# Elastic solids with strain-gradient elastic boundary surfaces

C. Rodriguez<EMAIL_ADDRESS>
###### Abstract.

Recent works have shown that in contrast to classical linear elastic fracture
mechanics, endowing crack fronts in a brittle solid with Steigmann-Ogden
surface elasticity yields a model that predicts bounded strains at the crack
tips for plane-strain problems. However, a logarithmic singularity is still
present in general for anti-plane shear (mode-III fracture) even when
Steigmann-Ogden surface elasticity is incorporated.

Motivated by obtaining a model of brittle fracture capable of predicting
bounded strains for all modes of loading, we formulate an exact general theory
of a bulk solid containing a boundary surface with strain-gradient surface
elasticity. For planar reference surfaces parameterized by flat coordinates,
the form of surface elasticity reduces to that introduced by Hilgers and
Pipkin, and when the surface energy is independent of the surface gradient of
the stretching, the theory reduces to that of Steigmann and Ogden. We give a
full discussion of material symmetry using Murdoch and Cohen’s extension of
Noll’s theory. We present a model quadratic surface energy that incorporates
resistance to geodesic distortion, satisfies strong ellipticity, and requires
the same material constants found in the Steigmann-Ogden theory.

Finally, we derive and apply the linearized theory to mode-III fracture in an
infinite plate. We prove that there always exists a unique classical solution
to the governing integro-differential equation, and in contrast to using
Steigmann-Ogden surface elasticity, our model is consistent with the
linearization assumption in predicting finite strains at the crack tips.

##  1\. Introduction

###  1.1. Surface stressed solid bodies

The study of surface tension for solids was initiated by Gibbs in 1857, and it
is now widely accepted that surface tension and more general surfaces stresses
must be accounted for when modeling mechanical structures at small length
scales. In particular, interfaces111By an interface, we mean a thin region
separating either two distinct materials or two distinct phases of a material.
between a bulk solid and its environment can form due to various mechanisms
including coating or atomic rearrangement during fracture. A way to
mathematically model such an interface is by endowing part of the three-
dimensional body’s two-dimensional boundary with it’s own thermodynamic
properties that are distinct from the bulk (such as energy).

Motivated by modeling the apparent compressive surface stresses in certain
cleaved crystals, Gurtin and Murdoch [4, 5] developed a rigorous general
theory of material surface stresses that accounts for the surface’s resistance
to stretching (but not flexure). Their celebrated theory has been used to
model a wide arrange of phenomena over the past 45 years, especially recently
due to advances in nanoscience and nanotechnology. In the special case of a
hyperelastic, three-dimensional bulk solid with reference configuration
$\mathcal{B}$ and material surface $\mathcal{S}\subseteq\partial\mathcal{B}$,
the field equations for the current configuration
$\boldsymbol{\chi}:\mathcal{B}\rightarrow\boldsymbol{\chi}(\mathcal{B})$ are
the Euler-Lagrange equations associated to the total strain energy

$\displaystyle\boldsymbol{\Phi}[\boldsymbol{\chi}]=\int_{\mathcal{B}}W(\boldsymbol{C})\,dA+\int_{\mathcal{S}}U(\boldsymbol{\mathsf{C}})\,dA,$
(1.1)

where we omit the possible explicit dependence of the functions $W$ and $U$ on
points in $\mathcal{B}$ and $\mathcal{S}$ (see Figure 1). Here
$\boldsymbol{C}$ is the left Cauchy-Green stretch tensor and
$\boldsymbol{\mathsf{C}}$ is the left Cauchy-Green surface stretch tensor, the
pullbacks by $\boldsymbol{\chi}$ of the metric tensors on
$\boldsymbol{\chi}(\mathcal{B})$ and $\boldsymbol{\chi}(\mathcal{S})$,
respectively. In particular, the material surface stress tensor is derived
from the surface energy density $U(\boldsymbol{\mathsf{C}})$ in analogy with
the bulk’s Piola stress being derived from the bulk energy density
$W(\boldsymbol{C})$.

Figure 1. The reference configuration of a bulk solid $\mathcal{B}$ with
strain-gradient elastic surface $\mathcal{S}\subseteq\partial\mathcal{B}$.

However, Steigmann and Ogden showed in their seminal works [22, 23] that
equilibrium states under compressive surface stresses obtained from the
Gurtin-Murdoch theory do not satisfy an associated Legendre-Hadamard
condition, and thus, these equilibrium states cannot be local energy
minimizers.222The fact that a pure membrane, a special type of Gurtin-Murdoch
material surface, in equilbrium and under compressive surface stresses cannot
by locally energy minimizing is due to Pipkin [20]. Steigmann and Ogden [22,
23] rectified this inadequacy and incorporated the material surface’s
resistance to flexure by including curvature dependence in the surface energy:

$\displaystyle\boldsymbol{\Phi}[\boldsymbol{\chi}]=\int_{\mathcal{B}}W(\boldsymbol{C})\,dA+\int_{\mathcal{S}}U(\boldsymbol{\mathsf{C}},\boldsymbol{\kappa})\,dA,$
(1.2)

where $\boldsymbol{\kappa}$ is the pullback of the second fundamental form on
$\boldsymbol{\chi}(\mathcal{S})$. Since $\boldsymbol{\kappa}$ depends on the
second derivatives of $\boldsymbol{\chi}$, the surface energy is of _strain-
gradient_ type. In recent years, the Steigmann-Ogden theory has attracted
considerable interest from various perspectives including the study of contact
problems [31, 13, 32, 10, 11, 12, 37] and inclusion problems [6, 35, 15, 2,
14, 27]. The theory has also been used to model fracture in brittle materials
[30, 34, 33], the main phenomenon motivating this work.

