# Rotational Symmetry of Solutions of Mean Curvature Flows Coming Out of A
Double Cone

Letian Chen  Department of Mathematics, Johns Hopkins University, 3400 N.
Charles Street, Baltimore, MD 21218<EMAIL_ADDRESS>
(Date: Jan. 24, 2021)

###### Abstract.

We show that any smooth solution to the mean curvature flow equations coming
out of a rotationally symmetric double cone is also rotationally symmetric.

##  1\. Introduction

We say a family of properly embedded smooth hypersurface
$\\{\Sigma_{t}\\}_{t\in I}\subset\mathbb{R}^{n+1}$ is a solution of the mean
curvature flow (MCF) equations if

$\displaystyle\left(\frac{\partial x}{\partial
t}\right)^{\perp}=H_{\Sigma_{t}}(x).$

Here $H_{\Sigma_{t}}(x)$ denotes the mean curvature vector of $\Sigma_{t}$ at
$x$, and $x^{\perp}$ is the normal component of $x$.

In this article we are interested in solutions of MCF coming out of a
rotationally symmetric double cone, by which we mean a (hyper)cone
$\mathcal{C}\subset\mathbb{R}^{n+1}$ whose link
$\mathcal{L}(\mathcal{C})=\mathcal{C}\cap\mathbb{S}^{n}$ is a smooth
hypersurface of $\mathbb{S}^{n}$ and has two connected components lying in two
separate hemispheres. More explicitly, we consider a cone of the form (up to
an ambient rotation so that the axis of symmetry is $x_{1}$-axis)

$x_{1}^{2}=\begin{cases}m_{1}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2})&x_{1}\geq
0\\\ m_{2}(x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2})&x_{1}<0\end{cases}$ (1.1)

where $m_{1},m_{2}>0$ are constants related to the aperture of the one.
Solutions coming out of cones arise naturally in the singularity analysis of
MCF. In particular the self-expanders, which are special solutions of the MCF
satisfying $\Sigma_{t}=\sqrt{t}\Sigma$ for some hypersurface
$\Sigma\subset\mathbb{R}^{n+1}$, are often thought of as models of MCF flowing
out of a conical singularity (see for example [1]). Self-expanders satisfy the
elliptic equation

$\displaystyle H_{\Sigma}(x)=\frac{x^{\perp}}{2},$

which is the Euler-Lagrange equation of the functional
$\int_{\Sigma}e^{\left|x\right|^{2}/4}d\mathcal{H}^{n}$. We can therefore talk
about the Morse index of a given self-expander, and the Morse flow lines
between two self-expanders (asymptotic to the same cone $\mathcal{C}$) are
examples of non self-similar solutions coming out of the cone.

We show that, given a smooth double cone $\mathcal{C}\subset\mathbb{R}^{n+1}$
and a solution to the MCF, $\\{\Sigma_{t}\\}_{t\in[0,T]}$, asymptotic to
$\mathcal{C}$, then the flow inherits the rotational symmetry of $\mathcal{C}$
at all times. More precisely we prove:

######  Theorem 1.1.

Let $\mathcal{C}\subset\mathbb{R}^{n+1}$ be a smooth, rotationally symmetric
double cone. Suppose $\\{\Sigma_{t}\\}_{t\in[0,T]}$ is a smooth solution to
the mean curvature flow asymptotic to $\mathcal{C}$, in the sense that

$\displaystyle\lim_{t\to
0^{+}}\mathcal{H}^{n}\llcorner\Sigma_{t}=\mathcal{H}^{n}\llcorner\mathcal{C}$

as Radon measures, then $\Sigma_{t}$ is also rotationally symmetric (with the
same axis of symmetry) for any $t\in[0,T]$.

######  Remark.

It is likely that only a finite number of such solutions exist. These include
self-expanders and Morse flow lines between two self-expanders asymptotic to
the same cone, some of which can be constructed using methods from [6]. In
particular the latter solutions might develop singularities. Indeed, when the
parameters $m_{1}$ and $m_{2}$ in Equation 1.1 are sufficiently small, by [14]
we can find an unstable (connected) catenoidal self-expander and a
disconnected self-expander whose two components are given by the unique self-
expanders asymptotic to the top part and bottom part of the cone. One expects
that there exists a Morse flow line connecting these two self-expanders. Such
a flow line will necessarily develop a neck pinch in order to become
disconnected.

As an easy corollary we obtain the following rotational symmetry result:

######  Corollary 1.2.

Let $\mathcal{C}\subset\mathbb{R}^{n+1}$ be a smooth, rotationally symmetric
double cone, then any smooth self-expander $\Sigma$ asymptotic to
$\mathcal{C}$ is also rotationally symmetric (with the same axis of symmetry).

######  Remark.

It is expected that no singular self-expander asymptotic to $\mathcal{C}$
exists, but our theorem only applies in the smooth case. The smoothness
assumption is in place to avoid further technicality introduced by the moving
plane method, see Section 2.4.

The rotational symmetry is known in many other cases. Fong and McGrath [13]
showed that same conclusion holds if the cone is rotationally symmetric and
the expander is mean convex. Bernstein-Wang (Lemma 8.3 in [4]) later showed
that same conclusion holds if the cone is rotationally symmetric and the
expander is weakly stable (in particular, mean convexity implies weak
stability so this generalizes the Fong–McGrath result). In contrast, our
result applies to all solutions coming out of the cone and does not assume any
extra condition about the flow other than smoothness. For other geometric
flows, Chodosh [10] proved rotational symmetry of expanding asymptotically
conical Ricci solitons with positive sectional curvature.

It is also worth mentioning that, although in general given a rotationally
symmetric smooth cone $\mathcal{C}$ there could be multiple self-expanders
asymptotic to $\mathcal{C}$ , if there exists a unique self-expander
asymptotic to $\mathcal{C}$, it must inherit the rotational symmetry.
Uniqueness holds, for example, when the link of $\mathcal{C}$,
$\mathcal{L}(C)$, is connected, or, in the double cone case, when the
parameters $m_{1},m_{2}$ in Equation 1.1 are sufficiently large [4]. It is
interesting to determine whether the rotational symmetry holds when the link
$\mathcal{L}(C)$ has 3 or more connected components. We suspect that
counterexamples exist. We refer to [3], [4], [5], [7], and [11] for more
information on self-expanders.

The proof of 1.1 relies on the moving plane method pioneered by Alexandrov to
prove that embedded compact constant mean curvature hypersurfaces are round
spheres. The method was further employed to minimal surfaces by Schoen [20] to
prove certain uniqueness theorems for catenoids. More recently, Martín–Savas-
Halilaj–Smoczyk [19] showed uniqueness of translators (that is, solutions of
the MCF equation that evolve by translating along one fixed direction) with
one asymptotically paraboloidal end. Choi–Haslhofer–Hershkovitz [8] and
Choi–Haslhofer–Hershkovitz–White [9] used a parabolic variant of the method to
deduce rotational symmetry of certain ancient solutions to the MCF equation
(that is, solutions of the MCF equation which exist on $(-\infty,0)$). These
methods were further generalized to non-smooth settings very recently by
Haslhofer–Hershkovitz–White [15] and by Bernstein–Maggi [2].

Although a self-expander $\Sigma$ satisfies an elliptic PDE, the hypersurface
obtained after reflecting a self-expander with respect to a hyperplane does
not satisfy the above equation anymore (it is rather a translated self-
expander). For this reason we could not directly apply the usual elliptic
maximum principle and Hopf lemma, and we need to work in spacetime
$\mathbb{R}^{n+1}\times[0,T]$ and use the MCF equations directly with a
parabolic version of the maximum principles, which will lead to the more
general 1.1. Consequently, our method is in spirit closer to that used by [8].

### Acknowledgment

The author would like to thank his advisor, Jacob Bernstein, for numerous
helpful advice and constant encouragement, especially during a period of
extreme difficulty around the globe. The author would also like to thank Rory
Martin-Hagemeyer for useful discussions.

##  2\. Preliminaries

###  2.1. Notations

Throughout the paper, $B_{r}(x)$ will denote the Euclidean ball of radius $r$
centered at a point $x\in\mathbb{R}^{n+1}$. By a (smooth) MCF in
$\mathbb{R}^{n+1}$ we mean a family of embedded hypersurfaces
$\\{\Sigma_{t}\\}_{t\in I}$ for some interval $I$ such that

$\displaystyle\left(\frac{\partial x}{\partial
t}\right)^{\perp}=H_{\Sigma_{t}}(x)$

for all $x\in\Sigma_{t}$, $t\in I$. Given an open set
$U\subset\mathbb{R}^{n+1}$, we say $\\{\Sigma_{t}\\}_{t\in I}$ is a MCF in $U$
if the above equation is satisfied locally (given a local parametrization of
the hypersurface) at every $x\in\Sigma_{t}\cap U$ and $t\in I$.

