# On self-gravitating
polytropic elastic balls

Simone Calogero
Department of Mathematical Sciences
Chalmers University of Technology, University of Gothenburg
Gothenburg, Sweden
<EMAIL_ADDRESS>
###### Abstract

A new four-parameters family of constitutive functions for spherically
symmetric elastic bodies is introduced which extends the two-parameters class
of polytropic fluid models widely used in several applications of fluid
mechanics. The four parameters in the polytropic elastic model are the
polytropic exponent $\gamma$, the bulk modulus $\kappa$, the shear exponent
$\beta$ and the Poisson ratio $\nu\in(-1,1/2]$. The two-parameters class of
polytropic fluid models arises as a special case when $\nu=1/2$ and
$\beta=\gamma$. In contrast to the standard Lagrangian approach in elasticity
theory, the polytropic elastic model in this paper is formulated directly in
physical space, i.e., in terms of Eulerian state variables, which is
particularly useful for the applications e.g. to astrophysics where the
reference state of the bodies of interest (stars, planets, etc.) is not
observable. After discussing some general properties of the polytropic elastic
model, the steady states and the homologous motion of Newtonian self-
gravitating polytropic elastic balls are investigated. It is shown numerically
that static balls exist when the parameters $\gamma,\beta$ are contained in a
particular region $\mathcal{O}$ of the plane, depending on $\nu$, and proved
analytically for $(\gamma,\beta)\in\mathcal{V}$, where
$\mathcal{V}\subset\mathcal{O}$ is a disconnected set which also depends on
the Poisson ratio $\nu$. Homologous solutions describing continuously
collapsing balls are constructed numerically when $\gamma=4/3$. The radius of
these solutions shrinks to zero in finite time, causing the formation of a
center singularity with infinite density and pressure. Expanding self-
gravitating homologous elastic balls are also constructed analytically for
some special values of the shear parameter $\beta$.

##  1 Introduction

The problem of self-gravitating bodies in astrophysics is the source of many
interesting questions in mathematical physics. An important example is the
motion of self-gravitating fluids described by the Euler-Poisson system

$\displaystyle\partial_{t}\rho+\nabla\cdot(\rho u)=0,$
$\displaystyle\rho(\partial_{t}u+u\cdot\nabla u)+\nabla p=-\rho\nabla\phi,$
$\displaystyle\Delta\phi=4\pi G\rho,\quad\lim_{|x|\to\infty}\phi(t,x)=0.$

Here $\rho,u,p$ are respectively the mass density, the velocity field and the
pressure of the fluid, $\phi$ denotes the fluid own self-generated Newtonian
gravitational field and $G$ is Newton’s gravitational constant. The system is
completed by adding a constraint between the variables $\rho,u,p$, e.g.,
$p=F(\rho)$ for barotropic fluids. When applied to gaseous stars, the
polytropic equation of state

$p=c\rho^{\gamma},\quad c,\gamma>0,$ (1.1)

is a standard choice in astrophysics [12] and remains today the most studied
stellar model in the literature [15].

The present paper is part of a series which aims to extend the theory of self-
gravitating fluids to self-gravitating elastic bodies. The articles [1, 2, 3]
focus on self-gravitating spherically symmetric elastic bodies in static
equilibrium. In the present paper some first results on the problem of self-
gravitating elastic balls in motion will be established.

Even though elastic matter models have long been used in astrophysics [14, 16,
17], and despite important applications to e.g. describe the deformation of
planets [20] and neutron stars crusts [10], the problem of self-gravitating
elastic bodies has received so far scarce attention compared to the analogous
problem for fluid bodies. This could in part be the result of an apparent
conceptual discrepancy between (non-linear) elasticity and astrophysics: the
former is originally a theory in Lagrangian form [11, 19, 21] describing the
state of bodies in terms of spatial deformations from a given reference
configuration, whereas the latter is an Eulerian type theory concerned with
the properties of planets, stars, galaxies, etc., in their current (physical)
configuration. While it is possible to study the problem of self-gravitating
elastic bodies in the Lagrangian formulation of continuum mechanics [4, 7, 8,
9], for astrophysical applications the obtained results should eventually be
translated in terms of Eulerian state variables, such as $\rho,u,p$ in the
fluid case, which can be determined by model-fitting observational data.
However this transition from the reference to the physical space is in general
not possible because the deformation map solution of the field equations in
Lagrangian coordinates is typically neither globally injective nor a very
regular function.

In [1] it was shown that these two different approaches of astrophysics and
elasticity can be reconciled, namely that the equation of state of spherically
symmetric elastic bodies can be defined directly in physical space as a
constraint between Eulerian state variables. This formulation has been used to
prove the existence of static self-gravitating elastic matter distributions
with arbitrarily large strain [1, 2], albeit only in the spherically symmetric
case. This is the formulation of the theory of self-gravitating elastic balls
used in the present paper. It is described, and expanded, in Section 2. The
(formally) equivalent Lagrangian formulation is discussed in Section 2.3.

Another reason for the hard interplay between astrophysics and elasticity
theory is the absence of a systematic study on which elastic constitutive
functions may be relevant for the applications to the problem of self-
gravitating elastic bodies. Some classical elastic material models, such as
the Saint-Venant-Kirchhoff model, have been used to describe planet
deformations [20]. However, as shown in Section 2.4 below, see also [11], for
large deformations the Saint-Venant-Kirchhoff model—and others—violates
important physical and mathematical properties, such as global hyperbolicity
and the Baker-Ericksen inequality.

One aim of this paper is to introduce a new elastic material model for
spherically symmetric bodies that generalises the polytropic equation of state
for fluids. More precisely, the new constitutive function implies the
following properties: (i) the equations of motion (conservation of mass and
balance of momentum) are scale invariant and (ii) the boundary of isolated
balls is subject to constant shear. While in the case of barotropic fluids
these two properties uniquely characterize the polytropic equation of state,
in the elastic case they still leave the freedom to choose an arbitrary
function of the shear in the constitutive function. In Section 3 a well-
motivated choice for this shear function is made, which results in a
particular elastic constitutive function satisfying the properties (i) and
(ii) above and which, in addition, is globally hyperbolic and verifies the
Baker-Ericksen inequality. Each model in this class will be referred to as
“polytropic elastic model” and depends on four parameters
$\kappa,\nu,\beta,\gamma$. The parameters $\gamma,\kappa$ correspond to the
polytropic exponent and the bulk modulus, $\nu$ is the Poisson ratio of the
elastic material ($\nu=1/2$ for fluid matter), while $\beta$ is a new
parameter expressing the dependence of the elastic constitutive function on
shear deformations.

The last section of the paper is devoted to the analysis of self-gravitating
polytropic elastic balls. In Section 4.1 static balls are constructed
numerically when $\beta,\gamma$ are contained in a particular connected set
$\mathcal{O}$ depending on the Poisson ratio $\nu$, see Figure 3. A conjecture
is put forward that $\mathcal{O}$ is the largest region in the
$(\gamma,\beta)$ plane for which static self-gravitating polytropic elastic
balls exist. In the fluid limit $\nu\to 1/2$, $\beta=\gamma$, the conjecture
reduces to the well-known fact that static self-gravitating polytropic fluid
balls exist if and only if $\gamma>6/5$. In Appendix a special case of the
conjecture is proved analytically, namely that static self-gravitating
polytropic elastic balls form when the parameters $\gamma,\beta$ are contained
in a certain disconnected region $\mathcal{V}\subset\mathcal{O}$, depending on
the Poisson ratio $\nu$. Finally in Section 4.2 the homologous motion of self-
gravitating polytropic elastic balls is investigated. Homologous solutions
describe self-gravitating balls which, after a time $t_{0}>0$, are either
continuously expanding or continuously collapsing and which, in the second
case, develop a singularity at the center in finite time. In the fluid case
homologous collapsing balls were constructed numerically in [13] and
analytically in [18]. In this paper similar collapsing solutions are found
numerically in the elastic case. Subsequently the existence of expanding
elastic balls is proved analytically for some special values of the shear
parameter $\beta$.

##  2 Eulerian representation of elastic balls

A spherically symmetric matter distribution is a quadruple
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$ of real-valued functions of time
$t\in\mathbb{R}$ and radius $r>0$, where $\rho(t,r)$ is the mass density,
$p_{\mathrm{rad}}(t,r),p_{\mathrm{tan}}(t,r)$ are the radial and tangential
pressure and $u(t,r)$ is the radial component of the velocity field. The
Cauchy stress tensor of spherically symmetric matter distributions is

$\sigma_{ij}(t,r)=\sigma_{\mathrm{rad}}(t,r)\frac{x_{i}x_{j}}{r^{2}}+\sigma_{\mathrm{tan}}(t,r)\left(\delta_{ij}-\frac{x_{i}x_{j}}{r^{2}}\right),$

where $x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$ are spatial Cartesian
coordinates such that $r=|x|$ and

$\sigma_{\mathrm{rad}}(t,r)=-p_{\mathrm{rad}}(t,r),\quad\sigma_{\mathrm{tan}}(t,r)=-p_{\mathrm{tan}}(t,r)$

are the radial and tangential stress in the matter interior.

A spherically symmetric matter distribution
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$ is called a spherically symmetric
body if its domain in the radial variable is, at all times, non-empty,
connected and bounded. Any such body is therefore either a ball or a shell of
matter. The focus of this paper is on balls of matter with strongly regular
center, whose precise definition is the following.

######  Definition 1.

Let $T>0$. A quadruple
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u):[0,T]\times[0,\infty)\to\mathbb{R}^{4}$
is said to be a ball of matter with strongly regular center if there exists
$R:[0,T]\to(0,\infty)$ such that $R\in C^{1}([0,T])$ and

  * (i)

$\rho(t,r),p_{\mathrm{rad}}(t,r)$ and $p_{\mathrm{tan}}(t,r)$ are positive for
$r\in[0,R(t))$, for all $t\in[0,T]$;

  * (ii)

$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)\in C^{0}(\Omega_{T})\cap
C^{1}(\Omega^{*}_{T})$, where

$\Omega_{T}:=\cup_{t\in[0,T]}[0,R(t)],\quad\Omega^{*}_{T}:=\cup_{t\in[0,T]}[0,R(t));$

  * (iii)

$\rho=p_{\mathrm{rad}}=p_{\mathrm{tan}}=u=0$, for
$(t,r)\in[0,T]\times[0,\infty)\diagdown\Omega_{T}$;

  * (iv)

$p_{\mathrm{rad}}(t,0)=p_{\mathrm{tan}}(t,0)$,
$\partial_{t}p_{\mathrm{rad}}(t,0)=\partial_{t}p_{\mathrm{tan}}(t,0)$ and
$u(t,0)=\partial_{t}u(t,0)=0$;

  * (v)

$\lim_{r\to 0^{+}}\partial_{r}\rho(t,r)=\lim_{r\to
0^{+}}\partial_{r}p_{\mathrm{rad}}(t,r)=\lim_{r\to
0^{+}}\partial_{r}p_{\mathrm{tan}}(t,r)=0$, and

$\lim_{r\to 0^{+}}\partial_{r}u(t,r)=\lim_{r\to
0^{+}}u(t,r)/r:=\omega(t),\quad\omega\in C([0,T]).$

The fundamental equations describing the evolution of balls of matter are the
continuity equation for the mass density and the balance of momentum equation:

$\displaystyle\partial_{t}\rho+\frac{1}{r^{2}}\partial_{r}(r^{2}\rho u)=0,$
(2.1a)
$\displaystyle\rho(\partial_{t}u+u\partial_{r}u)=-\partial_{r}p_{\mathrm{rad}}+\frac{2}{r}(p_{\mathrm{tan}}-p_{\mathrm{rad}})+\rho
f$ (2.1b)

where $f(t,r)$ denotes the force (per unit of mass) acting on the ball and
which is not due to the internal strain. In this paper we are mostly
interested in self-gravitating balls, for which

$f(t,r)=-G\frac{m}{r^{2}}$ (2.2a) where $G$ is Newton’s gravitational constant
and $m(t,r)=4\pi\int_{0}^{r}\rho(t,s)\,s^{2}\,ds.$ (2.2b)

The system (2.1) is posed in the matter interior and has to be supplemented by
boundary conditions at $r=R(t)$. We assume

$u(t,R(t))=\dot{R}(t),\quad p_{\mathrm{rad}}(t,R(t))=0.$ (2.3)

The boundary condition $u(t,R(t))=\dot{R}(t)$ means that the boundary of the
ball is comoving with the matter and implies, using (2.1a), that the total
mass of the ball

$m(t,R(t))=4\pi\int_{0}^{R(t)}\rho(t,r)r^{2}\,dr$

is conserved. The boundary condition $p_{\mathrm{rad}}(t,R(t))=0$ signifies
that the ball is surrounded by vacuum; see [15] for other examples of boundary
conditions used in astrophysics.

The system (2.1) is completed by adding an equation of state on the variables
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$. As mentioned in the
Introduction, a popular example is the barotropic fluid equation of state
$p_{\mathrm{rad}}(t,r)=p_{\mathrm{tan}}(t,r)=F(\rho)$. In [1] it was shown
that a similar equation of state holds for spherically symmetric elastic
bodies.

######  Definition 2.

Let $\kappa>0$ and $1<\nu\leq 1/2$ be given. An elastic constitutive function
for spherically symmetric bodies with bulk modulus $\kappa$ and Poisson ratio
$\nu$ is a $C^{2}$ map
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}}):(0,\infty)^{2}\to\mathbb{R}^{2}$
satisfying

$\widehat{p}_{\mathrm{rad}}(\delta,\delta)=\widehat{p}_{\mathrm{tan}}(\delta,\delta),\quad\partial_{\eta}(\widehat{p}_{\mathrm{rad}}+2\widehat{p}_{\mathrm{tan}})(\delta,\delta)=0,\quad\text{for
all $\delta>0$,}$ (2.4)

as well as

$\widehat{p}_{\mathrm{rad}}(1,1)=\widehat{p}_{\mathrm{tan}}(1,1)=0,$ (2.5a)
$\kappa^{-1}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(1,1)=\frac{3(1-\nu)}{1+\nu},\quad\kappa^{-1}\partial_{\eta}\widehat{p}_{\mathrm{rad}}(1,1)=-\frac{2(1-2\nu)}{1+\nu},$
(2.5b)
$\kappa^{-1}\partial_{\delta}\widehat{p}_{\mathrm{tan}}(1,1)=\frac{3\nu}{1+\nu},\quad\kappa^{-1}\partial_{\eta}\widehat{p}_{\mathrm{tan}}(1,1)=\frac{1-2\nu}{1+\nu}.$
(2.5c)

If there exists a $C^{3}$ function $\widehat{w}:(0,\infty)^{2}\to\mathbb{R}$
such that

$\widehat{w}(1,1)=0$ (2.6)

and

$\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\delta^{2}\partial_{\delta}\widehat{w}(\delta,\eta),\quad\widehat{p}_{\mathrm{tan}}(\delta,\eta)=\widehat{p}_{\mathrm{rad}}(\delta,\eta)+\frac{3}{2}\delta\eta\partial_{\eta}\widehat{w}(\delta,\eta),$
(2.7)

then the constitutive function is said to be hyperelastic with stored energy
function $\widehat{w}$.

######  Definition 3.

A ball of matter $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$ is said to be an
(homogeneous) elastic ball with reference density $\mathcal{K}>0$ and
reference pressure $\mathcal{P}\geq 0$ if there exists an elastic constitutive
function $(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for
spherically symmetric bodies such that the following equation of state holds:

$\displaystyle\rho(t,r)=\mathcal{K}\delta(t,r),$ (2.8a) $\displaystyle
p_{\mathrm{rad}}(t,r)=\mathcal{P}+\widehat{p}_{\mathrm{rad}}(\delta(t,r),\eta(t,r)),$
(2.8b) $\displaystyle
p_{\mathrm{tan}}(t,r)=\mathcal{P}+\widehat{p}_{\mathrm{tan}}(\delta(t,r),\eta(t,r)),$
(2.8c) where $\eta(t,r)=\frac{m(t,r)}{\frac{4\pi}{3}\mathcal{K}r^{3}},\quad
r\in(0,R(t)).$ (2.8d)

Remark. The $(\delta,\eta)$-space in Definition 2 corresponds to the Eulerian
(or physical) space of configurations of the spherically symmetric elastic
body. The point $(\delta,\eta)=(1,1)$ is called reference, or unstrained,
configuration; in terms of the principal stretches $\lambda_{1},\lambda_{2}$
we have $(\delta,\eta)=(1,1)\Leftrightarrow(\lambda_{1},\lambda_{2})=(1,1)$,
see Section 2.3. The reference configuration is used in Definition 3 to define
the reference state of elastic balls. In particular, according to Definition
3, elastic balls have constant mass density $\mathcal{K}$ and constant
(isotropic) pressure $\mathcal{P}$ in the reference state. When
$\mathcal{P}=0$, as it is assumed in [1], the reference state is said to be a
natural state

Remark. As a consequence of (2.5), all constitutive functions
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for a given material
(i.e., for given parameters $\kappa,\nu$) have the same linear approximation
around the reference configuration $(\delta,\eta)=(1,1)$. This property
expresses the postulate of compatibility with linear elasticity in our
formulation of (non-linear) elasticity theory for spherically symmetric
bodies.

