Expansion and collapse of spherically symmetric isotropic elastic bodies surrounded by vacuum

A class of isotropic and scale invariant strain energy functions is given for which the corresponding spherically symmetric elastic motion includes bodies whose diameter becomes infinite with time or collapses to zero in finite time, depending on the sign of the residual pressure. The body is surrounded by vacuum so that the boundary surface forces vanish, while the density remains strictly positive. The body is subject only to internal elastic stress.


Introduction
We shall be concerned with C 2 spherically symmetric and separable motions of a three-dimensional hyperelastic material based on a class of isotropic and scale invariant strain energy functions.The solid elastic body is surrounded by vacuum so that the boundary surface force vanishes, while the boundary density remains strictly positive.The body is subject only to internal elastic stress.Depending on the sign of the residual pressure, we shall show that the diameter of a spherical body can expand to infinity with time or it can collapse to zero in finite time.
In addition to the assumptions of objectivity and isotropy, we shall impose the more severe restriction of scale invariance on the strain energy function W .That is, W is a homogeneous function of degree h in the deformation gradient F .In the next section, we will show that these basic assumptions imply that W has the form is the shear strain tensor, see [26].The factor (det F ) h/3 accounts for compressibility, and the quantity κ(h) = h The function Φ measures the resistance of the material to shear.The special case of a polytropic fluid arises when Φ is constant and h/3 = −(γ − 1), where γ > 1 is the adiabatic index.Here, however, we shall focus on the case where Φ is far from a constant.This will be measured by the size of a parameter β which is proportional to the shear modulus.For example, an admissible choice would be in a neighborhood of Σ(F ) = I.More generally, higher order terms of the form O (β|Σ(F ) − I| 3 ) may be included.We will return to this example in Section 11.Little is known about the long-time behavior of solutions to the initial free boundary value problem in elastodynamics.In order to gain some insight into the possible behavior, we shall investigate the restricted class of separable motions, the existence of which is dependent upon the scale invariance hypothesis mentioned above.We shall call a motion separable if its material description has the form x(t, y) = a(t)φ(y), in which a : [0, τ ) → R + is a scalar function and φ : B → R 3 is a time-independent deformation of the reference domain B ⊂ R 3 .Separable motions are self-similar in spatial coordinates.In order to have nonconstant shear Φ, the function a(t) must be a scalar.This contrasts with the the case of polytropic fluids where there exist affine motions with a(t) taking values in GL + (3, R).
Under the separability assumption, the spatial configuration of the body evolves by simple dilation, whereby the scalar a(t) controls the diameter.The equations of motion split into an ordinary differential equation for the scalar a(t) and an eigenvalue problem for a nonlinear partial differential equation involving the deformation φ(y).The evolution of a(t) depends on the sign of the residual pressure, which turns out to be − sgn h.When h < 0, the body continuously expands for all time with a(t) ∼ t, as t → ∞.On the other hand, when h > 3, we have a(t) → 0, as t → τ < ∞, so that the body collapses to a point in finite time.
The main effort, then, will be devoted to solving the nonlinear eigenvalue problem for the deformation φ in C 2 .This will be carried out under the assumption of spherical symmetry, consistent with the objectivity and isotropy of W , whence the PDE for φ reduces to an ODE.
For spherically symmetric bodies, the boundary surface force is a pressure, and the nonlinear vacuum (traction) boundary condition requires that the pressure vanishes on ∂B.The vacuum boundary condition shall be fulfilled with the material in a nongaseous phase, i.e. with strictly positive density on ∂B.This also contrasts with the results on affine compressible fluid motion where the vacuum boundary condition holds in a gaseous phase, i.e. both pressure and density vanish on the boundary.
The existence of a family of spherically symmetric eigenfunctions {φ µ } close to the identity deformation with eigenvalue |µ| ≪ 1 will be established in Section 9 by a perturbative fixed point argument, for every value of the elastic moduli κ(h) > 0 and β > 0. The behavior of W (Σ(F )) restricted to the set of spherically symmetric deformation gradients plays a decisive role, see Section 8.If β is sufficiently large, then there exists an eigenvalue for which the eigenfunction satisfies the nongaseous vacuum boundary condition.The positivity of the shear parameter β rules out the hydrodynamical case.A detailed statement of the existence results for expanding and collapsing spherically symmetric separable motion follows in Section 10.
In the final section, we aim to persuade the reader that the assumptions imposed on the strain energy function are physically plausible.We show that any self-consistent choice for the values of W (Σ(F )), restricted to the spherically symmetric deformation gradients, can be extended to all deformation gradients, and we also show that the assumptions are consistent with the Baker-Ericksen condition [2].
Related literature.The equations of motion for nonlinear elastodynamics with a vacuum boundary condition are locally well-posed in Sobolev spaces under appropriate coercivity conditions, see for example [25], [21], [22], [16].Local well-posedness for compressible fluids with a liquid boundary condition was examined in [17], [6] and with a gaseous vacuum boundary condition in [7], [15] respectively.
Affine motion for compressible hydrodynamical models has been studied extensively, see for example [1], [9], [14], [20], but without explicit discussion of boundary conditions.Global in time expanding affine motions for compressible ideal fluids satisfying the gaseous vacuum boundary conditions were constructed and analyzed in [24].
An interesting recent article [4] considers the separable (the term homologous is used instead) motion of self-gravitating elastic balls in the mass critical case h = −1.Expanding solutions are constructed with a solid vacuum boundary condition, and collapsing solutions with a gaseous vacuum boundary condition are predicted on the basis of numerical simulations.
We emphasize that the present work neglects self-gravitation and external forces.The sign of the residual pressure alone determines whether the body collapses or expands.