One of the most successful and practical theories for modeling fracture in
brittle materials is classical linear elastic fracture mechanics. The
governing linear partial differential equations are derived from finite
elasticity under the assumption of _infinitesimal_ strains (linearized
elasticity), but the theory predicts _unbounded_ singular strains at the crack
tips, a striking inconsistency and physically impossible prediction. There
have been a vast number of suggestions for supplementing classical linear
elastic fracture mechanics to correct this defect in the theory (see, e.g.,
[1]).

A more recent approach aimed at eliminating these crack tip singularities is
to modify the boundary conditions of classical linear elastic fracture
mechanics by endowing the crack fronts with material surface stresses.
Beginning with [19] and developed further by Sendova and Walton [21], one line
of thought has been to prescribe the crack front’s surface stresses as a
curvature dependent surface tension. Although able to remove the singularities
completely in a diverse range of settings (see [36, 28, 29, 26, 3]), it is
unclear if the surface stresses from the Sendova-Walton theory can be derived
from a surface energy density,333Equivalently, it is unclear if the governing
field equations from the Sendova-Walton model can be derived from a Lagrangian
energy functional. a reasonable definition of “elastic-like” behavior.
Deriving the material stresses for the crack fronts from a Gurtin-Murdoch
surface energy _does not_ completely remove the crack tip singularities (see
[25, 9]), but however, endowing the crack fronts with Steigmann-Ogden surface
energy _does_ completely remove the singularities for plane-strain problems
[30, 33] and axisymmetric penny shaped cracks [34]. Unfortunately, for anti-
plane shear (mode-III loading), Steigmann-Ogden surface energy reduces to
Gurtin-Murdoch surface energy and crack tip singularities _persist_. The fact
that the Steigmann-Ogden theory reduces to the Gurtin-Murdoch theory for mode-
III loading is due to the fact that the linearized curvature vanishes for
anti-plane shear, i.e., for displacement fields tangent to the material
surface.

###  1.2. Main results and outline

Inspired by [3] and motivated by obtaining a model of brittle fracture capable
of predicting bounded crack tip strains for _all modes_ of loading, this work
proposes an augmentation of the Steigmann-Ogden theory that is of strain-
gradient type and includes the derivative of stretching in the surface energy:

$\displaystyle\boldsymbol{\Phi}[\boldsymbol{\chi}]=\int_{\mathcal{B}}W(\boldsymbol{C})\,dA+\int_{\mathcal{S}}U(\boldsymbol{\mathsf{C}},\boldsymbol{\kappa},\nabla\boldsymbol{\mathsf{C}})\,dA,$
(1.3)

where $\nabla$ is the Levi-Cevita connection on $\mathcal{S}$. For
$\mathcal{S}$ contained in a plane and parameterized by flat coordinates, the
surface energy is equivalent to that introduced by Hilgers and Pipkin [7].

In Section 2, the relevant kinematics of the boundary surface $\mathcal{S}$
convected by a deformation of the bulk solid $\mathcal{B}$ are first
summarized. In particular, we introduce a third order tensor
$\boldsymbol{\mathsf{L}}$ that is obtained from certain transposes of
$\nabla\boldsymbol{\mathsf{C}}$ and has components in terms of the difference
of Christoffel symbols on $\mathcal{S}$ and $\boldsymbol{\chi}(\mathcal{S})$.
This tensor is used in a model surface energy proposed in Section 3.
Physically, the tensor $\boldsymbol{\mathsf{L}}$ furnishes the rate of
stretching of convected geodesics and locally characterizes _geodesic
distortion_ , i.e., how convected geodesics fail to be geodesics on
$\boldsymbol{\chi}(\mathcal{S})$ (see Proposition 2.1). The general form of
(1.3) that we consider is then introduced (see (2.47)), and inspired by
Murdoch and Cohen’s extension to surfaces of Noll’s classical theory of
material symmetry, we introduce material symmetry for the surface energy
density $U$ (see 2.74). In the final subsection of Section 2, we derive the
field equations (2.82) governing equilibrium states of the solid $\mathcal{B}$
with strain-gradient elastic surface $\mathcal{S}\subseteq\partial\mathcal{B}$
from a Lagrangian energy functional (2.75) including the boundary relations
between the boundary tractions and the surface stresses.