Remark. The Eulerian characterization of elastic balls given in Definition 3
has an equivalent representation in the Lagrangian space, i.e., in the space
of spatial deformations of the ball from the reference configuration, which is
discussed in Section 2.3. In the Lagrangian formulation the radius of the ball
is constant and given by

$Z=\left(\frac{3M}{4\pi\mathcal{K}}\right)^{1/3},$

where $M$ is the total mass of the ball, which is preserved by deformations.

Remark. It is important to keep in mind the difference between the elastic
constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ and the equation of
state (2.8). The former depends only on material properties, such as $\kappa$
and $\nu$, while the latter depends also on the reference state parameters
$\mathcal{K},\mathcal{P}$ as well as, in general, on geometrical properties of
the body. For instance, the equation of state for elastic shells is given by
(2.8) with the following alternate expression for the function $\eta$:

$\eta(t,r)=\left(\frac{\mathcal{S}}{r}\right)^{3}+\frac{3}{r^{3}}\int_{R_{\mathrm{in}}(t)}^{r}\delta(t,s)s^{2}ds,\quad
r\in(R_{\mathrm{in}}(t),R(t)),$

where $R_{\mathrm{in}}(t)\in(0,R(t))$ is the inner radius of the shell and
$\mathcal{S}$ is a physically dimensional constant which corresponds to the
inner radius of the shell in the reference state, see [1]. In the case of
fluid bodies, the equations of state of balls and shells are the same, because
the fluid constitutive function is independent of $\eta$, see Section 2.1
below.

Remark. Let the shear variable $y$ be defined as

$y=\frac{\delta}{\eta};$ (2.9)

in terms of the principal stretches $\lambda_{1},\lambda_{2}$, we have
$y=\lambda_{2}/\lambda_{1}$, see Section 2.3. The condition (2.4) signifies
that the state $y=1$ is isotropic and ensures in particular that the center of
static elastic balls is strongly regular. Moreover it can be shown that for
hyperelastic constitutive functions the second condition in (2.4) follows by
the first one, see [2, Thm. 1].

Remark. In [1, 2] the Lamé parameters $\lambda,\mu$ are used instead of
$\kappa,\nu$. The relation between these two pairs of material constants is

$\kappa=\lambda+\frac{2\mu}{3},\quad\nu=\frac{\lambda}{2(\lambda+\mu)}.$

In this paper the Poisson ratio plays an important role, which is the reason
to deviate from the notation used in [1, 2].

Remark. Letting

$h_{1}(\delta,\eta)=\delta^{-2}\widehat{p}_{\mathrm{rad}}(\delta,\eta),\quad
h_{2}(\delta,\eta)=\frac{2}{3}(\delta\eta)^{-1}(\widehat{p}_{\mathrm{tan}}(\delta,\eta)-\widehat{p}_{\mathrm{rad}}(\delta,\eta)),$
(2.10a) an elastic constitutive equation is hyperelastic if and only if the
form $h_{1}(\delta,\eta)d\delta+h_{2}(\delta,\eta)d\eta$ is exact. Hence we
have the following equivalency:
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})\
\text{hyperelastic}\Leftrightarrow\partial_{\eta}h_{1}=\partial_{\delta}h_{2}.$
(2.10b)

###  2.1 Fluid constitutive functions

In this section we discuss how the special case of (barotropic) fluid balls
fits into our formulation of elasticity theory for spherically symmetric
bodies.

######  Definition 4.

An elastic constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for spherically
symmetric bodies is said to be a (barotropic) fluid constitutive function if
there exists $\widehat{p}:(0,\infty)\to\mathbb{R}$ such that

$\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\widehat{p}_{\mathrm{tan}}(\delta,\eta)=\widehat{p}(\delta).$

A ball of matter $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$ is said to be a
fluid ball with reference density $\mathcal{K}>0$ and reference pressure
$\mathcal{P}\geq 0$ if there exists a fluid constitutive function
$\widehat{p}$ such that the following equation of state holds

$\rho(t,r)=\mathcal{K}\delta(t,r),\quad
p_{\mathrm{rad}}(t,r)=p_{\mathrm{tan}}(t,r)=p(t,r)=\mathcal{P}+\widehat{p}(\delta(t,r)).$
(2.11)

Remark. By (2.5), $\nu=1/2$ and $\widehat{p}\,^{\prime}(1)=\kappa$ must hold
for a fluid constitutive function. Moreover all fluid constitutive functions
are hyperelastic.

Remark. Definition 4 is perfectly meaningful even for shells of matter, or
even if the body is not spherically symmetric, but in the following it will be
applied only to fluid balls.

The important example of polytropic fluid balls, i.e., fluid balls with
equation of state (1.1), is obtained by choosing the constitutive function and
the reference pressure as

$\widehat{p}(\delta)=\frac{\kappa}{\gamma}\,(\delta^{\gamma}-1),\quad\mathcal{P}=\frac{\kappa}{\gamma}=-\widehat{p}(0),$
(2.12)

which we call respectively the polytropic fluid constitutive function and the
polytropic reference pressure. The constant $c>0$ in (1.1) is given by
$c=\kappa/(\gamma\mathcal{K}^{\gamma})$; the stored energy function of
polytropic fluid balls is

$\widehat{w}_{\mathrm{pf}}(\delta,\eta)=\widehat{w}_{\mathrm{pf}}(\delta)=\frac{\kappa}{\gamma}\left(\frac{\delta^{\gamma-1}-1}{\gamma-1}+\delta^{-1}-1\right).$
(2.13)

The advantage of this re-formulation of the polytropic fluid model is that it
arises from a unified description of fluid and elastic bodies.

###  2.2 Dynamically equivalent constitutive functions

The system (2.1) for elastic balls becomes

$\displaystyle\partial_{t}\delta+\frac{1}{r^{2}}\partial_{r}(r^{2}\delta
u)=0,$ (2.14a)
$\displaystyle\mathcal{K}\delta(\partial_{t}u+u\partial_{r}u)=-\widehat{a}(\delta,\eta)\partial_{r}\delta+\widehat{b}(\delta,\eta)\frac{\eta-\delta}{r}+\mathcal{K}\delta
f,$ (2.14b) where
$\widehat{a}(\delta,\eta)=\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta),\quad\widehat{b}(\delta,\eta)=2\frac{\widehat{p}_{\mathrm{tan}}(\delta,\eta)-\widehat{p}_{\mathrm{rad}}(\delta,\eta)}{\eta-\delta}+3\partial_{\eta}\widehat{p}_{\mathrm{rad}}(\delta,\eta).$
(2.14c) Moreover $\eta(t,r)$ satisfies the equations
$\partial_{t}\eta+u\partial_{r}\eta=-3\frac{u}{r}\eta,\quad\partial_{r}\eta=-\frac{3}{r}(\eta-\delta),\quad\partial_{t}\eta=-3\frac{u}{r}\delta.$
(2.14d)

The quantity $\widehat{b}(\delta,\eta)$ in (2.14b) is the new term that
distinguishes the equations of motion of elastic balls from those of fluid
balls. As
$\lim_{\delta\to\eta}\widehat{b}(\delta,\eta)=\partial_{\eta}(\widehat{p}_{\mathrm{rad}}+2\widehat{p}_{\mathrm{tan}})(\delta,\delta)$,
then, by (2.4),

$\lim_{\delta\to\eta}\widehat{b}(\delta,\eta)=0,$ (2.15)

i.e., the function $\widehat{b}$ vanishes on the isotropic state
$\delta=\eta$.

The system (2.14) depends only on the elastic constitutive function and the
reference density $\mathcal{K}$; it is independent of the reference pressure
$\mathcal{P}$. However different elastic constitutive functions give rise to
the same system (2.14) on the variables $(\delta,\eta,u)$ when the conditions
in the following definition are satisfied.

######  Definition 5.

Two constitutive functions
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$,
$(\widetilde{p}_{\mathrm{rad}},\widetilde{p}_{\mathrm{tan}})$ are said to be
dynamically equivalent if
$\widehat{a}(\delta,\eta)=\widetilde{a}(\delta,\eta)$ and
$\widehat{b}(\delta,\eta)=\widetilde{b}(\delta,\eta)$, for all
$\delta,\eta>0$.

In the fluid case, dynamically equivalent constitutive functions are
necessarily identical, i.e., $\widehat{p}(\delta)=\widetilde{p}(\delta)$. In
the more general case of elastic materials, we have the following simple
result.

######  Lemma 1.

Two elastic constitutive functions
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$,
$(\widetilde{p}_{\mathrm{rad}},\widetilde{p}_{\mathrm{tan}})$ for spherically
symmetric bodies are dynamically equivalent if and only if there exists a
$C^{2}$ function $q:(0,\infty)\to\mathbb{R}$ such that $q(1)=q^{\prime}(1)=0$
and

$\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\widetilde{p}_{\mathrm{rad}}(\delta,\eta)+q(\eta),\quad\widehat{p}_{\mathrm{tan}}(\delta,\eta)=\widetilde{p}_{\mathrm{tan}}(\delta,\eta)+q(\eta)-\frac{3}{2}(\eta-\delta)q^{\prime}(\eta).$
(2.16)

###### Proof.

The condition $\widehat{a}(\delta,\eta)=\widetilde{a}(\delta,\eta)$ is
equivalent to
$\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\widetilde{p}_{\mathrm{rad}}(\delta,\eta)+q(\eta)$,
where, by (2.5), $q(1)=q^{\prime}(1)=0$. Substituting in the condition
$\widehat{b}(\delta,\eta)=\widetilde{b}(\delta,\eta)$ gives the second
equation in (2.16).∎

######  Lemma 2.

If the constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ is hyperelastic and
$\widehat{b}(\delta,\eta)\equiv 0$, then
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ is dynamically
equivalent to a fluid constitutive function.

###### Proof.

We prove the result by showing that, under the stated assumptions,
$\partial_{\delta}\widehat{p}_{\mathrm{rad}}$ is independent of $\eta$. By
(2.7) we have

$\displaystyle{\textstyle\frac{1}{3}}\partial_{\delta}\widehat{b}(\delta,\eta)$
$\displaystyle=\partial_{\eta}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)+\partial_{\delta}\left(\frac{\delta\eta}{\eta-\delta}\partial_{\eta}\widehat{w}(\delta,\eta)\right)$
$\displaystyle=\partial_{\eta}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)+\frac{\eta^{2}}{(\eta-\delta)^{2}}\partial_{\eta}\widehat{w}(\delta,\eta)+\frac{\delta\eta}{\eta-\delta}\partial_{\delta}\partial_{\eta}\widehat{w}(\delta,\eta)$
$\displaystyle=\partial_{\eta}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)+\frac{2}{3}\frac{\eta}{\delta(\eta-\delta)^{2}}(\widehat{p}_{\mathrm{tan}}(\delta,\eta)-\widehat{p}_{\mathrm{rad}}(\delta,\eta))+\frac{\eta}{\delta(\eta-\delta)}\partial_{\eta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)$
$\displaystyle=\partial_{\eta}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)+\frac{\eta}{3\delta(\eta-\delta)}\partial_{\delta}\widehat{b}(\delta,\eta),$

by which the claim follows. ∎

###  2.3 Lagrangian formulation

The purpose of this section is to derive the Lagrangian formulation for the
evolution problem of elastic balls implied by the Eulerian formulation in
physical space described so far. The relation between the two formulations in
the static case is discussed in [1]. For simplicity we set $f\equiv 0$ in this
section.

Let $(\delta,\eta,u)$ be a smooth solution of the system (2.14)f≡0 describing
the motion of an elastic ball with radius $R(t)$. We assume in addition that
$u$ satisfies the boundary condition $u(t,R(t))=\dot{R}(t)$, so that in
particular the total mass $M$ of the ball is conserved. Define

$Z=\left(\frac{3M}{4\pi\mathcal{K}}\right)^{1/3},\quad\phi_{t}:[0,R(t)]\to[0,Z],\quad\phi_{t}(r)=(\eta(t,r)r^{3})^{1/3}.$
(2.17a) $\phi_{t}$ is monotonically increasing and satisfies $\phi_{t}(0)=0$,
$\phi_{t}(R(t))=Z$, $\phi^{\prime}_{t}(0)=1$. Hence
$\psi_{t}=\phi_{t}^{-1}:[0,Z]\to[0,R(t)]$ (2.17b) is also monotonically
increasing and satisfies $\psi_{t}(0)=0$, $\psi_{t}(Z)=R(t)$,
$\psi^{\prime}_{t}(0)=1$. Letting
$\mathcal{B}=\\{X\in\mathbb{R}^{3}:|X|=z\in[0,Z)\\},$ (2.17c) the deformation
function is defined as
$\Psi_{t}:\mathcal{B}\to\mathbb{R}^{3},\quad\Psi_{t}(X)=\frac{\psi_{t}(z)}{z}X.$
(2.17d) The region $\mathcal{B}$ is the interior of the elastic ball in the
reference configuration, while
$\Psi_{t}(\mathcal{B})=\\{x\in\mathbb{R}^{3}:|x|=r\in[0,R(t))\\}$ is the
interior of the elastic ball in physical space. $\Psi_{t}$ is a diffeomorphism
from $\overline{\mathcal{B}}$ onto $\overline{\Psi_{t}(\mathcal{B})}$. The
deformation gradient is
$F_{ij}(X)=(\nabla\Psi_{t})_{ij}(X)=\psi_{t}^{\prime}(z)\frac{X_{i}X_{j}}{z^{2}}+\frac{\psi_{t}(z)}{z}\left(\delta_{ij}-\frac{X_{i}X_{j}}{z^{2}}\right),\quad
X\in\mathcal{B},\ z=|X|.$ Letting
$\det(F(X))=\psi^{\prime}_{t}(z)\left(\frac{\psi_{t}(z)}{z}\right)^{2}=J_{t}(z),$
and using the identity $\phi^{\prime}_{t}(r)=\delta(t,r)/\eta(t,r)^{2/3}$ we
find $\delta(t,r)=J_{t}(\phi_{t}(r))^{-1}$. Hence the last equation in (2.14d)
gives
$u(t,r)=-\frac{r\partial_{t}\eta(t,r)}{3\delta(t,r)}=-\left(\frac{\phi_{t}(r)}{r}\right)^{2}J_{t}(\phi_{t}^{-1}(r))\partial_{t}\phi_{t}(r)=-\psi^{\prime}_{t}(\phi_{t}(r))\partial_{t}\phi_{t}(r)=\partial_{t}\psi_{t}(\phi_{t}(r)),$
where in the last step we used $\psi_{t}(\phi_{t}(r))=r$. It follows that
$\partial_{t}\Psi_{t}:\mathcal{B}\to\mathbb{R}^{3},\quad\partial_{t}\Psi_{t}=\frac{u_{t}\circ\psi_{t}^{-1}(z)}{z}X.$
(2.17e)

Equations (2.17) define the deformation function and its time derivative in
terms of the solution $(\delta,\eta,u)$ of the system (2.14). Conversely,
given a deformation $\Psi_{t}:\mathcal{B}\to\mathbb{R}^{3}$ of the form
$\Psi_{t}(X)=\psi_{t}(z)X/z$, where $\psi_{t}:[0,Z]\to\mathbb{R}$ is $C^{1}$
and monotonically increasing, the functions

$u(t,r)=\partial_{t}\psi_{t}(z),\quad\delta(t,r)=\frac{z^{2}}{\psi_{t}(z)^{2}\psi_{t}^{\prime}(z)},\quad\eta(t,r)=\left(\frac{z}{\psi_{t}(z)}\right)^{3},\quad
z=\psi^{-1}_{t}(r)$ (2.18)

define a smooth solution $(\delta,\eta,u)$ of (2.14a) and (2.14d) for
$r\in(0,R(t))$, where $R(t)=\psi_{t}(Z)$. If in addition
$\psi(t,z)=\psi_{t}(z)$ satisfies the non-linear wave equation

$\mathcal{K}\partial_{t}^{2}\psi-A(z,\psi,\partial_{z}\psi)\partial_{z}^{2}\psi+\frac{B(z,\psi,\partial_{z}\psi)}{\psi}=0,\quad
t>0,\ z\in(0,Z),$ (2.19a) where $\displaystyle
A(z,\psi,\partial_{z}\psi)=\frac{1}{(\partial_{z}\psi)^{2}}\Big{\\{}\widehat{a}(\delta,\eta)\Big{\\}}_{\displaystyle{(\delta=z^{2}/(\psi^{2}\partial_{z}\psi),\eta=(z/\psi)^{3}})}$
(2.19b) $\displaystyle
B(z,\psi,\partial_{z}\psi)=\Big{\\{}(\delta-\eta)\Big{(}\frac{2\,\widehat{a}(\delta,\eta)}{\eta}+\frac{\widehat{b}(\delta,\eta)}{\delta}\Big{)}\Big{\\}}_{\displaystyle{(\delta=z^{2}/(\psi^{2}\partial_{z}\psi),\eta=(z/\psi)^{3}})}$
(2.19c)

then $(\delta,\eta,u)$ given by (2.18) satisfies the balance of momentum
equation (2.14b)f≡0 as well. The boundary condition $u(t,R(t))=\dot{R}(t)$ is
identically satisfied, since $R(t)=\psi(t,Z)$ and
$u(t,R(t))=\partial_{t}\psi(t,Z)$.