Notation and basic assumptions
We denote by M 3 the set of 3 × 3 matrices over R with the Euclidean inner product ⟨A, B⟩ = tr AB ⊤ .
We define the groups be a smooth strain energy function.We shall assume that W is objective: and isotropic: Conditions (2.1a), (2.1b), (2.1c) allow for spherically symmetric motion.Finally, we assume that W is scale invariant, that is, it is homogeneous 1 of degree h in F for some h ∈ R: 1 Use of the term homogeneous here should not be confused with the distinct notion of a homogeneous material which in continuum mechanics refers to the independence of the strain energy function with respect to the material coordinates in some reference configuration.This has been tacitly assumed in (2.1a).
Homogeneity of W in F is necessary in order to obtain separable motions.Since W (I) = σ −h W (σI), σ > 0, and since we expect on physical grounds that W (σI) > 0, for σ ̸ = 1, we assume that W (I) = 1.
Using the polar decomposition, it follows from objectivity (2.1b) that The positive definite symmetric matrix A(F ) = (F F ⊤ ) 1/2 is called the left stretch tensor, and its eigenvalues are the principal stretches.
Associated to W , we define its (first) Piola-Kirchhoff stress (2.4a) and Cauchy stress (2.4b) If W satisfies (2.1d), then by differentiation with respect to F we find

