# Equations of motion for cracked beams and shallow arches

Semion Gutman, Junhong Ha and Sudeok Shon  1 Department of Mathematics,
University of Oklahoma, Norman, Oklahoma 73019, USA, e-mail<EMAIL_ADDRESS>1School of Liberal Arts, Korea University of Technology and Education, Cheonan
31253, South Korea, e-mail<EMAIL_ADDRESS>3Department of Architectural
Engineering, Korea University of Technology and Education, Cheonan 31253,
South Korea, e-mail<EMAIL_ADDRESS>
###### Abstract.

Cracks in beams and shallow arches are modeled by massless rotational springs.
First, we introduce a specially designed linear operator that ”absorbs” the
boundary conditions at the cracks. Then the equations of motion are derived
from the first principles using the Extended Hamilton’s Principle, accounting
for non-conservative forces. The variational formulation of the equations is
stated in terms of the subdifferentials of the bending and axial potential
energies. The equations are given in their abstract (weak), as well as in
classical forms.

###### Key words and phrases:

Shallow arch, beam, subdifferential, cracks, Hamilton’s principle

###### 2010 Mathematics Subject Classification:

47J35, 35Q74, 35D30, 70G75

##  1\. Introduction

Modeling dynamic behavior of cracked beams and arches has important
engineering applications. The main goal of this paper is to develop a rigorous
mathematical framework for such problems.

The theory of uniform beams and shallow arches is well developed. An early
exposition can be found in [1]. More general models in the multidimensional
setting, and a literature survey are presented in [11]. A review for vibrating
beams is given in [16]. Motion of uniform arches and a related parameter
estimation problem are studied in [14]. These results are extended to point
loads in [12]. The existence of a compact, uniform attractor is established in
[13].

For a theory of cracked Bernoulli-Euler beams see [9]. A significant effort
has been directed at the vibration analysis of cracked beams. Representation
of a crack by a rotational spring has been proven to be accurate, and it is
often used, see [4, 5] and the extensive bibliography there. Determination of
the beam natural frequencies is discussed in [18, 19, 20]. S. Caddemi and his
colleagues have further developed the theory using energy functions in [5].
Substantial reviews of cracked elements are presented in [6, 10, 8].

The transverse motion of a beam or an arch is described by the function
$y(x,t),\,x\in[0,\pi],\,t\geq 0$, which represents the deformation of the
beam/arch measured from the $x$-axis. For definiteness, the boundary
conditions are of the hinged type

(1.1) $y(0,t)=y^{\prime\prime}(0,t)=0,\quad
y(\pi,t)=y^{\prime\prime}(\pi,t)=0,\quad t\in(0,T).$

Other types of boundary conditions, can be treated similarly.

Figure 1. Crack parameters.

A crack is fully described by its position along the axis, and the crack depth
ratio $\hat{\mu}$, as shown in Figure 1. According to the common practice in
the field, see [7], a crack is modeled by a massless rotational spring. The
spring flexibility $\theta=\theta(\hat{\mu})$ depends on the crack depth ratio
$\hat{\mu}$, and on whether the crack is one-sided or two-sided, open or
closed, and so on. The flexibility $\theta$ is equal to $0$ if there is no
crack, and it increases with the crack depth. Explicit expressions for the
functions $\theta(\hat{\mu})$ are provided in Section 3.

######  Remark 1.1.

The following discussion is applicable to both arches and beams, but to avoid
repetitions we will refer just to arches.

Suppose that there are $m$ cracks along the length of the arch, located at
$0<x_{1}<\dots<x_{m}<\pi$. For convenience, we denote $x_{0}=0$, and
$x_{m+1}=\pi$. Consequently, the cracked arch is modeled as a collection of
$m+1$ uniform arches over the intervals
$l_{i}=(x_{i-1},x_{i}),\,i=1,\dots,m+1$, as shown in Figure 2(b).

We consider only the transverse motion of the arch, so its position can be
described by the function $y=y(x,t)$, $0\leq x\leq\pi$, $t\geq 0$. The
boundary conditions at the cracks enforce the continuity of the displacement
field $y$, the bending moment $y^{\prime\prime}$, and the shear force
$y^{\prime\prime\prime}$. Condition
$y^{\prime}(x_{i}^{+},t)-y^{\prime}(x_{i}^{-},t)=\theta_{i}y^{\prime\prime}(x_{i}^{+},t)$
expresses the discontinuity of the arch slope at the $i$-th crack, where
$\theta_{i}=\theta(\hat{\mu}_{i})$, see Figure 2(b).

Figure 2. Beam or shallow arch: (a) uniform, (b) with two cracks.

To simplify the statement of the boundary conditions at the cracks, we
introduce the notion of the _jump_ $J[u](x)$ of a function $u=u(x)$ at any
$x\in(0,\pi)$, as follows

(1.2) $J[u](x)=u(x^{+})-u(x^{-}).$

With this notation the conditions at the cracks (joint conditions) are

(1.3) $J[y](x_{i},t)=0,\quad J[y^{\prime\prime}](x_{i},t)=0,\quad
J[y^{\prime\prime\prime}](x_{i},t)=0,$

and

(1.4) $J[y^{\prime}](x_{i},t)=\theta_{i}y^{\prime\prime}(x_{i}^{+},t),$

where $\theta_{i}=\theta(\hat{\mu}_{i})$, $i=1,2,\dots,m$ and $t\geq 0$. Note
that $y^{\prime\prime}(x_{i}^{+},t)=y^{\prime\prime}(x_{i}^{-},t)$ by (1.3).

In Section 2 we review our recent results from [15] on the variational setting
for cracked beams and arches. First, special Hilbert spaces $V,H_{0}^{1},H$
are defined satisfying

(1.5) $V\subset H_{0}^{1}\subset H\subset(H_{0}^{1})^{\prime}\subset
V^{\prime},$

with continuous and dense embeddings. These spaces are broad enough to contain
continuous functions with discontinuous derivatives at the joint points.

Then we introduce the operator $\mathcal{A}:V\to V^{\prime}$, by

(1.6)
$\langle\mathcal{A}u,v\rangle_{V}=\sum_{i=1}^{m+1}(u^{\prime\prime},v^{\prime\prime})_{i}+\sum_{i=1}^{m}\frac{1}{\theta}_{i}J[u^{\prime}](x_{i})J[v^{\prime}](x_{i}),$

for any $u,v\in V$, where
$J[u^{\prime}](x)=u^{\prime}(x^{+})-u^{\prime}(x^{-})$.

Our main result in [15] is that the solution $u$ of the equation
$\mathcal{A}u=f$ in $H$ satisfies the joint conditions, including (1.4). Thus
the operator $\mathcal{A}$ ”absorbs” the boundary conditions, as expected of
the weak formulation of the steady state problem. This result allows us to
mathematically justify the existence of the eigenvalues and the eigenfunctions
of $\mathcal{A}$.

In Section 3 we consider relevant physical quantities, including the potential
energy $U_{b}$, due to bending, and the potential energy $U_{a}$, due to the
axial force. This is the only section in the paper, that contains physical
variables. Their non-dimensional equivalents are used in all the other
sections.

Typically, equations of motion are derived from the Hamilton’s Principle
$\delta I=0$, which seeks the stationary paths of the action $I$. A closer
examination of this statement in the framework of the Hilbert spaces reveals
the importance of the subdifferential $\partial\phi$ of a convex lower-
semicontinuous function $\phi$. The potential energies $U_{b}$, and $U_{a}$
are two examples of such functions.

Section 4 presents a brief review of these concepts. One can view the
subdifferential, which can be multi-valued, as a generalization of the
derivative. For example, let $\phi(x)=|x|,\,x\in{\mathbb{R}}$. Then
$\partial\phi(x)=-1$, for $x<0$, $\partial\phi(x)=1$, for $x>0$, and
$\partial\phi(0)=[-1,1]$. Geometrically, it means that the ”tangent” line to
the graph of $\phi$ at $x=0$ can have any slope between $-1$ and $1$. We
conclude Section 4 by deriving various expressions for the subdifferentials
$\partial U_{b}$, and $\partial U_{a}$. In particular, we show that $\partial
U_{b}=\mathcal{A}$.

Section 5 uses the Extended Hamilton’s Principle, which accommodates non-
conservative forces, to derive the equations of motion. It contains our main
result: the abstract equation of motion for cracked beams and arches is

(1.7) $\ddot{y}+\partial U_{b}(y)+\partial U_{a}(y)+c_{d}\dot{y}=p,$

where $\dot{y},\ddot{y}$ denote the time derivatives.

For beams, it is assumed that the influence of the axial force can be
neglected. This case is considered in Section 6. The abstract equation for
cracked beams is

(1.8) $\ddot{y}+\mathcal{A}y+c_{d}\dot{y}=p.$

If the beam contains no cracks, then (1.8) becomes
$\ddot{y}+y^{\prime\prime\prime\prime}+c_{d}\dot{y}=p$, which is consistent
with the classical Euler-Bernoulli theory.

Our main result in Section 7 is that the ”classical” equation for a cracked
shallow arch is

$\displaystyle\ddot{y}+y^{\prime\prime\prime\prime}$
$\displaystyle-\frac{1}{\pi}\left(\beta+\frac{1}{2}\int_{0}^{L}|y^{\prime}(x,t)|^{2}\,dx\right)\left(y^{\prime\prime}+\sum_{i=1}^{m}\theta_{i}y^{\prime\prime}(x_{i},t)\delta(x-x_{i})\right)+c_{d}\dot{y}=p,$

where $\delta=\delta(x)$ is the delta function.

Motion in viscous media results in the additional term
$\mu\mathcal{A}\dot{y}$, $\mu>0$ in the governing equations. Such a case is
referred to as the strong damping motion. If the viscous effects are neglected
($\mu=0$), we have the weak damping case.

##  2\. Variational setting for cracked beams and arches

This section contains a brief review of our results from [15], to which we
refer for further details. Let $H$ be the Hilbert space

(2.1) $H=\bigoplus_{i=1}^{m+1}L^{2}(l_{i}).$

Let the inner product and the norm in $L^{2}(l_{i})$ be denoted by
$(\cdot,\cdot)_{i}$ and $|\cdot|_{i}$ correspondingly. The inner product and
the norm in $H$ are defined by

(2.2)
$(u,v)_{H}=\sum_{i=1}^{m+1}(u,v)_{i},\quad|u|^{2}_{H}=\sum_{i=1}^{m+1}|u|^{2}_{i}.$

Consider the Sobolev space $H^{2}(a,b)$ on a bounded interval
$(a,b)\subset{\mathbb{R}}$, and let $u\in H^{2}(a,b)$. Then $u,u^{\prime}$ are
continuous functions on $[a,b]$, up to a set of measure zero, and
$u^{\prime\prime}\in L^{2}(a,b)$. Therefore, for such $u$, we will always
assume that $u,u^{\prime}\in C[a,b]$.

Define the linear space

(2.3) $V=\left\\{u\in\bigoplus_{i=1}^{m+1}H^{2}(l_{i})\,:\,u(0)=u(\pi)=0,\
J[u](x_{i})=0,\ i=1,\dots,m\right\\}.$

We interpret $u\in V$ as a continuous function on $[0,\pi]$, such that
$u(0)=u(\pi)=0$, with $u^{\prime}\in L^{2}(0,\pi)$, i.e. $u\in
H_{0}^{1}(0,\pi)$. Furthermore, $u|_{l_{i}}\in H^{2}(l_{i})$, and
$u^{\prime}|_{l_{i}}\in C[x_{i-1},x_{i}]$ for $i=1,2,\dots,m+1$.