Remark. While the transition $(\delta,\eta,u)\to(\Psi,\partial_{t}\Psi)$ is
possible for all sufficiently smooth solutions of (2.14), the opposite
transition $(\Psi,\partial_{t}\Psi)\to(\delta,\eta,u)$ is only possible if the
configuration map solution of (2.19) satisfies $\partial_{z}\psi(t,z)>0$, for
all $z\in[0,Z]$.

#### Stored energy function

Let $(\Lambda_{1}^{2},\Lambda_{2}^{2},\Lambda_{3}^{2})$ be the eigenvalues of
the (right) Cauchy-Green tensor $C=F^{T}F$. In the Lagrangian formulation of
elasticity, the stored energy function of homogeneous, isotropic and frame
indifferent hyperelastic materials depends only the principal stretches
$(\Lambda_{1},\Lambda_{2},\Lambda_{3})$. In the case of spherically symmetric
deformations we have

$C_{ij}(X)=(\psi_{t}^{\prime}(z))^{2}\frac{X_{i}X_{j}}{z^{2}}+\left(\frac{\psi_{t}(z)}{z}\right)^{2}\left(\delta_{ij}-\frac{X_{i}X_{j}}{z^{2}}\right),\quad
X\in\mathcal{B},\ z=|X|,$

hence
$\Lambda_{1}(t,z)=\psi^{\prime}_{t}(z),\Lambda_{2}(t,z)=\Lambda_{3}(t,z)=\psi_{t}(z)/z$,
or equivalently

$\Lambda_{i}(t,z)=\lambda_{i}(t,\psi_{t}(z)),\quad\lambda_{1}(t,r)=\frac{\eta(t,r)^{2/3}}{\delta(t,r)}\quad\lambda_{2}(t,r)=\lambda_{3}(t,r)=\frac{1}{\eta(t,r)^{1/3}},$

from which we derive in particular $y=\delta/\eta=\lambda_{2}/\lambda_{1}$.
Letting $\widehat{W}(\Lambda_{1},\Lambda_{2})$ be a stored energy function for
spherically symmetric deformations in the Lagrangian space, the expression

$\widehat{w}(\delta,\eta)=\widehat{W}(\delta^{-1}\eta^{2/3},\eta^{-1/3}),$

defines the stored energy function in the Eulerian space used in Definition 2.
For instance the stored energy function of Saint-Venant Kirchhoff (SVK)
materials for spherically symmetric deformations is given by

$\kappa^{-1}\widehat{W}_{SVK}(\Lambda_{1},\Lambda_{2})=-\frac{3}{4}(\Lambda_{1}^{2}+2\Lambda_{2}^{2})+\frac{3}{8}\frac{1-\nu}{1+\nu}(\Lambda_{1}^{4}+2\Lambda_{2}^{4})+\frac{3}{4}\frac{\nu}{1+\nu}(\Lambda_{2}^{4}+2\Lambda_{1}^{2}\Lambda_{2}^{2})+\frac{9}{8},$

see [11, p. 183]. Hence in physical space we find

$\displaystyle\kappa^{-1}\widehat{w}_{\mathrm{SVK}}(\delta,\eta)$
$\displaystyle=\eta^{-4/3}\left(\frac{3(1-\nu)}{8(1+\nu)}\left(\frac{\delta}{\eta}\right)^{-4}+\frac{3\nu}{2(1+\nu)}\left(\frac{\delta}{\eta}\right)^{-2}+\frac{3}{4(1+\nu)}\right)$
$\displaystyle\quad+\eta^{-2/3}\left(-\frac{3}{4}\left(\frac{\delta}{\eta}\right)^{-2}-\frac{3}{2}\right)+\frac{9}{8}.$
(2.20)

The constitutive function for the principal pressures in the Lagrangian space
is given in terms of the stored energy function by

$\widehat{P}_{\mathrm{rad}}(\Lambda_{1},\Lambda_{2})=-\partial_{1}\widehat{W}(\Lambda_{1},\Lambda_{2}),\quad\
\widehat{P}_{\mathrm{tan}}(\Lambda_{1},\Lambda_{2})=-\frac{1}{2}\partial_{2}\widehat{W}(\Lambda_{1},\Lambda_{2}),$
(2.21a) see [1], which are equivalent to the formulas (2.7) in the Eulerian
space, with
$\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\eta^{2/3}\widehat{P}_{\mathrm{rad}}(\delta^{-1}\eta^{2/3},\eta^{-1/3}),\quad\widehat{p}_{\mathrm{tan}}(\delta,\eta)=\frac{\delta}{\eta^{1/3}}\widehat{P}_{\mathrm{tan}}(\delta^{-1}\eta^{2/3},\eta^{-1/3}).$
(2.21b)

###  2.4 Constitutive inequalities

The constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ will now be required
to satisfy three important inequalities in continuum mechanics. The first one
follows by demanding that the system (2.14) with $f\equiv 0$ be strictly
hyperbolic. Letting $q=\delta u$ be the local (radial) momentum and

$v=\begin{pmatrix}\delta\\\ \eta\\\ q\end{pmatrix},\quad
A(v)=\begin{pmatrix}0&0&1\\\ 0&\frac{q}{\delta}&0\\\
-(\frac{q}{\delta})^{2}+\frac{\widehat{a}(\delta,\eta)}{\mathcal{K}}&\frac{\widehat{b}(\delta,\eta)}{3\mathcal{K}}&\frac{2q}{\delta}\end{pmatrix},\quad
F(r,v)=-\frac{1}{r\delta}\begin{pmatrix}2q\delta\\\ {\displaystyle 3q\eta}\\\
2q^{2}\end{pmatrix},$

the system (2.14)f≡0 can be written in the form

$\partial_{t}v+A(v)\partial_{r}v=F(r,v);$

the function $F(r,v)$ is bounded at $r=0$ for solutions with strongly regular
center. The eigenvalues of $A$ are

$\mathrm{Eigen}(A)=(\frac{q}{\delta},\frac{q}{\delta}-\sqrt{\widehat{a}(\delta,\eta)},\frac{q}{\delta}+\sqrt{\widehat{a}(\delta,\eta)})=(u,u-\sqrt{\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)},u+\sqrt{\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)}).$

Hence the system (2.14)f≡0 is strictly hyperbolic if
$\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)>0$.

######  Definition 6.

An elastic constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for spherically
symmetric bodies is said to be globally strictly hyperbolic if

$\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)>0,\quad\text{for all
$(\delta,\eta)\in(0,\infty)^{2}$.}$ (2.22)

If the constitutive function is hyperelastic then the stored energy function
itself is said to be globally strictly hyperbolic.

A further condition that is expected to be satisfied by all reasonable elastic
constitutive functions is the Baker-Ericksen inequality [5]. In the Lagrangian
formulation of elasticity theory, the Baker-Ericksen inequality is an
assumption on the stored energy function implying that the deformation in the
interior of the body along each principal direction increases with the
intensity of the applied force, see [19]. More precisely, let
$\psi:\mathcal{B}\to\mathbb{R}^{3}$ be the deformation function of a body from
a reference configuration in which it occupies the region
$\mathcal{B}\subset\mathbb{R}^{3}$. Let $F=\nabla\psi$ be the deformation
gradient, $C=F^{T}F$ the (right) Cauchy-Green tensor and
$\Lambda_{1}^{2},\Lambda_{2}^{2},\Lambda_{3}^{2}$ the eigenvalues of $C$. Let
$\Phi(\Lambda_{1},\Lambda_{2},\Lambda_{3})$ be a stored energy function for
homogeneous, isotropic and frame indifferent elastic materials. Then the
Baker-Ericksen inequality reads

$\frac{\Lambda_{i}\partial_{i}\Phi-\Lambda_{j}\partial_{j}\Phi}{\Lambda_{i}-\Lambda_{j}}>0,\quad\Lambda_{i}\neq\Lambda_{j}.$
(2.23)

As shown in [19], the Baker-Ericksen inequality is implied by the strong
ellipticity condition on the stored energy function in the Lagrangian space.

The form of the Baker-Ericksen inequality in our Eulerian formulation of
elasticity theory for spherically symmetric bodies is derived using (2.21)
with
$\widehat{W}(\Lambda_{1},\Lambda_{2})=\Phi(\Lambda_{1},\Lambda_{2},\Lambda_{2})$
and reads
$(\delta-\eta)^{-1}(\widehat{p}_{\mathrm{tan}}(\delta,\eta)-\widehat{p}_{\mathrm{rad}}(\delta,\eta))>0$,
for all $\eta\neq\delta$. However for our purpose it is convenient to slightly
modify this inequality to the following.

######  Definition 7.

The inequality

$\frac{\widehat{p}_{\mathrm{tan}}(\delta,\eta)-\widehat{p}_{\mathrm{rad}}(\delta,\eta)}{\eta-\delta}\geq
0,$ (2.24)

on the elastic constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for spherically
symmetric bodies is called strong Baker-Ericksen inequality if it is satisfied
for all $(\delta,\eta)\in(0,\infty)^{2}$, and weak Baker-Ericksen inequality
if it is satisfied for all $\eta\geq\delta>0$.

The inequality in Definition 7 differs from (2.23) in three ways. Firstly, we
introduced a weak and a strong version of the Baker-Ericksen inequality; the
reason for this is that the weak version suffices in the important case of
static bodies, see Lemma 4. Secondly we replaced the strict inequality in
(2.23) with the non-strict one, so that (2.24) can also be applied to fluid
modes. Finally the condition $\Lambda_{i}\neq\Lambda_{j}$, which in our
formulation is equivalent to $\eta\neq\delta$, has not been imposed in (2.24).
Hence we assume that the Baker-Ericksen inequality still holds in the limit
$\eta\to\delta$.

To justify the next constitutive inequality, we remark that, by (2.5),
$\nabla\widehat{w}(1,1)=0$ and

$\kappa^{-1}\nabla^{2}\widehat{w}(1,1)=\frac{1}{1+\nu}\begin{pmatrix}3(1-\nu)&-2(1-2\nu)\\\
-2(1-2\nu)&2(1-2\nu)\end{pmatrix},$

which is positive definite for $\nu\in(-1,1/2)$. It follows that the stored
energy function has a local minimum at the reference configuration
$(\delta,\eta)=(1,1)$, which we shall now require to be global (but not
necessarily unique). As for the Baker-Ericksen inequality, we introduce a weak
and a strong version of this condition, the former being sufficient in the
case of static solutions.

######  Definition 8.

An hyperelastic constitutive function with stored energy function
$\widehat{w}$ is said to satisfy the weak, respectively strong, non-negative
stored energy condition if

$\widehat{w}(\delta,\eta)\geq 0,\quad\text{for all $\eta\geq\delta>0$,
respectively for all $(\delta,\eta)\in(0,\infty)^{2}$.}$ (2.25)

Remark. Definitions 6–8 do not exclude the possibility that the constitutive
inequalities in this section only hold for some, and not all, values of the
Poisson ratio $\nu$, i.e., only for specific materials.

Remark. The polytropic fluid constitutive function (2.12) is globally strictly
hyperbolic and satisfies the strong non-negative stored energy condition as
well as the strong Baker-Ericksen inequality.

It turns out that several elastic models used in materials science satisfy the
non-negative stored energy condition, but are neither globally hyperbolic nor
satisfy the Baker-Ericksen inequality. For instance, the region in the
$(\delta,\eta)$ plane where the SVK model (2.3) is hyperbolic and verifies the
Baker-Ericksen inequality is depicted in Figure 1. In particular, these two
important constitutive inequalities are violated for large deformations.

Figure 1: The region in the $(\delta,\eta)$ plan where the hyperbolicity
condition (left) and the Baker-Ericken inequality (right) are satisfied by the
SVK model (2.3). Both are verified in a neighbourhood of the reference state
$(\delta,\eta)=(1,1)$. The Poisson ratio is $\nu=1/4$ in this picture.

To conclude this section we show that the Baker-Ericksen inequality entails a
simple condition for the existence of a global minimum of the stored energy
function.

######  Lemma 3.

Let $(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ be an
hyperelastic constitutive function for spherically symmetric bodies satisfying
the strong, respectively weak, Baker-Ericksen inequality. Then the stored
energy function satisfies

$\widehat{w}(\delta,\eta)\geq\widehat{w}(\delta,\delta),\quad\text{for all
$(\delta,\eta)\in(0,\infty)^{2}$, respectively for all $\eta\geq\delta>0$.}$

###### Proof.

The proof is straightforward, because the inequality (2.24) applied to
hyperelastic materials is equivalent to the property that, for all given
$\delta>0$, the function $\eta\to\widehat{w}(\delta,\eta)$ is non-increasing
for $\eta<\delta$ and non-decreasing for $\eta>\delta$. ∎

Lemma 3 reduces the problem of proving the non-negativity of the stored energy
function $\widehat{w}$ to the much simpler problem of proving the same
property for the function $F$ of one variable given by
$F(\delta)=\widehat{w}(\delta,\delta)$.

###  2.5 Constant boundary shear condition

In this paper we are particularly interested in (self-gravitating) elastic
balls with constant boundary shear, i.e., elastic balls for which there exists
a constant $y_{\mathrm{b}}\geq 0$ such that

$y(t,R(t))=y_{\mathrm{b}}$ (2.26)

for all times. As we shall see, the constant boundary shear condition
generalizes to the elastic case the property that the mass density vanishes on
the boundary of polytropic fluid balls (surrounded by vacuum). We show now
that (2.26) is also necessary for the existence of homologous motions for
elastic balls. A ball of matter $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$
is said to be in homologous motion if the velocity field $u(t,r)$ has the form

$u(t,r)=\frac{\dot{\omega}(t)}{\omega(t)}r,\quad t\geq 0,\ 0\leq r\leq R(t),$
(2.27)

for some $C^{1}$ function $\omega$ such that $\omega(0)=1$. Replacing (2.27)
into (2.14a) we obtain

$\delta(t,r)=\frac{1}{\omega(t)^{3}}\delta_{0}\left(\frac{r}{\omega(t)}\right),\quad\eta(t,r)=\frac{1}{\omega(t)^{3}}\eta_{0}\left(\frac{r}{\omega(t)}\right),\quad\eta_{0}(r)=\frac{3}{r^{3}}\int_{0}^{r}\delta_{0}(s)s^{2}\,ds,$

where $\delta_{0}=\delta(0,r)$. The boundary condition $\dot{R}(t)=u(t,R(t))$
defines the radius of the ball as $R(t)=Z\omega(t)$, where $Z>0$ is a
constant. Hence

$y(t,R(t))=\frac{\delta(t,R(t))}{\eta(t,R(t))}=\frac{\delta_{0}(Z)}{\eta_{0}(Z)}=y_{\mathrm{b}},$

and thus, as claimed, the boundary of elastic balls in homologous motion must
have constant shear.