Equations of motion for separable solutions
We shall be concerned with the problem of constructing certain smooth motions of an elastic body whose reference configuration B is the unit sphere A motion is a time-dependent family of orientation-preserving deformations x(t, y) x The image, Ω t , of B under the deformation x(t, •) represents the spatial configuration of an elastic body at time t.The spatial description of the body can be given in terms of the velocity vector u(t, x) = D t x(t, y(t, x)) and density ϱ(t, x) = ρ/ det D y x(t, y(t, x)) where y(t, •) = x −1 (t, •) is the reference map taking the spatial domain Ω t back to the material domain B and ρ > 0 is the constant reference density.
The governing equations of elastodynamics, in the absence of external forces, can be written in the form where ω = y/|y| is the normal at y ∈ ∂B.The initial conditions x(0, y), D t x(0, y), y ∈ B are also prescribed.Local well-posedness for this system was studied in [16], [21], [22], [25].
Remark.In the case of polytropic fluids, the Cauchy stress is The vacuum boundary condition can only be fulfilled with vanishing density, i.e. (det F ) −1 = 0 on ∂B.In the sequel, we shall solely consider the case of nonvanishing density on ∂B, in order that F ∈ GL + (3, R) on B.
We shall now impose the major restriction of separability, namely, that the motion can be written in the form (3.2) x(t, y) = a(t)φ(y), for some scalar function and a time-independent orientation-preserving deformation Thus, the spatial configuration of an elastic body under a separable motion evolves by dilation, Ω t = {x = a(t)y : y ∈ B}.
In spatial coordinates, the reference map, velocity, and density of a separable motion are self-similar for x ∈ Ω t .We shall, however, continue to work in material coordinates.
Suppose that a ∈ C 2 ([0, τ )) is a positive solution of ) is an orientation-preserving deformation which solves and satisfies the boundary condition In addtion, φ satisfies and −hµ ≥ 0.
Proof.Since a is assumed to be positive and φ is assumed to be a deformation, x(t, y), as defined, is a motion.By (2.4c), (3.3), (3.4a), the motion x satisfies the system (3.1a): The boundary condition (3.1b) is similarly verified using (3.4b) and the homogeneity of S in F : Finally, by (2.1d),We shall consider the initial value problem for (3.3) in the next section.Sections 5-9 will be devoted to the solution of eigenvalue problem (3.4a),(3.4b).
The deformation gradient of φ, D y φ : B → GL + (3, R), is given by , and Proof.By writing we see that the function is strictly positive and C 1 on (0, 1] with Thus, λ 2 (r) extends to a strictly positive function in C 1 ([0, 1]).It follows that the function ), so to prove that φ ∈ C 2 (B), we need only show that the derivatives up to second order extend continuously to the origin.
It remains to show that D y φ ∈ C 1 (B).For y ̸ = 0, we find from (5.3a) that Since χ(0) = χ ′ (0) = 0, we have that χ ′ (r), χ(r)/r → χ ′ (0) = 0, as r → 0. Thus, we see that the second derivatives of φ extend to continuous functions on B which vanish at the origin.□ A deformation of the form φ(y) = ϕ(r)ω is said to be spherically symmetric.From now on, we focus exclusively upon C 2 spherically symmetric orientation-preserving deformations of the reference domain B where ϕ satisfies (5.1).
Remark.The spatial configuration of a body at time t under a spherically symmetric separable motion x(t, y) = a(t)ϕ(r)ω, defined on [0, τ )× B, is a sphere of radius a(t)ϕ(1).
The next result summarizes the properties of the gradient of a spherically symmetric deformation.
Proof.Fix a vector ω 0 ∈ S 2 .For an arbitrary vector ω ∈ S 2 , choose From (2.1b) and (2.1c), we obtain Using the result of Lemma 6.1, we may define a C 2 function L by (6.1a) In other words, L is the restriction of W to the set of spherically symmetric deformation gradients.By (5.2b), (2.1d), L scales like W : We now obtain expressions for the stresses restricted to the set of spherically symmetric deformation gradients.Lemma 6.2.If W satisfies (2.1a), (2.1b), (2.1c), and L is defined by (6.1a), then the Piola-Kirchhoff stress defined in (2.4a) satisfies λ)P 2 (ω), and the Cauchy stress defined in (2.4b) satisfies λ)P 2 (ω) .Proof.Differentiation of (6.1a) with respect to λ yields It is a standard fact ( [19], Theorem 4.2.5) that the Cauchy stress T (F ) associated to an objective and isotropic strain energy function satisfies By Lemma 5.2, we have Since By (2.4b), there also holds Hence, we may write Taking the M 3 -inner product with P 1 (ω) and P 2 (ω), we have from (6.3) This proves (6.2a), and (6.2b) now follows from (2.4b) and Lemma 5.2.□ Corollary 6.3.Under the assumptions of Lemma 6.2, there holds Proof.For any α > 0, the map φ(y) = αy = αrω is a smooth spherically symmetric deformation, and D y φ(y) = αI.Since S : GL + (3, R) → M 3 is C 1 , the map S(αI) is C 1 in α.By (6.2a), we have α, α)I Now P 1 (ω) is bounded and discontinuous at the origin, so (6.4) must hold.□ Corollary 6.4.Under the assumptions of Lemma 6.2, there holds Proof.This follows directly from Lemma 6.2 and Corollary 6.4.□ We define the residual stress to be S(I) = T (I) = −P(1)I.We shall refer to P(1) as the residual pressure.Corollary 6.5.Under the assumptions and notation of Lemmas 5.1 and 6.2, we have for any spherically symmetric orientation-preserving deformation φ
Remark.We have by (2.2) That is, f is the restriction of W (Σ(F )) to the spherically symmetric deformation gradients.
Next, we explore the implications of (8.2) for the equation (7.1a).
This shows that the problems (8.5a), (8.5b) and (7.1a), (7.1b) are equivalent.By Lemma 7.1, φ(y) = ϕ(r)ω is a C 2 spherically symmetric orientation-preserving deformation which solves (3.4a), (3.4b).□ Remark.The quantity κ(h) defined in (8.4b) corresponds to the bulk modulus, as we shall explain in Lemma 11.3.Materials with a negative bulk modulus are uncommon, and therefore it is reasonable physically to assume κ(h) > 0. In the next lemma, we will also see that positivity of κ(h) relates to the coercivity of the differential operator in (8.5a), and hence the hyperbolicity of the equations of motion.
We now introduce the class to which function f in (8.2) will belong.For a > 0, let us denote The important parameter f ′′ (1) will appear frequently, and for convenience we shall label it as β(f ) = f ′′ (1).We shall see in Lemma 11.3, that this parameter is proportional to the shear modulus.The family {C(M )} M ≥0 is increasing with respect to M .Note that for every B > 0, there exists f B ∈ C(M ) with β(f B ) = B, as illustrated by the functions f B (u) = 1 + 1 2 B(u − 1) 2 , B > 0. We have discussed the assumption that f (1) = W (I) = 1 in Section 2. We have also seen in Lemma 8.1 that the condition f ′ (1) = 0 is necessary.Along with κ(h), the positivity of β(f ) relates to the coercivity condition for (8.5a).The restriction on the third derivative will enable us to establish estimates for the coefficients in (8.4a) uniform with respect to β(f ) for any f ∈ C(M ) in the following lemma, and this, in turn, will prove essential in establishing existence of solutions.Lemma 8.3.Fix h with κ(h) > 0, M ≥ 0, and let f ∈ C(M ).Define (8.9) δ ≡ min{1/8, 1/(8|h|), 1/(8M )}.
Let the functions U i (u), i = 1, 2, be defined according to (8.4a).Then , for some constant C 0 depending only on h and M .