###  2.2. Pseudolocality for MCF

We will be frequently using the following pseudolocality result of Ilmanen-
Neves-Schulze (see also [12]):

######  Theorem 2.1 (Theorem 1.5 of [16]).

Let $\\{\Sigma_{t}\\}_{t\in(0,T]}$ be a mean curvature flow in
$\mathbb{R}^{n+1}$. Given any $\eta>0$, there is $\delta,\varepsilon>0$ such
that: if $x\in\Sigma_{0}$ and $\Sigma_{0}\cap C_{1}(x)$ is a graph over
$C^{n}_{1}(x)$ with Lipschitz constant bounded by $\varepsilon$, then
$\Sigma_{t}\cap C_{\delta}(x)$ can be written as a graph over
$C^{n}_{\delta}(x)$ with Lipschitz bounded by $\eta$ for any
$t\in[0,\delta^{2})\cap[0,T)$.

Here

$\displaystyle
C_{r}^{n}(x)=\\{(y,y_{n+1})\in\mathbb{R}^{n+1}\mid\left|y-x\right|<r\\}$

for $(x,x_{n+1})\in\mathbb{R}^{n+1}$ is the open cylinder in
$\mathbb{R}^{n+1}$ centered at $x$ and

$\displaystyle C_{r}(x)=\\{(y,y_{n+1})\in
C_{r}^{n}(x)\mid\left|y_{n+1}-x_{n+1}\right|<r\\}$

is the closed cylinder. Roughly speaking, this theorem says that if the
initial data of our MCF is graphical in some cylinder centered at $x$, then at
least for a short time the evolution of the hypersurface stays graphical in a
possibly smaller cylinder. We will primarily use this theorem to show that our
flow is graphical outside of a large ball for a short time, although strictly
speaking we sometimes need to apply the above theorem in the context of
integral Brakke flow.

###  2.3. Parabolic Maximum Principles

In this section $Z_{r}(x,t)$ will denote the spacetime cylinder of radius $r$
centered at $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$; that is,

$\displaystyle
Z_{r}(x,t)=\\{(y,s)\in\mathbb{R}^{n}\times\mathbb{R}\mid\left|y-x\right|<r,\left|t-s\right|<r^{2}\\}.$

$Z_{r}^{-}(x,t)$ will denote the part of cylinder $Z_{r}(x,t)$ whose time
component is smaller than $t$. To carry out the moving plane method, the most
important ingredients are the maximum principle and Hopf lemma. In our case we
need a version of those theorems applicable to graphical solutions of MCF;
that is, functions $u:Z_{r}^{-}(0,0)\to\mathbb{R}$ satisfying the following
parametrized PDE:

$\displaystyle u_{t}=\sqrt{1+\left|\nabla
u\right|^{2}}\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+\left|\nabla
u\right|^{2}}}\right).$

Observe that the difference of two graphical solutions to MCF satisfies a
second-order linear parabolic PDE (provided the gradients are bounded a
priori, which will be the case since our solutions are asymptotically
conical), so by standard theory of linear parabolic PDEs [17] we have (cf.
Section 6.2 in [8]):

######  Lemma 2.2 (Maximum Principle).

Suppose $u,v$ are graphical solutions to the MCF in a parabolic cylinder
$Z_{r}^{-}(0,0)$ with $u(0,0)=v(0,0)$. If $u\leq v$ in $Z_{r}^{-}(0,0)$, then
$u=v$ in $Z_{r}(0,0)$.

######  Lemma 2.3 (Hopf Lemma).

Suppose $u,v$ are graphical solutions to the MCF in a half parabolic cylinder
$Z_{r}^{-}(0,0)\cap\\{x_{1}\geq 0\\}$ with $u(0,0)=v(0,0)$ and $\frac{\partial
u}{\partial x_{1}}(0,0)=\frac{\partial v}{\partial x_{1}}(0,0)$. If $u\leq v$
in $Z_{r}^{-}(0,0)\cap\\{x_{1}\geq 0\\}$, then $u=v$ in
$Z_{r}^{-}(0,0)\cap\\{x_{1}\geq 0\\}$.

###  2.4. Asymptotically Conical Mean Curvature Flow

Here we will briefly discuss the class of MCFs we consider in 1.1. We need our
MCF to be at least $C^{2,\alpha}$-asymptotically conical in order to apply the
maximum principle and Hopf Lemma. This will almost not be an issue if we
assume our cone is at least $C^{3}$, as the following proposition shows.

######  Proposition 2.4 (cf. Proposition 3.3 in [5]).

Let $\mathcal{C}$ be a $C^{3}$ cone and suppose $\\{\Sigma_{t}\\}_{t\in(0,T]}$
is a MCF such that

$\displaystyle\lim_{t\to
0^{+}}\mathcal{H}^{n}\llcorner\Sigma_{t}=\mathcal{H}^{n}\llcorner\mathcal{C},$

then we have for $\alpha\in[0,1)$ and $t\in(0,T)$,

$\displaystyle\lim_{\rho\to 0^{+}}\rho\Sigma_{t}=\mathcal{C}\text{ in
}C^{2,\alpha}_{loc}(\mathbb{R}^{n+1}\setminus\\{0\\}).$

###### Proof.

It is enough to prove that locally $\Sigma_{t}$ is a $C^{2,\alpha}$ normal
graph over $\mathcal{C}$ outside of a large ball. By 2.1 (strictly speaking we
need to consider the flow together with the initial condition
$\Sigma_{0}=\mathcal{C}$ as an integral Brakke flow and apply the theorem for
Brakke flows), there is $\delta>0$ such that $\Sigma_{t}\cap
C_{\delta}(x_{0})$ can be written as a normal graph over $C_{\delta}^{n}(x)$
with Lipschitz constant bounded by $\eta$ for $t\in(0,\delta^{2})$. This
induces a map $u_{x_{0}}:[0,\delta^{2})\times
C_{\delta}^{n}(x_{0})\to\mathbb{R}$ whose graph describes part of the flow in
the space-time of the flow:

$\displaystyle\mathcal{M}=\mathcal{C}\times\\{0\\}\cup\bigcup_{t\in(0,T)}\Sigma_{t}\times\\{t\\}.$

Since $\\{\Sigma_{t}\\}$ is a MCF, $u_{x}$ satisfies the parametrized equation

$\displaystyle\frac{\partial u_{x_{0}}}{\partial
t}=\sqrt{1+\left|\nabla_{x}u_{x_{0}}\right|^{2}}\operatorname{div}\left(\frac{\nabla_{x}u_{x_{0}}}{\sqrt{1+\left|\nabla_{x}u_{x_{0}}\right|^{2}}}\right)$

with initial conditions
$u_{x_{0}}(0,x_{0})=\left|\nabla_{x}u_{x_{0}}(0,x_{0})\right|=0$. Since we did
not introduce the various Hölder and weighted Hölder norms (to account for the
cone structure), we will only briefly outline the proof, which is essentially
the same as the proof of Proposition 3.3 in [5]. This is a quasilinear
parabolic PDE in divergence form and we may use Schauder estimates (Theorem
5.1 in Chapter 5 of [18]) to get

$\displaystyle\left|\nabla_{x}u_{x_{0}}(t,x)\right|\leq
C(\left|x-x_{0}\right|+\sqrt{t})\leq C\delta.$

For the pointwise estimate on $u_{x_{0}}(t,x)$ we have to combine the above
and the MCF equation to bound the time derivative
$\left|\partial_{t}u_{x_{0}}(t,x)\right|$, giving an estimate of the form

$\displaystyle\left|u_{x_{0}}(t,x)\right|\leq
C(\left|x-x_{0}\right|^{2}+t)\leq C\delta^{2}.$

Finally, Schauder estimates (Theorem 1.1 in Chapter 6 of [18]) gives Holder
semi-norm bounds for any $\alpha\in(0,1)$. Combining these estimates yield a
weighted-$C^{2,\alpha}$ estimate on the cone, which is what we needed. ∎

Unfortunately pseudolocality only gives normal graphicality outside of a large
compact set, and so we can not conclude that the entire flow will be of class
$C^{2,\alpha}$. For this reason it is assumed that the MCF is smooth to begin
with in 1.1. We note that it might be possible to remove this assumption using
a moving plane method in non-smooth settings such as those presented in [9],
[2] or [15].

##  3\. Rotational Symmetry

In this section we prove 1.1. As claimed before, a direct consequence of the
pseudolocality theorem 2.1 is the graphicality of the immortal solution
outside of a large ball. For the next lemma we denote
$\Sigma^{+}=\Sigma\cap\\{x_{n+1}>0\\}$ and
$\Sigma^{-}=\Sigma\cap\\{x_{n+1}<0\\}$ for $\Sigma\subset\mathbb{R}^{n+1}$.