The constitutive function for the radial pressure must be consistent with
(2.26) in order for elastic balls with constant boundary shear to satisfy the
boundary condition $p_{\mathrm{rad}}(t,R(t))=0$. As

$p_{\mathrm{rad}}(t,R(t))=\mathcal{P}+\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}}\eta(t,R(t)),\eta(t,R(t))),\quad\eta(t,R(t))=\frac{3M}{4\pi\mathcal{K}}R(t)^{-3},$

where $M$ is the total mass of the ball, a necessary condition on the
constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for the existence of
elastic balls with constant boundary shear is the following.

######  Definition 9.

An elastic constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for spherically
symmetric bodies is said to satisfy the Constant Boundary Shear (CBS)
condition if there exists a (possibly not unique) $y_{b}\geq 0$ such that
$\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}}\eta,\eta)$ is constant, or
equivalently,

$(y_{\mathrm{b}}\partial_{\delta}\widehat{p}_{\mathrm{rad}}+\partial_{\eta}\widehat{p}_{\mathrm{tan}})(y_{\mathrm{b}}\eta,\eta)=0,\quad\text{for
all $\eta>0$.}$ (2.28)

The CBS condition is only necessary and by no means sufficient for the
existence of elastic balls with constant boundary shear. An additional
necessary condition is that the reference pressure in the equation of state
(2.8) of the elastic ball is given by

$\mathcal{P}=-\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}},1).$ (2.29)

Globally strictly hyperbolic fluid constitutive functions $\widehat{p}$
satisfy the CBS condition with $y_{\mathrm{b}}=0$ and thus in this case the
reference pressure (2.29) is $\mathcal{P}=-\widehat{p}(0)$. For polytropic
fluid balls this leads to the choice $\mathcal{P}=\kappa/\gamma$, which
characterizes the equation of state of polytropic fluid balls, see (2.12).

Remark. By (2.18), the deformation function on the boundary of an elastic ball
with constant boundary shear satisfies

$\psi(t,Z)-y_{\mathrm{b}}Z\partial_{z}\psi(t,Z)=0,$ (2.30)

where $Z$ is the constant radius of the ball in the Lagrangian space, see
Section 2.3.

###  2.6 Remarks on static self-gravitating elastic balls

This section contains some general remarks on static self-gravitating elastic
balls with strongly regular center. A spherically symmetric matter
distribution is said to be in static equilibrium if $u=0$ and
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}})$ are time independent. For self-
gravitating static balls the system (2.1) becomes

$p_{\mathrm{rad}}^{\prime}=\frac{2}{r}(p_{\mathrm{tan}}-p_{\mathrm{rad}})-G\rho\frac{m}{r^{2}}.$
(2.31)

When the static ball is elastic, (2.31) is equivalent to (2.14) with $u=0$ and
$f=-\frac{4\pi G}{3}\mathcal{K}r\eta$, namely

$\widehat{a}(\delta,\eta)\delta^{\prime}=\frac{\widehat{b}(\delta,\eta)}{r}(\eta-\delta)-\frac{4\pi
G}{3}\mathcal{K}^{2}r\,\eta\,\delta,\quad\eta(r)=\frac{3}{r^{3}}\int_{0}^{r}\delta(s)s^{2}\,ds.$
(2.32)

A solution $\delta$ of (2.32) in the interval $[0,R)$ will be called regular
if $\delta\in C^{0}([0,R))\cap C^{1}((0,R))$ and $\delta(r)>0$, for all
$r\in[0,R)$; the identity $\eta(0)=\delta(0)$ holds for these solutions. If in
addition $\delta\in C^{1}([0,R))$ and $\lim_{r\to 0^{+}}\delta^{\prime}(r)=0$,
then $\delta$ is said to be a strongly regular solution.

######  Lemma 4.

Let $\delta$ be a a regular solution of (2.32) defined on the interval
$[0,R)$, $R\in(0,\infty]$.

  * (i)

If $\widehat{a}(\delta(0),\delta(0))\neq 0$, then $\delta$ is strongly regular
and $|\delta^{\prime}(r)|\leq Cr$, for a positive constant $C>0$ and $r<1$.
Moreover $\delta\in C^{2}([0,R))$ and at $r=0^{+}$ there holds

$\delta^{\prime\prime}(0)=-\frac{4\pi
G}{3}\mathcal{K}^{2}\frac{\delta(0)^{2}}{a(\delta(0),\delta(0))},\quad\eta^{\prime\prime}(0)=\frac{3}{5}\delta^{\prime\prime}(0).$

  * (ii)

If $\widehat{a}(\delta,\delta)>0$, for all $\delta>0$, then
$\eta(r)>\delta(r)$ and $\eta^{\prime}(r)<0$, for all $r\in(0,R)$.

  * (iii)

If $\widehat{a}(\delta,\delta)>0$, for all $\delta>0$, and the constitutive
function satisfies the weak Baker-Ericksen inequality, then

$\widehat{p}_{\mathrm{tan}}(\delta(r),\eta(r))\geq\widehat{p}_{\mathrm{rad}}(\delta(r),\eta(r)),$
(2.33)

for all $r>0$.

###### Proof.

(i) The claim that $\delta$ is strongly regular and the bound on
$\delta^{\prime}$ are proved in [2, Thm. 1]. As to the second statement, it is
clear that $\delta$ is $C^{2}$ for $r>0$, hence we only need to check that
$\delta^{\prime\prime}$ extends continuously at $r=0$ (from the right). For
this purpose it can be assumed that $(\delta,\eta)$ lies within an arbitrarily
small disk $D$ centered in $(\delta(0),\eta(0)=\delta(0))$. Then, by Taylor
expanding the right hand side of (2.32), we obtain

$\widehat{a}(\delta(r),\eta(r))\delta^{\prime}(r)=\mathcal{R}(\delta(r),\eta(r))\frac{(\delta(r)-\eta(r))^{2}}{r}-\frac{4\pi
G}{3}\mathcal{K}^{2}\delta(r)\eta(r)r,$

where $\mathcal{R}(\delta,\eta)$ is bounded in $D$ and thus in particular
$\mathcal{R}(\delta(r),\eta(r))$ is bounded for $r$ small. Dividing by $r$ and
taking the limit $r\to 0^{+}$, using that $(\delta(r)-\eta(r))^{2}/r^{2}\to 0$
as $r\to 0^{+}$, we find

$\delta^{\prime\prime}(0)=\lim_{r\to
0^{+}}\frac{\delta^{\prime}(r)}{r}=-\frac{4\pi
G}{3}\mathcal{K}^{2}\frac{\delta(0)^{2}}{a(\delta(0),\delta(0))}.$

Moreover by the second equation in (2.32) we have

$\frac{\eta^{\prime}(r)}{r}=\frac{3\delta(r)}{r^{2}}-\frac{9\int_{0}^{r}\delta(s)s^{2}}{s^{5}}.$

Replacing
$\delta(r)=\delta(0)+\frac{1}{2}\delta^{\prime\prime}(0)r^{2}+o(r^{2})$,
carrying out the integral and taking the limit $r\to 0^{+}$ we find

$\eta^{\prime\prime}(0)=\lim_{r\to
0+}\frac{\eta^{\prime}(r)}{r}=\frac{3}{5}\delta^{\prime\prime}(0).$

(ii) By (i), $v(r)=\eta(r)-\delta(r)$ is increasing for small $r>0$, hence
$v(r)>0$ holds for small radii. Let $r_{*}>0$ be the supremum radius in the
interval $(0,R)$ for which $v(r)>0$ for $r\in(0,r_{*})$. If $r_{*}<R$, then
$v(r_{*})=0$ and $v^{\prime}(r_{*})\leq 0$. However this is not possible
because

$\displaystyle\lim_{r\to
r_{*}^{-}}\widehat{a}(\delta(r),\eta(r))v^{\prime}(r)$
$\displaystyle=-\widehat{a}(\delta(r_{*}),\eta(r_{*}))\left(3\frac{\eta(r_{*})-\delta(r_{*})}{r_{*}}\right)$
$\displaystyle\quad-\frac{\widehat{b}(\delta(r_{*}),\eta(r_{*}))}{r_{*}}(\eta(r_{*})-\delta(r_{*}))+\frac{4\pi}{3}\mathcal{K}^{2}r_{*}\,\eta(r_{*})\,\delta(r_{*})$
$\displaystyle=\frac{4\pi G}{3}\mathcal{K}^{2}r_{*}\,\eta(r_{*})^{2}>0,$

where for the second equality we used that $\delta(r_{*})=\eta(r_{*})$. Hence
$\eta(r)>\delta(r)$, and thus $\eta^{\prime}(r)<0$, must hold for $r\in(0,R)$.

(iii) Since the weak Baker-Ericksen inequality means that (2.24) is satisfied
for all $\eta\geq\delta$, then (3.22) follows from (ii). ∎

Remark. Thanks to the result in Lemma 4(iii), when the constitutive function
satisfies the weak Baker-Ericksen inequality the positivity of the tangential
pressure in the interior of static balls follows from the positivity of the
radial pressure.

In Section 4.1 it will be shown that static self-gravitating elastic balls can
be constructed by truncating strongly regular solutions of (2.32) at the first
radius $R>0$ where
$p_{\mathrm{rad}}(r)=\mathcal{P}+\widehat{p}_{\mathrm{rad}}(\delta(r),\eta(r))$
vanishes. It is convenient to distinguish two types of static self-gravitating
balls.

######  Definition 10.

Let $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}})$ be a static self-gravitating
elastic ball obtained by truncating a strongly regular solution of (2.32) with
maximal interval of existence $[0,R_{\mathrm{max}})$. If
$R_{\mathrm{max}}<\infty$, $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}})$ is said
to be of type $\mathrm{A}$, while if $R_{\mathrm{max}}=\infty$ then
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}})$ is said to be of type $\mathrm{B}$.

Remark. The distinction between types $\mathrm{A}$ and $\mathrm{B}$ may be
important to determine the number of shells which can surround a static
elastic ball if the different bodies are all made of the same elastic
material. In [1] it was shown that in the particular case of the Seth elastic
model, for which all static self-gravitating balls are of type $\mathrm{B}$,
there is no limit on the number of shells that can form around the ball. An
interesting open question is whether the same is possible for static self-
gravitating elastic balls of type $\mathrm{A}$. In the polytropic fluid case,
only type $\mathrm{A}$ are admissible and no shell can form around static
self-gravitating balls, because the interior pressure is a decreasing function
of the radius.

###  2.7 Final comments on the constitutive function of elastic bodies

This section contains a list of three properties which, in the author opinion,
should be satisfied by elastic material models in the applications to the
problem of self-gravitating bodies. The following list of conditions is
inspired partially by physical considerations and mostly by the wish to avoid
unnecessary mathematical complications not related to the problem.

  1. 1.

The constitutive function should not depend on additional material parameters
other than the bulk modulus $\kappa$ and the Poisson ratio $\nu$. Some of the
elastic models used in materials science, e.g. the Signorini model, do not
satisfy this condition, see [3]. It should be emphasized that the extra
parameters in these models are not proper material constants because their
value depends on having assumed a specific constitutive function for the body.
While considering additional phenomenological parameters may be needed for
engineering applications, it does not seem a priority at the present stage in
the analysis of self-gravitating elastic bodies.

  2. 2.

The constitutive function should be globally strictly hyperbolic. When the
material model is not globally strictly hyperbolic the existence theory for
the equations of motion of elastic bodies becomes considerably more technical,
since one has to show that solutions launched by initial data in the
hyperbolicity region
$\mathcal{H}=\\{(\delta,\eta):\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)>0\\}$
of the state space remain in this region for all times. As the region
$\mathcal{H}$ is known explicitly only for rather simple constitutive
functions, this task is often very hard, if not impossible, to be done
analytically. It turns out that many popular elastic models found in the
literature, such as the Saint Venant-Kirchhoff model, the Hadamard model, and
others, do not satisfy this condition, see [2].

  3. 3.

The constitutive function should satisfy the Baker-Ericksen inequality and (in
the hyperelastic case) the non-negative stored energy condition. The are many
constitutive inequalities proposed in the literature, some of which however
have unclear physical interpretation (e.g., polyconvexity [6]). An important
condition on elastic constitutive functions, and with a clear physical
meaning, is the Baker-Ericksen inequality, see Definition 7. This inequality
has already played a role in Lemma 4 and will show up again in the following
sections.

In the next section a new four-parameters family of elastic models is
introduced that satisfies the properties 1-3 and which at the same time may be
seen as a natural generalisation of the polytropic fluid constitutive function
(2.12).

##  3 Polytropic hyperelastic constitutive function

The ultimate goal of this section is to justify the definition of polytropic
elastic constitutive function for spherically symmetric bodies, see Definition
12 below. To this regard we remark that the polytropic fluid constitutive
function (2.12) is uniquely characterised by the scaling invariance property

$\varepsilon^{1-\gamma}\widehat{p}\,^{\prime}(\varepsilon\delta)=\widehat{p}\,^{\prime}(\delta),\quad\text{for
all $\varepsilon,\gamma>0$}.$ (3.1)

The identity (3.1) entails that the Euler equations are scale invariant and
thus it is the reason for the existence of self-similar solutions [22]. As
(3.1) plays a fundamental role in the mathematical and physical study of fluid
dynamics, it is natural to look for an elastic constitutive function that
satisfies a similar scaling property. The minimal scaling condition on
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ required for the
existence of self-similar solutions to (2.14) (with $f\equiv 0$) is

$\varepsilon^{1-\gamma}\,\widehat{a}(\varepsilon\delta,\varepsilon\eta)=\widehat{a}(\delta,\eta),\quad\varepsilon^{1-\gamma}\,\widehat{b}(\varepsilon\delta,\varepsilon\eta)=\widehat{b}(\delta,\eta),\
\text{for all $\varepsilon,\gamma>0$},$ (3.2)

where $\widehat{a}(\delta,\eta)$ and $\widehat{b}(\delta,\eta)$ are given by
(2.14c). Another property of the polytropic fluid constitutive function that
we want to generalize to the elastic case is the constant boundary shear
condition, see Definition 9.

######  Definition 11.

An elastic constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ for spherically
symmetric bodies is said to be a generalized polytropic constitutive function
if:

  1. 1.

$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ is hyperelastic,
globally strictly hyperbolic and satisfies (3.2);

  2. 2.

$\widehat{p}_{\mathrm{rad}}$ satisfies the CBS condition (2.28), for some
$y_{\mathrm{b}}\geq 0$.

An elastic ball with constitutive function
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ satisfying the
conditions 1-2 and with reference pressure
$\mathcal{P}=-\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}},1)$ will be called a
generalized polytropic elastic ball.

######  Proposition 1.

Let $f_{\mathrm{rad}},f_{\mathrm{tan}},g:(0,\infty)\to\mathbb{R}$ be $C^{2}$
functions satisfying

$f_{\mathrm{rad}}(1)=f_{\mathrm{tan}}(1)=0,\quad
f^{\prime}_{\mathrm{rad}}(1)=f^{\prime}_{\mathrm{tan}}(1)=\frac{1-\nu}{1+\nu},\quad
g(1)=0,\ g^{\prime}(1)=1.$ (3.3)

Then

$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\delta,\eta)=3f_{\mathrm{rad}}(\delta/\eta)\eta^{\gamma}+g(\eta),$
(3.4a)
$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{tan}}(\delta,\eta)=-{\textstyle\frac{3}{2}}f_{\mathrm{tan}}(\delta/\eta)\eta^{\gamma}+g(\eta)-{\textstyle\frac{3}{2}}(\eta-\delta)g^{\prime}(\eta)$
(3.4b)

define an elastic constitutive function for spherically symmetric bodies that
satisfies (3.2). Conversely, any such constitutive function must be of the
form (3.4) for some $C^{2}$ functions
$f_{\mathrm{rad}},f_{\mathrm{tan}},g:(0,\infty)\to\mathbb{R}$ satisfying
(3.3).