Define the inner product on $V$ by

(2.4)
$((u,v))_{V}=\sum_{i=1}^{m+1}(u^{\prime\prime},v^{\prime\prime})_{i}+\sum_{i=1}^{m}J[u^{\prime}](x_{i})J[v^{\prime}](x_{i}),\quad\text{for
any}\ u,v\in V,$

where
$(u^{\prime\prime},v^{\prime\prime})_{i}=\int_{l_{i}}u^{\prime\prime}(x)v^{\prime\prime}(x)\,dx$.

The corresponding norm in $V$ is

(2.5)
$\|u\|_{V}^{2}=\sum_{i=1}^{m+1}|u^{\prime\prime}|^{2}_{i}+\sum_{i=1}^{m}|J[u^{\prime}](x_{i})|^{2},\quad\text{for
any}\ u\in V,$

where $|\cdot|_{i}$ is the norm in $L^{2}(l_{i})$. It can be shown that $V$ is
a Hilbert space.

At this time we introduce the Hilbert space $H_{0}^{1}=H_{0}^{1}(0,\pi)$, with
the inner product and the norm given by

(2.6)
$(u,v)_{1}=(u^{\prime},v^{\prime})_{H},\quad\|u\|_{1}^{2}=|u^{\prime}|_{H}^{2},\quad
u,v\in H_{0}^{1}.$

The norm in $(H_{0}^{1})^{\prime}$ will be denoted by $\|\cdot\|_{-1}$. It can
be shown that the identity embedding $i:V\to H_{0}^{1}$ is linear, continuous,
with a dense range in $H_{0}^{1}$. Furthermore, it is compact.

This allows us to define the Gelfand triple $V\subset H\subset V^{\prime}$,
with the pairing $\langle\cdot,\cdot\rangle_{V}$ between $V$ and $V^{\prime}$
extending the inner product in $H$. This means that given $f\in
H=H^{\prime}\subset V^{\prime}$, and $v\in V$, we have $\langle
f,v\rangle_{V}=(f,v)_{H}$.

Also, we have

(2.7) $V\subset H_{0}^{1}\subset
H\subset\left(H_{0}^{1}\right)^{\prime}\subset V^{\prime},$

with dense embeddings. Furthermore, the embeddings $V\subset H_{0}^{1}\subset
H$ are compact.

Now we can introduce the operator $\mathcal{A}:V\to V^{\prime}$ that ”absorbs”
the junction boundary conditions. This operator is central to the variational
setting of problems for cracked beams and arches.

######  Definition 2.1.

Define the operator $\mathcal{A}$ on $V$ by

(2.8)
$\langle\mathcal{A}u,v\rangle_{V}=\sum_{i=1}^{m+1}(u^{\prime\prime},v^{\prime\prime})_{i}+\sum_{i=1}^{m}\frac{1}{\theta}_{i}J[u^{\prime}](x_{i})J[v^{\prime}](x_{i}),$

for any $u,v\in V$. We will also write $\langle\mathcal{A}u,v\rangle$ for
$\langle\mathcal{A}u,v\rangle_{V}$, if it does not cause a confusion.

Recall, that a linear operator $A:V\to V^{\prime}$ is called coercive, if
there exists $c>0$, such that $\langle Au,u\rangle\geq c\|u\|^{2}_{V}$ for any
$u\in V$. We have

######  Lemma 2.2.

Let $\mathcal{A}$ be defined by (2.8). Then $\mathcal{A}$ is a symmetric,
continuous, linear, and coercive operator from $V$ onto $V^{\prime}$.

As was mentioned in Section 1, functions $u=u(x)$ modeling an arch with cracks
are expected to satisfy certain boundary conditions. For convenience, we
restate them here:

(2.9) $u(0)=u(\pi)=0,\quad u^{\prime\prime}(0)=u^{\prime\prime}(\pi)=0,$

and

(2.10) $J[u](x_{i})=0,\quad J[u^{\prime\prime}](x_{i})=0,\quad
J[u^{\prime\prime\prime}](x_{i})=0,\quad
J[u^{\prime}](x_{i})=\theta_{i}u^{\prime\prime}(x_{i}^{+}),$

for $i=1,\dots,m$.

The next theorem is the main result of this section.

######  Theorem 2.3.

Let the domain of $\mathcal{A}$ be $D(\mathcal{A})=\\{v\in V:\mathcal{A}v\in
H\\}$.

  1. (i)

If $u\in D(\mathcal{A})$, then $u|_{l_{i}}\in H^{4}(l_{i})$,
$\mathcal{A}u=u^{\prime\prime\prime\prime}$ a.e. on $l_{i}$, $i=1,\dots,m+1$,
and $u$ satisfies conditions (2.9)–(2.10).

  2. (ii)

Let $f\in H$, then equation $\mathcal{A}u=f$ in $V^{\prime}$ has a unique
solution $u\in D(\mathcal{A})$.

Remark. The fact that $u^{\prime\prime\prime\prime}=f$ a.e. on $(0,\pi)$ in
Theorem 2.3 does not imply that $u\in H^{4}(0,\pi)$. This is similar to the
fact that the strong derivative $p^{\prime}$ of a step function $p$ on
$(0,\pi)$ is zero a.e. on $(0,\pi)$. However, $p\not\in H^{1}(0,\pi)$.

Finally in this section we discuss the eigenvalues and the eigenfunctions of
the operator $\mathcal{A}$. It was shown in Lemma 2.2 that $\mathcal{A}$ is a
continuous, linear, symmetric, and coercive operator from $V$ onto
$V^{\prime}$. Following [21, Section 2.2.1], $\mathcal{A}$ can also be
considered as an unbounded operator in $H$.

Since the embedding $V\subset H$ is compact, the standard spectral theory for
Sturm-Liouville boundary value problems is applicable. The eigenfunctions
belong to $H$. Therefore, by Theorem 2.3, they are in the domain
$D(\mathcal{A})\subset V$, thus continuous on $[0,\pi]$, and satisfy
conditions (2.9)–(2.10).

We summarize these results in the following lemma.

######  Lemma 2.4.

Let $\mathcal{A}$ be the operator defined in (2.8). Then

  1. (i)

There exists an increasing sequence of its real positive eigenvalues
$\lambda^{4}_{1},\lambda^{4}_{2},\dots$, with
$\lim_{k\to\infty}\lambda^{4}_{k}=\infty$.

  2. (ii)

The corresponding eigenfunctions $\varphi_{k}\in D(\mathcal{A})\subset V$,
$k\geq 1$, and they satisfy the junction conditions (2.9)–(2.10).

  3. (iii)

The eigenfunctions $\varphi_{k}$ satisfy
$\mathcal{A}\varphi_{k}=\lambda^{4}_{k}\varphi_{k}$ in $H$, $k\geq 1$. That
is, $\varphi_{k}^{\prime\prime\prime\prime}(x)=\lambda^{4}_{k}\varphi_{k}(x)$
a.e. on every interval $l_{i}$, $i=1,\dots,m+1$.

  4. (iv)

The set $\\{\varphi_{k}\\}_{k=1}^{\infty}$ is a complete orthonormal basis in
$H$.

An efficient method for a computational determination of the eigenvalues and
the eigenfunctions of $\mathcal{A}$ (Modified Shifrin’s method) is discussed
in [15].

##  3\. Physical parameters and their non-dimensional equivalents

This is the only section in the paper that uses physical variables. All the
other sections use their non-dimensional equivalents. Since both the physical
variables and their equivalents use the same notation, such an arrangement
helps to avoid confusing them. The physical variables are contained in Table 1
together with their units. Recall that the newton $N=kg\cdot m\cdot s^{-2}$ is
the unit of force.

$L$ | Length of bar | $m$ |
---|---|---|---
$T$ | Final time | $s$ |
$y$ | Bar deflection | $m$ | $y=y(x,t)$
$p$ | Transverse load/unit of length | $N\cdot m^{-1}$ | $p=p(x,t)$
$c_{d}$ | Damping coefficient | $kg\cdot m^{-1}s^{-1}$ |
$\kappa$ | Bar curvature | $m^{-1}$ |
$\rho$ | Volume mass density | $kg\cdot m^{-3}$ |
$E$ | Young’s modulus | $N\cdot m^{-2}$ |
$A$ | Cross-sectional area | $m^{2}$ |
$I$ | Area moment of inertia | $m^{4}$ |
$r$ | Radius of gyration | $m$ | $r=\sqrt{I/A}$
$\omega_{0}$ |  $t$-scale | $s^{-1}$ |
$\omega$ | Natural frequency | $s^{-1}$ |
$k$ | Spring stiffness | $N\cdot m/rad$ |
$\theta$ | Flexibility | $m$ | $\theta=EI/k$
$T_{k}$ | Kinetic energy | $J=N\cdot m$ |
$U_{a}$ | Potential energy for axial force | $J$ |
$U_{b}$ | Potential energy due to bending | $J$ |
$M$ | Bending moment magnitude | $N\cdot m$ |
$S_{0}$ | Initial tensile axial force | $N$ |
$S_{1}$ | Tensile force due to deflection | $N$ |
$N$ | Total axial force | $N$ | $N=S_{0}+S_{1}$
$\mathcal{L}$ | Lagrangian | $J$ | $\mathcal{L}=T_{k}-U_{b}-U_{a}$
$\mathcal{W}_{ext}$ | External work | $J$ |
$\mathcal{W}_{d}$ | Dissipative work | $J$ |
$\mathcal{W}_{nc}$ | Non-conservative work | $J$ | $\mathcal{W}_{nc}=\mathcal{W}_{d}+\mathcal{W}_{ext}$
$e_{I}$ | Energy normalization factor | $J$ |
$\beta$ | Axial force renormalization | $1$ |
$\hat{\mu}$ | Crack depth ratio | $1$ |
$\mu$ | Normalized dynamic viscosity | $1$ |
Table 1. Nomenclature

To use the Extended Hamilton’s Principle we need to derive the Lagrangian
$\mathcal{L}=T_{k}-U=T_{k}-U_{b}-U_{a}$, where $U_{b},U_{a}$ and $T_{k}$ are
the potential and the kinetic energies. Also, we need the non-conservative
external work $\mathcal{W}_{nc}$, which is the sum of the external work
$\mathcal{W}_{ext}$ due to the load $p$, and the dissipative work
$\mathcal{W}_{d}$ due to the damping force $\mathcal{F}_{d}$.