In Section 3, we present a model hemitropic, quadratic surface energy that
requires the same number of material constants (with the same physical
interpretations) as found in the Steigmann-Ogden theory (see (3.2)). In
contradistinction, however, the surface energy we propose satisfies the strong
ellipticity condition (see (3.12), (3.28)). We then derive the linearized
field equations governing infinitesimal displacements of $\mathcal{B}$ and
$\mathcal{S}$, including the boundary relations connecting the linearized
boundary tractions and the linearized surface stresses.

Finally, in Section 4, we apply the linearized theory (3.46) to the problem of
a brittle infinite solid with a straight, non-interfacial crack of finite
length, under mode-III loading. Using the explicit Dirichlet-to-Neumann map,
the problem is reduced to solving a fourth order integro-differential equation
for the crack profile along the boundary crack front (see (4.14)).
Analytically, it is the surface energy satisfying the strong ellipticity
condition that implies that this integro-differential equation is fourth order
in the surface derivative of the displacement.444Physically, it is the model
incorporating resistance to geodesic distortion that implies that this
integro-differential equation is fourth order in the surface derivative of the
displacement. The dimensionless form of the equation implies that the behavior
of the displacement depends on the size of the crack, and for macro cracks, we
expect the displacement to be well-approximated by the solution given by
classical linear elastic fracture mechanics except in small regions near the
crack tips (boundary layers). Finally, using the Lax-Milgram theorem and
regularity afforded by the presence of the fourth order derivative, we prove
that there always exist a unique classical solution to the governing integro-
differential equation, and in contrast to using the Steigmann-Ogden theory,
our model predicts bounded strains up to the crack tips (see Theorem 4.4).

### Acknowledgments

The author would like to thank Jay R. Walton for several fruitful discussions
related to fracture, surface elasticity and interfacial physics. He would also
like to thank Jeremy Marzuola for his help generating numerical solutions to
the mode-III fracture problem discussed in Section 4.

##  2\. Kinematics and field equations

In this section formulate a general mathematical model for a hyperelastic bulk
solid containing a boundary surface with strain-gradient surface elasticity.
For planar reference surfaces, the form of surface elasticity reduces to that
introduced by Hilgers and Pipkin [7], and when the surface energy is
independent of the surface gradient of the stretching, the theory reduces to
that of Steigmann and Ogden [22, 23].

###  2.1. Kinematics

Let $\mathbb{E}^{3}$ be three-dimensional Euclidean space with Cartesian
coordinates $(X^{1},X^{2},X^{3})=\boldsymbol{X}\in\mathbb{E}^{3}$ and
coordinate vector fields given by a fixed orthonormal basis
$\\{\boldsymbol{e}_{i}\\}_{i=1}^{3}$ of $\mathbb{R}^{3}$. We define the
following operations for elementary tensor products of vectors in
$\mathbb{R}^{3}$,

$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2})\boldsymbol{b}=(\boldsymbol{a}_{2}\cdot\boldsymbol{b})\boldsymbol{a}_{1},\quad(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})\boldsymbol{b}=(\boldsymbol{a}_{3}\cdot\boldsymbol{b})\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2},$
(2.1)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})(\boldsymbol{b}_{1}\otimes\boldsymbol{b}_{2})=(\boldsymbol{a}_{3}\cdot\boldsymbol{b}_{1})\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{b}_{2},$
(2.2)
$\displaystyle(\boldsymbol{b}_{1}\otimes\boldsymbol{b}_{2})(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})=(\boldsymbol{b}_{2}\cdot\boldsymbol{a}_{1})\boldsymbol{b}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3},$
(2.3)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})[\boldsymbol{b}_{1}\otimes\boldsymbol{b}_{2}]=(\boldsymbol{a}_{2}\cdot\boldsymbol{b}_{1})(\boldsymbol{a}_{3}\cdot\boldsymbol{b}_{2})\boldsymbol{a}_{1},$
(2.4)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2})[\boldsymbol{b}_{1}\otimes\boldsymbol{b}_{2}]=(\boldsymbol{a}_{1}\cdot\boldsymbol{b}_{1})(\boldsymbol{a}_{2}\cdot\boldsymbol{b}_{2}),$
(2.5)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})[\boldsymbol{b}_{1}\otimes\boldsymbol{b}_{2}\otimes\boldsymbol{b}_{3}]=(\boldsymbol{a}_{1}\cdot\boldsymbol{b}_{1})(\boldsymbol{a}_{2}\cdot\boldsymbol{b}_{2})(\boldsymbol{a}_{3}\cdot\boldsymbol{b}_{3}),$
(2.6)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2})^{T}=\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{1},\quad(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})^{T}=\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{3}\otimes\boldsymbol{a}_{2},$
(2.7)
$\displaystyle(\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{3})^{\sim}=\boldsymbol{a}_{2}\otimes\boldsymbol{a}_{1}\otimes\boldsymbol{a}_{3}.$
(2.8)

These operations are extended to general second and third order tensors on
$\mathbb{R}^{3}$ by linearity.