$\Pi_{s}$$(\mathcal{M}_{s}^{+})^{*}$$\mathcal{C}\times[0,\infty)$$\mathcal{M}_{s}^{-}$

Figure 1. A typical picture of the moving plane

######  Lemma 3.1.

Let $\mathcal{C},\\{\Sigma_{t}\\}_{t\in(0,T)}$ be as in 1.1. For each
$t\in[0,T)$ there is $R=R(\mathcal{C},\Sigma,t)$ such that
$(\Sigma_{t})^{+}\setminus B_{R}(0)$ is graphical over $\Pi_{0}\setminus
B_{R}(0)$, where $\Pi_{0}=\\{x_{n+1}=0\\}$; that is, the projection
$\pi:(\Sigma_{t})^{+}\setminus B_{R}(0)\to\Pi_{0}$ is injective. The same
holds for $(\Sigma_{t})^{-}$.

###### Proof.

Let $\eta>0$. Consider the MCF $\\{\Sigma_{t}\\}_{t\in(0,T)}$ together with
the initial data $\Sigma_{0}=\mathcal{C}$ and apply 2.1. Since the cone is
smooth this can be treated as an integral Brakke flow (with no sudden mass
loss). Take any point $x\in\mathcal{C}^{+}\setminus B_{1}(0)$. Since
$\mathcal{C}^{+}$ is clearly graphical over itself, by pseudolocality theorem
2.1 there exists $t_{1}>0$ such that $0<t<t_{1}$ implies $(\Sigma_{t})^{+}\cap
C_{\sqrt{t_{1}}}(x)$ is a normal graph over $B_{\sqrt{t_{1}}}(x)$ with
Lipschitz constant bounded above by $\eta$. Since $x$ is arbitrary, this shows
that $(\Sigma_{t})^{+}\setminus B_{1}(0)$ is a normal graph over
$\mathcal{C}^{+}\setminus B_{1}(0)$ for $0<t<t_{1}$ with Lipschitz constant
bounded above by $\eta$. For a general $t>t_{1}$ parabolic rescaling shows
shows that for $R=tt_{1}^{-1}$ we have that $(\Sigma_{t})^{+}\setminus
B_{R}(0)$ is a normal graph over $(\mathcal{C}^{+})\setminus B_{R}(0)$ with
Lipschitz constant bounded above by $\eta$.

The cone $\mathcal{C}^{+}$ is graphical over $\Pi_{0}$ with a fixed angle
$\theta<\frac{\pi}{2}$. Moreover we have proved that given $\eta>0$ there is
$R>0$ such that $(\Sigma_{t})^{+}\setminus B_{R}(0)$ is a normal graph over
$\mathcal{C}^{+}\setminus B_{R}(0)$, so by choosing $\eta$ small enough we
have that the angle between the tangent planes of $\mathcal{C}^{+}\setminus
B_{R}(0)$ and $(\Sigma_{t})^{+}\setminus B_{R}(0)$ are as small as we want.
Hence $(\Sigma_{t})^{+}\setminus B_{R}(0)$ is graphical over $\Pi_{0}$ as
well. ∎

For the rest of the section, let

$\displaystyle\Pi_{s}=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\mid
x_{n+1}=s\\}\times[0,\infty)\subset\mathbb{R}^{n+1}\times[0,\infty)$

be the hyperplane at level $s$ in spacetime. Given a set
$A\subset\mathbb{R}^{n+1}\times[0,\infty)$ and $t,s\in[0,\infty)$ we let

$\displaystyle A^{t}=\\{(x,x_{n+1},t^{\prime})\in A\mid t^{\prime}=t\\}$

be the time $t$ slice of $A$,

$\displaystyle A_{s}^{+}=\\{(x,x_{n+1},t)\in A\mid x_{n+1}>s\\}$

be the part of $A$ lying above $\Pi_{s}$, $A_{s}^{-}$ be the part of $A$ lying
below $\Pi_{s}$ and finally

$\displaystyle A^{*}_{s}=\\{(x,x_{n+1},t)\mid(x,2s-x_{n+1},t)\in A\\}$

be the reflection of $A$ across $\Pi_{s}$, but we will often drop the
subscript $s$ when it is understood to avoid excessive subscripts. Given two
sets $A,B\subset\mathbb{R}^{n+1}\times[0,\infty)$ we say $A>B$ if for any
$(x,x_{n+1},t)\in A$ we have $x_{n+1}>y_{n+1}$ for any $(x,y_{n+1},t)\in B$
(if there is any such point).

###### Proof of 1.1.

Without loss of generality assume that $\mathcal{C}$’s axis of symmetry is the
$x_{1}$-axis. Evidently it suffices to show that the flow preserves the
reflection symmetry across any hyperplane containing the $x_{1}$-axis, which
without loss of generality we will take to be $\\{x_{n+1}=0\\}$.

We will use the moving plane method on the spacetime
$\mathbb{R}^{n+1}\times[0,\infty)$. The spacetime track

$\displaystyle\mathcal{M}=\bigcup_{t\in[0,T)}\Sigma_{t}\times\\{t\\}$

is a properly embedded hypersurface in $\mathbb{R}^{n+1}\times[0,T)$
asymptotic to $\mathcal{C}\times[0,T)$, in the sense that at each time slice
$t$, $\Sigma_{t}$ is $C^{2,\alpha}$-asymptotic to $\mathcal{C}$ as we have
demonstrated in 2.4. Let

$\displaystyle
S=\\{s\in[0,\infty)\mid(\mathcal{M}_{s}^{+})^{*}>\mathcal{M}_{s}^{-},(\mathcal{M}_{s}^{+})^{t}\text{
is graphical over $(\Pi_{s})^{t}$ for $t\in[0,T]$}\\}.$

Here by graphical we meant that $(\mathcal{M}_{s}^{+})^{t}$ can be written as
a normal graph over $(\Pi_{s})^{t}$. Alternatively, since our solution is
smooth we can require that the vertical vector $e_{n+1}=(0,\ldots,0,1)$ is not
contained in the tangent space of any point $p\in(\mathcal{M}_{s}^{+})^{t}$.
We first note that since the cone is symmetric across $\Pi_{0}$ we have
$(\mathcal{C}_{s}^{+})^{*}>\mathcal{C}_{s}^{-}$ for every $s>0$. It is not
hard to see that $S$ is an open set. In fact we just need to show that
$e_{n+1}$ is not in the tangent space at infinity for
$(\mathcal{M}_{s}^{+})^{t}$. By 2.4,

$\displaystyle\lim_{\rho\to 0^{+}}\rho(\mathcal{M}_{s}^{+})^{t}=\mathcal{C}$

in $C^{2,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n+1}\setminus\\{0\\})$, so
eventually the tangent space at a point $p\in(\mathcal{M}_{s}^{+})^{t}$ will
lie close to the tangent space of $\mathcal{C}$. Since the cone is not
vertical $e_{n+1}$ is clearly not contained in the tangent space of any point
on $\mathcal{C}$, so by the convergence there is $\varepsilon>0$ such that
$e_{n+1}$ is not in the tangent space of any point
$p\in(\mathcal{M}_{s-\varepsilon}^{+})^{t}$.

$\Pi_{s}$$(\mathcal{M}_{s}^{+})^{*}$$\mathcal{C}\times[0,\infty)$

Figure 2. Boundary touching

$\Pi_{s}$$(\mathcal{M}_{s}^{+})^{*}$$\mathcal{C}\times[0,\infty)$$\mathcal{M}_{s}^{-}$

Figure 3. Interior touching

By 3.1, there is $R>0$ such that $\Sigma_{1}\setminus
B_{R}(0)=\mathcal{M}^{1}\setminus B_{R}(0)$ is graphical over $(\Pi_{0})^{1}$.
Moreover this graphical scale scales parabolically, so for $s>T^{2}R$ we have
$(\mathcal{M}_{s}^{+})^{t}$ is graphical over $(\Pi_{s})^{t}$ for each
$t\in[0,T]$. It is also evident that $(\mathcal{M}_{s}^{+})^{t}$ is asymptotic
to the translated cone $\mathcal{C}+2se_{n+1}$, so when $s$ is large enough
the reflected part is disjoint from $(\mathcal{M}_{s}^{-})_{0}^{+}$ (that is,
the part of $\mathcal{M}$ that lies below level $s$ and above $0$). Together
with graphicality this implies $(\mathcal{M}_{s}^{+})^{*}>\mathcal{M}_{s}^{-}$
for sufficiently large $s$, so $S$ is not empty.