Let $y(x,t)$, $0\leq x\leq L$, be the position of the arch at the time $t\geq
0$. As usual, the dots denote the time derivatives, and the primes denote the
spatial ones. For the kinetic energy we have

(3.1) $T_{k}(y)=\frac{\rho A}{2}\int_{0}^{L}|\dot{y}(x,t)|^{2}\,dx.$

The external work $\mathcal{W}_{ext}$ by the non-conservative load $p(x,t)$ is

(3.2) $\mathcal{W}_{ext}(y)=\int_{0}^{L}p(x,t)y(x,t)\,dx.$

The dissipative force $\mathcal{F}_{d}(x,t)=-c_{d}\dot{y}(x,t)$, $c_{d}\geq 0$
is a uniformly distributed viscous damping force acting only in the transverse
direction. So, the dissipative work $\mathcal{W}_{d}$ done by the force is
given by

(3.3)
$\mathcal{W}_{d}(y)=\int_{0}^{L}\mathcal{F}_{d}(x,t)y(x,t)\,dx=-c_{d}\int_{0}^{L}\dot{y}(x,t)y(x,t)\,dx.$

The potential energy is

(3.4) $U(y)=U_{b}(y)+U_{a}(y),$

where $U_{b}$ is the potential energy due to the bending, and $U_{a}$ is the
potential energy due to the axial force. The particular form of these terms
depends on the presence of cracks and other factors.

To simplify the notations the $t$-dependency of $y$ is suppressed, if it does
not cause a confusion.

Beam. Suppose that we have a uniform beam with no cracks, as shown in Figure
2(a). The magnitude $M(x)$ of the bending moment vector $\vec{M}(x)$ of the
beam at any point $x\in(0,L)$ is given by $M(x)=EI\kappa(x)$, where
$\kappa(x)$ is the curvature of the beam at $x$. Assuming $|y^{\prime}|\ll 1$,
we get $\kappa(x)=y^{\prime\prime}(x)$, and $M(x)=EIy^{\prime\prime}(x)$.
Accordingly, the bending potential energy of the beam is given by

(3.5) $U_{b}(y)=\frac{EI}{2}\int_{0}^{L}(y^{\prime\prime})^{2}\,dx.$

Now suppose that the beam has cracks as in Section 1, and in Figure 2(b). The
cracks are at $0<x_{1}<\dots<x_{m}<L$. The standard approach to modeling a
crack is to represent it as a massless rotational spring with the spring
constant $k$, and the flexibility $\theta$.

The spring constant $k$ relates the torque ($N\cdot m$) to the angle of
rotation. In our case this relationship takes the form
$EIy^{\prime\prime}(x)=kJ[y^{\prime}](x)$, or $J[y^{\prime}](x)=\theta
y^{\prime\prime}(x)$, where

(3.6) $\theta=\frac{EI}{k}.$

Thus the unit of the flexibility $\theta$ is $m$.

If the beam has a rectangular cross-section, as shown in Figure 1, then the
area moment of inertia $I$ of the rectangle can be computed explicitly, and
(3.6) can be simplified further. If the crack is double-sided, then by [19,
Eq. (2.8)-(2.10)], the expression for the flexibility $\theta$ becomes

(3.7) $\theta=6\pi
H\hat{\mu}^{2}(0.535-0.929\hat{\mu}+3.500\hat{\mu}^{2}-3.181\hat{\mu}^{3}+5.793\hat{\mu}^{4}),$

where $H$ is the half-height of the beam cross-section, and $\hat{\mu}=a/H$.

If the crack is single-sided, then by [19, Eq. (2.8)-(2.10)]

(3.8) $\theta=6\pi
H\hat{\mu}^{2}(0.6384-1.035\hat{\mu}+3.7201\hat{\mu}^{2}-5.1773\hat{\mu}^{3}+7.553\hat{\mu}^{4}-7.332\hat{\mu}^{5}),$

where $H$ is the entire height of the beam cross-section, and $\hat{\mu}=a/H$.

The potential energy of the rotational spring with the spring constant $k$ is

$U_{crack}=\frac{1}{2}k\alpha^{2},$

where $\alpha$ is the angle of twist from its equilibrium position in radians.
Since $J[y^{\prime}](x)\approx\alpha$ for small jumps in the slope of the
beam, we conclude that the total bending potential energy of the cracked beam
with $m$ cracks is

(3.9)
$U_{b}(y)=\frac{EI}{2}\int_{0}^{L}(y^{\prime\prime})^{2}\,dx+\frac{EI}{2}\sum_{i=1}^{m}\frac{1}{\theta_{i}}|J[y^{\prime}](x_{i})|^{2}.$

Shallow arch. First, consider the uniform case. Following [22], the axial
force $N$ in the uniform arch shown in Figure 2(a) is represented as the sum
of two components $N=S_{0}+S_{1}$, where $S_{0}$ is the initial axial tensile
force, and $S_{1}$ is the axial tensile force due to deflection. That is, the
force is positive if it is tensile, and negative if it is compressing. The
value of $S_{0}$ is assumed to be given, and the unknown force $S_{1}$ can be
found through the deflection $y=y(x)$ as follows.

Let $\Delta L$ be the elongation of the arch due to the deflection. By the
definition of the Young’s modulus $E$

(3.10) $S_{1}=EA\frac{\Delta L}{L}.$

The elongation is

(3.11) $\displaystyle\Delta L$
$\displaystyle=\int_{0}^{L}\sqrt{1+|y^{\prime}(x)|^{2}}\,dx-L$
$\displaystyle\approx\int_{0}^{L}\left(1+\frac{1}{2}|y^{\prime}(x)|^{2}\right)dx-L=\frac{1}{2}\int_{0}^{L}|y^{\prime}(x)|^{2}\,dx.$

Then

(3.12) $S_{1}=\frac{EA}{2L}\int_{0}^{L}|y^{\prime}(x)|^{2}\,dx.$

The potential energy $U_{a}$ due to the axial force $N$ is

(3.13) $U_{a}(y)=\frac{EA}{2L}(\Delta L^{*})^{2},$

where $\Delta L^{*}=\frac{NL}{EA}$ is the elongation of the arch caused by the
total axial force $N=S_{0}+S_{1}$. Therefore

(3.14)
$U_{a}(y)=\frac{L}{2EA}(S_{0}+S_{1})^{2}=\frac{L}{2EA}\left(S_{0}+\frac{EA}{2L}\int_{0}^{L}|y^{\prime}(x)|^{2}\,dx\right)^{2}.$

Now suppose that the arch has cracks as in Section 1, and in Figure 2(b). To
derive its axial potential energy $U_{a}(y)$, note that there exists a
sequence of smooth functions $u_{n}$, such that $u_{n}\to y$ in
$H^{1}_{0}(0,L)$, as $n\to\infty$. For such functions, the potential energy
$U_{a}(u_{n})$ is given by the expression in (3.14), i.e.

(3.15)
$U_{a}(u_{n})=\frac{L}{2EA}\left(S_{0}+\frac{EA}{2L}\int_{0}^{L}|u_{n}^{\prime}(x)|^{2}\,dx\right)^{2},\quad
n=1,2,\dots$

Because of the continuity of this functional on $H^{1}_{0}(0,L)$, we conclude
that we can pass to the limit in (3.15), as $n\to\infty$, to get

(3.16)
$U_{a}(y)=\frac{L}{2EA}\left(S_{0}+\frac{EA}{2L}\int_{0}^{L}|y^{\prime}(x)|^{2}\,dx\right)^{2},$

i.e. the same expression as (3.14), even for an arch with cracks.

Non-dimensional variables. Now we find the non-dimensional equivalents for the
above physical quantities. Define the $t$-scale $\omega_{0}$, and the radius
of gyration $r$ by

(3.17) $\omega_{0}=\left(\frac{\pi}{L}\right)^{2}\sqrt{\frac{EI}{\rho
A}},\quad r=\sqrt{\frac{I}{A}}.$

Then make the change of variables

(3.18) $x\leftarrow\frac{\pi x}{L},\quad y\leftarrow\frac{y}{r},\quad
p\leftarrow\frac{p}{EIr}\left(\frac{L}{\pi}\right)^{4},\quad
t\leftarrow\omega_{0}t,\quad c_{d}\leftarrow\frac{c_{d}}{\rho A\omega_{0}}.$

Temporarily distinguish the notation for the original physical variables by
assigning them the $o$ subscript, and their non-dimensional equivalents by
assigning them the $n$ subscript. For example, (3.18) transformation for $t$
says that $t_{n}=\omega_{0}t_{o}$. Then

(3.19) $y_{o}(x_{o},t_{o})=ry_{n}(x_{n},t_{n})=ry_{n}\left(\frac{\pi
x_{o}}{L},\omega_{0}t_{o}\right).$

Therefore

(3.20)
$y_{o}^{\prime}(x_{o},t_{o})=r\frac{\pi}{L}y_{n}^{\prime}(x_{n},t_{n}),\quad
y_{o}^{\prime\prime}(x_{o},t_{o})=r\left(\frac{\pi}{L}\right)^{2}y_{n}^{\prime\prime}(x_{n},t_{n}),$

and

(3.21)
$\dot{y}_{o}(x_{o},t_{o})=r\omega_{0}\dot{y}_{n}(x_{n},t_{n}),\quad\ddot{y}_{o}(x_{o},t_{o})=r\omega_{0}^{2}\ddot{y}_{n}(x_{n},t_{n}).$

Let the energy normalization factor $e_{I}$ be defined by

(3.22) $e_{I}=\frac{L}{\pi}\rho Ar^{2}\omega_{0}^{2}=\frac{L}{\pi}\rho
I\omega_{0}^{2}=\left(\frac{\pi}{L}\right)^{3}r^{2}EI.$

Then the kinetic energy $T_{k,o}$ from (3.1) is transformed according to

(3.23) $\displaystyle T_{k,o}(y_{o})$ $\displaystyle=\frac{\rho
A}{2}\int_{0}^{L}|\dot{y}_{o}(x_{o},t_{o})|^{2}\,dx_{o}=\frac{L}{\pi}\rho
Ar^{2}\omega_{0}^{2}\frac{1}{2}\int_{0}^{\pi}|\dot{y}_{n}(x_{n},t_{n})|^{2}\,dx_{n}$
$\displaystyle=e_{I}\frac{1}{2}|\dot{y}_{n}|_{H}^{2}=e_{I}T_{k,n}(y_{n}).$

Consistent with the definition of the flexibility $\theta$ in (3.6), we use
the transformation $\theta\leftarrow\pi\theta/L$. Therefore, for the bending
potential energy (3.9) we have

(3.24) $\displaystyle
U_{b,o}(y_{o})=\frac{EI}{2}\int_{0}^{L}|y^{\prime\prime}_{o}(x_{o})|^{2}\,dx_{o}+\frac{EI}{2}\sum_{i=1}^{m}\frac{1}{\theta_{i,o}}|J[y^{\prime}_{o}](x_{i,o})|^{2}$
$\displaystyle=\left(\frac{\pi}{L}\right)^{3}r^{2}EI\left[\frac{1}{2}\int_{0}^{\pi}|y^{\prime\prime}_{n}(x_{n})|^{2}\,dx_{n}+\frac{1}{2}\sum_{i=1}^{m}\frac{1}{\theta_{i,n}}|J[y^{\prime}_{n}](x_{i,n})|^{2}\right]$
$\displaystyle=e_{I}U_{b,n}(y_{n}).$

The potential energy $U_{a}$ due to the axial force is given by (3.16). After
the substitution (3.18) we get

$\displaystyle
U_{a,o}(y_{o})=\frac{L}{2EA}\left(S_{0}+\frac{EA}{2L}r^{2}\frac{\pi}{L}\int_{0}^{\pi}|y^{\prime}_{n}(x_{n})|^{2}\,dx_{n}\right)^{2}$
$\displaystyle=\frac{L}{2EA}\frac{(EA)^{2}\pi^{2}r^{2}r^{2}}{L^{4}}\left(\frac{L^{2}S_{0}}{EA\pi
r^{2}}+\frac{1}{2}\int_{0}^{\pi}|y^{\prime}_{n}(x_{n})|^{2}\,dx_{n}\right)^{2}$
$\displaystyle=\frac{e_{I}}{2\pi}\left(\beta+\frac{1}{2}\int_{0}^{\pi}|y^{\prime}_{n}(x_{n})|^{2}\,dx_{n}\right)^{2},$

where the non-dimensional $\beta\in{\mathbb{R}}$ is a renormalization of the
axial force $S_{0}$.