The constant depends on h and M , but not β(f ).
With the aid of (8.13a), (8.13b), we can now verify that the estimates (8.10c) also hold.
Finally, we prove to the statement concerning the function g defined in (8.5b).Returning to (8.12c), we have From the first of these inequalities, it follows that if |h|/β(f ) < δ, then Thus, g has a zero u 0 ∈ U(|h|/β(f )).It follows from the second inequality, that g is strictly increasing on U(δ) and since g(1) = h/3, that sgn(u 0 − 1) = − sgn h.□ Remark.The homogeneous solutions of (7.2) are given by for any c 0 , h ∈ R. Thus, spherically symmetric null Lagrangians may be homogeneous of any degree in F .Null Lagrangians are necessarily homogeneous of degree h = 1, 2, or 3 in F , see [3].For example, the classical null Lagrangian W 0 (F ) = det F has h = 3 and

Existence of eigenfunctions
We shall now address the question of existence of solutions to the problem (8.11b), (8.5b).
Theorem 9.1.Fix h with κ(h) > 0, M ≥ 0, and let f ∈ C(M ).There exists a small constant R > 0 depending only on h and M such that if and The map from {µ : |µ|/(κ(h) is sufficiently large, then there exists an eigenvalue µ ̸ = 0 with sgn µ = − sgn h such that the solution ϕ µ satisfies the boundary condition (8.5b).

Remark.
The assumption that D r ϕ µ (0) = 1 in (9.1a) does not restrict the possible initial data of the motion in (3.2).
Making the substitution (9.3) in equation (8.11b) and using (9.4a), we find that (9.5) Recall that the functions V i (u) depend on f ∈ C(M ) and satisfy the conditions (8.10b), (8.10c).The operator L is an isomorphism on C([0, 1]) with bounded inverse From (9.5), we arrive at the reformulation In order to solve (8.11b), (9.1a), it is sufficient to find a solution ζ of (9.6) in N R , with R ≤ δ.

Let us define
since this expression appears repeatedly.The claim is that for |ε| ≤ R ≪ 1 the map As a consequence of (8.10b), (8.10c), (9.4b), there exists a constant C 1 , independent of R and β(f ), such that For the contraction estimate, we have from (9.8a) Since L −1 ε = 3ε/5 for any constant function ε, by (9.8b), (9.7), there holds . This shows that the map leaves N R invariant.
By assumption, f (u) is the restriction of an appropriate strain energy function to the spherically symmetric deformation gradients.Therefore, Lemma 8.2 yields the desired deformation φ(y) = ϕ(r)ω.
Since φ(B) is a sphere of radius ϕ(1) = λ 2 (1), the final statements follow from (9.2a), (9.2b).□ Remark.Since we have assumed that the reference density is ρ = 1, the density of the deformed configuration in material coordinates is ϱ for then, by (11.1a), (11.1b), (11.1c), the function automatically satisfies the requirements (2.1a), (2.1b), (2.1c), (2.1d), (8.2).The construction of Φ is not entirely routine because the curve has a cusp at u = 1, as shown in Figure 1.Equivalently, (11.2) will be proven if we can find a C 2 function Φ : The function ℓ 2 : R + → R is a homeomorphism, and it can be written as where l2 is a smooth positive function.
Although ξ is not differentiable at w = 0, it follows from (11.5a), (11.5b) Since f > 0, by assumption, the resulting function Φ is positive in a neighborhood of the curve H, and it can modified away from H, if necessary, to ensure positivity on its entire domain without changing its values along the curve H.More generally, we could take If G : R 2 + → R is any smooth function with ∥G(ℓ 1 , ℓ 2 )∥ C 3 uniformly bounded independent of c 1 , c 2 , then there exists an M > 0, such that f (u) ∈ C(M ), for all c 1 , c 2 > 0.
As a final result, we discuss the physical significance of the parameters κ(h) and β(f ).

Figure 1 .
Figure 1.The region R and its boundary H.