Finally we show $S$ is closed. Suppose for a contradiction that
$(s,\infty)\subset S$ (clearly $s\in S$ implies $[s,\infty)\subset S$) but
$s\not\in S$. At level $s$, either the graphicality condition or the set
comparison condition $(\mathcal{M}_{s})^{*}>\mathcal{M}_{s}^{-}$ is violated.
In the first case, by parabolic rescaling we may assume for simplicity that
the nongraphicality happens first at time $t=1$. This means that there is
$p\in(\mathcal{M}_{s}^{+})^{1}$ such that $e_{n+1}\in T_{p}(M_{s}^{+})^{1}$.
Thus tangent planes of $(\mathcal{M}_{s}^{+})^{*}$ and $\mathcal{M}_{s}^{-}$
at the point $(p,1)$ must coincide. If we choose $r$ small enough we can
ensure that $(\mathcal{M}_{s}^{+})^{*}$ and $\mathcal{M}_{s}^{-}$ are
graphical over $Z_{r}^{-}(p,1)\cap\\{x_{n+1}\leq s\\}$. Since the tangent
planes coincide we can apply Hopf Lemma 2.3 to $(\mathcal{M}_{s}^{+})^{*}$ and
$\mathcal{M}_{s}^{-}$ to conclude that these hypersurfaces agree on an open
neighborhood of $(p,1)$. Moreover, the set
$(\mathcal{M}_{s}^{+})^{*}\cap\mathcal{M}_{s}^{-}$ is closed by definition and
open by the maximum principle, so at least a connected component of
$(\mathcal{M}_{s}^{+})^{*}$ must coincide with a component of
$\mathcal{M}_{s}^{-}$. This implies that $(\mathcal{M}_{s}^{+})^{*}$ is
asymptotic to both the cones $\mathcal{C}\times[0,\infty)$ and
$(\mathcal{C}+2se_{n+1})\times[0,\infty)$, a contradiction. In the second
case, $s$ is necessarily the first level such that
$(\mathcal{M}_{s})^{+}\cap\mathcal{M}_{s}^{-}\neq\emptyset$, and the
graphicality condition implies that $(\mathcal{M}_{s}^{+})^{*}$ and
$\mathcal{M}_{s}^{-}$ must touch at an interior point $(p,t)$ of the flow.
Again for $r$ small enough they are both graphical solutions of the MCF, so
the maximum principle 2.2 implies that $(\mathcal{M}_{s}^{+})^{*}$ and
$\mathcal{M}_{s}^{-}$ agree on an open neighborhood of $(p,t)$. Since we can
do this for any point
$(p,t)\in(\mathcal{M}_{s}^{+})^{*}\cap\mathcal{M}_{s}^{-}$, at least a
connected component of $(\mathcal{M}_{s}^{+})^{*}$ coincides with a component
of $\mathcal{M}_{s}^{-}$, a contradiction.

We have thus proved that $S$ is open, non-empty and closed, and thus
$S=(0,\infty)$ (note that since we have a strict inequality in our set up so
we cannot conclude directly that $0\in S$). Note that we can run a similar
argument starting from the bottom half, yielding
$(\mathcal{M}_{s}^{-})^{*}>\mathcal{M}_{s}^{+}$ for any $s<0$. Hence there
must be a point of touching at $s=0$. If the intersection is in the interior
we can apply the maximum principle 2.2 to conclude that
$(\mathcal{M}_{0}^{+})^{*}=\mathcal{M}_{0}^{-}$, i.e. $\mathcal{M}$ is
symmetric across the reflection with respect to $\Pi_{0}$. The same conclusion
holds if the intersection is along the boundary by using the Hopf lemma 2.3
instead. ∎

######  Remark.

In the proof of 1.1 we actually proved the stronger result that reflection
symmetry is preserved for self-expanders asymptotic to a double cone. This
yields, for example, that when $m_{1}=m_{2}$ in Equation 1.1, any self-
expander asymptotic to $\mathcal{C}$ is symmetric across the reflection with
respect to the plane $\\{x_{1}=0\\}$. We do not expect this to hold for cones
whose links have more than 2 components.

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# Lifetime predictions for virgin and recycled high-density polyethylene under
creep conditions

A.D. Drozdov,111E–mail<EMAIL_ADDRESS>R. Høj Jermiin, and J.
deClaville Christiansen
Department of Materials and Production
Aalborg University
Fibigerstraede 16, Aalborg 9220, Denmark

###### Abstract

Recycling has become a predominant subject in industry and science due to a
rising concern for the environment driven by high production volume of
plastics. Replacement of virgin polymers with their recycled analogs is not
always possible because recyclates cannot met the same property profiles as
their virgin counterparts. To avoid deterioration of the mechanical
properties, it is proposed to replace a virgin polymer with a recycled polymer
of another grade whose characteristics (measured in tensile tests) are close
to those of the virgin material. This approach opens a way for the use of
recycled polymers in short-term application, but its suitability for long-term
applications has not yet been assessed. A thorough experimental investigation
is conducted of the mechanical response of virgin high-density polyethylene
(HDPE) used for insulation of pipes and recycled HDPE manufactured from post-
consumer plastic waste (their stiffness, strength and elongation to break
adopt similar values). A model is presented in viscoelastoplasticity of
semicrystalline polymers. Its parameters are determined by matching
experimental data in short-term relaxation and creep tests. The lifetime of
virgin and recycled HDPE under creep conditions is evaluated by means of
numerical simulation. It is shown that the stress–time to failure diagrams for
virgin and recycled HDPE practically coincide.

Key-words: High-density polyethylene; Secondary recycling; Creep rupture;
Lifetime assessment; Modeling.

##  1 Introduction

Although polymers have become an indispensable part of our life, an extreme
increase in their production has lead to a steadily growing proportion of
plastic waste going into landfill. Due to considerable environmental concerns,
recycling and reuse of polymers has become a predominant subject in industry
and science [1, 2, 3].

Four approaches to recycling of plastic wastes are conventionally
distinguished [4]:

  1. 1.

At primary recycling, clean, uncontaminated, single-type waste with a
controlled history is blended in-plant with virgin polymers.

  2. 2.

At secondary recycling, plastic waste is sorted, separated from contaminants
and other waste materials, cleaned, and compounded, before being mixed with
virgin polymers.

  3. 3.

Under tertiary recycling, solid plastic materials are converted into smaller
molecules by means of chemical treatment. These intermediates are used as feed
stocks for the production of new plastics.

  4. 4.

The quaternary recycling consists in clean incineration (thermal and/or
catalytic pyrolysis) to recover the energy content of plastic wastes.

This study focuses on the secondary recycling of high-density polyethylene
(HDPE) and application of recycled HDPE for production of pipes. Due to
degradation of post-consumer plastics during their service life and their
thermo-mechanical degradation induced by recycling processes [5, 6],
mechanical characteristics of recycled polymers, in particular, polyolefins,
are lower than those of their virgin analogs [7]. Although deterioration of
the mechanical properties observed in short-term tests on recycled HDPE is not
very pronounced compared to the virgin polymer [8, 9, 10], it leads to a
catastrophic reduction in the lifetime measured in long-term experiments. For
example, the lifetime under creep conditions is reduced by one to two orders
of magnitude [11, 12], the number of cycles to failure under fatigue
conditions decreases by an order of magnitude [13, 14], and time to failure
caused by crack propagation diminishes by one to three orders of magnitude
[15, 16, 17].

To ensure comparable long-term properties of virgin and recycled plastics, it
appears natural to replace a virgin HDPE with a recycled polymer (prepared
from a different waste feedstock) whose mechanical properties after recycling
are close to those of the virgin material. This approach has recently been
proposed in [18, 19] where its environmental and economical advantages are
discussed. The objective of this study is to demonstrate that HDPE samples
manufactured from virgin and recycled polymers with similar mechanical
characteristics (the Young modulus, yield stress, and strain at break) have
practically identical lifetimes under creep conditions (which guarantees that
they withstand 50 years of service life under tensile creep with the same
stresses). This result is rather unexpected because creep resistance of solid
polymers is determined to a large extent by their viscoelastic and
viscoplastic properties (which differ noticeably for the virgin and recycled
HDPE).

For the analysis, two grades of high-density polyethylenes were chosen: a
virgin HDPE used for insulation of pipes for district heating and a recycled
HDPE manufactured from post-consumer household plastic waste. The similarity
of their mechanical responses was confirmed by observations in uniaxial
tensile tests with a constant cross-head speed. A thorough investigation of
the viscoplastic and viscoplastic properties of these polymers was conducted
in short-term tensile relaxation tests with various strains and creep tests
with various stresses. Predictions of their lifetime in long-term uniaxial
creep tests was performed with the help of the model recently developed in
[20] and validated by comparison of observations in short-term tests with
results of simulation.

The exposition is organized as follows. Experimental data on recycled HDPE in
mechanical tests are reported in Section 2, where they are compared with
observations on virgin HDPE presented in [20]. A brief description of the
model in viscoelastoplasticity of semicrystalline polymers is provided in
Section 3. Fitting of experimental data is conducted in Section 4.1.
Validation of the model is performed in Section 4.2 where results of
simulation are compared with observations. In Section 4.3, the stress–time to
failure diagrams are discussed for virgin and recycled HDPE under tensile
creep conditions. Concluding remarks are formulated in Section 5.