###### Proof.

It is straightforward to verify that the functions (3.4) satisfy the
properties in Definition 2, as well as the scaling identity (3.2), for all
given $C^{2}$ functions $f_{\mathrm{rad}},f_{\mathrm{tan}},g$ verifying (3.3).
Conversely, assume that
$(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ is a constitutive
function that satisfies (3.2). Then

$\left(\frac{d}{d\varepsilon}\varepsilon^{1-\gamma}\,\widehat{a}(\varepsilon\delta,\varepsilon\eta)\right)_{\varepsilon=1}=(1-\gamma)\widehat{a}(\delta,\eta)+\delta\partial_{\delta}\widehat{a}(\delta,\eta)+\eta\partial_{\eta}\widehat{a}(\delta,\eta)=0,$

hence
$\widehat{a}(\delta,\eta)=3\kappa(1-\nu)\delta^{\gamma-1}h(\delta/\eta)/(1+\nu)$,
where $h$ is an arbitrary $C^{1}$ function which, by the first equation in
(2.5b), satisfies $h(1)=1$. It follows that

$\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\delta,\eta)=3f_{\mathrm{rad}}(\delta/\eta)\eta^{\gamma}+g(\eta),$

where

$f_{\mathrm{rad}}(y)=\frac{1-\nu}{1+\nu}\int_{1}^{y}z^{\gamma}h(z)\,dz,\quad
g(\eta)=\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\eta,\eta).$

Thus $f_{\mathrm{rad}}(1)=0$,
$f^{\prime}_{\mathrm{rad}}(1)=\frac{1-\nu}{1+\nu}$ and, by the second equation
in (2.5b), $g^{\prime}(1)=1$. Likewise

$\left(\frac{d}{d\varepsilon}\varepsilon^{1-\gamma}\,\widehat{b}(\varepsilon\delta,\varepsilon\eta)\right)_{\varepsilon=1}=(1-\gamma)\widehat{b}(\delta,\eta)+\delta\partial_{\delta}\widehat{b}(\delta,\eta)+\eta\partial_{\eta}\widehat{b}(\delta,\eta)=0$

implies
$\widehat{b}(\delta,\eta)=3\kappa(1-\nu)\delta^{\gamma-1}s(\delta/\eta)/(1+\nu)$,
where $s$ is $C^{1}$ and $s(1)=0$ by (2.15). By the definition (2.14c) of
$\widehat{b}$ this entails that $\widehat{p}_{\mathrm{tan}}$ has the form
stated in (3.4) with

$f_{\mathrm{tan}}(y)=[3\gamma(1-y)-2]f_{\mathrm{rad}}(y)+3y(y-1)f^{\prime}_{\mathrm{rad}}(y)+\frac{1-\nu}{1+\nu}(y-1)y^{\gamma-1}s(y).$

As $f_{\mathrm{rad}}(1)=0$ and
$f^{\prime}_{\mathrm{tan}}(1)=f^{\prime}_{\mathrm{rad}}(1)$, the proof is
completed. ∎

The elastic constitutive function (3.4) contains the three arbitrary functions
$f_{\mathrm{rad}},f_{\mathrm{tan}},g$ and thus comprises a huge class of
models. However using Lemma 1 with $q(\eta)=g(\eta)+1-\eta$ we obtain that all
these constitutive functions are dynamically equivalent when
$f_{\mathrm{rad}}$ and $f_{\mathrm{tan}}$ are fixed. It is easy to see that
there is at most one representative in this dynamical equivalent class that
could satisfy the constant shear condition. In fact, assuming that there
exists $y_{\mathrm{b}}$ such that
$y_{\mathrm{b}}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}}\eta,\eta)+\partial_{\eta}\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}}\eta,\eta)=0$,
for all $\eta>0$, then necessarily $g^{\prime}(\eta)=-3\gamma
f_{\mathrm{rad}}(y_{\mathrm{b}})\eta^{\gamma-1}$ and thus the conditions
$g(1)=0,g^{\prime}(1)=1$ give

$f_{\mathrm{rad}}(y_{\mathrm{b}})=-(3\gamma)^{-1},\quad
g(\eta)=\frac{\eta^{\gamma}-1}{\gamma}.$ (3.5)

Hence any generalised elastic polytropic constitutive function has to have to
form

$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\delta,\eta)=3f_{\mathrm{rad}}(\delta/\eta)\eta^{\gamma}+\frac{\eta^{\gamma}-1}{\gamma},$
(3.6a)
$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{tan}}(\delta,\eta)=-\frac{3}{2}(f_{\mathrm{tan}}(\delta/\eta)+1-\delta/\eta)\eta^{\gamma}+\frac{\eta^{\gamma}-1}{\gamma},$
(3.6b)

where $\gamma>0$ and $f_{\mathrm{rad}},f_{\mathrm{tan}}$ are $C^{2}$ functions
that satisfy (3.3). Furthermore the constitutive function (3.6) is globally
strictly hyperbolic if and only if $f^{\prime}_{\mathrm{rad}}(y)>0$, for all
$y>0$, and thus if a solution of
$f_{\mathrm{rad}}(y_{\mathrm{b}})=-(3\gamma)^{-1}$ exists then it is
necessarily unique. We also obtain that the reference pressure of generalized
polytropic elastic balls must be given by
$\mathcal{P}=-\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}},1)=\kappa/\gamma$, as
in the polytropic fluid case, see (2.12).

The number of free functions in (3.6) can be further reduced by imposing that
the constitutive function should be hyperelastic.

######  Proposition 2.

The constitutive function (3.6) is hyperelastic if and only if

$f_{\mathrm{tan}}(y)=f_{\mathrm{rad}}(y)+3(1-\gamma)y\int_{1}^{y}\frac{f_{\mathrm{rad}}(s)}{s^{2}}\,ds.$
(3.7)

When (3.7) holds, the stored energy function is given by

$\kappa^{-1}\widehat{w}(\delta,\eta)=\left\\{\begin{array}[]{ll}{\displaystyle\eta^{\gamma-1}\left(3S(\delta/\eta)-\frac{1}{\gamma}(\delta/\eta)^{-1}+\frac{1}{\gamma-1}\right)+\frac{\delta^{-1}}{\gamma}-\frac{1}{(\gamma-1)}}&\text{for
$\gamma\neq 1$}\\\\[14.22636pt]
3S(\delta/\eta)-(\delta/\eta)^{-1}-\log(\delta/\eta)+\log\delta+\delta^{-1}&\text{for
$\gamma=1$}\end{array}\right.$ (3.8)

where

$S(y)=\int_{1}^{y}\frac{f_{\mathrm{rad}}(s)}{s^{2}}\,ds.$

Conversely, given any $C^{3}$ function $S:(0,\infty)\to\mathbb{R}$ such that

$S(1)=S^{\prime}(1)=0,\quad S^{\prime\prime}(1)=\frac{1-\nu}{1+\nu},$ (3.9)

(3.8) is the stored energy function for the elastic constitutive function
(3.6) where $f_{\mathrm{tan}}(y)$ is given by (3.7) and

$f_{\mathrm{rad}}(y)=y^{2}S^{\prime}(y).$ (3.10)

###### Proof.

Using (2.10), it is easy to show that the constitutive equation (3.4) is
hyperelastic if and only if (3.7) holds. The stored energy function is given
by

$\displaystyle\kappa^{-1}\widehat{w}({\delta,\eta})$
$\displaystyle=\kappa^{-1}\int_{\eta}^{\delta}\frac{\widehat{p}_{\mathrm{rad}}(s,\eta)}{s^{2}}\,ds+\kappa^{-1}W(\eta)$
$\displaystyle=\eta^{\gamma-1}\left(3S(\delta/\eta)-\frac{1}{\gamma}(\delta/\eta)^{-1}+\frac{1}{\gamma}\right)+\frac{\delta^{-1}-\eta^{-1}}{\gamma}+\kappa^{-1}W(\eta),$
(3.11)

where, imposing
$\widehat{w}(1,1)=\widehat{p}_{\mathrm{rad}}(1,1)=\widehat{p}_{\mathrm{tan}}(1,1)=0$
and
$\widehat{p}_{\mathrm{tan}}(\eta,\eta)=\widehat{p}_{\mathrm{rad}}(\eta,\eta)$,
the integration function $W$ is given by

$\displaystyle\kappa^{-1}W(\eta)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{1}{\gamma}\left(\frac{\eta^{\gamma-1}-1}{\gamma-1}+\eta^{-1}-1\right)}&\text{if
$\gamma\neq 1$}\\\\[14.22636pt] \log\eta+\eta^{-1}-1&\text{if
$\gamma=1$}\end{array}\right..$ (3.14)

Replacing (3.14) into (3.11) gives (3.8). Finally, given a $C^{3}$ function
$S$ satisfying (3.9) it is straightforward to verify that the elastic
constitutive function associated to the stored energy (3.8) has precisely the
form (3.4) where $f_{\mathrm{rad}},f_{\mathrm{tan}}$ satisfying (3.3) are
given by (3.10) and (3.7). ∎

So far it has been proved that the stored energy function of any generalized
polytropic elastic constitutive function for spherically symmetric bodies is
to be of the form (3.8) for some function $S$ that satisfies (3.9) and
$(y^{2}S^{\prime}(y))^{\prime}>0$, for all $y>0$. The function $S$ will be
called the shear function, as it expresses how the stored energy function
depends on the shear variable $y=\delta/\eta$. Next a specific choice will be
made for the shear function in such a way that the only freedom left in its
definition consists in fixing the value of a single parameter $\beta$.

We begin by choosing the shear function to be a linear combination of power-
laws, that is

$S(y)=\sum_{j=1}^{n}\alpha_{j}y^{\beta_{j}}.$

This choice seems rather natural for a polytropic model and agrees with the
general form of most stored energy functions found in the literature. By (3.9)
the coefficients $\alpha_{1},\dots,\alpha_{n}$ must satisfy the linear system

$\sum_{j=1}^{n}\alpha_{j}=0,\quad\sum_{j=1}^{n}\alpha_{j}\beta_{j}=0,\quad\sum_{j=1}^{n}\alpha_{j}\beta_{j}(\beta_{j}-1)=\frac{1-\nu}{1+\nu}.$

If this system has more than one solution, the stored energy function (3.8)
would depend on additional dimensionless constants besides the Poisson ratio
$\nu$, thereby violating Property 1 in Section 2.7. Hence we set $n=3$ and
thus obtain the following shear function

$S(y)=\frac{1-\nu}{1+\nu}\frac{(\beta_{2}-\beta_{3})y^{\beta_{1}}-(\beta_{1}-\beta_{3})y^{\beta_{2}}+(\beta_{1}-\beta_{2})y^{\beta_{3}}}{(\beta_{1}-\beta_{2})(\beta_{1}-\beta_{3})(\beta_{2}-\beta_{3})},$
(3.15)

where $\beta_{1},\beta_{2},\beta_{3}$ are distinct real numbers. Replacing
(3.15) in (3.8)γ≠1 we find that

$\widehat{w}(\delta,\eta)=\widehat{w}_{\mathrm{pf}}(\delta),\text{ for
$\nu=1/2$ and $(\beta_{1},\beta_{2},\beta_{3})=(\gamma-1,-1,0)$ (or
permutations),}$ (3.16)

where $\widehat{w}_{\mathrm{pf}}$ is the stored energy function of polytropic
fluids, see (2.13). Moreover (3.6) gives

$f_{\mathrm{rad}}(y)=y^{2}S^{\prime}(y)=\frac{1-\nu}{1+\nu}\frac{\beta_{1}(\beta_{2}-\beta_{3})y^{\beta_{1}+1}-\beta_{2}(\beta_{1}-\beta_{3})y^{\beta_{2}+1}+\beta_{3}(\beta_{1}-\beta_{2})y^{\beta_{3}+1}}{(\beta_{1}-\beta_{2})(\beta_{1}-\beta_{3})(\beta_{2}-\beta_{3})}.$
(3.17)

From this expression it is clear that proving the hyerbolicity condition
$f^{\prime}_{\mathrm{rad}}(y)>0$, for all $y>0$, and solving (3.5) for the
boundary value $y_{\mathrm{b}}$ of the shear variable is not possible without
imposing further conditions on the parameters $\beta_{1},\beta_{2},\beta_{3}$.
In the following we choose to simplify the model by fixing the value of two of
the parameters $\beta_{1},\beta_{2},\beta_{3}$ and letting only one be free.
The final condition that we require on the stored energy function is that it
should reduce to the polytropic fluid stored energy function in a suitable
limit when $\nu=1/2$. In view of (3.16) this condition is achieved by choosing
the fixed parameters in the shear function to be $-1$ and $0$. Denoting the
free parameter as $\beta-1$, we obtain the following final form of the shear
function

$S(y)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{1-\nu}{1+\nu}\left(\frac{y^{\beta-1}}{\beta(\beta-1)}+\frac{y^{-1}}{\beta}-\frac{1}{\beta-1}\right)}&\text{if
$\beta\neq 1$}\\\\[14.22636pt] {\displaystyle\frac{1-\nu}{1+\nu}\left(\log
y+y^{-1}-1\right)}&\text{if $\beta=1$}\end{array}\right.,$ (3.18)

where $\beta\neq 0$; the expression for $\beta=1$ is obtained in the limit of
the shear function for $\beta\neq 1$. With this choice of the shear function,
(3.17) simplifies to

$f_{\mathrm{rad}}(y)=y^{2}S^{\prime}(y)=\frac{1-\nu}{1+\nu}\frac{y^{\beta}-1}{\beta},$
(3.19)

and thus the CBS condition is satisfied with

$y_{\mathrm{b}}=\left[1-\frac{\beta}{3\gamma}\left(\frac{1+\nu}{1-\nu}\right)\right]^{1/\beta},$
(3.20)

provided the expression within the square brackets is non-negative, that is

$\beta\leq 3\frac{1-\nu}{1+\nu}\gamma.$ (3.21)

Note that $0\leq y_{\mathrm{b}}<1$, for all $\gamma>0$, $\beta\neq 0$, and
that $y_{\mathrm{b}}=0$ in the fluid limit $(\nu,\beta)=(1/2,\gamma)$.

$\gamma,\beta$ | $\kappa^{-1}\widehat{w}(\delta,\eta)$
---|---
$\gamma\neq 1$ | ${\displaystyle\eta^{\gamma-1}\Big{(}\frac{3(1-\nu)}{\beta(\beta-1)(1+\nu)}(\delta/\eta)^{\beta-1}+\Big{(}\frac{3(1-\nu)}{\beta(1+\nu)}-\frac{1}{\gamma}\Big{)}(\delta/\eta)^{-1}+\frac{1}{\gamma-1}-\frac{3(1-\nu)}{(\beta-1)(1+\nu)}\Big{)}}$
$\beta\neq 1$ | ${\displaystyle+\frac{\delta^{-1}}{\gamma}-\frac{1}{\gamma-1}}$
$\gamma=1$ | ${\displaystyle\frac{3(1-\nu)}{\beta(\beta-1)(1+\nu)}(\delta/\eta)^{\beta-1}+\left(\frac{3(1-\nu)}{\beta(1+\nu)}-1\right)(\delta/\eta)^{-1}-\frac{3(1-\nu)}{(1+\nu)(\beta-1)}-\log(\delta/\eta)}$
$\beta\neq 1$ | ${\displaystyle+\delta^{-1}+\log\delta}$
$\gamma\neq 1$ | ${\displaystyle\eta^{\gamma-1}\Big{(}\Big{(}\frac{3(1-\nu)}{1+\nu}-\frac{1}{\gamma}\Big{)}(\delta/\eta)^{-1}+\frac{3(1-\nu)}{1+\nu}(\log(\delta/\eta)-1)+\frac{1}{\gamma-1}\Big{)}+\frac{\delta^{-1}}{\gamma}-\frac{1}{\gamma-1}}$
$\beta=1$
$\gamma=1$ | ${\displaystyle\frac{2(1-2\nu)}{1+\nu}\left((\delta/\eta)^{-1}+\log(\delta/\eta)\right)-\frac{3(1-\nu)}{1+\nu}+\delta^{-1}+\log\delta}$
$\beta=1$
Table 1: Polytropic stored energy function.