Similarly, we conclude that $\mathcal{W}_{ext}$, and $\mathcal{W}_{d}$ are
transformed by (3.18) into their non-dimensional equivalents in the same way:

(3.25) $\displaystyle T_{k}\leftarrow\frac{T_{k}}{e_{I}},\quad
U_{a}\leftarrow\frac{U_{a}}{e_{I}},\quad
U_{b}\leftarrow\frac{U_{b}}{e_{I}},\quad\mathcal{W}_{ext}\leftarrow\frac{\mathcal{W}_{ext}}{e_{I}},\quad\mathcal{W}_{d}\leftarrow\frac{\mathcal{W}_{d}}{e_{I}}.$

Therefore, the expressions for the non-dimensional quantities are

(3.26)
$T_{k}(y)=\frac{1}{2}|\dot{y}|^{2}_{H},\quad\mathcal{W}_{ext}(y)=(p,y)_{H},\quad\mathcal{W}_{d}(y)=-c_{d}(\dot{y},y)_{H},$
(3.27)
$U_{b}(y)=\frac{1}{2}\left(|y^{\prime\prime}|^{2}_{H}+\sum_{i=1}^{m}\frac{1}{\theta}_{i}|J[y^{\prime}](x_{i})|^{2}\right),$
(3.28)
$U_{a}(y)=\frac{1}{2\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)^{2},$

where $\beta\in{\mathbb{R}}$.

Natural beam frequencies. Equation of harmonic transverse oscillations
$v=v(x)$ of a uniform beam defined on interval $(0,L)$ is

(3.29) $EIv^{\prime\prime\prime\prime}(x)=\omega^{2}\rho Av(x),\quad 0<x<L.$

For a cracked beam, equation (3.29) is satisfied on every subinterval
$(x_{i-1},x_{i})$, $i=1,\dots,m+1$.

Using the transformations to the non-dimensional variables (3.17)–(3.21), we
get

$EIr\left(\frac{\pi}{L}\right)^{4}v_{n}^{\prime\prime\prime\prime}(x_{n},t_{n})=\omega^{2}\rho
Arv_{n}(x_{n},t_{n}),$

where $v_{n}(x_{n},t_{n})$ is $v(x,t)$ in the new (non-dimensional) variables.
Dropping the subscript $n$, we obtain the equation for $v$ in the non-
dimensional quotients

$v^{\prime\prime\prime\prime}=\omega^{2}\left(\frac{L}{\pi}\right)^{4}\frac{\rho
A}{EI}v.$

Comparing this equation with the definition of the eigenvalues and the
eigenfunctions
$\varphi_{k}^{\prime\prime\prime\prime}=\lambda_{k}^{4}\varphi_{k}$, we
conclude that the natural beam frequencies are given by

(3.30)
$\omega_{k}=\lambda_{k}^{2}\left(\frac{\pi}{L}\right)^{2}\sqrt{\frac{EI}{\rho
A}},\quad k\geq 1.$

##  4\. Convex functions and subdifferentials

Subdifferentials provide the proper mathematical framework for the abstract
formulation of equations of motion. Following [3, Section 1.2], let $X$ be a
Hilbert space. A function $\phi:X\to(-\infty,+\infty]$ is called _proper and
convex_ on $X$, if $\phi$ is not identically $+\infty$, and

(4.1) $\phi((1-\lambda)x+\lambda y)\leq(1-\lambda)\phi(x)+\lambda\phi(y),$

for any $x,y\in X$, and $\lambda\in[0,1]$. The function $\phi$ is called
_lower-semicontinuous_ on $X$, if every level set $\\{x\in X:\phi(x)\leq
c\\},\,c>-\infty$, is closed in $X$.

Given a proper, convex, lower-semicontinuous function $\phi$ on $X$, the
_subdifferential_ $\partial\phi:X\to X^{\prime}$ is defined by

(4.2) $\partial\phi(x)=\\{x^{*}\in X^{\prime}\ :\ \phi(y)\geq\phi(x)+\langle
x^{*},y-x\rangle\\},$

for any $y\in X$. Thus $\partial\phi\subset X\times X^{\prime}$.

In general, $\partial\phi$ does not have to be defined everywhere on $X$.
Furthermore, the mapping $x\to\partial\phi(x)\subset X^{\prime}$ can be multi-
valued. Geometrically, if $\partial\phi(x)$ is single-valued at $x\in X$, then
$y\to\phi(x)+\langle\partial\phi(x),y-x\rangle$ is the tangent plane to the
graph of $\phi$ at $x$.

Let $D(\phi)=\\{x\in X:\phi(x)<\infty\\}$, and $D(\partial\phi)=\\{x\in
X:\partial\phi(x)\not=\emptyset\\}$. Note that if $x\in D(\partial\phi)$, then
$x\in D(\phi)$. Indeed, $\phi$ is a proper function. Therefore
$D(\phi)\not=\emptyset$. Choose $y\in D(\phi)$, and $x^{*}\in\partial\phi(x)$.
Then $\phi(x)\leq\phi(y)-\langle x^{*},y-x\rangle<\infty$, as claimed.

Let $f:X\to{\mathbb{R}}$. The _directional derivative_ $f^{\prime}(x;y)$ of
$f$ at $x\in X$ in the direction $y\in X$ is defined by

(4.3) $f^{\prime}(x;y)=\lim_{\alpha\to 0+}\frac{f(x+\alpha y)-f(x)}{\alpha}.$

A function $f$ is said to be Gâteaux differentiable at $x\in X$, if there
exists $(\nabla f)(x)\in X^{\prime}$, such that

(4.4) $f^{\prime}(x;y)=\langle\nabla f(x),y\rangle,$

for any $y\in X$. The linear functional $(\nabla f)(x)$ is called the Gâteaux
derivative of $f$ at $x\in X$.

If the convergence in (4.3) is uniform in $y$ on bounded subsets, then $f$ is
said to be _Fréchet differentiable_.

The Gâteaux differentiability of $f$ is a more restrictive condition, than $f$
is having a subdifferential. The following lemma is proved in [3, Section
1.2]:

######  Lemma 4.1.

If $\phi$ is convex and Gâteaux differentiable at $x\in X$, then
$\partial\phi(x)=\nabla\phi(x)$, and $\partial\phi(x)$ is a singleton.

Recall that an operator $A:X\to X^{\prime}$ is called symmetric, if $\langle
Au,v\rangle=\langle Av,u\rangle$, for any $u,v\in D(A)$.

######  Theorem 4.2.

Let $X$ be a Hilbert space, and $A:X\to X^{\prime}$ be a linear, continuous,
and symmetric operator, such that $D(A)=X$, and $\langle Au,u\rangle\geq 0$
for any $u\in X$. Then function $\phi:X\to{\mathbb{R}}$ defined by

(4.5) $\phi(u)=\frac{1}{2}\langle Au,u\rangle,\quad u\in X,$

is convex, proper, and lower-semicontinuous on $X$. Moreover, it is Fréchet
differentiable on $X$ with $\nabla\phi(u)=\partial\phi(u)=Au$ for any $u\in
X$, and $D(\phi)=D(\partial\phi)=X$.

###### Proof.

To see that $\varphi$ is convex, let $0\leq\lambda\leq 1$, and $u,v\in X$.
Then

$\displaystyle\phi(\lambda u+(1-\lambda)v)=\frac{1}{2}\left[\lambda^{2}\langle
Au,u\rangle+2\lambda(1-\lambda)\langle Au,v\rangle+(1-\lambda)^{2}\langle
Av,v\rangle\right]$ $\displaystyle=\frac{1}{2}\left[\lambda\langle
Au,u\rangle+(1-\lambda)\langle Av,v\rangle-\lambda(1-\lambda)\langle
A(u-v),(u-v)\rangle\right]$
$\displaystyle\leq\lambda\phi(u)+(1-\lambda)\phi(v).$

Since $A$ is continuous, function $\phi$ is continuous on $X$. In particular,
it is lower-semicontinuous on $X$. From the continuity of $A$, we have
$\|Av\|_{X^{\prime}}\leq C\|v\|_{X}$, for some $C>0$. Since

$\phi(v)-\phi(u)-\langle Au,v-u\rangle=\frac{1}{2}\langle A(v-u),v-u\rangle,$

we conclude that

(4.6) $|\phi(v)-\phi(u)-\langle Au,v-u\rangle|\leq\frac{C}{2}\|v-u\|_{X}^{2}.$

Therefore $Au=(\nabla\phi)(u)$ is the Gâteaux (even Fréchet) derivative of
$\phi$ at $u\in X$, and $\partial\phi=A$ is its single-valued subdifferential,
with $D(\partial\phi)=D(\varphi)=X$. ∎

######  Example 4.3.

Let $V$ be the Hilbert space defined in (2.3), and $\mathcal{A}$ be the linear
operator defined in (2.8). Let

(4.7) $\varphi(u)=\frac{1}{2}\langle\mathcal{A}u,u\rangle,\quad u\in V.$

By Theorem 4.2 with $X=V$, function $\varphi:V\to{\mathbb{R}}$ is proper,
convex, and lower-semicontinuos on $V$. Furthermore, $D(\partial\varphi)=V$,
and $\partial\varphi(u)=\mathcal{A}u$, for any $u\in V$.

For a general $u\in V$, the expression for $\partial\varphi(u)\in V^{\prime}$
is complicated. However, if we assume that $u$ is somewhat more regular, then
we can get a simpler expression for it.

Suppose that $u\in D(\mathcal{A})\subset V$. Then, by Theorem 2.3, we have
$\mathcal{A}u=u^{\prime\prime\prime\prime}$ a.e. on every interval $l_{i}$,
$i=1,\dots,m+1$. Thus, we can say that
$\partial\varphi(u)=u^{\prime\prime\prime\prime}$ a.e. on every such interval.

Suppose further, that $u\in H_{0}^{1}(0,\pi)\cap H^{4}(0,\pi)\subset
D(\mathcal{A})$. Then $u^{\prime\prime\prime\prime}\in L^{2}(0,\pi)$, and
$u^{\prime}$ is smooth. Thus $J[u^{\prime}](x_{i})=0$ for any $i=1,\dots,m$,
and we have

(4.8)
$\langle\partial\varphi(u),v\rangle=\langle\mathcal{A}u,v\rangle=\int_{0}^{\pi}u^{\prime\prime}(x)v^{\prime\prime}(x)\,dx=\int_{0}^{\pi}u^{\prime\prime\prime\prime}(x)v(x)\,dx,$

for any $v\in V$. Therefore, in this case
$\partial\varphi(u)=u^{\prime\prime\prime\prime}$ a.e. on $[0,\pi]$.