##  2 Materials and Methods

As a virgin material, we chose HDPE Borsafe HE3490-LS purchased from Borealis.
It has density 959 kg/m3, melt flow rate 0.25 g/10 min, and melting
temperature 130 ∘C. For comparison, we used high-density polyethylene PEHD-R-
E-GREY supplied by Aage Vestergaard Larsen A/S (Denmark). This polymer is
manufactured from post-consumer household plastic waste, and it has density
960 kg/m3, melt flow rate 1.2 g/10 min, and melting temperature 130 ∘C.

Dumbbell-shaped specimens (ISO 527-2-1B) with the total length 145 mm, the
gauge length 65 mm, and the cross sectional area 9.81 mm $\times$ 3.95 mm were
prepared by using injection-moulding machine Ferromatik Milacron K110.

Uniaxial tensile tests were performed by means of the testing machine Instron
5568 equipped with an extensometer and a 5 kN load cell. Each test was
repeated at least by twice.

Three series of tests were performed: (i) tensile tests with cross-head speeds
$\dot{d}=50$ mm/min up to breakage of samples, (ii) short-term stress
relaxation tests (with the duration of 30 min) with strains $\epsilon$ ranging
from 0.01 and 0.05, and (iii) short-term creep tests (with the durations up to
6 h) with various stresses $\sigma$ belonging to the interval between 9 and
20.5 MPa. All experiments were conducted at room temperature ($T=21$ ∘C) in a
climate-controlled room.

###  2.1 Tensile tests

Uniaxial tensile tests on virgin and recycled HDPE were conducted with cross-
head speed $\dot{d}=50$ (which corresponded to the strain rate
$\dot{\epsilon}=0.0088$ s-1) until breakage of samples. This cross-head speed
is conventionally used for quality assessment of HDPE pipes under quasi-static
loading [21]. The tests were repeated five times on samples of virgin HDPE and
six times on samples of recycled HDPE. The specimens were prepared by
injection molding under the same conditions.

01020304050$\epsilon$%0.010.020.030.0$\sigma$MPaA

01020304050$\epsilon$%0.010.020.030.0$\sigma$MPaB

Figure 1: Tensile stress $\sigma$ versus tensile strain $\epsilon$. Circles:
experimental data in tensile tests with strain rate 0.0088 s-1 on virgin (A)
and recycled (B) HDPE.

Experimental data on virgin and recycled HDPE are depicted in Figure 1, where
the engineering tensile stress $\sigma$ is plotted versus the engineering
strain $\epsilon$. For both polymers under consideration, the stress $\sigma$
increases with strain $\epsilon$ below the yield point $\epsilon_{\rm y}$,
reaches its maximum (the yield stress $\sigma_{\rm y}$) at $\epsilon_{\rm y}$,
and decreases slightly afterwards. The strain at break $\epsilon_{\rm b}$ is
calculated as the maximum strain along the stress–strain diagram. Figure 1
reveals good repeatability and reliability of measurements. For example, the
standard deviation $\delta_{E}$ of the Young’s modulus $E$ and the standard
deviation $\delta_{\sigma_{\rm y}}$ of the yield stress $\sigma_{\rm y}$ do
not exceed 1% of their average values $\bar{E}$ and $\bar{\sigma}_{\rm y}$,
respectively. The standard deviation $\delta_{\epsilon_{\rm b}}$ of the strain
at break $\epsilon_{\rm b}$ is relatively large, which may be explained by
significant toughness of samples. The average values and the standard
deviations of parameters $E$, $\sigma_{\rm y}$ and $\epsilon_{\rm b}$ are
collected in Table 1. This table shows that the stiffness (characterized by
$\bar{E}$) of the recycled polymer exceeds that of the virgin HDPE by 35%,
their strengths (estimated by $\bar{\sigma}_{\rm y}$) coincide practically,
whereas the toughness of the recycled HDPE (characterized by
$\bar{\epsilon}_{\rm b}$) is lower than that of virgin HDPE by 25%.

Table 1: Material parameters for virgin and recycled HDPE determined in
tensile tests.

|  $\bar{E}$ GPa |  $\delta_{E}$ GPa |  $\bar{\sigma}_{\rm y}$ MPa |  $\delta_{\sigma_{\rm y}}$ MPa | $\bar{\epsilon}_{\rm b}$ | $\delta_{\epsilon_{\rm b}}$
---|---|---|---|---|---|---
Virgin | 0.966 | 0.072 | 25.429 | 0.216 | 0.397 | 0.180
Recycled | 1.305 | 0.014 | 26.166 | 0.224 | 0.297 | 0.073

###  2.2 Relaxation tests

Short-term stress relaxation tests on recycled HDPE were performed at strains
$\epsilon_{0}=0.01$, 0.02 and 0.05. Tests were repeated by twice on different
specimens manufactured under the same conditions. In each test, a specimen was
stretched with a cross-head speed of 50 mm/min up to the required strain
$\epsilon_{0}$. Then, the strain was fixed, and the decay in stress $\sigma$
was measured as a function of relaxation time $t_{\rm rel}=t-t_{0}$, where
$t_{0}$ stands for the instant when the strain $\epsilon$ reaches its ultimate
value $\epsilon_{0}$.

-101234$\log t_{\rm rel}$s0.05.010.015.020.025.0$\sigma$MPa Figure 2: Tensile stress $\sigma$ versus relaxation time $t_{\rm rel}$. Circles: experimental data in relaxation tests with strains $\epsilon_{0}=0.01$ (blue and green), $\epsilon_{0}=0.02$ (lime and yellow), and $\epsilon_{0}=0.05$ (orange and red). Solid lines: results of numerical analysis.

Experimental data in relaxation tests on recycled HDPE are reported in Fig. 2
together with their fits by the model. In this figure, the engineering stress
$\sigma$ is plotted versus the logarithm ($\log=\log_{10}$) of relaxation time
$t_{\rm rel}$. Circles stand for the experimental data, and solid lines denote
their fits by the model described in Section 3.

###  2.3 Creep tests

Two series of uniaxial creep tests on recycled HDPE were performed with “low”
($\sigma=9.0$, 10.0, 12.0, and 14.5 MPa) and “high” ($\sigma=17.5$, 18.0,
19.0, 20.0 and 20.5 MPa) tensile stresses. Each test was repeated by twice on
specimens prepared under the same condition. In a creep test, a sample was
stretched up to the required stress $\sigma$ with a cross-head speed of 50
mm/min. Then, the stress was fixed, and an increase in tensile strain
$\epsilon$ was measured as a function of creep time $t_{\rm cr}=t-t_{0}$,
where $t_{0}$ stands for the instant when the required stress $\sigma$ is
reached. The duration of creep tests with low stresses (6 h) was fixed. Creep
tests with high stresses were conducted until breakage of samples.

0246$t_{\rm cr}$h0.00.040.080.120.160.20$\epsilon$ Figure 3: Tensile strain
$\epsilon$ versus creep time $t_{\rm cr}$. Circles: experimental data in creep
tests with tensile stresses $\sigma=9.0$ (blue), 10.0 (green), 12.0 (orange)
and 14.5 (red) MPa. Solid lines: prediction of the model in
viscoelastoplasticity. 0123$t_{\rm cr}$h0.00.20.40.6$\epsilon$ Figure 4:
Tensile strain $\epsilon$ versus creep time $t_{\rm cr}$. Circles:
experimental data in creep tests with tensile stresses $\sigma=17.5$ (blue),
18.0 (green), 19.0 (orange), 20.0 (red) and 20.5 (brown) MPa. Solid lines:
prediction of the model in viscoelastoplasticity.

Experimental data in creep tests on recycled HDPE are reported in Figures 3
(low stresses) and 4 (high stresses). In these figures, tensile strain
$\epsilon$ is plotted versus creep time $t_{\rm cr}$. For each stress
$\sigma$, circles denote the mean (over observations on two samples) values of
tensile strain, and the solid line stands for their approximation by the model
described in Section 3.

To assess the repeatability of measurements and the accuracy of their fitting,
experimental data in creep tests with two highest stresses ($\sigma=20.0$ and
20.5 MPa) are reported in Figure 5 together with results of numerical
simulation. This figure reveals good reproducibility of observations. Figures
4 and 5 show that for all stresses under consideration, transition to the
stage of tertiary creep (characterized by a rapid growth of tensile strain
induced by damage of samples) occurs at the critical strain $\epsilon_{\ast}$
close to 0.4.

0.00.10.20.30.40.5$t_{\rm cr}$h0.00.20.40.6$\epsilon$ Figure 5: Tensile strain
$\epsilon$ versus creep time $t_{\rm cr}$. Circles: experimental data in creep
tests with tensile stresses $\sigma=20.0$ (blue, green) and $\sigma=20.5$
(yellow, red) MPa. Solid lines: prediction of the model in
viscoelastoplasticity.

