κt−2 , we find that (6.4) Ψ̂2 (κt−2 , t) = φ′ (t) + κt−2 ψ ′ (κt−1 ) + tψ ′ (t2 ). In order to obtain (En3) we take the absolute value of (6.4) and use the triangle inequality and (Og2) to conclude that |Ψ̂2 (κt−2 , t)| ≤ B[(t−γ + tβ ) + (κ1−γ tγ−2 + κ(β+1)/2 t−(β+3)/2 ) + (t1−2γ + tβ )], which implies (En3). Similarly, if we take q = αt−2 in (6.2)1 and use the triangle inequality and (Og2), we obtain |t−2 Ψ̂1 (αt−2 , t)| ≤ B[α−γ t2(γ−1) + αβ t−2(β+1) + 2α−γ tγ−1 + 2α(β−1)/2 t−(β+1)/2 ], which approaches zero as t → ∞ since γ < 1. This implies (En2). In order to obtain (En4) we first use (6.2)1 and (6.3) to get (6.5) R(q, t) = tφ′ (t) − qφ′ (q) t2 ψ ′ (t2 ) − qtψ ′ (qt) + . t−q t−q We then fix µ > λ∗ and consider two cases: λ∗ < t < µ. 1 2 λ∗ ≤ q < t < µ and 0 < q < 1 2 λ∗ < MYRIAD RADIAL CAVITATING EQUILIBRIA 1471 Case I. 0 < q < 21 λ∗ < λ∗ < t < µ. Then |t − q|−1 ≤ 2/λ∗ , and hence (6.5) together with (Og2) and the triangle inequality yield |R(q, t)| ≤ 2B 1−γ [t + tβ+1 + q 1−γ + q β+1 + t2(1−γ) + tβ+1 + (qt)1−γ + (qt)(β+1)/2 ], λ∗ which is bounded for 0 < q < t < µ since γ < 1. This implies (En4) for small q. Case II. 12 λ∗ ≤ q < t < µ. Then (6.5) together with the mean-value theorem applied to the functions φ̃(s) := sφ′ (s) and ψ̃(s) := sψ ′ (s) yield R(q, t) = φ̃′ (c∗ ) + tψ̃ ′ (ĉ) (6.6) for some c∗ ∈ (q, t) and ĉ ∈ (qt, t2 ). Thus, since φ and ψ are C 2 , the function |R| is bounded when 12 λ∗ ≤ q < t < µ. Therefore (En4) is also valid for larger q. 6.2. Deformations. It is easy to construct radial cavitating deformations: the specification of a strictly monotone increasing radial Jacobian J(R) that satisfies (3.1) suffices. The content of Theorem 3.3 is that such a deformation will satisfy the radial equilibrium equation for every stored energy function of the form (6.1) that satisfies (Og1) and (Og2), provided h is defined by (3.4). The difficulty is then ensuring that this equilibrium deformation is the unique radial minimizer of the energy, i.e., that the combination of deformation and stored energy satisfies (5.4). One method of obtaining a family of such deformations is to perturb from a known solution. We illustrate this idea when the initial solution is isochoric. The resulting deformations will be nearly incompressible, as is expected in many elastomers (see, e.g., [2] or [8]), since the resulting energy will heavily penalize even small changes in volume. First, let’s restrict our attention to the energy (6.7) Ψ(λ1 , λ2 , λ3 ) := φ(λ1 ) + φ(λ2 ) + φ(λ3 ), where φ satisfies (Og1), (Og2) and φ′ is a convex function. (For example, φ(t) = tβ+1 with β ∈ [1, 2).) Then (5.4) reduces to    ρ(R) d  2 ′ ′ R φ (ρ (R)) < 2Rφ′ dR R or, equivalently, (6.8)     ρ(R) − φ′ (ρ′ (R)) . Rφ′′ (ρ′ (R))ρ′′ (R) < 2 φ′ R However, the mapping t 7→ φ′ (t) is convex:     ρ(R) ρ(R) ≥ φ′ (ρ′ (R)) + φ′′ (ρ′ (R)) − ρ′ (R) , φ′ R R and so, in view of (6.8), a sufficient condition for (5.4) is that the function ρ satisfies   ρ(R) ′′ ′ Rρ (R) < 2 (6.9) − ρ (R) R (or, equivalently, d dR trace(F) > 0). 1472 J. SIVALOGANATHAN AND S. J. SPECTOR Now, let λ0 > 0, and suppose that θ : [0, 1] → [0, 1] is continuous with θ > 0 on (0, 1] and that θ(s) = o(s6 ) (6.10) as s → 0+ . Then for each s ∈ [0, 1] define 3 (6.11) ρ(R, s) := 1 + 3 Z R J(t, s)t2 dt, 0 J(R, s) := λ30 + θ(s)(1 − e−sR ) (6.12) and note that −sR JR (R, s) = sθ(s)e (6.13) , sθ(s) Z ∞ t3 e−st dt = 0 6θ(s) , s3 where the subscript R denotes the partial derivative with respect to R. (If λ0 = 1, the deformation at s = 0 is isochoric.) By (6.13), for each s > 0, R 7→ J(R, s) is strictly monotone increasing and satisfies (3.1)3 , provided 0 < θ(s) ≤ s3 /6, which is a consequence of (6.10) for all sufficiently small s. Next, by (4.2), (4.3), and (6.13)1 , 2     ρ(R) ρ(R) ρ(R) ′ −sR ′ ′′ ρ (R) − , = sθ(s)Re − 2ρ (R) Rρ (R) R R R which shows that (6.9) is equivalent to  2  ρ(R) ρ(R) (6.14) − ρ′ (R) . sθ(s)Re−sR < 2 R R However, in view of (4.1), (4.2), and (6.13)1 , R 6  ρ(R) R 4  #2 2 " Z R ρ(R) 3 −st ′ t e dt , − ρ (R) = 1 − sθ(s) R 0 so that (6.14) is the same as (6.15) sθ(s)R 7  ρ(R) R 3 sR < 2e " 1 − sθ(s) Z 0 R 3 −st t e dt #2 . Finally, we note that in view of (6.10), θ(s) ≤ s3 /12 for s sufficiently small, and consequently, by (6.13)2 , Z R 1 (6.16) sθ(s) t3 e−st dt ≤ 2 0 for small s. In addition, et ≥ (1+t7 )/7! and hence, upon multiplying (6.15) by (sR)−7 and making use of (6.16), it suffices to show 3   θ(s) ρ(R) 1  2 6 < (6.17) (sR)−7 + 1 s R 7! in order to obtain (6.15). 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