Let $\mathcal{B}\subseteq\mathbb{E}^{3}$ be a domain with smooth boundary
$\partial\mathcal{B}$, the reference configuration of a three-dimensional
body, and let $\mathcal{S}\subseteq\partial\mathcal{B}$ be a closed surface
with smooth boundary. Let
$\boldsymbol{\chi}:\mathcal{B}\rightarrow\boldsymbol{\chi}(\mathcal{B})\subseteq\mathbb{E}^{3}$
be a smooth, invertible deformation of $\mathcal{B}$. We denote the current
position of the reference particle $\boldsymbol{X}\in\mathcal{B}$ by

$\displaystyle\boldsymbol{x}=\boldsymbol{\chi}(\boldsymbol{X})=(\chi^{1}(\boldsymbol{X}),\chi^{2}(\boldsymbol{X}),\chi^{3}(\boldsymbol{X})).$
(2.9)

The deformation gradient
$\boldsymbol{F}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is the second order
tensor field

$\displaystyle\boldsymbol{F}$ $\displaystyle=\frac{\partial\chi^{i}}{\partial
X^{a}}(\boldsymbol{X})\boldsymbol{e}_{i}\otimes\boldsymbol{e}^{a}$ (2.10)

where $\boldsymbol{e}^{a}:=\boldsymbol{e}_{a}$, $a=1,2,3$. We denote the left
Cauchy-Green stretch tensor by
$\boldsymbol{C}=\boldsymbol{F}^{T}\boldsymbol{F}$.

Let $\boldsymbol{Y}=\hat{\boldsymbol{Y}}(\theta^{1},\theta^{2})$ be a local
parameterization of the reference surface
$\mathcal{S}\subseteq\partial\mathcal{B}$. Then

$\displaystyle\boldsymbol{y}=\hat{\boldsymbol{y}}(\theta^{1},\theta^{2})=\boldsymbol{\chi}(\hat{\boldsymbol{Y}}(\theta^{1},\theta^{2}))$
(2.11)

is a local parameterization of the current surface
$\boldsymbol{\chi}(\mathcal{S})\subseteq\boldsymbol{\chi}(\partial\mathcal{B})$.
The (local) tangent vector fields on the reference and current surfaces are
then given by

$\displaystyle\boldsymbol{Y}_{,\alpha}\in
T_{\boldsymbol{Y}}\mathcal{S},\quad\boldsymbol{y}_{,\alpha}=\boldsymbol{F}\boldsymbol{Y}_{,\alpha}\in
T_{\boldsymbol{y}}\boldsymbol{\chi}(\mathcal{S}),$ (2.12)

where $\cdot_{,\alpha}:=\frac{\partial}{\partial\theta^{\alpha}}$. The dual
tangent vector fields on the reference and current surfaces are denoted by
$\boldsymbol{Y}^{,\beta}$ and $\boldsymbol{y}^{,\beta}$ respectively and
satisfy

$\displaystyle\boldsymbol{Y}^{,\beta}\cdot\boldsymbol{Y}_{,\alpha}=\tensor{\delta}{{}^{\beta}_{\alpha}},\quad\boldsymbol{y}^{,\beta}\cdot\boldsymbol{y}_{,\alpha}=\tensor{\delta}{{}^{\beta}_{\alpha}}$

We remark that we may also write

$\displaystyle\boldsymbol{y}_{,\alpha}=\boldsymbol{\mathsf{F}}\boldsymbol{Y}_{,\alpha}$
(2.13)

where
$\boldsymbol{\mathsf{F}}=\tensor{F}{{}^{k}_{\beta}}\boldsymbol{e}_{k}\otimes\boldsymbol{Y}^{,\beta}:=y^{k}_{,\beta}\boldsymbol{e}_{k}\otimes\boldsymbol{Y}^{,\beta}=\boldsymbol{y}_{,\beta}\otimes\boldsymbol{Y}^{,\beta}$
is the _surface deformation gradient_. The first fundamental forms for the
reference and current surfaces are then given by

$\displaystyle\boldsymbol{\mathsf{G}}=\mathsf{G}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta},\quad\mathsf{G}_{\alpha\beta}=\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{Y}_{,\beta},$
(2.14)
$\displaystyle\boldsymbol{\mathsf{g}}=\mathsf{g}_{\alpha\beta}\boldsymbol{y}^{,\alpha}\otimes\boldsymbol{y}^{,\beta},\quad\mathsf{g}_{\alpha\beta}=\boldsymbol{y}_{,\alpha}\cdot\boldsymbol{y}_{,\beta},$
(2.15)

and we note that

$\displaystyle\boldsymbol{Y}^{,\alpha}=(\mathsf{G}^{-1})^{\alpha\beta}\boldsymbol{Y}_{,\beta},\quad\boldsymbol{y}^{,\alpha}=(\mathsf{g}^{-1})^{\alpha\beta}\boldsymbol{y}_{,\beta}$
(2.16)