Using (3.18) in (3.8), the stored energy function reads as in Table 1,
depending on the values of the parameters $\gamma,\beta$. The constitutive
function $(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ is given by

$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\delta,\eta)=\frac{3(1-\nu)}{\beta(1+\nu)}\big{(}(\delta/\eta)^{\beta}-y_{\mathrm{b}}^{\beta}\big{)}\eta^{\gamma}-\frac{1}{\gamma}$
(3.22a)
$\displaystyle\kappa^{-1}\widehat{p}_{\mathrm{tan}}(\delta,\eta)=\kappa^{-1}\widehat{p}_{\mathrm{rad}}(\delta,\eta)+3\big{(}1-\delta/\eta\big{)}Q(\delta/\eta)\eta^{\gamma},$
(3.22b) where
$Q(y)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{3(\beta-\gamma)(1-\nu)}{2\beta(\beta-1)(1+\nu)}\frac{1-y^{\beta}}{1-y}+\frac{3(\gamma-1)(1-\nu)}{2(\beta-1)(1+\nu)}-\frac{1}{2}}&\text{if
$\beta\neq 1$}\\\\[14.22636pt]
{\displaystyle\frac{3(\gamma-1)(1-\nu)}{2(1+\nu)}\frac{y\log
y}{1-y}+\frac{3\gamma(1-\nu)}{2(1+\nu)}-\frac{1}{2}}&\text{if
$\beta=1$}\end{array}\right..$ (3.22c)

Moreover

$\displaystyle\kappa^{-1}\widehat{a}(\delta,\eta)=\kappa^{-1}\partial_{\delta}\widehat{p}_{\mathrm{rad}}(\delta,\eta)=3\frac{1-\nu}{1+\nu}(\delta/\eta)^{\beta-1}\eta^{\gamma-1}$
(3.23a)
$\displaystyle\kappa^{-1}\widehat{b}(\delta,\eta)=9\frac{1-\nu}{1+\nu}(\beta-\gamma)(1-\delta/\eta)^{-1}B(\delta/\eta)\eta^{\gamma-1},$
(3.23b) $\displaystyle\text{where}\quad
B(y)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{y+(\beta-1)y^{1+\beta}-\beta
y^{\beta}}{\beta(\beta-1)}}&\text{if $\beta\neq 1$}\\\\[14.22636pt] y(y-\log
y-1)&\text{if $\beta=1$}\end{array}\right..$ (3.23e)

In particular the constitutive function is globally strictly hyperbolic and,
by Lemma 2, for $\beta=\gamma$ it is dynamically equivalent to a fluid
constitutive function.

Our search for a hyperelastic constitutive function resembling the polytropic
fluid model (2.12) ends here and has led to the following definition.

######  Definition 12.

Given $\kappa>0$, $\nu\in(-1,1/2]$, $\gamma>0$, and $\beta\neq 0$ satisfying
(3.21), the function $(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$
defined by (3.22) is called polytropic elastic constitutive function for
spherically symmetric bodies with bulk modulus $\kappa$, Poisson ratio $\nu$,
polytropic exponent $\gamma$ and shear exponent $\beta$.

Remark. For $\nu=1/2$ and $\beta=\gamma$, the polytropic elastic constitutive
function becomes the polytropic fluid model (2.12), for all $\gamma>0$, while
for $\beta=\gamma$ and for all $\nu\in(-1,1/2)$ it is dynamically equivalent
to such fluid. In all other cases the constitutive function (3.22) is
“genuinely” elastic.

To conclude this section we show that the polytropic elastic constitutive
function satisfies the weak Baker-Ericksen inequality and the weak non-
negative stored energy condition for all admissible values of the parameters
$\gamma,\beta$ and the strong version of these inequalities if and only if
$\gamma,\beta$ verify an additional bound.

######  Proposition 3.

The polytropic elastic constitutive function (3.22) satisfies the weak Baker-
Ericksen inequality and the weak non-negative stored energy condition for all
$\gamma>0$, and $\beta\neq 0$ satisfying (3.21). If and only if in addition

$\beta\geq\min\left(\gamma,3\frac{1-\nu}{1+\nu}\gamma-\frac{2(1-2\nu)}{1+\nu}\right)=\left\\{\begin{array}[]{ll}\gamma&\text{if
$\gamma\geq 1$}\\\ {\displaystyle
3\frac{1-\nu}{1+\nu}\gamma-\frac{2(1-2\nu)}{1+\nu}}&\text{if
$0<\gamma<1$}\end{array}\right.,$ (3.24)

then (3.22) satisfies the strong Baker-Ericksen inequality and the strong non-
negative stored energy condition as well.

###### Proof.

For the polytropic elastic constitutive function the inequality (2.24) reads
$Q(y)\geq 0$, where $Q$ is defined in (3.22). We compute

$Q^{\prime}(y)=\frac{3(\beta-\gamma)(1-\nu)}{2(1-y)^{2}(1+\nu)}\frac{B(y)}{y},$

where $B$ is defined in (3.23). A simple analysis shows that

$B(y)>0,\quad\text{for all $0<y\neq 1$},\quad B(1)=0.$

It follows that $Q(y)$ is strictly increasing for $\beta>\gamma$ and strictly
decreasing for $\beta<\gamma$, while for $\beta=\gamma$ we have
$Q(y)=\frac{1-2\nu}{1+\nu}\geq 0$. Hence
$\inf_{y>0}Q(y)=\inf_{y\in(0,1)}Q(y)=\lim_{y\to 0^{+}}Q(y)$, for
$\beta>\gamma$, while $\inf_{y>0}Q(y)=\lim_{y\to\infty}Q(y)$, and
$\inf_{y\in(0,1)}Q(y)=\lim_{y\to 1}Q(y)=\frac{1-2\nu}{1+\nu}\geq 0$, for
$\beta<\gamma$. The proof of the claims on the weak and the strong Baker-
Ericksen inequality can now easily be completed by using that

$\lim_{y\to
0^{+}}Q(y)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{3\gamma(1-\nu)}{2\beta(1+\nu)}-\frac{1}{2}}&\text{if
$\beta>0$}\\\\[11.38092pt]
{\displaystyle\frac{\gamma-\beta}{\beta(\beta-1)}\cdot\infty}&\text{if
$\beta<0$}\end{array}\right.,\quad\lim_{y\to\infty}Q(y)=\left\\{\begin{array}[]{ll}{\displaystyle\frac{3(\gamma-1)(1-\nu)}{2(\beta-1)(1+\nu)}-\frac{1}{2}}&\text{if
$\beta<1$}\\\\[11.38092pt]
{\displaystyle\frac{\beta-\gamma}{\beta(\beta-1)}\cdot\infty}&\text{if
$\beta>1$}\\\\[11.38092pt]
{\displaystyle\frac{1-\nu}{1+\nu}(1-\gamma)\cdot\infty}&\text{if
$\beta=1$.}\end{array}\right..$

As to the claims on the weak and strong non-negative stored energy condition,
we first observe that, by Lemma 3, for all $\gamma>0$, and $\beta\neq 0$
satisfying (3.21) there holds

$\widehat{w}(\delta,\eta)\geq\inf_{\delta>0}\widehat{w}(\delta,\delta),\quad\text{for
all $\eta\geq\delta>0$}$

and if in addition $(\beta,\gamma)$ satisfy (3.24) there also holds

$\widehat{w}(\delta,\eta)\geq\inf_{\delta>0}\widehat{w}(\delta,\delta),\quad\text{for
all $(\delta,\eta)\in(0,\infty)^{2}$}.$

As $\widehat{w}(\delta,\delta)=\widehat{w}_{\mathrm{pf}}(\delta)$ and
$\widehat{w}_{\mathrm{pf}}(\delta)>\widehat{w}_{\mathrm{pf}}(1)=0$, for all
$0<\delta\neq 1$, see (2.13), the “ if ” part of the claim on the non-negative
stored energy condition is proved. It remains to prove that when
$(\beta,\gamma)$ do not satisfy (3.24) the strong non-negative stored energy
condition is violated, namely, there exists $\eta<\delta$ such that
$\widehat{w}(\delta,\eta)<0$. We present the details only for
$(\beta,\gamma)\neq(1,1)$, which is actually the less trivial case. Assume
first $\beta<1$. Then (3.24) is violated for all $\gamma>1$ and for $\gamma<1$
when

$\beta<3\frac{1-\nu}{1+\nu}\gamma-\frac{2(1-2\nu)}{1+\nu}.$ (3.25)

Moreover for all $\eta>0$ fixed we have

$\lim_{\delta\to\infty}\widehat{w}(\delta,\eta)=\eta^{\gamma-1}\left(\frac{1}{\gamma-1}-\frac{3(1-\nu)}{(\beta-1)(1+\nu)}\right)-\frac{1}{\gamma-1}:=F(\eta)$

and it is straightforward to verify that for all $\gamma>1$ and $\beta<1$, or
for all $\gamma<1$, $\beta<1$ satisfying (3.25), $F(\eta)$ is negative when
$\eta$ is small enough. In the case $\beta>1$, (3.24) is violated for
$\gamma>\beta$. Under the latter condition, for all $\delta$ fixed we have

$\lim_{\eta\to
0^{+}}\widehat{w}(\delta,\eta)=\frac{\delta^{-1}}{\gamma}-\frac{1}{\gamma-1},$

which is negative for $\delta>(\gamma-1)/\gamma$. This concludes the proof of
the proposition. ∎

The region in the $(\gamma,\beta)$ plane where (3.22) satisfies the strong
Baker-Ericksen inequality and the strong non-negative energy stored condition
for $\nu\in(-1,1/2)$ fixed is depicted in Figure 2. In the limit $\nu\to
1/2^{-}$ this region reduces to the line $\beta=\gamma$.

Figure 2: The region on the $(\gamma,\beta)$ plane where the polytropic
elastic constitutive function satisfies the strong Baker-Ericksen inequality
and the strong non-negative stored energy condition for a fixed value of
$\nu\in(-1,1/2)$ ($\nu=1/4$ in the picture). The dashed line is the curve
(3.24), the continuous line is the curve $\beta=3\frac{1-\nu}{(1+\nu)}\gamma$.
For $\nu=1/2$ the two lines both reduce to $\beta=\gamma$. The weak Baker-
Ericksen inequality and the weak non-negative stored energy condition are
verified everywhere below the continuous line.

##  4 Self-gravitating polytropic elastic balls

The main topic of this section is the numerical analysis of the existence of
self-gravitating polytropic elastic balls in static equilibrium or homologous
motion. The constitutive function of the ball is then given by (3.22), with
$\gamma>0$, $\beta\neq 0$ satisfying (3.21), while the reference pressure in
the equation of state is given by
$\mathcal{P}=-\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}},1)=\kappa/\gamma$,
where $y_{\mathrm{b}}$ is given by (3.20). Hence the equation of state of the
ball is

$p_{\mathrm{rad}}(t,r)=F_{\mathrm{rad}}\left(\frac{\rho(t,r)}{\mathcal{K}},\frac{m(t,r)}{\frac{4\pi}{3}\mathcal{K}r^{3}}\right),\quad
p_{\mathrm{tan}}(t,r)=F_{\mathrm{tan}}\left(\frac{\rho(t,r)}{\mathcal{K}},\frac{m(t,r)}{\frac{4\pi}{3}\mathcal{K}r^{3}}\right),$
(4.1a) where $\displaystyle
F_{\mathrm{rad}}(\delta,\eta)=\frac{\kappa}{\gamma}+\widehat{p}_{\mathrm{rad}}(\delta,\eta)=3\kappa\frac{1-\nu}{1+\nu}\left(\frac{(\delta/\eta)^{\beta}-y_{\mathrm{b}}^{\beta}}{\beta}\right)\eta^{\gamma}$
(4.1b) $\displaystyle
F_{\mathrm{rad}}(\delta,\eta)=\frac{\kappa}{\gamma}+\widehat{p}_{\mathrm{tan}}(\delta,\eta)=3\kappa\frac{1-\nu}{1+\nu}\left[\left(\frac{(\delta/\eta)^{\beta}-y_{\mathrm{b}}^{\beta}}{\beta}\right)+\frac{1+\nu}{1-\nu}\big{(}1-\delta/\eta\big{)}Q(\delta/\eta)\right]\eta^{\gamma}$
(4.1c)

where $Q(y)$ is given by (3.22c). From these expressions it is clear that the
interior of the ball corresponds to the region where $y(t,r)>y_{\mathrm{b}}$,
while the boundary is the first radius $r=R(t)$ at which
$y(t,R(t))=y_{\mathrm{b}}$; recall that $y(t,0)=1$ and
$y_{\mathrm{b}}\in[0,1)$. Moreover if (and only if)
$\beta=3\frac{1-\nu}{1+\nu}\gamma$ we have $y_{\mathrm{b}}=0$, and thus the
mass density vanishes on the boundary of the ball. In this case (4.1) give
that the radial and tangential pressure coincide on the boundary, that is to
say, the boundary has zero shear stress. A special example of this situation,
but not the only possible one, is when $\nu=1/2$ and $\beta=\gamma$, i.e.,
when the elastic ball is actually a polytropic fluid ball.

###  4.1 Static solutions

Using (3.23) in (2.32) we obtain the following system on $(\delta,\eta)$:

$\displaystyle\delta^{\prime}=\left(\frac{\delta}{\eta}\right)^{1-\beta}\left(\frac{3(\beta-\gamma)}{r}B(\delta/\eta)\eta-\theta
r\eta^{2-\gamma}\delta\right),$ (4.2a)
$\displaystyle\eta^{\prime}=-\frac{3}{r}(\eta-\delta),$ (4.2b) where
$\theta=\frac{4\pi G\mathcal{K}^{2}}{9\kappa}\frac{1+\nu}{1-\nu}.$ (4.2c)

The system (4.2) is supplied with center data
$\delta(0)=\eta(0)=\delta_{c}>0$, i.e., $y(0)=1$. If within the interval of
existence of strongly regular solutions of (4.2) there is a first radius $R>0$
such that $y(R)=y_{\mathrm{b}}$, then a static self-gravitating polytropic
elastic ball with radius $R$ forms. The positivity of the tangential pressure
in the interior follows by the weak Baker-Ericksen inequality, see Lemma
4(iii). The values of the parameters $\gamma,\beta$ for which it is found
numerically that static self-gravitating polytropic elastic balls exist form
the domain $\mathcal{O}$, depending on $\nu$, depicted in Figure 3. The
particular shape of this region leads to the following conjecture:

Conjecture 1: There exists $\beta_{\star}=\beta_{\star}(\nu,\gamma)$ and
$\gamma_{\star}=\gamma_{\star}(\nu)$ satisfying

$\beta_{\star}(\nu,\gamma_{\star}(\nu))=3\frac{1-\nu}{1+\nu}\gamma_{\star}(\nu),\quad\text{
and }\quad\beta_{\star}(\nu,\gamma)<3\frac{1-\nu}{1+\nu}\gamma,\text{ for
$0<\gamma<\gamma_{\star}(\nu)$},$ (4.3)

such that static self-gravitating polytropic elastic balls exist if and only
if

$\mathrm{(a)}\ 0<\gamma\leq\gamma_{\star}(\nu)\text{ and
}\beta<\beta_{\star}(\nu,\gamma),\quad\text{or}\quad\mathrm{(b)}\
\gamma>\gamma_{\star}(\nu)\text{ and }\beta\leq 3\frac{1-\nu}{1+\nu}\gamma.$

Remark. When $\beta=3\frac{1-\nu}{1+\nu}\gamma$, i.e., when the boundary of
the polytropic elastic ball has zero shear, condition (a) is empty, while
condition (b) becomes $\gamma>\gamma_{\star}(\nu)$. These static balls are
necessarily of type $\mathrm{A}$. Moreover it is found numerically that
$\gamma_{\star}$ is increasing, $\gamma_{\star}(\nu)\to 0$ as $\nu\to-1^{+}$
and $\gamma_{\star}(\nu)\to 6/5$ as $\nu\to 1/2^{-}$, see Figure 4. In
particular, when $\nu\to 1/2^{-}$ the conjecture reduces to $\gamma>6/5$,
which is indeed the necessary and sufficient condition for the existence of
static self-gravitating polytropic fluid balls.

(a) $\nu=-0.5,\gamma_{\star}(\nu)\approx 0.50$

(b) $\nu=0,\gamma_{\star}(\nu)\approx 0.92$

(c) $\nu=0.25,\gamma_{\star}(\nu)\approx 1.08$

(d) $\nu=0.48,\gamma_{\star}(\nu)\approx 1.19$

Figure 3: The region $\mathcal{O}$ in the $(\gamma,\beta)$ for which it found
numerically that static self-gravitating polytropic elastic balls exist. The
darkest region is the domain
$\mathcal{V}=\mathcal{V}_{\mathrm{A}}\cup\mathcal{V}_{\mathrm{B}}$ where
existence is proved analytically in Theorem 1. The set
$\mathcal{V}_{\mathrm{A}}$ is defined by (4.4) and corresponds to type
$\mathrm{A}$ balls; the set $\mathcal{V}_{\mathrm{B}}$ is defined by (4.5) and
corresponds to type $\mathrm{B}$ balls. The upper continuous line is
$\beta=3(1-\nu)/(1+\nu)\gamma$, which corresponds to polytropic elastic balls
with zero boundary shear, while the lower continuous line is the curve in
(3.24).  Figure 4: The region on the $(\nu,\gamma)$ plane for which it found
numerically that static self-gravitating polytropic elastic balls with zero
shear at the boundary exist. These static balls are necessarily of type
$\mathrm{A}$ and are characterized by the shear parameter
$\beta=3\gamma(1-\nu)/(1+\nu)$.

