On the Global Stability of Compressible Elastic Cylinders in Tension

Consider a three-dimensional, homogeneous, compressible, hyperelastic body that occupies a cylindrical domain in its reference configuration. We identify a variety of hypotheses on the structure of the stored-energy function under which there exists an axisymmetric, homogeneous deformation that globally minimizes the energy. For certain classes of energy functions the uniqueness of this minimizer is also established. The primary boundary condition considered is the extension of the cylinder via the prescription of its deformed axial length, but the biaxial extension of the curved surface is also briefly considered. In particular, the results contained in this paper give conditions on the stored-energy function under which material instabilities, such as necking or the formation of shear bands, are not energy favorable.

1. If we denote the gradient of this homogeneous minimizer by \(\mathbf{F}_{0}\in\mathrm{M}_{+}^{3\times3}\), then it follows that the stored-energy function is quasiconvex at the boundary at (F ₀,n), in the sense of Ball and Marsden [7], for any choice of unit vector n which is normal to some part of the lateral boundary ∂Ω×[0,L] of the cylinder \(\mathcal{C}\).

2. Hill also allows m _i <1, but such values are not covered by our results. See, also, Storåkers [35].

3. Strict rank-one convexity is itself a consequence of the global strong ellipticity of the linearized equations, i.e., \(\mathbf{a}\otimes\mathbf{b}:\mathbb{C}(\mathbf {F})[\mathbf{a}\otimes\mathbf{b}]>0\) for every \(\mathbf{F}\in\mathrm{M}^{3\times3}_{+}\), \(\mathbf{a}\in \mathbb{R}^{3}\), and \(\mathbf{b}\in\mathbb{R}^{3}\) with a≠0≠b. Here \(\mathbb{C}(\mathbf {F}):=\mathrm{d}^{2}W/\mathrm{d}\mathbf{F}^{2}\).

4. See, e.g., [20, p. 220] or [31, p. 408].

5. Φ(a,b,c)=Φ(b,a,c)=Φ(c,b,a) and hence Φ,₁(a,b,c)=Φ,₂(b,a,c)=Φ,₃(c,b,a).

6. The fact that (3.33) implies ∇v is constant for v∈W ^1,3 was first proven by Reshetnyak [27].

7. The flat surfaces of the cylinder, z=−L and z=L, are left free.

8. The existence of such a δ follows from the continuity of W together with the growth conditions (2.8). The uniqueness follows either from the strict rank-one convexity of W or, more simply, from the strict tension-extension inequality Φ,₃₃>0.

The authors thank the referees and R. Fosdick for their helpful comments.

Authors and Affiliations

1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

   Jeyabal Sivaloganathan

2. Department of Mathematics, Southern Illinois University, Carbondale, IL, 62901, USA

   Scott J. Spector

Corresponding author

Correspondence to Scott J. Spector.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1107899.

Appendix A

We here gather a couple of the technical results used in this manuscript.

Proposition A.1

Fix \(N\in\mathbb{N}\). Let \({\varPsi}:(0,\infty)^{N+1}\to\mathbb{R}\) be (strictly) convex with

increasing for i=1,2,…,N. Suppose that v:(0,∞)→(0,∞)^N with v _i convex for i=1,2,…,N. Then

Proof

Let s,t∈(0,∞) and λ∈(0,1). Then the convexity of each component of v implies that

Next, (A.1) and the monotonicity of Ψ in its first N-arguments yields

Finally, the strict convexity of Ψ gives us

which together with (A.2) yields the strict convexity of ψ. □

Remark A.2

If N=1 and Ψ is independent of its second argument then Proposition A.1 reduces to the classical result that the composition of 2 convex functions is convex whenever the outer function is also increasing.

A proof of the following well-known result on convex functions can be found in, for example, [21, Lemma 2.1] or [31, Lemma A.1].

Lemma A.3

Let \(\vartheta:\mathbb{R}^{+}\to\mathbb{R}\) be (strictly) convex. Then

is (strictly) convex on \(\mathbb{R}^{+}\times\mathbb{R}^{+}\).

Appendix B: The Principal Stretches; Other Invariants

In this section we show that slight variants of our method will allow for a variety of constitutive relations that depend on the principal stretches. The key to these results is the following classical lemma, a proof of which can be found in the appendix of [33]. (See [22, p. 693] for a proof in the case ω(t)=t ^p.)