The Christoffel symbols of the second kind for the reference and current
surfaces are denoted by

$\displaystyle\tensor{\Gamma}{{}^{\alpha}}{{}_{\beta\delta}}=\boldsymbol{Y}^{,\alpha}\cdot\boldsymbol{Y}_{,\beta\delta},\quad\tensor{\gamma}{{}^{\alpha}}{{}_{\beta\delta}}=\boldsymbol{y}^{,\alpha}\cdot\boldsymbol{y}_{,\beta\delta}.$
(2.17)

The _left Cauchy-Green surface stretch tensor_ on $\mathcal{S}$ is the second
order tensor

$\displaystyle\boldsymbol{\mathsf{C}}:=\boldsymbol{\mathsf{F}}^{T}\boldsymbol{\mathsf{F}}=\mathsf{g}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta},$
(2.18)

and the _s_ urface Green-St. Venant tensor is the second order tensor
$\boldsymbol{\mathsf{E}}=\frac{1}{2}(\boldsymbol{\mathsf{C}}-\boldsymbol{\mathsf{G}})$.
We assume that the reference and current surfaces are orientable with unit
normals to the reference and current surfaces (locally) given by

$\displaystyle\boldsymbol{N}=|\boldsymbol{Y}_{,1}\times\boldsymbol{Y}_{,2}|^{-1}\boldsymbol{Y}_{,1}\times\boldsymbol{Y}_{,2},\quad\boldsymbol{n}=|\boldsymbol{y}_{,1}\times\boldsymbol{y}_{,2}|^{-1}\boldsymbol{y}_{,1}\times\boldsymbol{y}_{,2}$
(2.19)

respectively. The second fundamental forms on the reference and current
surfaces are given by

$\displaystyle\boldsymbol{\mathsf{B}}=\mathsf{B}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta},\quad\mathsf{B}_{\alpha\beta}:=\boldsymbol{Y}_{,\alpha\beta}\cdot\boldsymbol{N},$
(2.20)
$\displaystyle\boldsymbol{\mathsf{b}}=\mathsf{b}_{\alpha\beta}\boldsymbol{y}^{,\alpha}\otimes\boldsymbol{y}^{,\beta},\quad\mathsf{b}_{\alpha\beta}:=\boldsymbol{y}_{,\alpha\beta}\cdot\boldsymbol{n}.$
(2.21)

The _relative curvature tensor_ on $\mathcal{S}$ is the second order tensor
$\boldsymbol{\mathsf{K}}=\mathsf{K}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}$
given by

$\displaystyle\boldsymbol{\mathsf{K}}=\boldsymbol{\mathsf{F}}^{T}\boldsymbol{\mathsf{b}}\boldsymbol{\mathsf{F}}-\boldsymbol{\mathsf{B}}=[\mathsf{b}_{\alpha\beta}-\mathsf{B}_{\alpha\beta}]\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}.$
(2.22)

As discussed by Steigmann and Ogden [23], $\boldsymbol{\mathsf{E}}$ and
$\boldsymbol{\mathsf{K}}$ furnish local differences in length and scaled
extrinsic normal curvature between a given curve on $\mathcal{S}$ and the
convected curve on $\boldsymbol{\chi}(\mathcal{S})$, where
$\dot{\mbox{}}:=\frac{d}{ds}$. Indeed, if $\boldsymbol{Z}(s)$ is a a curve on
$\mathcal{S}$ with unit tangent vector field $\boldsymbol{\mathsf{T}}$ and
$\boldsymbol{z}(s)=\boldsymbol{\chi}(\boldsymbol{Z}(s))$ is the convected
curve, then the tangent to the convected curve is given by
$\boldsymbol{\mathsf{t}}=\dot{\boldsymbol{z}}=\boldsymbol{F}\boldsymbol{\mathsf{T}}=\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}$.
The _stretch_ $\nu$ of the convected curve is

$\displaystyle\nu=|\boldsymbol{\mathsf{t}}|^{2}-1=\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}-1=\boldsymbol{\mathsf{C}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}-1=2\boldsymbol{\mathsf{E}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}.$
(2.23)

The extrinsic normal curvature of the convected curve is

$\displaystyle\kappa=\boldsymbol{\mathsf{b}}\boldsymbol{\mathsf{t}}\cdot\boldsymbol{\mathsf{t}}=|\boldsymbol{\mathsf{t}}|^{-2}\boldsymbol{\mathsf{b}}\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}=(\nu+1)^{-1}(\boldsymbol{\mathsf{F}}^{T}\boldsymbol{\mathsf{b}}\boldsymbol{\mathsf{F}})\boldsymbol{\mathsf{T}},$
(2.24)

and thus, the difference in scaled extrinsic normal curvature of the convected
curve and the original curve is given by

$\displaystyle|\boldsymbol{\mathsf{t}}|^{2}\kappa-|\boldsymbol{\mathsf{T}}|^{2}\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}=\boldsymbol{\mathsf{K}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}.$
(2.25)