######  Example 4.4.

Let $H_{0}^{1}=H_{0}^{1}(0,\pi)$ be the Hilbert space defined in (2.6), and
$\langle\cdot,\cdot\rangle_{1}$ be the duality pairing between $H_{0}^{1}$ and
$(H_{0}^{1})^{\prime}$. Let $\mathcal{B}$ be the linear operator defined by

(4.9) $\langle\mathcal{B}u,v\rangle_{1}=(u^{\prime},v^{\prime})_{H},\quad
u,v\in H_{0}^{1}.$

Then $\mathcal{B}:H_{0}^{1}\to(H_{0}^{1})^{\prime}$ is continuous, symmetric
and coercive on $H_{0}^{1}$. In particular, $\mathcal{B}$ is positive, and its
range is $(H_{0}^{1})^{\prime}$.

Let

(4.10) $\psi(u)=\frac{1}{2}\langle\mathcal{B}u,u\rangle,\quad u\in H_{0}^{1}.$

Theorem 4.2 is applicable with $X=H_{0}^{1}$, and $A=\mathcal{B}$. We conclude
that the function $\psi:H_{0}^{1}\to{\mathbb{R}}$ is proper, convex, and
lower-semicontinuos on $H_{0}^{1}$. Furthermore, $D(\partial\psi)=H_{0}^{1}$,
and $\partial\psi(u)=\mathcal{B}u\in(H_{0}^{1})^{\prime}$, for any $u\in
H_{0}^{1}$.

As in Example 4.3, a simpler expression for the subdifferential
$\partial\psi(u)$ can be obtained assuming an additional regularity of $u\in
H_{0}^{1}$.

Suppose that $u\in V\subset H_{0}^{1}$. Then we have

(4.11) $\displaystyle\langle\mathcal{B}u,v\rangle_{1}$
$\displaystyle=(u^{\prime},v^{\prime})_{H}=\int_{0}^{\pi}u^{\prime}(x)v^{\prime}(x)\,dx=$
$\displaystyle-\sum_{i=1}^{m}J[u^{\prime}(x_{i})]v(x_{i})-\sum_{i=1}^{m+1}(u^{\prime\prime},v)_{i},$

for any $v\in H_{0}^{1}$. Therefore, in this case,

(4.12)
$\partial\psi(u)=\mathcal{B}u=-\sum_{i=1}^{m}J[u^{\prime}](x_{i})\delta(x-x_{i})-u^{\prime\prime},$

where $\delta(x-a),\,a\in[0,\pi]$ is the element of $(H_{0}^{1})^{\prime}$,
defined by $\langle\delta(x-a),v\rangle_{1}=v(a)$, for any $v\in H_{0}^{1}$.

Suppose further, that $u\in D(\mathcal{A})\subset V$. Then, by Theorem 2.3,
$u$ satisfies conditions (2.9)–(2.10). In particular,
$J[u^{\prime}](x_{i})=\theta_{i}u^{\prime\prime}(x_{i})$. Thus, with this
additional assumption on $u$, we have

(4.13)
$\partial\psi(u)=\mathcal{B}u=-\sum_{i=1}^{m}\theta_{i}u^{\prime\prime}(x_{i})\delta(x-x_{i})-u^{\prime\prime},$

which is still an element of $(H_{0}^{1})^{\prime}$.

##  5\. Extended Hamilton’s principle

To derive the governing equations for beams and arches, we use the Extended
Hamilton’s Principle, which accommodates non-conservative forces, see [17].

The Principle states that the motion $y=y(x,t)$ of the system gives a
stationary value to the _action_ of the system, i.e. to the integral

(5.1) $I(y)=\int_{t_{1}}^{t_{2}}(\mathcal{L}+\mathcal{W}_{nc})\,dt,$

where the Lagrangian $\mathcal{L}=T_{k}-U$ is the difference between the
kinetic energy $T_{k}$ and the potential energy $U=U_{b}+U_{a}$. The term
$\mathcal{W}_{nc}=\mathcal{W}_{d}+\mathcal{W}_{ext}$ for the non-conservative
external work $\mathcal{W}_{nc}$ is the sum of the external work
$\mathcal{W}_{ext}$, due to the load $p$, and the dissipative work
$\mathcal{W}_{d}$, due to a damping force $\mathcal{F}_{d}$.

Thus (5.1) becomes

(5.2)
$I(y)=\int_{t_{1}}^{t_{2}}\left(T_{k}(y)-U_{b}(y)-U_{a}(y)+\mathcal{W}_{d}(y,\dot{y})+\mathcal{W}_{ext}(y)\right)\,dt,$

where the components of the action are given in (3.26)–(3.28).

The traditional way of expressing the fact that the motion of the system gives
a stationary value to the functional $I$ defined in (5.2), is to say that its
_variation_ $\delta I=0$. Let us examine this statement in more detail within
the framework of the Hilbert spaces.

Recall that the Hilbert spaces $V,H_{0}^{1}$, and $H$ were defined in Section
2. Define

$W=\\{y\ :\ y\in L^{2}(0,T;V),\ \dot{y}\in L^{2}(0,T;H),\ \ddot{y}\in
L^{2}(0,T;V^{\prime})\\},$

where the derivatives are taken in the sense of distributions, [21, Chapter
2]. This is a Hilbert space with the standard inner product and the norm.

Given $\eta\in W$, the directional derivative $I^{\prime}(y,\eta)$ is defined
by

(5.3) $I^{\prime}(y,\eta)=\lim_{\alpha\to
0+}\frac{I(y+\alpha\eta)-I(y)}{\alpha},$

which is consistent with (4.3).

The Gâteaux derivative $\nabla I(y)$ (if exists) is an element of the dual
$W^{\prime}$, such that

(5.4) $I^{\prime}(y,\eta)=\langle\nabla I(y),\eta\rangle_{W},$

for any $\eta\in W$, see Section 4, and [3, Section 1.2].

Functional $I$ has a stationary value at $y$ means that $\nabla I(y)=0$, which
gives the meaning to the infinitesimal variation equation $\delta I=0$.

To derive the governing equation, one has to transform the stationary value
equation $\nabla I(y)=0$ into a more explicit form by computing the
directional derivatives of $I$ along the specially chosen $\eta\in W$.

Let $\eta(x,t)=\zeta(t)v(x)$, where $\zeta:[t_{1},t_{2}]\to{\mathbb{R}}$ is a
smooth function satisfying $\zeta(t_{1})=\zeta(t_{2})=0$, and $v\in V$.

To obtain $\nabla I(y)$, we compute the corresponding directional derivatives
for every term in (5.2) for the same $\eta=\zeta v$. First, notice that

(5.5)
$T_{k}(y+\alpha\eta)-T_{k}(y)=\alpha(\dot{y},v)_{H}\dot{\zeta}+\frac{\alpha^{2}}{2}|\dot{\zeta}|^{2}\,|v|_{H}^{2}.$

By the definition of the directional derivative,
$T_{k}^{\prime}(y,\eta)=\int_{t_{1}}^{t_{2}}(\dot{y},v)_{H}\dot{\zeta}\,dt$.
Integrating by parts in $t$, and using $\zeta(t_{1})=\zeta(t_{2})=0$, we get

(5.6)
$T_{k}^{\prime}(y,\eta)=\int_{t_{1}}^{t_{2}}(\dot{y}(t),v)_{H}\dot{\zeta}(t)\,dt=-\int_{t_{1}}^{t_{2}}\langle\ddot{y}(t),v\rangle_{V}\zeta(t)\,dt.$

Similarly,

(5.7)
$\mathcal{W}_{ext}^{\prime}(y,\eta)=\int_{t_{1}}^{t_{2}}(p(t),v)_{H}\zeta(t)\,dt,$

where the dependency of $p=p(x,t)$ on $x$ is suppressed.

The work $\mathcal{W}_{d}$ done by the non-conservative force
$\mathcal{F}_{d}$ is not covered by the classical Hamilton’s Principle. To
accommodate such a case, the Principle is extended by introducing the
Rayleigh’s dissipation functional, for which the action is _defined_ in a non-
variational manner, see [17] for details. In our case it implies that

(5.8)
$\mathcal{W}_{d}^{\prime}(y,\eta)=-c_{d}\int_{t_{1}}^{t_{2}}(\dot{y}(t),v)_{H}\zeta(t)\,dt.$

The expressions for the potential energies $U_{b}$ and $U_{a}$ depend on
whether the system is a beam or an arch, and on the presence of cracks. They
are given in (3.27)–(3.28). In every case, $U_{b}(u)$ is a convex, lower-
semicontinuous function on $V$, and $U_{a}(u)$ is a convex, lower-
semicontinuous function on $H_{0}^{1}$. Therefore they admit subdifferentials
$\partial U_{b}$ and $\partial U_{a}$, on $V$ and $H_{0}^{1}$ correspondingly,
see Section 4.

If $U_{b}$ and $U_{a}$ are Gateaux differentiable, then the subdifferentials
are equal to their Gateaux derivatives. Thus, we have for $\eta=\zeta v$

(5.9) $\displaystyle U_{b}^{\prime}(y,\eta)$
$\displaystyle=\int_{t_{1}}^{t_{2}}\langle\partial
U_{b}(y(t)),v\rangle_{V}\zeta(t)\,dt,$ (5.10) $\displaystyle
U_{a}^{\prime}(y,\eta)$ $\displaystyle=\int_{t_{1}}^{t_{2}}\langle\partial
U_{a}(y(t)),v\rangle_{1}\zeta(t)\,dt.$

The stationary value equation $\nabla I(y)=0$, for $I(y)$ given by (5.2),
implies

(5.11) $0=(\nabla I(y),\eta)_{W}=I^{\prime}(y,\eta)\\\
=\int_{t_{1}}^{t_{2}}\left[(-c_{d}\dot{y}+p(t),v)_{H}-\langle\ddot{y}+\partial
U_{b}(y(t)),v\rangle_{V}-\langle\partial
U_{a}(y(t)),v\rangle_{1}\right]\zeta(t)\,dt.$

Since $\zeta$ is an arbitrary smooth function on $[t_{1},t_{2}]$, we get

$(-c_{d}\dot{y}+p(t),v)_{H}-\langle\ddot{y}+\partial
U_{b}(y(t)),v\rangle_{V}-\langle\partial U_{a}(y(t)),v\rangle_{1}=0,$

for any $v\in V$, and almost any $t\in[0,T]$. This can further be written as

(5.12) $\langle-\ddot{y}-\partial U_{b}(y(t))-\partial
U_{a}(y(t))-c_{d}\dot{y}+p(t),v\rangle_{V}=0.$

Thus (5.12) implies the following main result: equation

(5.13) $\ddot{y}+\partial U_{b}(y)+\partial U_{a}(y)+c_{d}\dot{y}=p,$

is the abstract governing equation of the system in $V^{\prime}$, a.e. for
$t\in[0,T]$. Note: $\partial U_{b}:V\to V^{\prime}$, and $\partial
U_{a}:H_{0}^{1}\to(H_{0}^{1})^{\prime}$.