Lemma B.1

Let P∈M^3×3 be symmetric and strictly positive definite with eigenvalues λ ₁, λ ₂, and λ ₃. Suppose that \(\omega:(0,\infty)\to\mathbb{R}\) is convex. Then

for any orthonormal basis {f ₁,f ₂,f ₃} of \(\mathbb{R}^{3}\).

We also made use of a slight generalization of the arithmetic-geometric mean inequality. Again see, e.g., the appendix of [33] for a proof.

Lemma B.2

Let \(\omega:(0,\infty)\to\mathbb{R}\) be monotone increasing and convex. Then, for every \(\mathbf{a}\in\mathbb{R}^{n}\) with a _i >0, i=1,2,…,n,

Moreover, this inequality is strict if either ω is strictly monotone or strictly convex unless all of the a _i are equal.

Let q≥1 and r≥1. Suppose that \(\phi,\psi:(0,\infty)\to\mathbb{R}\) are monotone increasing with ψ convex and \(s\mapsto\phi(\sqrt{s^{q}})\) convex. In this section we will consider the energy functions

where α, β, and γ are the eigenvalues of \(\mathbf{U}=\sqrt{\mathbf{F}^{\mathrm{T}}\mathbf{F}}\) and \(\mathbf{F}\in\mathrm{M}^{3\times3}_{+}\).

The main result of this section is the following. The proof is contained in the subsections that follow.

Proposition B.3

Suppose that Λ is given by (B.1), (B.2), (B.3), or (B.4). Let λ>0, \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\), and p∈[1,p _max]. Define μ=μ(λ,p,u)>0 to be the unique positive real number that satisfies (3.2). Then

(B.5)

Moreover, if μ≤λ then

(B.6)

where \({\mathbf{u}_{\lambda}^{\mu}}\) is given by (3.1) with ν:=μ. If, in addition, ϕ is strictly increasing and p>1, then (B.6) will be a strict inequality unless u∈PS _λ .

Remark B.4

1. 1.

   Since \(\omega(s):=\phi(\sqrt{s^{q}})\) in (B.1) is convex and increasing with q∈[1,2], so is ϕ(t)=ω(t ^2/q) (see Remark A.2).

2. 2.

   For \(\phi\in C^{2}((0,\infty);\mathbb{R})\), the hypothesis \(s\mapsto\phi(\sqrt{s^{q}})\) is convex is equivalent to

   $$t\phi''(t)\ge \biggl(1-\frac{q}{2} \biggr)\phi'(t)\quad \mbox{for all } t>0. $$
3. 3.

   Suppose that the energy is given by (B.1) with q>2. Then one can rewrite (B.1) as

   $${\varLambda}(\mathbf{F})=\hat{\phi} \bigl(\alpha^2 \bigr) +\hat{ \phi} \bigl(\beta^2 \bigr) +\hat{\phi} \bigl(\gamma^2 \bigr), \qquad\hat{\phi}(t):=\phi \bigl(t^{q/2} \bigr), $$

   where \(\omega(s):=\hat{\phi}(s^{2/q})=\phi(s)\) is increasing and convex. Thus the restriction q≤2 is not essential.

4. 4.

   If r=1 the energy (B.3) is the same as the energy in Sect. 3.1.

5. 5.

   If r=1 the energy (B.4) can also be written:

   $${\varLambda}(\mathbf{F})= \biggl[\frac{|\mathrm{adj}\,\mathbf{F}|^2}{\det \mathbf{F}} \biggr]^{q}. $$

In order to establish the existence of a homogeneous minimizer we will also need the following result.

Proposition B.5

Suppose that Λ is given by (B.1), (B.2), (B.3), or (B.4). Let p∈[1,p _max]. Then \(\sigma(t):={\varLambda}(\sqrt[p]{t}\mathbf{I})\) is increasing and convex.

Proof

If Λ is given by (B.1), then σ(t):=3ϕ(t ^q/p), which is monotone increasing and convex since q≥p>0 and ϕ is increasing and convex (see Remark A.2). If Λ is given by (B.2), then σ(t):=3ψ(t ^q/p), which is increasing and convex since q≥p>0 and ψ is increasing and convex. If Λ is given by (B.3), then σ(t):=3^q/(2r) t ^q/p, which is increasing and convex since q≥p>0. Finally, if Λ is given by (B.4), then σ(t):=3^q/r t ^q/p, which is increasing and convex since q≥p>0. □

2.1 B.1 The Energy Λ(F)=ϕ(α ^q)+ϕ(β ^q)+ϕ(γ ^q) for q∈[1,2]

In this subsection we prove Proposition B.3 when Λ is given by (B.1).