The Levi-Cevita connection on $\mathcal{S}$ is denoted by $\nabla$, so

$\displaystyle\nabla\boldsymbol{\mathsf{E}}=\frac{1}{2}\nabla\boldsymbol{\mathsf{C}}=\frac{1}{2}\nabla_{\delta}\mathsf{g}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}\otimes\boldsymbol{Y}^{,\delta},$
(2.26)

where

$\displaystyle\nabla_{\delta}\mathsf{g}_{\alpha\beta}:=\mathsf{g}_{\alpha\beta,\delta}-\tensor{\Gamma}{{}^{\mu}}{{}_{\alpha\delta}}\mathsf{g}_{\mu\beta}-\tensor{\Gamma}{{}^{\mu}}{{}_{\beta\delta}}\mathsf{g}_{\mu\alpha}.$
(2.27)

For later use, we define the following third order tensor on $\mathcal{S}$,

$\displaystyle\boldsymbol{\mathsf{L}}=\nabla\boldsymbol{\mathsf{E}}+(\nabla\boldsymbol{\mathsf{E}})^{T}-(\nabla\boldsymbol{\mathsf{E}}^{\sim})^{T},$
(2.28)

with components

$\displaystyle\mathsf{L}_{\alpha\beta\delta}=\frac{1}{2}(\nabla_{\delta}\mathsf{g}_{\alpha\beta}+\nabla_{\beta}\mathsf{g}_{\alpha\delta}-\nabla_{\alpha}\mathsf{g}_{\beta\delta})=(\tensor{\gamma}{{}^{\mu}_{\beta}{}_{\delta}}-\tensor{\Gamma}{{}^{\mu}_{\beta}{}_{\delta}})\mathsf{g}_{\mu\alpha}.$
(2.29)

Physically, the tensor $\boldsymbol{\mathsf{L}}$ (and thus
$\nabla\boldsymbol{\mathsf{E}}$) furnishes the rate of stretching of convected
geodesics, and more generally, it quantifies _geodesic distortion_ , i.e. how
convected geodesics fail to be geodesics on $\boldsymbol{\chi}(\mathcal{S})$.
More precisely, we have the following.

######  Proposition 2.1.

Let $\boldsymbol{Z}(s):I\rightarrow\mathcal{S}$ be a geodesic on $\mathcal{S}$
with unit tangent vector field $\boldsymbol{\mathsf{T}}=\dot{\boldsymbol{Z}}$.
Then $\boldsymbol{\mathsf{L}}$ yields the rate of stretching of the convected
curve $\boldsymbol{z}(s)=\boldsymbol{\chi}(\boldsymbol{Z}(s))$,

$\displaystyle
2\boldsymbol{\mathsf{L}}[\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}]=\frac{d}{ds}|\dot{\boldsymbol{z}}|^{2}.$
(2.30)

Moreover, the convected curve $\boldsymbol{z}(s)$ is a geodesic on
$\boldsymbol{\chi}(\mathcal{S})$ if and only if for each $s_{0}\in I$,

$\displaystyle\forall\boldsymbol{\mathsf{U}}\in
T_{\boldsymbol{Z}(s_{0})}\mathcal{S},\quad\boldsymbol{\mathsf{L}}[\boldsymbol{\mathsf{U}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}]\big{|}_{s=s_{0}}=0.$
(2.31)

###### Proof.

To prove (2.30), we simply note that since $\boldsymbol{Z}$ is a geodesic,
$\nabla_{\boldsymbol{\mathsf{T}}}\boldsymbol{\mathsf{T}}=\boldsymbol{0}$ so

$\displaystyle\frac{d}{ds}|\dot{\boldsymbol{z}}|^{2}$
$\displaystyle=\frac{d}{ds}[\boldsymbol{\mathsf{C}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}]$
(2.32)
$\displaystyle=\nabla\boldsymbol{\mathsf{C}}[\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}]+2\boldsymbol{\mathsf{C}}\nabla_{\boldsymbol{\mathsf{T}}}\boldsymbol{\mathsf{T}}\cdot\boldsymbol{\mathsf{T}}$
(2.33)
$\displaystyle=\nabla\boldsymbol{\mathsf{C}}[\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}]=2\boldsymbol{\mathsf{L}}[\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}].$
(2.34)

We now prove (2.31). The convected curve $\boldsymbol{z}$ on
$\boldsymbol{\chi}(\mathcal{S})$ has tangent vector field
$\boldsymbol{\mathsf{t}}=\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{T}}=\mathsf{t}^{\mu}\boldsymbol{y}_{,\mu}$
and acceleration vector field

$\displaystyle\boldsymbol{\mathsf{a}}=\bigl{(}\dot{\mathsf{t}}^{\mu}+\tensor{\gamma}{{}^{\mu}_{\beta}{}_{\delta}}\mathsf{t}^{\beta}\mathsf{t}^{\delta}\bigr{)}\boldsymbol{y}_{,\mu}.$
(2.35)

The acceleration is zero and $\boldsymbol{z}$ is a geodesic if and only if for
each $s_{0}\in I$,