##  6\. Beam equation

Uniform beam. Suppose that we have a uniform beam with no cracks, as shown in
Figure 2(a). The classical Euler-Bernoulli beam theory [16] assumes that the
beam motion is due mainly to its bending, while the influence of the axial
force is negligible. Thus we let $U_{a}=0$.

By (3.27) the bending potential energy of the uniform beam is given by

(6.1) $U_{b}(y)=\frac{1}{2}|y^{\prime\prime}|^{2}_{H}.$

Define operator $\mathcal{A}:V\to V^{\prime}$ by

(6.2) $\langle\mathcal{A}u,v\rangle=(u^{\prime\prime},v^{\prime\prime})_{H},$

for any $u,v\in V$. The operator $\mathcal{A}$ is linear, symmetric, and
coercive on $V$. Then

(6.3) $U_{b}(u)=\frac{1}{2}\langle\mathcal{A}u,u\rangle,$

for any $u\in V$. By Theorem 4.2, $U_{b}$ is a convex, lower-semicontinuous
function on $V$. Furthermore, $\partial U_{b}(u)=\mathcal{A}u$. From (5.13),
the abstract Uniform Beam equation in $V^{\prime}$ is

(6.4) $\ddot{y}+\mathcal{A}y+c_{d}\dot{y}=p.$

Assume that $u\in D(\mathcal{A})$. By Example 4.3,
$\mathcal{A}u=u^{\prime\prime\prime\prime}$ for such $u$. Therefore, with this
assumption, equation (6.4) can be written as

(6.5) $\ddot{y}+y^{\prime\prime\prime\prime}+c_{d}\dot{y}=p,$

which is the classical Euler-Bernoulli form of the Beam equation for beams
without cracks. To obtain this equation in physical variables, reverse the
substitutions in Section 3.

Beam with cracks. Now suppose that the beam has cracks as in Section 1, and in
Figure 2(b). Continuing with the classical Euler-Bernoulli beam theory we
disregard the axial force, and let $U_{a}=0$.

By (3.27), the bending potential energy of the cracked beam with $m$ cracks is

(6.6)
$U_{b}(y)=\frac{1}{2}\left(|y^{\prime\prime}|^{2}_{H}+\sum_{i=1}^{m}\frac{1}{\theta}_{i}|J[y^{\prime}](x_{i})|^{2}\right).$

Arguing as in Example 4.3, we get that $U_{b}$ is a convex, lower-
semicontinuous function on $V$.

Define operator $\mathcal{A}:V\to V^{\prime}$ as in equation (2.8), i.e.

(6.7)
$\langle\mathcal{A}u,v\rangle=\sum_{i=1}^{m+1}(u^{\prime\prime},v^{\prime\prime})_{i}+\sum_{i=1}^{m}\frac{1}{\theta}_{i}J[u^{\prime}](x_{i})J[v^{\prime}](x_{i}),$

for any $u,v\in V$. By Lemma 2.2, the operator $\mathcal{A}$ is linear,
symmetric, and coercive on $V$. Then

(6.8) $U_{b}(u)=\frac{1}{2}\langle\mathcal{A}u,u\rangle,$

for any $u\in V$.

By Theorem 4.2, $\partial U_{b}(u)=\mathcal{A}u$. Therefore the abstract
equation for the beam with cracks is

(6.9) $\ddot{y}+\mathcal{A}y+c_{d}\dot{y}=p,$

which is satisfied in $V^{\prime}$, a.e. for $t\in[0,T]$.

Furthermore, using Example 4.3, if $u\in D(\mathcal{A})$, then $u$ satisfies
the boundary conditions of the problem, i.e. (2.9) and (2.10), as well as
$\mathcal{A}u=u^{\prime\prime\prime\prime}$ a.e. on every interval $l_{i}$,
$i=1,\dots,m+1$.

Then equation (6.9) can be written as

(6.10) $\ddot{y}+y^{\prime\prime\prime\prime}+c_{d}\dot{y}=p.$

We can call it the classical Beam equation for beams with cracks. While this
equation looks the same as the Beam equation for beams with no cracks (6.5),
its abstract formulation (6.9) uses the operator $\mathcal{A}$ defined by
(6.7), rather than by (6.2). In particular, $y(\cdot,t)\in V$, a.e.
$t\in[0,T]$.

Strong damping. Viscous effects on the beam and arch motion are discussed in
[2, 11]. Considerations based on the Voigt model for viscoelasticity result in
the additional term $\mu\mathcal{A}\dot{y}$ in the governing equations. Here
$\mu>0$ is a non-dimensional normalized dynamic viscosity coefficient.

If such a term is present, we refer to the model as having the _strong
damping_. Otherwise, if $\mu=0$, the model is for the _weak damping_. In
particular, equations (6.9) and (6.10) describe the weak beam damping motion
case. The corresponding non-dimensional abstract and classical equations in
the presence of the strong damping $\mu>0$ are

(6.11) $\ddot{y}+\mathcal{A}y+\mu\mathcal{A}\dot{y}+c_{d}\dot{y}=p,$

and

(6.12)
$\ddot{y}+y^{\prime\prime\prime\prime}+\mu\dot{y}^{\prime\prime\prime\prime}+c_{d}\dot{y}=p.$

##  7\. Arch equation

Uniform shallow arch. The potential energy for the arch contains the term
$U_{a}$, due to the axial force. By (3.28)

(7.1)
$U_{a}(y)=\frac{1}{2\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)^{2}.$

To compute the subdifferential of $U_{a}$, note that for $u\in V$,

(7.2) $\partial
U_{a}(u)=\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)\partial\psi(u),$

where $\psi$ is

(7.3)
$\psi(u)=\frac{1}{2}|u^{\prime}|^{2}_{H}=\frac{1}{2}\langle\mathcal{B}u,u\rangle,\quad
u\in V,$

see Example 4.4. Since there are no cracks in the arch, we have
$\partial\psi(u)=-u^{\prime\prime}$, per (4.12).

Using the subdifferential $\partial U_{b}(u)=\mathcal{A}u$, for $\mathcal{A}$
defined by (6.2), the abstract form of the uniform shallow arch equation is

(7.4)
$\ddot{y}+\mathcal{A}y-\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)y^{\prime\prime}+c_{d}\dot{y}=p,$

in $V^{\prime}$, a.e. on $[0,T]$.

Its classical form is

(7.5)
$\ddot{y}+y^{\prime\prime\prime\prime}-\frac{1}{\pi}\left(\beta+\frac{1}{2}\int_{0}^{\pi}|y^{\prime}(x,t)|^{2}\,dx\right)y^{\prime\prime}+c_{d}\dot{y}=p,$

cf. [22, eq. (6)].

Arch with cracks. Now suppose that the arch has cracks as in Section 1, and in
Figure 2(b). Its axial potential energy $U_{a}(y)$ has the same expression as
(7.1), even if the arch has cracks.

However, the subdifferential $\partial U_{a}(u)$ is different from the one in
the smooth case. It has been computed in Example 4.4 as

(7.6)
$\partial\psi(u)=\mathcal{B}u=-\sum_{i=1}^{m}J[u^{\prime}](x_{i})\delta(x-x_{i})-u^{\prime\prime},$

for $u\in V$. The bending potential $U_{b}$ is given by (6.6). Then its
subdifferential $\partial U_{b}(u)=\mathcal{A}u$, and we get from (5.13) that

(7.7)
$\ddot{y}+\mathcal{A}y-\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)\left(\sum_{i=1}^{m}J[y^{\prime}](x_{i})\delta(x-x_{i})-y^{\prime\prime}\right)+c_{d}\dot{y}=p,$

is the abstract equation for a shallow arch with cracks in $V^{\prime}$, a.e.
$t\in[0,T]$.

Then, assuming that the function $y$ is smooth as discussed in Example 4.4, we
can use (4.13) for the subdifferential $\partial U_{a}(u)$, and $\partial
U_{b}(u)=\mathcal{A}u=u^{\prime\prime\prime\prime}$. This results in

(7.8) $\displaystyle\ddot{y}+y^{\prime\prime\prime\prime}$
$\displaystyle-\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)\left(\sum_{i=1}^{m}\theta_{i}y^{\prime\prime}(x_{i},t)\delta(x-x_{i})-y^{\prime\prime}\right)+c_{d}\dot{y}=p,$

which can be called the ”classical” form of the shallow arch equation with
cracks. These equations are also referred to as describing the weak arch
damping motion.

Strong damping. Equations (7.7) and (7.8) describe the weak arch damping
$\mu=0$ motion case. The corresponding non-dimensional abstract and
”classical” equations in the presence of the strong damping $\mu>0$ are

(7.9)
$\ddot{y}+\mathcal{A}y+\mu\mathcal{A}\dot{y}-\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)\left(\sum_{i=1}^{m}J[y^{\prime}](x_{i})\delta(x-x_{i})-y^{\prime\prime}\right)+c_{d}\dot{y}=p,$

and

(7.10)
$\displaystyle\ddot{y}+y^{\prime\prime\prime\prime}+\mu\dot{y}^{\prime\prime\prime\prime}$
$\displaystyle-\frac{1}{\pi}\left(\beta+\frac{1}{2}|y^{\prime}|^{2}_{H}\right)\left(\sum_{i=1}^{m}\theta_{i}y^{\prime\prime}(x_{i},t)\delta(x-x_{i})-y^{\prime\prime}\right)+c_{d}\dot{y}=p,$

see Section 6.

## References

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# Activation of magnetic moments in CVD-grown graphene by annealing

Hyungki Shin<EMAIL_ADDRESS>Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, British Columbia, V6T1Z4, Canada
Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia, V6T1Z1, Canada  Ebrahim Sajadi  Stewart Blusson
Quantum Matter Institute, University of British Columbia, Vancouver, British
Columbia, V6T1Z4, Canada  Department of Physics and Astronomy, University of
British Columbia, Vancouver, British Columbia, V6T1Z1, Canada  Ali Khademi
Stewart Blusson Quantum Matter Institute, University of British Columbia,
Vancouver, British Columbia, V6T1Z4, Canada  Department of Physics and
Astronomy, University of British Columbia, Vancouver, British Columbia,
V6T1Z1, Canada  Present address: National Research Council Canada, Edmonton,
Alberta T6G 2M9, Canada  Silvia Lüscher  Stewart Blusson Quantum Matter
Institute, University of British Columbia, Vancouver, British Columbia,
V6T1Z4, Canada  Department of Physics and Astronomy, University of British
Columbia, Vancouver, British Columbia, V6T1Z1, Canada  Joshua A. Folk
<EMAIL_ADDRESS>Stewart Blusson Quantum Matter Institute, University of
British Columbia, Vancouver, British Columbia, V6T1Z4, Canada  Department of
Physics and Astronomy, University of British Columbia, Vancouver, British
Columbia, V6T1Z1, Canada

###### Abstract

Effects of annealing on chemical vapor deposited graphene are investigated via
a weak localization magnetoresistance measurement. Annealing at
$300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in inert gases, a common cleaning
procedure for graphene devices lin2011graphene ; cheng2011toward , is found to
raise the dephasing rate significantly above the rate from electron-electron
interactions, which would otherwise be expected to dominate dephasing at 4 K
and below PhysRevB.22.5142 ; kozikov2010electron ; tikhonenko2008weak ;
tikhonenko2009transition ; min2017asymmetric . This extra dephasing is
apparently induced by local magnetic moments activated by the annealing
process, and depends strongly on the backgate voltage applied.