Experimental data in relaxation and creep tests on virgin HDPE are reported in
[20]. Comparison of observations on virgin and recycled HDPE shows
similarities of their time-dependent behavior. A detailed discussion of
differences between their responses is provided in Section 4.

##  3 Model

The viscoelastoplastic response of HDPE under isothermal deformation with
small strains is described by a constitutive model based on the following
assumptions.

HDPE is a semicrystalline polymer consisting of two (amorphous and
crystalline) phases (for simplicity, the presence of the rigid amorphous phase
and inter-phases is disregarded [22]). The viscoplastic deformation of the
amorphous phase describes (i) chain slip through the crystals, (ii) sliding of
tie chains along and their detachment from lamellar blocks, and (iii)
detachment of chain folds and loops from surfaces of crystal blocks [23]. The
viscoplastic deformation in the crystalline phase reflects (i) inter-lamellar
separation, (ii) rotation and twist of lamellae, and (iii) fine and coarse
slip of lamellar blocks [24]. The viscoelastic response is associated with (i)
relaxation of stresses in chains located in the amorphous regions and (ii)
time-dependent decay in forces transmitted to the crystalline skeleton by tie
chains [25].

A detailed account for the evolution of micro-structure in semicrystalline
polymers under loading leads to a strong increase in the number of adjustable
parameters. To make the model tractable, we treat HDPE as an equivalent non-
affine network of polymer chains connected by permanent and temporary
junctions [26]. The non-affinity of the network means that junctions between
chains can slide with respect to their reference positions under deformation.
Keeping in mind that the sliding process reflects viscoplastic deformations in
the amorphous and crystalline regions, the strain tensor for plastic
deformation $\bm{\epsilon}_{\rm p}$ consists of two components

$\bm{\epsilon}_{\rm p}=\bm{\epsilon}_{\rm pa}+\bm{\epsilon}_{\rm pc},$

where the strain tensors $\bm{\epsilon}_{\rm pa}$ and $\bm{\epsilon}_{\rm pc}$
describe viscoplastic deformations in the amorphous phase and the crystalline
skeleton, respectively. A similar decomposition of the plastic strain into two
components was introduced in [27] to describe the viscoplastic deformation of
semicrystalline polymers at finite strains.

Chains in the polymer network are bridged by permanent and temporary bonds.
Both ends of a permanent chain are merged with the network by permanent bonds.
At least one end of a temporary chain is connected with the network by a
temporary (physical) bond that can break and reform. When both ends of a
temporary chain are attached to the network, the chain is in its active state.
When an end of an active chain separates from its junction at some instant
$\tau_{1}$, the chain is transformed into the dangling state. When the free
end of a dangling chain merges with the network at an instant
$\tau_{2}>\tau_{1}$, the chain returns into the active state. Attachment and
detachment events occur at random times being driven by thermal fluctuations
[28].

The network consist of meso-regions with various activation energies for
breakage of temporary bonds. The rate of separation of an active chain from
its junction in a meso-domain with activation energy $u$ is governed by the
Eyring equation

$\Gamma=\gamma\exp\Bigl{(}-\frac{u}{k_{\rm B}T}\Bigr{)},$

where $\gamma$ stands for the attempt rate, $T$ is the absolute temperature,
and $k_{\rm B}$ denotes the Boltzmann constant. For isothermal processes at a
fixed temperature $T$, we introduce the dimensionless energy $v=u/(k_{\rm
B}T)$ and find that

$\Gamma=\gamma\exp(-v).$ (1)

Distribution of meso-domains with various activation energies $v$ is described
by the quasi-Gaussian formula

$f(v)=f_{0}\exp\Bigl{(}-\frac{v^{2}}{2\Sigma^{2}}\Bigr{)}\quad(v\geq 0),$ (2)

where $f$ stands for the distribution function. This function is characterized
by the only parameter $\Sigma>0$ that serves as a measure of inhomogeneity of
a polymer network [25]. The pre-factor $f_{0}$ is determined from the
normalization condition

$\int_{0}^{\infty}f(v){\rm d}v=1.$ (3)

According to Equation (1), the average rate of rearrangement of temporary
bonds in a transient network reads

$\bar{\Gamma}=\gamma\int_{0}^{\infty}\exp(-v)f(v){\rm d}v.$ (4)

Focusing on experimental data in uniaxial tests, we disregard volume
deformation and treat of HDPE as an incompressible medium (according to [29,
30], the Poisson ratio of polyethylene $\nu$ belongs to the interval between
0.48 and 0.49).

Constitutive equations for a semicrystaline polymer under an arbitrary three-
dimensional deformation with small strains were derived in [20] by means of
the Clausius–Duhem inequality. Under uniaxial tensile relaxation with a fixed
strain $\epsilon_{0}$, the governing relation reads

$\sigma(t)=\sigma_{0}\Bigl{[}1-\kappa\int_{0}^{\infty}f(v)\Bigl{(}1-\exp(-\Gamma(v)t)\Bigr{)}{\rm
d}v\Bigr{]},$ (5)

where $\sigma(t)$ denotes tensile stress at an arbitrary instant $t\geq 0$,

$\sigma_{0}=E\epsilon_{0}$ (6)

stands for the stress at the initial instant $t=0$, and $\kappa$ denotes the
ratio of the number of active chains to the total number of (active and
permanent) chains per unit volume.

Given a strain $\epsilon_{0}$, Equations (5), (6) together with Equation (1)
for $\Gamma(v)$ and Equation (2) for $f(v)$ involve four material parameters:
(i) the Young’s modulus $E$, (ii) the rate of separation of active chains from
their junctions $\gamma$, (iii) the measure of inhomogeneity of the equivalent
network $\Sigma$, and (iv) the fraction of temporary bonds in the network
$\kappa$. These quantities are found in Section 4 by matching experimental
data on virgin and recycled HDPE.

Under uniaxial tensile creep with a fixed stress $\sigma$, the tensile strain
$\epsilon$ is given by

$\epsilon=\epsilon_{\rm e}+\epsilon_{\rm pa}+\epsilon_{\rm pc},$ (7)

where $\epsilon_{\rm e}$ stands for the strain caused by the viscoelastic
response (rearrangement of temporary bonds between polymer chains in the
network), and $\epsilon_{\rm pa}$, $\epsilon_{\rm pc}$ are the strains induced
by the viscoplastic flows (sliding of junctions between chains) in the
amorphous and crystalline regions, respectively.

The strain $\epsilon_{\rm e}(t)$ is determined by the formula

$\epsilon_{\rm e}(t)=\frac{\sigma}{E}+\kappa\int_{0}^{\infty}f(v)z(t,v){\rm
d}v,$ (8)

where the function $z(t,v)$ obeys the differential equation

$\frac{\partial z}{\partial t}=\Gamma(v)(\epsilon_{\rm e}-z),\qquad z(0,v)=0.$
(9)

The viscoplastic flow in the amorphous phase is governed by the equation

$\epsilon_{\rm pa}(t)=A\Bigl{[}1-\exp(-\alpha\sqrt{t})\Bigr{]},$ (10)

where $A$ and $\alpha$ are adjustable parameters. The coefficient $A$ stands
for the maximum viscoplastic strain induced by sliding of junctions between
chains, and $\alpha$ characterizes the rate of sliding of junctions with
respect to their initial positions. An analog of Equation (10) (with
$\sqrt{t}$ replaced with $t$) is conventionally used to describe at the
initial stage of flow of dislocations in crystalline materials [31].

The viscoplastic flow in the crystalline phase is described by the Norton
equation [32]

$\epsilon_{\rm pc}(t)=Bt,$ (11)

where $B$ is an adjustable parameter.

Given a stress $\sigma$, the viscoplastic flow under creep conditions is
determined by three material parameters: (i) the rate of sliding of junctions
in the amorphous phase with respect to their initial positions $\alpha$, (ii)
the maximum viscoplastic strain $A$ induced by sliding of junctions in the
amorphous phase, and (iii) the rate of plastic flow in the crystalline phase
$B$.

The influence of stress on the rate of sliding of junctions in amorphous
regions $\alpha$ is described the power-law relation

$\alpha=\alpha_{0}\qquad(\sigma<\sigma_{\ast}),\qquad\alpha=\alpha_{0}+\alpha_{1}(\sigma-\sigma_{\ast})^{m},$
(12)

where $\alpha_{0}$, $\alpha_{1}$ and $m$ are material parameters, and
$\sigma_{\ast}$ denotes the threshold stress below which the viscoplastic flow
is not observed.