$\displaystyle\forall\boldsymbol{\mathsf{u}}\in
T_{\boldsymbol{z}(s_{0})}\boldsymbol{\chi}(\mathcal{S}),\quad\boldsymbol{\mathsf{u}}\cdot\boldsymbol{\mathsf{a}}|_{s=s_{0}}=0.$
(2.36)

We now show that (2.36) is equivalent to (2.31). We assume without loss of
generality that $s_{0}=0$, and we choose normal coordinates
$(\theta^{1},\theta^{2})$ centered at $\boldsymbol{Z}(0)$. Then for all
$\alpha,\beta,\delta=1,2$,

$\displaystyle\mathsf{G}_{\alpha\beta}|_{s=0}=\delta_{\alpha\beta},\quad\tensor{\Gamma}{{}^{\mu}_{\beta}{}_{\delta}}|_{s=0}=0,$
(2.37)

and there exist $\mathsf{T}^{1},\mathsf{T}^{2}\in\mathbb{R}$ with
$\delta_{\alpha\beta}\mathsf{T}^{\alpha}\mathsf{T}^{\beta}=1$ such that
$\boldsymbol{Z}(s)=\hat{\boldsymbol{Y}}(s\mathsf{T}^{1},s\mathsf{T}^{2})$. In
particular, we conclude that

$\displaystyle\boldsymbol{\mathsf{a}}=\tensor{\gamma}{{}^{\mu}_{\beta}{}_{\delta}}\mathsf{T}^{\beta}\mathsf{T}^{\delta}\boldsymbol{y}_{,\mu}.$
(2.38)

Since
$\boldsymbol{\mathsf{F}}|_{\boldsymbol{Z}(0)}:T_{\boldsymbol{Z}(0)}\mathcal{S}\rightarrow
T_{\boldsymbol{z}(0)}\boldsymbol{\chi}(\mathcal{S})$ is an isomorphism, (2.36)
is equivalent to

$\displaystyle\forall\boldsymbol{\mathsf{U}}=\mathsf{U}^{\mu}\boldsymbol{Y}_{,\mu}|_{\boldsymbol{Z}(0)}\in
T_{\boldsymbol{Z}(0)}\mathcal{S},\quad\boldsymbol{\mathsf{F}}\boldsymbol{\mathsf{U}}\cdot\boldsymbol{\mathsf{a}}\big{|}_{s=0}=0$
(2.39)
$\displaystyle\iff\forall\boldsymbol{\mathsf{U}}=\mathsf{U}^{\mu}\boldsymbol{Y}_{,\mu}|_{\boldsymbol{Z}(0)}\in
T_{\boldsymbol{Z}(0)}\mathcal{S},\quad\mathsf{g}_{\mu\alpha}\mathsf{U}^{\alpha}\tensor{\gamma}{{}^{\mu}_{\beta}{}_{\delta}}\mathsf{T}^{\beta}\mathsf{T}^{\delta}\big{|}_{s=0}=0$
(2.40)
$\displaystyle\iff\forall\boldsymbol{\mathsf{U}}=\mathsf{U}^{\mu}\boldsymbol{Y}_{,\mu}|_{\boldsymbol{Z}(0)}\in
T_{\boldsymbol{Z}(0)}\mathcal{S},\quad\boldsymbol{\mathsf{L}}[\boldsymbol{\mathsf{U}}\otimes\boldsymbol{\mathsf{T}}\otimes\boldsymbol{\mathsf{T}}]\big{|}_{s=0}=0.$
(2.41)

∎

As an illustrative example, consider

$\mathcal{S}=\\{(X^{1},X^{2},0)\mid
X^{1}\in[a,b],X^{2}\in[0,\pi]\\}\subseteq\mathcal{B}=[a,b]\times[0,\pi]\times[0,\infty),$

and the deformation

$\displaystyle\boldsymbol{\chi}(X^{1},X^{2},X^{3})=(e^{X^{1}}\cos
X^{2},e^{X^{1}}\sin X^{2},X^{3}).$ (2.42)

Then

$\displaystyle\boldsymbol{\mathsf{E}}=\frac{1}{2}(e^{2X^{1}}-1)\bigl{(}\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}\bigr{)},\quad\boldsymbol{\mathsf{K}}=\boldsymbol{0},$
(2.43)
$\displaystyle\boldsymbol{\mathsf{L}}=e^{2X^{1}}\bigl{(}\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}+\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{2}+\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{1}-\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}\bigr{)}.$
(2.44)

The image of the coordinate curve $X^{2}=d$ is a straight, radially outward
traveling curve in the $x^{1}x^{2}$-plane parameterized by $X^{1}\in[a,b]$
with

$\displaystyle\boldsymbol{\mathsf{L}}[\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}]=e^{2X^{1}},\quad\boldsymbol{\mathsf{L}}[\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{1}]=0.$
(2.45)