##  I I. Introduction

The first graphene samples made into electronic devices came from flakes
exfoliated from bulk graphite, and were only a few microns in size
neto2009electronic ; geim2010rise . The discovery that graphene could also be
grown over large areas, using a chemical vapour deposition (CVD) process on
metal films, opened up many technological possibilities that were unimaginable
with exfoliated flakes. One application proposed in the early days of graphene
research was spintronics, making use of potentially long spin lifetimes in
carbon-based materials. Although the hope of graphene spintronics persists,
experimental realizations remain less impressive than theoretical proposals
RevModPhys.92.021003 ; han2011spin .

Despite the promise of CVD-grown graphene for technology, this growth
technique tends to yield samples with more defects, such as carbon vacancies
and domain boundaries, compared to exfoliated graphene lee2017review . Here,
we present an unexpected characteristic of those defects, with implications
for any technology that relies on spin or quantum coherence. We show that a
typical annealing process used to remove residues from graphene devices
lin2011graphene ; ishigami2007atomic ; dan2009intrinsic ; cheng2011toward ;
pirkle2011effect ; dean2010boron ; leong2014does  significantly enhances the
quantum mechanical phase-breaking (dephasing) rate measured via weak
localization (WL)tikhonenko2008weak . Although data from phase coherent
measurements at cryogenic temperatures might seem unrelated to applications at
room temperature, our results suggest that these activated defects create a
fast mechanism for spin relaxation that would limit the use of CVD graphene in
spintronics. At the same time, this finding sheds light on the microscopic
origin of local moments in this form of graphene, which may help to devise
growth recipes to avoid them in the first place.

##  II II. Experimental methods

Multiple samples were prepared by transferring commercial CVD graphene (ACS
Materials Co., MA) onto n-doped Si wafers with a 295 nm SiOx dielectric. In
some samples, a 30 nm HfO2 film was grown by atomic layer deposition (ALD) at
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ on the SiOx before transferring
graphene [Fig. S1 inset]. Hall bars with a width and a length of 30 $\mu$m by
30 $\mu$m were defined by electron beam lithography. Cr/Au (5/80nm) ohmic
contacts were used to apply a bias current, $I$, and measure longitudinal and
transverse voltages, $V_{xx}$ and $V_{xy}$ [Fig. 1(a)], then $V_{xx}$ was
converted into resistivity $\rho$ or conductivity $\sigma\equiv\rho^{-1}$
using the geometric aspect ratio of the Hall bar. The carrier density,
$n_{s}$, was tuned using the backgate voltage, $V_{G}$, and measured via low-
field Hall effect.

Figure 1:  (a) Schematic of the sample geometry and measurement setup. Pink
indicates CVD graphene, yellow indicates Au/Cr electrodes. The dielectric
stack shown here includes the SiOx. (b) Schematic of the thermal anneling
setup. (c) Change in conductivity as a function of charge carrier density,
before and after Ar annealing. Inset: same data as a function of gate voltage.
(d) Magnetoconductivity before and after Ar annealing, with fits to WL theory.
Fit parameters before annealing: $\tau_{i}^{-1}$=160$\pm$45 ns-1,
$\tau_{\varphi}^{-1}$=33$\pm$2 ns-1; after annealing:
$\tau_{i}^{-1}$=215$\pm$90 ns-1, $\tau_{\varphi}^{-1}$=60$\pm$5 ns-1. Fits
included data over a larger field range, see Supplementary Information.

Annealing was performed in Ar, N2, forming gas (5% H2/95% N2, hereafter
H2/N2), or under vacuum, using a simple thermal annealer [Fig. 1(b)]. Samples
wire-bonded into ceramic chip carriers were loaded into the annealer, then
pumped to a base pressure of $3\times 10^{-3}$ mbar (gas anneal) or $<10^{-5}$
mbar (vacuum anneal). For gas annealing, the chamber was pumped-and-flushed,
then the flow rate adjusted using a needle valve until a presure of 200 mbar
was attained. Samples were annealed for 1 hour at
$300\text{\,}\mathrm{\SIUnitSymbolCelsius}$, then cooled to room temperature
in the same atmosphere as the annealing step, and immediately transferred to
the cryostat. After evacuating the cryostat and adding He exchange gas,
samples were cooled to 4.2 K for measurement. The total air exposure time
after annealing was less than one minute.

##  III III. Result and discussion

Figure 1(c) compares $\sigma(n_{s})$ of a sample before and after annealing in
Ar. The conductivity itself rose slightly, while the gate voltage required to
reach the charge neutrality point decreased from $V_{n=0}$=88 V to
$V_{n=0}$=18 V. The quantum correction to the perpendicular
magnetoconductivity, $\sigma(B_{\bot})$, also changed after annealing [Fig.
1(d)]. To interpret this change, phase ($\tau_{\varphi}^{-1}$) and intervalley
($\tau^{-1}_{i}$) scattering rates were extracted from
$\Delta\sigma(B_{\perp})\equiv\sigma(B_{\perp})-\sigma(0)$ using fits to the
standard WL expression for graphene mccann2006weak ; tikhonenko2008weak ,

$\Delta\sigma(B_{\perp})=\frac{e^{2}}{\pi\hbar}\left[F\left(\frac{\tau^{-1}_{B}}{\tau_{\varphi}^{-1}}\right)-F\left(\frac{\tau^{-1}_{B}}{\tau_{\varphi}^{-1}+2\tau^{-1}_{i}}\right)\right],$
(1)

where $\tau_{B}^{-1}=4eDB_{\perp}/\hbar$, $F(z)=ln(z)+\psi(0.5+z^{-1})$,
diffusion constant $D=\frac{\sigma\pi\hbar v_{f}}{2e^{2}\sqrt[]{\pi n_{s}}}$
and $v_{f}=10^{6}$ m/s for graphene. A third term reflecting intravalley
scattering is omitted here for clarity as it did not affect the extracted
$\tau_{\varphi}^{-1}$ or $\tau_{i}^{-1}$, but was included in the fitting
(Supplementary Information).

Both $\tau^{-1}_{i}$ and $\tau_{\varphi}^{-1}$ rose due to annealing. The rise
in $\tau_{i}^{-1}$ reflects increased short-range scattering due to changes in
the interaction between graphene and SiOx dielectric, caused by the annealing
itself cheng2011toward . The significant rise in $\tau_{\varphi}^{-1}$ is more
surprising, and is the central focus of this work.

Electron-electron (e-e) and electron-phonon (e-p) interactions are well known
depahsing mechanisms in metallic systems. Below 50 K in graphene the e-e rate
dominates, and at 4 K the e-p rate is negligible in comparison
tikhonenko2008weak ; tikhonenko2009transition ; min2017asymmetric . The e-e
dephasing rate in graphene is Altshuler_1982 ; aleiner1999interaction ;
mccann2006weak ; tikhonenko2008weak ; min2017asymmetric :

$\displaystyle\tau_{ee}^{-1}=\frac{k_{B}T}{\hbar}\left(\frac{ln\left(\frac{g}{2}\right)}{g}\right),~{}g=\frac{\sigma
h}{e^{2}}.$ (2)

Figure 2:  The change of magnetoconductivity
$\Delta\sigma(B)\equiv\sigma(B)-\sigma(B=0)$ with carrier density before (a)
and after (b) annealing in Ar. Dephasing rates $\tau_{\varphi}^{-1}$,
extracted from $\Delta\sigma(B)$ for each carrier density (filled/solid)
compared with e-e rates calculated using Eq. 2 (open/dashed), before (c) and
after (d) annealing. Error bars indicates range of possible values from
fitting.

For the data in Fig. 1(d), Eq. 2 predicts $\tau_{ee}^{-1}$= 33 ns-1 before
annealing, matching the measured value $\tau_{\varphi}^{-1}$=33$\pm$2 ns-1.
After annealing, the calculated value from Eq. 2 is barely changed,
$\tau_{ee}^{-1}$= 32 ns-1, but the measured value rises to
$\tau_{\varphi}^{-1}$=60$\pm$5 ns-1. Fig. 2 extends this analysis to a range
of $n_{s}$. Before annealing, measured $\tau_{\varphi}^{-1}$ coincide almost
exactly with calculated $\tau_{ee}^{-1}$ [Fig. 2(c)]. After annealing, on the
other hand, $\tau_{\varphi}^{-1}$ exceeds $\tau_{ee}^{-1}$ everywhere, with
the difference growing larger with $n_{s}$ [Fig. 2(d)]. Taken together, Figs.
2(c) and 2(d) indicate that e-e interaction dominates dephasing in samples as
deposited, but new interactions emerge in Ar-annealed samples adding 10’s of
ns-1 of excess dephasing.

Figure 3:  The excess dephasing rate
($\tau_{xs}^{-1}\equiv\tau_{\varphi}^{-1}-\tau_{ee}^{-1}$) as a function of
$n_{s}$, for various annealing conditions: no anneal; anneal in vacuum for 1
or 5 hours; and anneal 1 hour in Ar, N2, or H2/N2 . Samples are directly on
SiOx except where noted (Ar:HfO2, black diamonds). Larger symbols indicate the
datapoint corresponding to $V_{G}$=0 for each recipe (for Ar:HfO2, that point
is off the graph and not shown).

As an inert gas, Ar would not be expected to create chemical modifications to
graphene at temperatures of only a few hundred ∘C. Indeed, the choice of Ar,
versus other inert gases, does not seem to be important in the appearance of
excess dephasing. Fig. 3 compiles the excess dephasing rate
$\tau_{xs}^{-1}\equiv\tau_{\varphi}^{-1}-\tau_{ee}^{-1}$ for a variety of
annealing recipes, and leads to several observations. When annealing is
performed in vacuum, $\tau_{xs}^{-1}$ is very small, rising only to $\sim$10
ns-1 (constant in $n_{s}$) even after 5 hours of annealing. For anneals in a
gas environment, $\tau_{xs}^{-1}$ is significantly larger, and grows with
$n_{s}$ by an amount that does not depend on the choice of gas.
$\tau_{xs}^{-1}$ also increases at low $n_{s}$, most noticably for samples
annealed in H2/N2 .

Before attributing this rise at low $n_{s}$ to H2/N2 in particular, we note
that $V_{n=0}$ was much larger for H2/N2 ($V_{n=0}$=60 V) compared to Ar
($V_{n=0}$=18 V) or N2 ($V_{n=0}$=31 V), so larger negative $V_{G}$ was
applied to H2/N2 -annealed samples to reach the low density regime. Further
insight into the importance of $V_{G}$, independent of $n_{s}$, is gained by
measuring the $n_{s}$-dependent dephasing rate for a particular annealed
sample after repeatedly shifting the Dirac point through exposure to ambient
atmosphere [Fig. 4] C0JM02922J . ① represents $\tau_{\varphi}^{-1}$ for the
sample immediately after an hour of annealing in H2/N2 ($V_{n=0}$=60 V), ②-④
represent subsequent one hour exposures to ambient atmosphere
($V_{n=0}$=92,109,140 V), then ⑤ represents an additional one hour anneal in
H2/N2 ($V_{n=0}$=70 V). Calculated $\tau_{ee}^{-1}$ for the different
exposures fall along a single curve [Fig. 4(a)], reflecting the fact that
$\sigma(n_{s})$ does not change through these different steps. Measured
$\tau_{\varphi}^{-1}$ also fall along a single curve for the low-$n_{s}$ part
of each scan, for ①-④. However, the onset of the upturn in $\tau_{xs}^{-1}$ at
high $n_{s}$ shifts to higher and higher densities with exposure to air, while
the minimum in $\tau_{xs}^{-1}$ remains pinned to $V_{G}=0$ [arrows in Fig.
4(b)]. This seems to indicate that the upturns in $\tau_{\varphi}^{-1}$ and
$\tau_{xs}^{-1}$ at high density are connected to the electric field between
the graphene and the backgate rather than to the carrier density itself.