The effect of stress $\sigma$ on the coefficient $A$ is determined by the
formula

$A=\frac{A_{1}}{1+\exp[a(\sigma-\sigma_{\ast})]}+A_{2},$ (13)

where $A_{1}$, $A_{2}$ and $a$ are material parameters. Equation (13) implies
that $A$ adopts different values $A_{1}+A_{2}$ and $A_{2}$ far below and far
above some threshold stress $\sigma_{\ast}$. Transition from the limiting
values occurs in the close vicinity of $\sigma_{\ast}$ only. The rate of
transition is characterized by the coefficient $a$.

The influence of stress $\sigma$ on the rate of viscoplastic flow in the
crystalline phase $B$ is governed by the equation

$B=0\qquad(\sigma<\sigma_{\ast}),\qquad
B=B_{1}(\sigma-\sigma_{\ast})^{n}\qquad(\sigma\geq\sigma_{\ast}),$ (14)

where $B_{1}$ and $n$ are adjustable parameters. The coefficient
$\sigma_{\ast}$ denotes the threshold stress below which the viscoplastic flow
in spherulites does not occur.

Equations (12) and (14) imply that sliding of junctions is weakly affected by
tensile stresses when these stresses remain relatively small, but the rate of
sliding increases pronouncedly when $\sigma$ exceeds its threshold value
$\sigma_{\ast}$.

##  4 Results and Discussion

###  4.1 Fitting of experimental data

Material parameters for recycled HDPE are determined by matching observations
presented in Figures 2 to 4.

####  4.1.1 Relaxation tests

We begin with the analysis of experimental data in relaxation tests. Each set
of data in Figure 2 is fitted separately by means of Equation (5), where
$\Gamma(v)$ is given by Equation (1), $f(v)$ is determined by Equation (2),
and $\sigma_{0}$ is found from Equation (6). Given $\gamma$ and $\Sigma$, the
coefficients $\sigma_{0}$ and $\kappa$ in Equation (5) are calculated by using
the least-squares technique. The parameters $\gamma$ and $\Sigma$ are
determined by the nonlinear regression method from the condition of minimum
for the expression

$\sum_{k}\Bigl{[}\sigma_{\rm exp}(t_{k})-\sigma_{\rm sim}(t_{k})\Bigr{]}^{2},$

where summation is performed over all instants $t_{k}$ at which the data are
reported, $\sigma_{\rm exp}$ denotes the tensile stress measured in the
corresponding relaxation test, and $\sigma_{\rm sim}$ is given by Equstion
(5).

Table 2: Material parameters for recycled HDPE (relaxation tests with various
strains $\epsilon_{0}$)

$\epsilon_{0}$ |  $\sigma_{0}$ MPa |  $\gamma$ s | $\Sigma$ | $\kappa$
---|---|---|---|---
0.01 | 12.99 | 1.5 | 5.6 | 0.810
0.02 | 16.91 | 1.1 | 5.6 | 0.770
0.05 | 22.37 | 0.9 | 5.7 | 0.711

Figure 2 demonstrates good agreement between the experimental data in
relaxation tests and their approximation of the model with the material
constants collected in Table 2. This table shows that the coefficients
$\gamma$, $\Sigma$ and $\kappa$ are weakly affected by tensile strain
$\epsilon_{0}$ ($\gamma$ and $\kappa$ are reduced slightly with
$\epsilon_{0}$, whereas $\Sigma$ remains practically constant). In what
follows, changes in these coefficients with $\epsilon_{0}$ are disregarded.
Taking the value $\sigma_{0}$ at $\epsilon_{0}=0.01$ from Table 2 and using
Equation (6), we calculate the Young’s modulus $E=1.299$ GPa. This value is
very close to the value $\bar{E}=1.305$ GPa found by approximation of the
stress–strain curves under tensile deformation (Figure 1) and reported in
Table 1.

Fitting observations in relaxation tests on virgin HDPE was performed in [20].
For comparison, the best-fit values of $E$, $\gamma$, $\Sigma$, and $\kappa$
for virgin and recycled HDPE are collected in Table 3.

Table 3: Material parameters for virgin and recycled HDPE (relaxation tests
with strain $\epsilon_{0}=0.01$)

HDPE |  $E$ GPa |  $\gamma$ s | $\Sigma$ | $\kappa$
---|---|---|---|---
Virgin | 0.947 | 2.1 | 7.2 | 0.862
Recycled | 1.299 | 1.5 | 5.6 | 0.810

Table 3 shows that all parameters characterizing the linear viscoelastic
behavior of virgin and recycled HDPE adopt similar values. The Young’s modulus
$E$ of the recycled HDPE is higher (by 37%), whereas its attempt rate
$\gamma$, measure of inhomogeneity of the temporary network $\Sigma$, and
fraction of temporary junctions in the network $\kappa$ are lower (by 29, 22,
and 6 %, respectively). This difference may be explained by a higher degree of
crystallinity of the recycled HDPE [33, 34], as well as a higher degree of
long-chain branching in this polymer caused by its degradation under in-
service conditions and recycling [35]. The average rates of rearrangement of
temporary bonds $\bar{\Gamma}$ (determined by Equation (4)) coincide
practically for virgin and recycled HDPE and read $\bar{\Gamma}=0.255$ and
$\bar{\Gamma}=0.231$ s-1, respectively.

####  4.1.2 Creep tests

We now approximate the experimental creep diagrams on recycled HDPE depicted
in Figures 3 to 5. Each set of data is matched separately by using the
following algorithm.

91215182124$\sigma$MPa0.010.020.030.040.0$\alpha$A

91215182124$\sigma$MPa0.010.020.030.040.0$\alpha$B

Figure 6: Parameter $\alpha$ versus tensile stress $\sigma$. Circles:
treatment of experimental data on virgin (A) and recycled (B) HDPE. Solid
lines: results of numerical analysis.

91215182124$\sigma$MPa0.00.020.040.06$A$A

91215182124$\sigma$MPa0.00.020.040.06$A$B

Figure 7: Parameter $A$ versus tensile stress $\sigma$. Circles: treatment of
experimental data on virgin (A) and recycled (B) HDPE. Solid lines: results of
numerical analysis.

912151824$\sigma$MPa0.00.51.01.52.0$B$A

912151824$\sigma$MPa0.00.51.01.52.0$B$B

Figure 8: Parameter $B$ versus tensile stress $\sigma$. Circles: treatment of
experimental data on virgin (A) and recycled (B) HDPE. Solid lines: results of
numerical analysis.

Given a stress $\sigma$, the viscoelastic strain $\epsilon_{\rm e}$ is
determined from Equations (8) and (9). These equations are integrated
numerically by the Runge–Kutta method with the material parameters reported in
Table 3. Afterwards, the viscoplastic strain $\epsilon_{\rm p}=\epsilon_{\rm
pa}+\epsilon_{\rm pc}$ is determined from Equations (10) and (11). Given
$\alpha$, the coefficients $A$ and $B$ are found by the least-squares
technique. The parameter $\alpha$ is calculated by the nonlinear regression
method from the condition of minimum for the expression

$\sum_{k}\Bigl{[}\epsilon_{\rm p\;exp}(t_{k})-\epsilon_{\rm
p\;sim}(t_{k})\Bigr{]}^{2},$

where summation is performed over all instants $t_{k}$ at which the data are
reported, $\epsilon_{\rm p\;exp}$ is determined from Equation (7), and
$\epsilon_{\rm p\;sim}$ is found from Equations (10) and (11).

Figures 3 and 4 demonstrate good agreement between results of numerical
analysis and the experimental creep diagrams along the intervals of primary
and secondary creep. Slight deviations between the data and results of
simulation are observed along the intervals of tertiary creep in Figures 4 and
5 only. These discrepancies arise because the model does not account for
damage accumulation at the final stage of creep flow.

The coefficients $\alpha$, $A$, and $B$ for recycled HDPE are plotted versus
tensile stress $\sigma$ in Figures 6 to 8. For comparison, we present also the
corresponding dependencies for virgin HDPE obtained in [20]. The data are
approximated by Equations (12)–(14) with the adjustable parameters reported in
Table 4. The coefficients $\alpha_{0}$, $\alpha_{1}$, $A_{1}$ and $B_{1}$ are
calculated by the least-squares technique. The parameters $m$, $n$ and $a$ are
determined by the nonlinear regression method.