Proof of Proposition B.3: Case 1

Let Λ be given by (B.1) with q∈[1,2], where \(\phi\in C^{1}(\mathbb{R}^{+};\mathbb{R})\) is increasing and \(s\mapsto\phi(\sqrt{s^{q}})\) is convex. Fix λ>0, \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\), and p∈[1,q]. Let μ be given by (3.2). Define F=F(x,y,z):=∇u(x,y,z), J _u :=det∇u(x,y,z), and \(\mathbf{U}:=\sqrt{\mathbf{F}^{\mathrm{T}}\mathbf{F}}\) with eigenvalues α, β, and γ. Then, since q≥p>0, Hölder’s inequality and (3.2) yield

(B.7)

Next, since ϕ is increasing and convex, Lemma B.2, Jensen’s inequality, (B.1), (B.7), and the identity J _u =αβγ imply that

which establishes (B.5).

Now assume that μ≤λ. Define \(\omega:\mathbb{R}^{+}\to\mathbb{R}\) by \(\omega(s):=\phi (\sqrt{s^{q}})\) so that ω is convex. If we then apply Lemma B.1 with P=F ^T F (with eigenvalues α ², β ², and γ ²) we conclude that, for any \((x,y,z)\in\mathcal{C}\),

or, equivalently,

Next, (B.8) together with (3.6), Lemma B.2, and the monotonicity of ϕ yield

Define

Then, in view of (B.9),

Since ϕ is convex (see Remark B.4.1) and increasing it follows that \(H:\mathbb{R}^{+}\times\mathbb{R}^{+}\to \mathbb{R}\) is convex (see Lemma A.3 and Remark A.2). Moreover, since ϕ′ is positive and increasing, it follows that if t<s, then

thus, for any t>0, the mapping s↦H(s,t) is increasing on [t,∞). The remainder of the proof of (B.6), when μ≤λ, is then similar to the proof of (3.3) in Proposition 3.1. □

2.2 B.2 The Energy Λ(F)=ψ(α ^q β ^q/γ ^q)+ψ(β ^q γ ^q/α ^q)+ψ(γ ^q α ^q/β ^q) with q≥1

In this subsection we prove Proposition B.3 when Λ is given by (B.2).

Proof of Proposition B.3: Case 2

Let Λ be given by (B.2) with q≥1, where \(\psi\in C^{1}(\mathbb{R}^{+};\mathbb{R})\) is increasing and convex. Fix λ>0, \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\), and p∈[1,q]. Let μ be given by (3.2). Define F=F(x,y,z):=∇u(x,y,z), J _u :=det∇u(x,y,z), and \(\mathbf {U}:=\sqrt{\mathbf{F}^{\mathrm{T}}\mathbf{F}}\), with eigenvalues α, β, and γ.

We first note that, since ψ is increasing and convex, Lemma B.2, Jensen’s inequality, (B.7), and (B.2) imply that

which establishes (B.5).

Now suppose that μ≤λ. Note that \(\mathbf{P}:=(\operatorname {adj}\mathbf{F})(\operatorname {adj}\mathbf{F})^{\mathrm{T}}/(\det \mathbf{F})\) is symmetric and strictly positive definite with eigenvalues αβ/γ, βγ/α, and γα/β. Then, in view of (B.2), Lemma B.1 (with ω(t)=ψ(t ^q)) yields, for any \((x,y,z)\in\mathcal{C}\),

where J _u :=det∇u=αβγ. As in the proof in Sect. B.1, Lemma B.2 now implies that

However,

and Fe _z =u _z .

If we now combine (B.10), (B.11), and (B.12) we find, with the aid of the monotonicity of ψ, that

Define

Then, in view of (B.13),

Since ψ is convex and increasing and q≥p>0, it follows that \(H:\mathbb{R}^{+}\times\mathbb{R}^{+}\to\mathbb{R}\) is convex (see Lemma A.3 and Remark A.2). Moreover, since ψ′ is positive and increasing, it follows that if t<s, then

thus, for any t>0, the mapping s↦H(s,t) is increasing on [t,∞). The remainder of the proof of (B.6) is then similar to the proof of (3.3) in Proposition 3.1. □

2.3 B.3 The Energy Λ(F)=(α ^2r+β ^2r+γ ^2r)^q/(2r) with q≥1 and r≥1

In this subsection we prove Proposition B.3 when Λ is given by (B.3).