The the dark region
$\mathcal{V}=\mathcal{V}_{\mathrm{A}}\cup\mathcal{V}_{\mathrm{B}}$ in Figure 3
contains the values of the parameters $\beta,\gamma$ for which the existence
of static self-gravitating polytropic elastic balls is proved in the following
theorem.

######  Theorem 1.

Let $\delta_{c}>0$, $\kappa>0$ and $\nu\in(-1,1/2)$ be given. If

$\gamma>2\quad\text{and}\quad 1<\beta\leq\gamma,$ (4.4)

there exists a unique static self-gravitating polytropic elastic ball of type
$\mathrm{A}$ with central density $\delta(0)=\delta_{c}$. If

$(0<\gamma\leq\beta<1\text{ or }\beta<\gamma\leq
1)\quad\text{and}\quad\frac{4-3\gamma}{3(2-\gamma)}<\left(1-\frac{\beta}{3\gamma}\frac{1+\nu}{1-\nu}\right)^{1/\beta},$
(4.5)

there exists a unique static self-gravitating polytropic elastic ball of type
$\mathrm{B}$ with central density $\delta(0)=\delta_{c}$. Moreover in both
cases

$p_{\mathrm{tan}}(r)\geq p_{\mathrm{rad}}(r),\text{ for $r\in[0,R]$},$ (4.6)

where $R$ is the radius of the ball.

The proof of Theorem 1 can be found in Appendix. It can be seen that the
proved result is still quite far from the conjectured one and does not apply
to the important case of polytropic elastic balls with zero boundary shear;
this problem requires further investigation.

###  4.2 Homologous solutions

In this final section we discuss briefly self-gravitating polytropic elastic
balls in homologous motion. As mentioned in Section 2.5, the solution
$(\delta,\eta,u)$ of the system (2.14) for such balls has the form

$u(t,r)=\frac{\dot{\omega}(t)}{\omega(t)}r,\quad\delta(t,r)=\frac{1}{\omega(t)^{3}}\delta_{0}\left(\frac{r}{\omega(t)}\right),\quad\eta(t,r)=\frac{1}{\omega(t)^{3}}\eta_{0}\left(\frac{r}{\omega(t)}\right),$
(4.7a) where
$\eta_{0}(r)=\frac{3}{r^{3}}\int_{0}^{r}\delta_{0}(s)s^{2}\,ds,\quad\delta_{0}(r)=\delta(0,r),$
(4.7b) for some $C^{1}$ function $\omega:\omega(0)=1$.

The system (2.14) for solutions of the form (4.7) reads

$\displaystyle\mathcal{K}z\delta_{0}(z)\omega(t)\ddot{\omega}(t)$
$\displaystyle=-\widehat{a}(\omega(t)^{-3}\delta_{0}(z),\omega(t)^{-3}\eta_{0}(z))\delta_{0}^{\prime}(z)$
$\displaystyle\quad+\widehat{b}(\omega(t)^{-3}\delta_{0}(z),\omega(t)^{-3}\eta_{0}(z))\frac{\eta_{0}(z)-\delta_{0}(z)}{z}$
$\displaystyle\quad-\frac{4\pi
G}{3}\mathcal{K}^{2}z\delta_{0}(z)\eta_{0}(z)\omega(t)^{-1},$ (4.8)

where $z=r/\omega(t)$. Provided the functions $\widehat{a},\widehat{b}$
satisfy the scaling condition

$\varepsilon^{-1/3}\,\widehat{a}(\varepsilon\delta,\varepsilon\eta)=\widehat{a}(\delta,\eta),\quad\varepsilon^{-1/3}\,\widehat{b}(\varepsilon\delta,\varepsilon\eta)=\widehat{b}(\delta,\eta),\
\text{for all $\varepsilon>0$,}$

the system (4.8) transforms into three separate equations on
$\omega(t),\delta_{0}(z),\eta_{0}(z)$, namely

$\displaystyle\omega(t)^{2}\ddot{\omega}(t)=\frac{4\pi
G}{3}\mathcal{K}\alpha,$ (4.9a)
$\displaystyle\widehat{a}(\delta_{0}(z),\eta_{0}(z))\delta_{0}^{\prime}(z)=\widehat{b}(\delta_{0}(z),\eta_{0}(z))\frac{\eta_{0}(z)-\delta_{0}(z)}{z}-\frac{4\pi
G}{3}\mathcal{K}^{2}z\delta_{0}(z)(\eta_{0}(z)+\alpha),$ (4.9b)
$\displaystyle\eta_{0}^{\prime}(z)=-\frac{3}{z}(\eta_{0}(z)-\delta_{0}(z)),$
(4.9c)

where $\alpha\neq 0$ is a (dimensionless) constant.

In the following we study the system (4.9) for the polytropic constitutive
function $(\widehat{p}_{\mathrm{rad}},\widehat{p}_{\mathrm{tan}})$ given by
(3.22) with $\gamma=4/3$. We are interested in solutions describing self-
gravitating balls with radius $R(t)$ and, as shown in Section 2.5, this
implies $R(t)=Z\omega(t)$, for some constant $Z>0$. Moreover the reference
pressure is given by
$\mathcal{P}=-\widehat{p}_{\mathrm{rad}}(y_{\mathrm{b}},1)=\kappa/\gamma$,
hence the equation of state of the ball is (4.1). In particular,
$y(t,R(t))=y_{\mathrm{b}}$.

The equation for $\omega$ is the same as in the polytropic fluid case [13,
18]. Positive solutions of (4.9a) with data $(\omega(0)=1,\dot{\omega}(0))$
have one of the following behaviors: either (a) $\omega\in C^{1}$ for all
$t\geq 0$ and $\omega(t)\to\infty$ as $t\to\infty$, or (b) $\omega\in
C^{1}([0,T))$ and $\omega(t)\to 0$ as $t\to T^{-}$, for some $T>0$. Without
loss of generality we assume $\dot{\omega}(0)=0$, i.e., the ball is initially
at rest, so that solutions of type (a) correspond to $\alpha>0$, while
solutions of type (b) correspond to $\alpha<0$. Hence the system (4.10)
describes continuously expanding balls when $\alpha>0$ and continuously
collapsing balls when $\alpha<0$, with $R(t)\to\infty$ as $t\to\infty$ in the
former case and $R(t)\to 0$ as $t\to T^{-}$ in the latter.

Next consider the equations for $(\delta_{0},\eta_{0})$, specifically

$\displaystyle\delta_{0}^{\prime}=\left(\frac{\delta_{0}}{\eta_{0}}\right)^{1-\beta}\left(\frac{3(\beta-4/3)}{z}B(\delta_{0}/\eta_{0})\eta_{0}-\theta
z\eta^{-1/3}\delta_{0}(\eta_{0}+\alpha)\right),$ (4.10a)
$\displaystyle\eta_{0}^{\prime}=-\frac{3}{z}(\eta_{0}-\delta_{0}),$ (4.10b)

where $\theta$ is given by (4.2c). Strongly regular solutions of (4.10) are
defined precisely as in the static case and satisfy $y_{0}(0)=1$, where
$y_{0}(z)=\delta_{0}(z)/\eta_{0}(z)$. Moreover since the difference between
(4.10) and (4.2) is a $O(z)$ term, it is clear that a unique strongly regular
solution of (4.10) with center data $\delta_{0}^{c}=\delta_{0}(0)=\eta_{0}(0)$
exists in a maximal interval $[0,Z_{\mathrm{max}})$. As

$\displaystyle
p_{\mathrm{rad}}(t,r)=F_{\mathrm{rad}}(\delta(t,r),\eta(t,r))=\omega(t)^{-4}F_{\mathrm{rad}}(\delta_{0}(r/\omega(t)),\eta_{0}(r/\omega(t)),$
(4.11a) $\displaystyle
p_{\mathrm{tan}}(t,r)=F_{\mathrm{tan}}(\delta(t,r),\eta(t,r))=\omega(t)^{-4}F_{\mathrm{tan}}(\delta_{0}(r/\omega(t)),\eta_{0}(r/\omega(t)),$
(4.11b)

the constant $Z$ in the radius $R(t)=Z\omega(t)$ of the ball is given by the
first value of $z\in(0,Z_{\mathrm{max}})$ at which
$F_{\mathrm{rad}}(\delta_{0}(z),\eta(z))=0$, that is
$y_{0}(Z)=y_{\mathrm{b}}$, where the boundary shear $y_{\mathrm{b}}$ is given
by

$y_{\mathrm{b}}=\left[1-\frac{\beta}{4}\left(\frac{1+\nu}{1-\nu}\right)\right]^{1/\beta},$
(4.12)

see (3.20). As in the static case, we can distinguish between type
$\mathrm{A}$ homologous balls, for which $0<Z<Z_{\mathrm{max}}<\infty$, and
type $\mathrm{B}$ homologous balls, for which $0<Z<Z_{\mathrm{max}}=\infty$;
in the case of polytropic elastic balls with zero boundary shear (including
fluid balls) only type $\mathrm{A}$ solutions are admissible.

The existence of a radius $Z$ with the properties above has been investigated
numerically in the case of collapsing balls with zero shear at the boundary,
i.e., $\beta=4(1-\nu)/(1+\nu)$; the results are summarized in Figure 5. It is
found that there exists $\delta_{\star}(\alpha,\nu)$ such that collapsing
homologous self-gravitating polytropic elastic balls with zero boundary shear
exist if and only if $\delta_{0}(0)\geq\delta_{\star}(\alpha,\nu)$. It can be
seen that $\delta_{\star}(\alpha,\nu)$ is increasing with respect to the
Poisson ratio, meaning that upon allowing shear interior deformations
collapsing homologous elastic balls can form with smaller initial central
densities than in the fluid case.

Figure 5: The region in the $(\alpha,\delta_{0}(0))$ plane for which it found
numerically that collapsing homologous self-gravitating polytropic elastic
balls with zero shear at the boundary exist. These balls exist only for
$\delta_{0}(0)\geq\delta_{\star}(\alpha,\nu)$; in this picture $\nu_{1}=0$,
$\nu_{2}=0.25$, $\nu_{3}=0.45$.

In the following simple final theorem the existence of expanding homologous
self-gravitating polytropic elastic balls is proved for some values of the
shear parameter $\beta$.

######  Theorem 2.

If $\alpha>0$ and $1<\beta\leq 4/3$, then for all $\delta_{0}^{c}>0$ there
holds $Z_{\mathrm{max}}<\infty$, $0<y_{0}(z)<1$, for
$z\in(0,Z_{\mathrm{max}})$, and $y_{0}(z)\to 0^{+}$ as $z\to
Z_{\mathrm{max}}^{-}$. In this case there exists a unique continuously
expanding homologous self-gravitating polytropic elastic ball
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)$ of type $\mathrm{A}$ with central
density $\rho(0,t)=\mathcal{K}\delta_{0}^{c}/\omega(t)$ and radius
$R(t)=Z\omega(t)$, where $Z\in(0,Z_{\mathrm{max}})$ is the first solution of
$y(Z)=y_{\mathrm{b}}$. Moreover $p_{\mathrm{tan}}(t,r)\geq
p_{\mathrm{rad}}(t,r)$ holds for all times in the interior of the ball.

###### Proof.

To start with we observe that by the exact same argument used in the proof of
Lemma 4(i), strongly regular solutions of (4.10) satisfy

$\delta_{0}^{\prime\prime}(0)=-\frac{4\pi
G}{3}\mathcal{K}^{2}\frac{\delta_{0}^{c}(\delta_{0}^{c}+\alpha)}{a(\delta_{0}^{c},\delta_{0}^{c})}=-\theta(\delta_{0}^{c})^{2/3}(\delta_{0}^{c}+\alpha),\quad\eta_{0}^{\prime\prime}(0)=\frac{3}{5}\delta^{\prime\prime}(0).$

Hence, for $\alpha>0$, and by the exact same proof as in Lemma 4(ii), the
inequalities $\eta_{0}(z)>\delta_{0}(z)$ and $\eta_{0}^{\prime}(z)<0$ hold for
$z\in(0,Z_{\mathrm{max}})$. Thus for $\alpha>0$ and $\beta\leq 4/3$ we have

$\delta_{0}^{\prime}\leq-\left(\frac{\delta_{0}}{\eta_{0}}\right)^{1-\beta}\theta
z\frac{\delta_{0}}{\eta_{0}^{1/3}}(\eta_{0}+\alpha)\leq-c\alpha
z\delta^{2-\beta},$

for a positive constant $c$. It follows that for $\alpha>0$ and $1<\beta\leq
4/3$, there hold $Z_{\mathrm{max}}<\infty$ and

$\lim_{z\to Z_{\mathrm{max}}^{-}}\delta_{0}(z)=\lim_{z\to
Z_{\mathrm{max}}^{-}}y_{0}(z)=0.$

As $y_{\mathrm{b}}\in(0,1)$, there exists a unique $Z\in(0,Z_{\mathrm{max}})$
such that $y(z)>y_{\mathrm{b}}$ for $z\in[0,Z_{\mathrm{max}})$ and
$y(Z)=y_{\mathrm{b}}$, or equivalently
$F_{\mathrm{rad}}(\delta_{0}(z),\eta_{0}(z))>0$ for $z\in(0,Z)$ and
$F_{\mathrm{rad}}(\delta_{0}(Z),\eta_{0}(Z))=0$. Moreover, by the Baker-
Ericksen inequality, $F_{\mathrm{tan}}(\delta_{0}(z),\eta_{0}(z))\geq
F_{\mathrm{rad}}(\delta_{0}(z),\eta_{0}(z))$, hence defining
$\rho(t,r)=\mathcal{K}\delta(t,r)$ and
$(p_{\mathrm{rad}}(t,r),p_{\mathrm{tan}}(t,r))$ as in (4.11), we obtain that
the quadruple $(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}},u)\mathbb{I}_{r<R(t)}$
is a homologous self-gravitating elastic ball with radius $R(t)=Z\omega(t)$,
for all $t\geq 0$, and $R(t)\to\infty$ as $t\to\infty$. ∎

##  Appendix A Appendix: Proof of Theorem 1

Recall that a solution of (4.2) in the interval $[0,R)$ is called regular if
$\delta\in C^{0}([0,R))\cap C^{1}((0,R))$ and $\delta(r)>0$, for all
$r\in[0,R)$, and strongly regular if in addition $\delta\in C^{1}([0,R))$ and
$\lim_{r\to 0^{+}}\delta(r)=0$; see Section 2.6. The following lemma contains
results on the limit $r\to R_{\mathrm{max}}^{-}$ of regular solutions defined
up to the maximal radius $R_{\mathrm{max}}$.

######  Lemma 5.

Let $\gamma\in\mathbb{R}$, $\beta\neq 0$ and $\delta_{c}>0$. Assume that there
exists a unique regular solution of (4.2) satisfying
$\eta(0)=\delta(0)=\delta_{c}>0$ and defined in a maximal radius interval
$[0,R_{\mathrm{max}})$, $R_{\mathrm{max}}>0$. Then the following holds:

  * (i)

If $R_{\mathrm{max}}<\infty$, then $\lim_{r\to
R_{\mathrm{max}}^{-}}\delta(r)=0$ and $\lim_{r\to
R_{\mathrm{max}}^{-}}\eta(r)>0$;

  * (ii)

If $R_{\mathrm{max}}=\infty$, then
$\lim_{r\to\infty}\delta(r)=\lim_{r\to\infty}\eta(r)=0$.

###### Proof.

The inequalities $\delta(r)<\eta(r)$ and $\eta^{\prime}(r)<0$, for
$r\in(0,R_{\mathrm{max}})$, will be used in the following arguments, see Lemma
4.