Table 4: Material parameters for virgin and recycled HDPE (creep tests with
various stresses $\sigma$)

Coefficient $\alpha$ |  |  |  |
---|---|---|---|---
HDPE | $\alpha_{0}$ | $\alpha_{1}$ |  $\sigma_{\ast}$ MPa | $m$
Virgin | 1.982 | $8.08\cdot 10^{-7}$ | 9.0 | 7.0
Recycled | 0.0 | $2.34\cdot 10^{-3}$ | 9.2 | 4.0
Coefficient $A$ |  |  |  |
HDPE | $A_{0}$ | $A_{1}$ |  $\sigma_{\ast}$ MPa |  $a$ MPa-1
Virgin | $4.67\cdot 10^{-2}$ | $-3.90\cdot 10^{-2}$ | 13.5 | 1.1
Recycled | $2.90\cdot 10^{-2}$ | $2.01\cdot 10^{-2}$ | 16.0 | 4.0
Coefficient $B$ |  |  |  |
HDPE | $B_{1}$ |  $\sigma_{\ast}$ MPa | $n$ |
Virgin | $3.51\cdot 10^{-9}$ | 9.0 | 8.0 |
Recycled | $1.28\cdot 10^{-7}$ | 9.0 | 6.6 |

Figures 6 and 8 demonstrate similar effects of tensile stress $\sigma$ on the
rates $\alpha$ and $B$ of the viscoplastic flow in amorphous and crystalline
regions. It should be noted, however, that the growth of $\alpha$ and $B$ with
$\sigma$ is stronger (the quantities $m$ and $n$ are higher) in virgin HDPE
compared with recycled HDPE.

Figure 7 shows a difference between the effects of stress $\sigma$ on the
coefficient $A$ for virgin and recycled HDPE. The parameter $A$ increases with
$\sigma$ in the virgin polymer and decreases with tensile stress in the
recycled HDPE. However, the stress-induced changes in $A$ are not substantial.
This coefficient varies in the interval between 0.01 and 0.05 for virgin HDPE
and in the interval between 0.03 and 0.05 for its recycled analog.

###  4.2 Validation of the model

To predict time to failure $t_{\rm f}$ of recycled HDPE in medium-term creep
tests (with the duration of several days), we integrate Equations (8) and (9)
with the parameters $E$, $\gamma$, $\Sigma$ and $\kappa$ reported in Table 3
and the parameters $\alpha$, $A$ and $B$ determined by Equations (12)–(14)
with the coefficients listed in Table 4. For each stress $\sigma$, evolution
of tensile strain $\epsilon$ with time $t$ is determined by Equation (7). Time
to failure $t_{\rm f}$ is calculated from the condition

$\epsilon(t_{\rm f})=\epsilon_{\ast},$ (15)

where the critical strain $\epsilon_{\ast}=0.4$ is found from Figures 4 and 5
(it corresponds to transition to the tertiary creep in short-term creep
tests). Equation (15) is based on the conventional assumption that the
duration of the interval of tertiary creep is negligible compared with the
durations of intervals of primary and secondary creep.

$-1$012$\log t_{\rm
f}$h0.05.010.015.020.025.0$\sigma$MPa$\star$$\star$$\star$$\star$$\star$$\star$$\star$$\star$$\star$$\star$
Figure 9: Tensile stress $\sigma$ versus time-to-failure $t_{\rm f}$ for
recycled HDPE under creep conditions. Asterisks: experimental data in creep
tests with various stresses $\sigma$. Circles: predictions of the model.

Results of numerical analysis are reported in Figure 9 where the stress
$\sigma$ is plotted versus time to failure $t_{\rm f}$. Results of simulation
(circles) and presented together with the experimental data (asterisks) in
creep tests with stresses $\sigma=17.5$, 18.0, 19.0, 20.0 and 20.5 MPa (Figure
4). Figure 9 confirms the ability of the model to predict the stress-time to
failure diagrams on recycled HDPE in medium-term creep tests.

###  4.3 Lifetimes of virgin and recycled HDPE

To compare the lifetimes of virgin and recycled HDPE under creep conditions,
their times to failure $t_{\rm f}$ in long-term creep tests (with the duration
up to 100 years) are calculated numerically. For this purpose, Equations (8)
and (9) are integrated with the parameters $E$, $\gamma$, $\Sigma$ and
$\kappa$ listed in Table 3 and the parameters $\alpha$, $A$ and $B$ determined
by Equations (12)–(14) with the coefficients collected in Table 4. For each
polymer under consideration, time to failure $t_{\rm f}$ is found from
condition (15).

$-1$0123456$\log t_{\rm f}$h0.05.010.015.020.025.0$\sigma$MPa50y10y1y1m1d
Figure 10: Tensile stress $\sigma$ versus time-to-failure $t_{\rm f}$ under
creep conditions. Circles: predictions of the model with $\epsilon_{\rm
cr}=0.4$ for virgin (red) and recycled (blue) HDPE. Solid lines: approximation
of the data by Equation (16).

Results of simulation are depicted in Figure 10 where tensile stress $\sigma$
is plotted versus time-to failure $t_{\rm f}$ (d, m, and y stand for day,
month and year, respectively). In this figure, circles denote results of
numerical analysis, and solid lines provide their approximations by the power-
law equation [37, 38]

$\sigma=\sigma_{0}+\sigma_{1}\Bigl{(}\frac{t_{\rm f0}}{t_{\rm
f}}\Bigr{)}^{\delta}$ (16)

with the characteristic time $t_{\rm f0}=1$ h. The best-fit values of the
material constants $\sigma_{0}$, $\sigma_{1}$ and $\delta$ in Equation (16)
are collected in Table 5.

Figure 10 provides the main result of this study. It shows that virgin and
recycled HDPE with similar mechanical characteristics (the Young modulus,
yield stress, and strain at break) have practically identical lifetimes under
tensile creep conditions. Unlike previous observations that revealed a
pronounced decay in creep resistance when the same polymer was recycled [11,
12], our analysis shows that that creep resistance of the recycled material
(prepared from post-consumer plastic waste of a different grade of HDPE) is
similar to that of the virgin HDPE. Figure 10 demonstrates that specimens
withstand 50 years of creep under tensile stresses lower than 10.9 and 10.3
MPa for virgin and recycled HDPE, respectively. Replacement of the virgin HDPE
with its recycled analog leads to an insignificant reduction in the critical
stress by 5.5% only. This may be explained by a higher stiffness of the
recycled polymer (Table 3) and a less pronounced growth of the rates of
viscoplastic flow in amorphous (Figure 6) and crystalline (Figure 8) regions.

Table 5: Material parameters for virgin and recycled HDPE (long-term creep
tests with various stresses $\sigma$)

HDPE |  $\sigma_{0}$ MPa |  $\sigma_{1}$ MPa | $\delta$
---|---|---|---
Virgin | 9.609 | 9.108 | 0.129
Recycled | 9.241 | 9.075 | 0.155

Although the above conclusion opens a new area of engineering applications for
recycled polymers, it should be treated with caution. It is conventionally
presumed that the stress–time to failure diagrams for polyethylene pipes
consist of three intervals [39]. The model under consideration describes
adequately the lifetime along the first interval (which corresponds to ductile
failure of HDPE). The other intervals are associated with quasi-brittle
failure driven by slow growth of micro-cracks and brittle failure caused by
chemical degradation of this polymer [32]. Analysis of crack resistance of
virgin and recycled HDPE under various loading programs will be conducted in a
subsequent study.

##  5 Conclusions

Replacement of virgin polymers with their recycled analogs is attractive from
the economical and environmental standpoints, but it is not always possible
because of substantial deterioration of the mechanical properties of
recyclates induced by their degradation during the service life and recycling
process. Due to a pronounced reduction in resistance to creep, fatigue, and
crack growth, recycled polymers cannot met the same property profiles for
long-term performance applications as their virgin counterparts.

To avoid a noticeable decline in mechanical characteristics of recycled
materials, it was proposed in [18, 19] to replace a virgin polymer of a
special grade with its recycled analog (a post-consumer recyclate prepared
from another grade) whose properties after recycling (evaluated in tensile
tests) are close to those of the virgin material. This approach opens a way
for the use of recycled polymers in short-term application. However, its
suitability for long-term applications has not yet been evaluated.

A thorough experimental investigation is performed of the mechanical response
of virgin HDPE used for insulation of pipes for district heating and recycled
HDPE manufactured from post-consumer household plastic waste. The Young’s
modulus, yield stress, and elongation to break of these two polymers adopt
similar values (Table 1). Their viscoelastic and viscoplastic responses in
short-term tensile relaxation and creep tests are adequately described by the
same model presented in Section 3 (Figures 2 to 5). However, the material
parameters of virgin and recycled HDPE determined by fitting experimental data
in relaxation and creep tests differ noticeably (Tables 3 and 4).

The lifetime of virgin and recycled HDPE under creep conditions is evaluated
by means of numerical simulation of the governing equations with the material
parameters found by matching observations in short-term tests. Validation of
the model is performed by comparison of its predictions with experimental data
in short-term creep tests (Figure 9). The main result of the study is that the
stress–time to failure diagrams for virgin and recycled HDPE practically
coincide (Figure 10 and Table 5). Numerical analysis reveals that recycled
HDPE withstands 50 years of creep under tensile stress which is lower by only
5.5% than that for virgin HDPE.

### Acknowledgement

Financial support by Innovationsfonden (Innovation Fund Denmark, project
9091-00010B) is gratefully acknowledged.

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