Let Λ be given by (B.3) with r≥1 and q≥1. Fix λ>0, \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\), and p≥1 with p≤q and p≤3r. Let μ be given by (3.2). Define F=F(x,y,z):=∇u(x,y,z), J _u :=det∇u(x,y,z), and \(\mathbf{U}:=\sqrt{\mathbf{F}^{\mathrm{T}}\mathbf{F}}\), with eigenvalues α, β, and γ. Then Lemma B.2, (B.7), and (B.3) yield

which establishes (B.5).

Now suppose that μ≤λ. If we then apply Lemma B.1 with ω(t)=t ^r and P=F ^T F (with eigenvalues α ², β ², and γ ²) we conclude that, for any \((x,y,z)\in\mathcal{C}\),

or, equivalently,

Next, (B.14) together with (3.6), the arithmetic-geometric mean inequality, and the monotonicity of t↦t ^r yield

and hence

Define

and note that (B.15) implies

Now

for s>0 and 3r≥p. Consequently, Lemma A.3 implies that G and hence H=G ^q/p is strictly convex. Moreover,

for τ:=t/s<1; thus for any t>0 the mappings s↦G(s,t) and s↦H(s,t) are strictly increasing on [t,∞). The remainder of the proof of (B.6) is then similar to the proof of (3.3) in Proposition 3.1.  □

2.4 B.4 The Energy Λ(F)=[(αβ/γ)^r+(βγ/α)^r+(γα/β)^r]^q/r with q≥1 and r≥1

In this subsection we prove Proposition B.3 when Λ is given by (B.4).

Proof of Proposition B.3: Case 4

Let Λ be given by (B.4) with r≥1 and q≥1. Fix λ>0, \(\mathbf{u}\in{\mathcal {A}_{\lambda}}\), and p≥1 with p≤q and p≤3r. Let μ be given by (3.2). Define F=F(x,y,z):=∇u(x,y,z), J _u :=det∇u(x,y,z), and \(\mathbf {U}:=\sqrt{\mathbf{F}^{\mathrm{T}}\mathbf{F}}\), with eigenvalues α, β, and γ. Then Lemma B.2, (B.7), and (B.4) yield

which establishes (B.5).

Now suppose that μ≤λ. Then, if we apply Lemma B.1 with ω(t)=t ^r and \(\mathbf{P}:=(\operatorname {adj}\mathbf{F})(\operatorname {adj}\mathbf{F})^{\mathrm{T}}/(\det \mathbf{F})\) (with eigenvalues αβ/γ, βγ/α, and γα/β) we conclude, with the aid of (B.4), that for any \((x,y,z)\in\mathcal{C}\)

where J _u :=det∇u=αβγ. The arithmetic-geometric mean inequality now implies that

However,

and Fe _z =u _z .

If we now combine (B.16), (B.17), and (B.18) we find, with the aid of the monotonicity of t↦t ^r, that

Define

so that (B.19) implies

Then

for s>0 and 3r≥p and hence, in view of Lemma A.3, H is strictly convex. Moreover,

for τ:=t/s<1; thus for any t>0 the mappings s↦G(s,t) and hence s↦H(s,t) are strictly increasing on [t,∞). The remainder of the proof of (B.6) is then similar to the proof of (3.3) in Proposition 3.1. □

Appendix C: The Homogeneity of Energy-Minimizing Deformations II

Fix \(N\in\mathbb{N}\) and define \({\boldsymbol{\varLambda}}:\mathrm {M}^{3\times3}_{+}\to\mathbb{R}^{N}\) by

where each component \({\varLambda}_{i}:\mathrm{M}^{3\times3}_{+}\to \mathbb{R}\) is one of the functions, Λ(F), given in this section and \(p^{i}_{\mathrm{max}}\) is the corresponding maximum value of the parameter p.