  * (i)

We begin by showing that $\delta(r)$ has a (finite) limit when $r\to
R_{\mathrm{max}}^{-}$. This is obvious when $\beta\leq\gamma$, as in this case
$\delta(r)$ is non-increasing. Let $\beta>\gamma$. For $\beta\leq 2$ we write
the $\delta$-equation in (4.2) as

$\delta^{\prime}=\frac{3(\beta-\gamma)}{r}\widetilde{B}(\delta/\eta)\eta-\theta
r\eta^{1+\beta-\gamma}\delta^{2-\beta},$ (A.1)

where $\widetilde{B}(y)=B(y)y^{1-\beta}$ is bounded for $y\in[0,1]$. Hence for
$r\in(0,R_{\mathrm{max}})$, there holds

$|\delta^{\prime}(r)|\leq\frac{c_{1}}{r}+c_{2}r,$

where $c_{1},c_{2}$ are positive constants. It follows that
$|\delta^{\prime}|$ is bounded in the interval $(0,R_{\mathrm{max}})$ and thus
$\delta(r)$ converges as $r\to R_{\mathrm{max}}^{-}$ when $\beta\leq 2$. For
$\beta>2$ we instead write the $\delta$-equation as

$\frac{1}{\beta}(\delta^{\beta})^{\prime}=\delta^{\beta-1}\delta^{\prime}=\frac{3(\beta-\gamma)}{r}B(\delta/\eta)\eta^{\beta}-\theta
r\eta^{1+\beta-\gamma}\delta$

and since $B(y)$ is bounded on $[0,1]$ for $\beta>2$, we obtain as before that
$|(\delta^{\beta})^{\prime}|$ is bounded on $(0,R_{\mathrm{max}})$. Hence
$\delta^{\beta}(r)$ converges as $r\to R_{\mathrm{max}}^{-}$, and so
$\lim_{r\to R_{\mathrm{max}}^{-}}\delta(r)$ exists for all $\beta\neq 0$. It
will now be proved that this limit is zero. Assume, by contradiction, that
$\lim_{r\to R_{\mathrm{max}}^{-}}\delta(r)>0$. Then there exists $c\in(0,1)$
such that $c<\delta(r)/\eta(r)<1$, for all $r\in(0,R_{\mathrm{max}}$), which
implies that the right hand side of (4.2) is bounded on
$(0,R_{\mathrm{max}})$. But then $|\delta^{\prime}|$ is bounded on
$(0,R_{\mathrm{max}})$, contradicting the hypothesis that
$R_{\mathrm{max}}<\infty$. Hence $\lim_{r\to R_{\mathrm{max}}^{-}}\delta(r)=0$
and

$\lim_{r\to
R_{\mathrm{max}}^{-}}\eta(r)=\frac{3}{R_{\mathrm{max}}^{3}}\int_{0}^{R_{\mathrm{max}}}\delta(s)s^{2}\,ds>0,$

must hold when $R_{\mathrm{max}}<\infty$.

  * (ii)

Let $\eta_{\infty}=\lim_{r\to\infty}\eta(r)$. If $\eta_{\infty}=0$, then
$\lim_{r\to\infty}\delta(r)=0$ and thus the claim follows. Assume
$\eta_{\infty}>0$; in particular $\eta_{\infty}<\eta(r)<\eta(0)=\delta_{c}$,
for all $r\in(0,\infty)$. As in (i) we start by showing that
$\lim_{r\to\infty}\delta(r)$ exists, which again is obvious for
$\beta\leq\gamma$. Let $\beta>\gamma$ and assume first $\beta\geq 1$. In this
case we rewrite the $\delta$-equation in (4.2) as

$\delta^{\prime}=\left(\frac{\delta}{\eta}\right)^{1-\beta}\eta^{2-\gamma}\delta\left(\frac{3(\beta-\gamma)}{r}\frac{B(\delta/\eta)}{\delta/\eta}\eta^{\gamma-2}-\theta
r\right)\leq\left(\frac{\delta}{\eta}\right)^{1-\beta}\eta^{2-\gamma}\delta(cr^{-1}-\theta
r),$

where $c$ is a positive constant and where we used that $B(y)/y$ is bounded
for $y\in(0,1)$ and $\beta\geq 1$. Hence $\delta^{\prime}(r)<0$ for
$r>\sqrt{c/\theta}$ and thus $\lim_{r\to\infty}\delta(r)$ exists when
$\beta\geq 1$. Assume now $\beta<1$. Then by (A.1) we can write

$\delta^{\prime}(r)=\frac{3(\beta-\gamma)}{r}\frac{\widetilde{B}(\delta/\eta)}{\delta/\eta}\delta-\theta
r\eta^{1+\beta-\gamma}\delta^{2-\beta}\leq\frac{c_{1}}{r}\delta-
c_{2}r\delta^{2-\beta},$

where $c_{1},c_{2}$ are positive constants and where we used that
$\widetilde{B}(y)/y=B(y)y^{-\beta}$ is bounded for $\beta<1$. Hence
$\delta(r)\leq u(r)$ for $r\geq 1$, where

$u(r)=\left(\frac{c_{2}(1-\beta)}{2+c_{1}(1-\beta)}r^{2}+\frac{\delta(1)^{\beta-1}(2+c_{1}(1-\beta))-c_{2}(1-\beta)}{2+c_{1}(1-\beta)}\frac{1}{r^{c_{1}(1-\beta)}}\right)^{-1/(1-\beta)}$

is the solution of $u^{\prime}(r)=c_{1}u(r)/r-c_{2}ru(r)^{2-\beta}$ with
$u(1)=\delta(1)$. We conclude that $\delta(r)\to 0$ as $r\to\infty$ when
$\beta<1$. We claim that $\delta(r)$ converges to zero at infinity even when
$\beta>1$. Assume $\lim_{r\to\infty}\delta(r)=\delta_{\infty}>0$; then

$\delta^{\prime}(r)\sim\left(\frac{\delta_{\infty}}{\eta_{\infty}}\right)^{1-\beta}\left(\frac{3(\beta-\gamma)}{r}B(\delta_{\infty}/\eta_{\infty})\eta_{\infty}-\theta
r\eta_{\infty}^{2-\gamma}\delta_{\infty}\right)\to-\infty,\quad\text{ as
$r\to\infty$},$

a contradiction. Hence $\lim_{r\to\infty}\delta(r)=0$ must hold and thus also
$\lim_{r\to\infty}\eta(r)=0$, by L’Hôpital’s rule.

∎

In the next proposition the existence of strongly regular solutions to (4.2)
in a maximal radius interval $[0,R_{\mathrm{max}})$ is proved and sufficient
conditions on the parameters $\gamma,\beta$ are given such that
$R_{\mathrm{max}}$ is finite or $R_{\mathrm{max}}=\infty$.

######  Proposition 4.

For all $\gamma\in\mathbb{R}$, $\beta\neq 0$ and $\delta_{c}>0$, the system
(4.2) admits a unique strongly regular local solution $(\delta,\eta)$ such
that $\delta(0)=\eta(0)=\delta_{c}$. Letting $[0,R_{\mathrm{max}})$ be the
maximal radius interval of definition of this solution the following holds:

  * $\mathrm{(A)}$

If $\gamma>2$ and $1<\beta\leq\gamma$, then $R_{\mathrm{max}}<\infty$;

  * $\mathrm{(B)}$

If $0<\gamma\leq\beta<1$ or $\beta<\gamma\leq 1$, then
$R_{\mathrm{max}}=\infty$ and
$\lim_{r\to\infty}y(r)=\frac{4-3\gamma}{3(2-\gamma)}$.

###### Proof.

Assume first $\gamma\neq 2$. The change of variables

$y=\delta/\eta,\quad v=\theta r^{2}\eta^{2-\gamma}y^{1-\beta},\quad\xi=\log r$

transforms (4.2) into the autonomous dynamical system

$\displaystyle\frac{dy}{d\xi}=[\Upsilon(y)-v]y,$ (A.2a)
$\displaystyle\frac{dv}{d\xi}=[(1-\beta)(\Upsilon(y)-v)+2-3(2-\gamma)(1-y)]v,$
(A.2b) where $\Upsilon(y)=3(1-y)+3(\beta-\gamma)B(y)y^{-\beta}.$ (A.2c)

As $(dy/d\xi)_{y=1}<0$, for all $v>0$, the open region
$\mathcal{U}=\\{v>0,y<1\\}$ of the state space is future invariant (which is
equivalent to the bound $\eta>\delta$ proved in Lemma 4(ii)). A simple local
stability analysis shows that the boundary point $\mathrm{O}$ with coordinates
$v=0,y=1$ is an hyperbolic saddle. The positive eigenvalue of the linearised
flow around $\mathrm{O}$ is equal to 2 and the corresponding eigenvector
$-5\partial_{y}+\partial_{v}$ points toward the interior of $\mathcal{U}$.
Hence there exists exactly one orbit
$\Gamma=(\Gamma_{y},\Gamma_{v})\subset\mathcal{U}$ such that
$\lim_{\xi\to-\infty}\Gamma(\xi)=\mathrm{O}$. Moreover $1-y(\xi)\sim
5Ce^{2\xi}$ and $v(\xi)\sim Ce^{2\xi}$ as $\xi\to-\infty$ along this orbit,
where $C$ is a positive constant. This orbit corresponds to a one parameter
family of regular solutions of (4.2) up to $\xi_{*}$ such that
$\Gamma_{y}(\xi)>0$ for $\xi<\xi_{*}$, and by Lemma 4(i) these solutions are
strongly regular. Uniqueness follows if we show that the constant $C$ is
uniquely determined by the center datum $\delta(0)=\eta(0)=\delta_{c}$. As
$\gamma\neq 2$, this follows by the definition of $v$, which gives
$C=\theta\delta_{c}^{2-\gamma}$. Assume now $\gamma=2$. In this case, the
system (4.2) is equivalent to the following decoupled system on
$y=\delta/\eta$ and $\eta$:

$y^{\prime}=\left(\frac{\Upsilon(y)}{r}-\theta
ry^{1-\beta}\right)y,\quad\eta^{\prime}=-\frac{3\eta}{r}(1-y).$ (A.3)

The same argument as above gives now a local unique regular solution of the
$y$-equation with asymptotic behavior $y=1+O(r^{2})$ as $r\to 0^{+}$, and thus
uniqueness for the system (A.3) follows by simply integrating the
$\eta$-equation with center datum $\eta(0)=\delta_{c}$. By Lemma 4 the unique
regular solution is strongly regular and thus the first part of the
proposition is proved.

Proof of $\mathrm{(A)}$: As $\beta\leq\gamma$ we have

$\delta^{\prime}\leq-\theta r\eta^{1+\beta-\gamma}\delta^{2-\beta}.$ (A.4)

For $\beta\leq\gamma-1$ we use $\eta\leq\eta(0)=\delta_{c}$ in (A.4) to obtain
$\delta^{\prime}\leq-cr\delta^{2-\beta}$, and thus

$\delta(r)\leq\left(\delta_{c}^{\beta-1}-\frac{c}{2}(\beta-1)r^{2}\right)^{\frac{1}{\beta-1}},$

which, as $\beta>1$, implies $R_{\mathrm{max}}<\infty$. For
$\gamma-1<\beta\leq\gamma$ we instead use $\eta>\delta$ in (A.4) to obtain
$\delta^{\prime}\leq-\theta r\delta^{3-\gamma}$, which, as $\gamma>2$, implies
again $R_{\mathrm{max}}<\infty$.

Proof of $\mathrm{(B)}$: To prove $\mathrm{(B)}$ we study the qualitative
behavior toward the future of the orbit $\Gamma$ of the dynamical system (A.2)
originating from the fixed point $\mathrm{O}$. As $\beta<1$, the function
$\Upsilon$ is bounded for $y\in[0,1]$ and

$\Upsilon(0)=3\frac{1-\gamma}{1-\beta}\geq 0.$

In fact, a straightforward analysis shows that $\Upsilon(y)>0$ for all
$y\in(0,1)$. In particular the region $\mathcal{V}=\\{v>0,0<y<1\\}$ is future
invariant and $\Gamma(\xi)\subset\mathcal{V}$, for all $\xi\in\mathbb{R}$.
Moreover $v^{\prime}(\xi)\leq(a-(1-\beta)v)v$, where
$a=\sup_{y\in(0,1)}[(1-\beta)\Upsilon(y)+2-3(2-\gamma)(1-y)]<\infty$ and thus
$v(\xi)\leq a/(1-\beta)$ along the orbit $\Gamma$. It follows that the
$\omega$-limit set $\omega(\Gamma)$ of $\Gamma$ is not empty. By Poincaré-
Bendixson theorem, $\omega(\Gamma)$ must be one of the following sets: (1) a
fixed point; (2) a periodic orbit; (3) a connected set consisting of a finite
number of fixed points $\\{P_{1},\dots,P_{n}\\}$ together with homoclinic and
heteroclinic orbits connecting $P_{1},\dots,P_{n}$. Let
$\phi(y,v)=v^{-1}y^{\beta-2}$ and let $F(v,y)$ denote the vector field in the
right hand side of (A.2). Since $\nabla\cdot(\phi
F)(y,v)=-3\phi(y,v)(1-\gamma(1-y))$ is negative for $\gamma\leq 1$ and
$(y,v)\in\mathcal{V}$, then, by Dulac-Bendixson theorem, no periodic orbits
exist in the region $\mathcal{V}$ and thus the alternative (2) above is not
possible. The alternative (3) can be ruled out by studying the stability
properties of the fixed points of the flow. Besides $\mathrm{O}$, the
dynamical system (A.2) admits the fixed points $\mathrm{Q}=(0,0)$ and

$\mathrm{P}=(\Upsilon(y_{\mathrm{P}}),y_{\mathrm{P}}),\quad\text{where}\quad\
y_{\mathrm{P}}=\frac{4-3\gamma}{3(2-\gamma)}.$

No other fixed points are present when $\gamma\leq 1$. A simple local
stability analysis shows that $\mathrm{P}$ is an hyperbolic sink, and thus it
is a local attractor for a one parameter family of interior orbits, while
$\mathrm{Q}$ is an hyperbolic saddle. The stable manifold of $\mathrm{Q}$ is
tangent to the axis $y=0$, while the unstable manifold is tangent to $v=0$. In
particular there is no interior orbit which converges to or emanates from
$\mathrm{Q}$. Putting this information together we conclude that none of the
structures mentioned in the alternative (3) above exists in the region
$\mathcal{V}$. Thus Poincaré Bendixson theorem entails that the fixed point
$\mathrm{P}$ is the $\omega$-limit set of all orbits entering the region
$\mathcal{V}$ and so in particular $\Gamma(\xi)\to\mathrm{P}$ as
$\xi\to\infty$, which is equivalent to
$\lim_{r\to\infty}y(r)=\frac{4-3\gamma}{3(2-\gamma)}$.

∎

###### Proof of Theorem 1.

When (4.4) holds, Proposition 4 gives that the maximal radius interval of
definition of strongly regular solutions of (4.2) is finite and by Lemma 5(i)

$y(r)=\frac{\delta(r)}{\eta(r)}\to 0,\quad\text{as $r\to
R_{\mathrm{max}}^{-}$}.$

As $y(0)=1$ and $y_{\mathrm{b}}\in(0,1)$, there exists a unique
$R\in(0,R_{\mathrm{max}})$ such that $y(r)>y_{\mathrm{b}}$ for $r\in[0,R)$ and
$y(R)=y_{\mathrm{b}}$. Thus, letting
$p_{\mathrm{rad}}(r)=F_{\mathrm{rad}}(\delta(r),\eta(r))$, we have
$p_{\mathrm{rad}}(R)=0$ and $p_{\mathrm{rad}}(r)>0$, for $r\in[0,R)$.
Moreover, by Lemma 4(iii),

$p_{\mathrm{tan}}(r)=F_{\mathrm{tan}}(\delta(r),\eta(r))\geq
p_{\mathrm{rad}}(r),\quad r>0.$

Hence, letting $\rho(r)=\mathcal{K}\delta(r)$, we obtain that
$(\rho,p_{\mathrm{rad}},p_{\mathrm{tan}})\mathbb{I}_{r\leq R}$ is a type
$\mathrm{A}$ static self-gravitating elastic ball. When (4.5) holds, strongly
regular solutions of (4.2) are global and
$y\to\frac{4-3\gamma}{3(2-\gamma)}=y_{\infty}>0$ as $r\to\infty$, see again
Proposition 4. Since the last inequality in (4.5) is equivalent to
$y_{\infty}<y_{\mathrm{b}}$, then there exists $R\in(0,R_{\mathrm{max}})$ such
that $y(r)>y_{\mathrm{b}}$ for $r\in[0,R)$ and $y(R)=y_{\mathrm{b}}$, and thus
the proof can be completed as before. ∎

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