It is worth noting that the persistent upturn in $\tau_{xs}^{-1}$ at large
$n_{s}$ (strongly negative $V_{G}$), even after extended exposure to air,
indicates that the activation of a dephasing mechanism by annealing is robust.
A second annealing step (⑤) following the multiple exposures to air results in
$V_{n=0}$ shifting back to 70 V, while $\tau_{xs}^{-1}$ increases above even
the values observed after the first anneal (①). As was the case for ①-④, the
minimum in $\tau_{xs}^{-1}$remains at $V_{G}=0$.

Figure 4:  (a) Evolution of the dephasing rate (solid lines) and e-e
interaction rate (dashed lines), shown as a function of carrier density, for
an H2/N2 annealed sample after air exposure. ① represents the sample
immediately after one hour H2/N2 anneal, ②-④ after consecutive air exposures,
then ⑤ after one more H2/N2 anneal (b) $\tau_{xs}^{-1}$calculated from the
data in (a), with arrows indicating $V_{G}$=0.

Just as the activated dephasing mechanism does not seem to depend on the
choice of annealing gas, it also does not depend on the substrate. In order to
test for a possible influence of defects in the SiOx substrate, Fig. 3
includes data from a sample, annealed in Ar gas, on a wafer where ALD-grown
HfO2 covers the SiOx [Fig. S1 inset]. HfO2 layers are amorphous when grown at
the low temperatures used here, presumably with a defect density and type
different from those found in thermal silicon oxide hausmann2003surface .
However, the measured $\tau_{xs}^{-1}$ for the HfO2 sample is nearly identical
to that of the SiOx sample [Section 4 in supplementary information]. From
this, we tentatively conclude that the defects relevant to this measurement
lie in or on the graphene itself, rather than in the substrate.

The role of annealing temperature was also explored (data not shown). Below
$200\text{\,}\mathrm{\SIUnitSymbolCelsius}$, annealing had minimal effect on
$\tau_{xs}^{-1}$ for any of the inert gases. At
$250\text{\,}\mathrm{\SIUnitSymbolCelsius}$ the effect was similar to, though
less strong than, the $300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ data shown
here. For anneals at $350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and above,
the excess dephasing was even larger but the graphene mobility was degraded.

Figure 5: Perpendicular field magnetoconductivity lineshape,
$\Delta\sigma(B_{\perp})$, for an H2N2-annealed sample changes dramatically
with in-plane magnetic field, $B_{\parallel}$. Solid lines are fits to Eq. 1.
Extracted $\tau_{\varphi}^{-1}$decreases from 95$\pm$10 ns-1 at
$B_{\parallel}$=0, to 24$\pm$3 ns-1 at $B_{\parallel}$=700 mT (T=100 mK).

The effect of an in-plane magnetic field on WL can reveal spin-related
mechanisms for dephasing. Magnetic fields, $B_{\parallel}$, applied in the
plane of a graphene sheet influence $\sigma(B_{\perp})$ in two ways. First,
ripples in the graphene create an out-of-plane local magnetic field,
fluctuating randomly across the graphene even when $B_{\parallel}$ is applied
exactly along the average plane of the sample. These out-of-plane components
break time reversal symmetry locally, yielding an additional dephasing
mechanism that grows quadratically with $B_{\parallel}$ lundeberg2010rippled .
Second, $B_{\parallel}$ aligns local magnetic moments with Lande g-factor $g$
once $g\mu_{B}B_{\parallel}\gg k_{B}T$. When free to rotate, these moments
contribute to dephasing through spin flip scattering of conduction electrons,
but that mechanism is suppressed when the moments are aligned by
$B_{\parallel}$ lundeberg2010rippled ; lundeberg2013defect ; kochan2014spin .

Cooling the graphene samples in a dilution refrigerator equipped with a two-
axis magnet enabled $\sigma(B_{\perp})$ measurements with finite
$B_{\parallel}$, at temperatures low enough that spins could be aligned by
$B_{\parallel}$ before dephasing due to ripples was significant. Fig. 5 shows
WL data for an H2/N2-annealed sample at 100 mK. (This sample was annealed at
$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to maximize $\tau_{xs}^{-1}$.) The
lineshape of $\sigma(B_{\perp})$ changes when in-plane field is applied, and
fits to Eq. 1 confirm that $\tau_{\varphi}^{-1}$ drops from 95 ns-1
($B_{\parallel}$=0) to 24 ns-1 at $B_{\parallel}$=0.7 T, where
$g\mu_{B}B_{\parallel}/k_{B}T\sim 10$ with $g$=2. Ripples would have induced
the opposite effect (increased $\tau_{\varphi}^{-1}$for higher
$B_{\parallel}$). This data shows that spin flips induced by magnetic moments
are the predominant mechanism for excess dephasing after annealing,
contributing at least 70 ns-1.

##  IV IV. conclusion

Magnetic moments are believed to emerge at graphene defects, including
vacancies yazyev2007defect ; chen2014magnetic , impurities han2011spin ;
jozsa2009linear , and strain fluctuations couto2014random ; neumann2015raman .
In most cases, the interactions of defect-bound moments with conduction
electrons are predicted to strengthen at low carrier density due to weaker
screening chen2014magnetic ; haase2011magnetic ; yazyev2007defect ;
engels2014limitations . We find, in contrast, that $\tau_{xs}^{-1}$ increases
at higher $n_{s}$ and, more importantly, depends more on the electric field
due to $V_{G}$ than on the carrier density itself since gas annealed samples
show the minimum near $V_{G}$=0. It may be, therefore, that the magnetic
moments being activated by gas annealing are of a different type than has been
previously reported in experiment or studied theoretically.

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##  V SUPPLEMENTARY INFORMATION

Section 1. Thermal annealer

The sample rests on a copper sheath covering a halogen projector bulb. The
temperature of the copper sheath is monitored with a thermocouple, and the
power of the bulb is adjusted with a variable transformer to keep the copper
at the desired temperature. Sample temperatures were confirmed to match those
of the copper using a mock sample with a fine-wire thermocouple glued on the
wafer surface. Temperataures of the thermocouple and copper were found to
match under both in gas and in vacuum annealing conditions.

Section 2. Weak localization fitting process and analysis

Weak localization fits were made to resistivity data, instead of conductivity
as expressed in Eq. 1 (main text), using

$\rho(B)=\alpha(B-B_{0})+\rho_{0}-\rho_{0}^{2}\frac{e^{2}}{\pi\hbar}\left[F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}}\right)-F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}+2\tau_{i}^{-1}}\right)-2F\left(\frac{\tau_{B}^{-1}}{\tau_{\phi}^{-1}+\tau_{\ast}^{-1}}\right)\right]$

The third term in the equation above, missing from Eq. 1, was found to be
strongly suppressed by the enormous $\tau_{*}^{-1}$ in CVD graphene and
therefore irrelevant to the extraction of $\tau_{i}^{-1}$ and
$\tau_{\varphi}^{-1}$. This equation adds several parameters to those two
rates, which appear in Eq. 1:

  * •

An offset field $B_{0}$ was included to take into account the residual field
in our superconducting magnet.

  * •

The zero-field resistivity $\rho_{0}\equiv\rho(B=B_{0})$.

  * •

A linear-in-B term, $\alpha(B-B_{0})$, was added to the resistivity,
accounting for any offset of voltage probes that would mix Hall and
longitudinal voltages.

Data were fit to this equation using a multi-step process involving two
magnetic field ranges: $\pm$10 mT [c.f. Fig. 1d] and $\pm$100 mT [Fig. S1]. In
general, fine scans over the smaller field range enabled a more accurate
determination of the smaller rate, $\tau_{\varphi}^{-1}$, whereas coarser data
over the larger field range was important for determining $\tau_{i}^{-1}$, and
the value of $\tau_{i}^{-1}$itself affected the determination of
$\tau_{\varphi}^{-1}$. The fitting process was:

  1. 1.

Determine $\rho_{0}$ and $B_{0}$ from Lorenzian fits to the peak of $\rho(B)$

  2. 2.

Fit $\pm$100 mT range to determine $\alpha,\tau_{\varphi}^{-1},\tau_{i}^{-1}$

  3. 3.

Fixing $\alpha,\rho_{0},B_{0}$ and using $\tau_{\varphi}^{-1}$and
$\tau_{i}^{-1}$from above as initial parameters, fit $\pm$10 mT range.

  4. 4.

Fixing $\alpha,\rho_{0},B_{0},\tau_{\varphi}^{-1}$ and using
$\tau_{i}^{-1}$from above as initial parameter, fit $\pm$100 mT range. If
there is a significant change in $\tau_{i}^{-1}$, repeat from step 2.

  5. 5.

Error bars for $\tau_{\varphi}^{-1}$were determined by trying different
combinations of $\tau_{i}^{-1}$and $\tau_{\varphi}^{-1}$to check how far
$\tau_{\varphi}^{-1}$could be off before the fit was visually bad.

Figure S1: Magnetoconductivity before and after Ar annealing, with fits to WL
theory (black solid lines), over the larger $\pm$100 mT range. Inset:
Schematic of the sample including a HfO2 ALD layer (grey) on top of the SiOx

Section 4. Graphene on HfO2

Figure 3 in the main text shows $\tau_{xs}^{-1}$ from a sample on HfO2-covered
SiOx, annealed in Ar. In addition to this data, a full set a measurements was
taken on this sample. Fig. S2 shows the measured $\tau_{\varphi}^{-1}$ and
expected $\tau_{ee}^{-1}$ before any annealing occured, confirming
$\tau_{\varphi}^{-1}$ is consistent with $\tau_{ee}^{-1}$. After annealing in
vacuum for 1 hour, $\tau_{\varphi}^{-1}$ and $\tau_{ee}^{-1}$ are nearly
unchanged, again consistent with each other. Only after annealing in Ar does
$\tau_{\varphi}^{-1}$ rise significantly above $\tau_{ee}^{-1}$, as it does
for samples directly sitting on SiOx. One difference between graphene on HfO2
and graphene on SiOx was that $V_{n=0}$ did not shift significantly due to the
annealing on HfO2, whereas on SiOx the shift was often several 10’s of volts.
Instead, $V_{n=0}$ for the HfO2 remained within 10 V of zero.

Figure S2: Measured ($\tau_{\varphi}^{-1}$) and expected ($\tau_{ee}^{-1}$)
dephasing rates for graphene on a HfO2-covered substrate (a) before annealing,
(b) after vacuum annealing, and (c) after Ar annealing.

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