Lemma C.1

Let Λ be given by (C.1). Suppose that the stored energy \(W\in C^{1}( \mathrm{M}_{+}^{3\times3};[0,\infty))\) satisfies

where p∈[1,p _max] with \(p_{\mathrm{max}}:=\min\{p^{i}_{\mathrm{max}}, 1\le i \le n \}\), \({\varPsi}: (0,\infty)^{N+1}\to\mathbb{R}\) is (strictly) convex, and Λ is given by (C.1). Then

Moreover, if in addition the reference configuration is stress free, then

Proof

To obtain convexity define

Then Proposition B.5 together with Proposition A.1 yield the (strict) convexity of ξ.

Next, if we take the derivative of ξ we find that

where S is the Piola-Kirchhoff stress tensor (2.6). Thus, if the reference configuration is stress free, \(\dot{\xi}(1)=0\). The (strict) convexity of ξ then implies that ξ and hence the mapping λ↦W(λ I) is (strictly) increasing on [1,∞). □

3.1 C.1 Energy Reduction by Symmetrization

Theorem C.2

Assume that the reference configuration is stress free. Suppose that the stored energy \(W\in C^{1}( \mathrm{M}_{+}^{3\times3};[0,\infty))\) satisfies

where p∈[1,p _max] with \(p_{\mathrm{max}}:=\min\{p^{1}_{\mathrm{max}},p^{2}_{\mathrm {max}},p^{3}_{\mathrm{max}},\ldots,p^{N}_{\mathrm{max}}\}\), Λ is given by (C.1), and \({\varPsi}:(0,\infty)^{N+1}\to\mathbb{R}\) is increasing in each of its first N-arguments and convex. Let λ≥1 and \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\). Define μ=μ(λ,p,u)>0 to be the unique positive real number that satisfies (3.2). Then

(C.3)

where \({\mathbf{u}_{\lambda}^{\nu}}\) is given by (3.1) with ν:=μ if μ≤λ and ν:=λ if μ≥λ. Moreover, if μ≤λ, p>1, and Ψ is strictly increasing in one of its first N-arguments, then (C.3) is a strict inequality unless u∈PS _λ .

Proof

Fix λ≥1, \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\), and define μ by (3.2). If we now let F=∇u in (C.2), integrate over the cylinder \(\mathcal{C}\), and apply Jensen’s inequality to the convex function Ψ we conclude, with the aid of (3.2), that

(C.4)

Now suppose that μ≤λ. Then we can make use of Proposition B.3, the monotonicity of Ψ in each of its first N-arguments, (3.1), and (C.2), to get

(C.5)

which when combined with (C.4) yields the desired inequality, (C.3) with ν=μ. Moreover, if Ψ is strictly increasing in, say, its i-argument, then the inequality in (C.5) is a strict inequality unless

Proposition B.3 now yields u∈PS _λ when p>1.

Finally, suppose that μ≥λ so that ν=λ. Then we can make use of Proposition B.3, the monotonicity of Ψ in each of its first N-arguments, (3.1), and (C.2), to obtain

(C.6)

and we have made use of (C.2). However, Lemma C.1 then implies that W(η I)≥W(λ I), which together with (C.4) and (C.6) yields the desired result, (C.3) with ν=λ. □

3.2 C.2 Existence of a Homogeneous Minimizer

Theorem C.3

Let W, Ψ, Λ, and p satisfy the hypotheses of Theorem  C.2. Suppose, in addition, that

Then for each λ≥1 there exists a κ=κ(λ)>0 such that the deformation

is an absolute minimizer of the energy among deformations in \({\mathcal{A}_{\lambda}}\).

Proof

Fix λ≥1. Then (C.7) and Proposition 2.5 yield α>0 and β>0 such that

Now apply Theorem C.2 with u=u ^∗, where

to get a κ:=ν>0 such that

where we have made use of (3.12) and (C.8).

We claim that the homogeneous deformation given by (C.8) with this value of κ is an absolute minimizer of E among deformations in \({\mathcal{A}_{\lambda}}\). To see this, let \(\mathbf{u}\in{\mathcal{A}_{\lambda}}\). Then, by Theorem C.2, there is ν>0 such that the homogeneous deformation \({\mathbf{u}_{\lambda}^{\nu}}\) given by (3.1) satisfies (see (3.12))

The desired minimality of \({\mathbf{u}_{\lambda}^{\kappa}}\) now follows from (C.10), (C.11), and (C.9) with s=t=ν. □

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