In particular, the stretch is not constant along the convected curve. The
image of the coordinate curve $X^{1}=c$ is the upper-half of the circle in the
$x^{1}x^{2}$-plane centered at $(0,0)$ of radius $e^{c}$. The convected curve
is parameterized by $X^{2}\in[0,\pi]$, has constant stretch but nonzero
curvature relative to the convected surface (i.e., the $x^{1}x^{2}$-plane),
and it satisfies

$\displaystyle\boldsymbol{\mathsf{L}}[\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}]=0,\quad\boldsymbol{\mathsf{L}}[\boldsymbol{e}_{1}\otimes\boldsymbol{e}_{2}\otimes\boldsymbol{e}_{2}]=-e^{2c}.$
(2.46)

###  2.2. Strain energy and material symmetry

In our mathematical model, a hyperelastic elastic body $\mathcal{B}$ with
strain-gradient elastic surface $\mathcal{S}\subseteq\partial\mathcal{B}$ is
prescribed a strain energy of the form

$\displaystyle\Phi[\boldsymbol{\chi}]=\int_{\mathcal{B}}W(\boldsymbol{C})\,dV+\int_{\mathcal{S}}U(\boldsymbol{\mathsf{E}},\boldsymbol{\mathsf{K}},\nabla\boldsymbol{\mathsf{E}})\,dA,$
(2.47)

where we omit listing the possible dependence of $W$ on
$\boldsymbol{X}\in\mathcal{B}$ and of $U$ on $\boldsymbol{Y}\in\mathcal{S}$.
We note that the strain energy is frame indifferent, i.e., it is invariant
with respect to super-imposed rigid motions.555In the case of $\mathcal{S}$
contained in the plane, Hilgers and Pipkin showed in Section 7 of [7] that
$U(\boldsymbol{\mathsf{E}},\boldsymbol{\mathsf{K}},\nabla\boldsymbol{\mathsf{E}})$
is the most general form of a surface energy density that is frame
indifferent.

Although our main motivation for considering a strain-gradient elastic surface
is when it forms part of the boundary of a bulk solid $\mathcal{B}$, we will
treat material symmetry of $\mathcal{S}$ independently of that of
$\mathcal{B}$. The notion of material symmetry for the energy density $W$ of
the bulk three-dimensional solid $\mathcal{B}$ is well-known, see [18, 24],
so, we will limit our discussion to that of $\mathcal{S}$.

Consider a material point with reference position
$\boldsymbol{Y}_{0}\in\mathcal{S}$. Our discussion of material symmetry for
the surface energy per unit reference area $U$ at $\boldsymbol{Y}_{0}$ is
inspired by the framework introduced by Murdoch and Cohen [16, 17] which was
later advocated for and summarized by Steigmann and Ogden [23] using local
coordinate parameterizations. We will follow Steigmann and Ogden’s style of
exposition, and unless specified otherwise, all quantities in what follows are
evaluated at $\boldsymbol{Y}_{0}$.

Let $\boldsymbol{\lambda}:\mathbb{E}^{3}\rightarrow\mathbb{E}^{3}$ be a rigid
motion with deformation gradient $\boldsymbol{R}\in SO(3)$ satisfying
$\boldsymbol{\lambda}(\boldsymbol{Y}_{0})=\boldsymbol{Y}_{0},$ and
$\boldsymbol{R}\boldsymbol{N}=\boldsymbol{N}$. We define a second reference
surface
$\mathcal{S}^{*}=\bigl{\\{}\boldsymbol{Y}^{*}=\boldsymbol{\lambda}^{-1}(\boldsymbol{Y})\mid\boldsymbol{Y}\in\mathcal{S}\bigr{\\}}$.
It follows that
$T_{\boldsymbol{Y}_{0}}\mathcal{S}^{*}=T_{\boldsymbol{Y}_{0}}\mathcal{S}$, at
$\boldsymbol{Y}_{0}\in\mathcal{S}^{*}$ the unit normal $\boldsymbol{N}^{*}$ to
$\mathcal{S}^{*}$ satisfies
$\boldsymbol{N}^{*}=\boldsymbol{N}=\boldsymbol{R}\boldsymbol{N}$, and

$\displaystyle\boldsymbol{\mathsf{R}}:=\boldsymbol{R}|_{\boldsymbol{T}_{\boldsymbol{Y}_{0}}\mathcal{S}}:T_{\boldsymbol{Y}_{0}}\mathcal{S}\rightarrow
T_{\boldsymbol{Y}_{0}}\mathcal{S}$ (2.48)

is a rotation. A local parameterization on $\mathcal{S}^{*}$ is given by
$\boldsymbol{Y}^{*}=\hat{\boldsymbol{Y}}^{*}(\theta^{1},\theta^{2}):=\boldsymbol{\lambda}^{-1}(\hat{\boldsymbol{Y}}(\theta^{1},\theta^{2}))$,
so then

$\displaystyle\boldsymbol{Y}_{,\alpha}=\boldsymbol{R}\boldsymbol{Y}^{*}_{,\alpha},\quad\boldsymbol{Y}^{,\alpha}=\boldsymbol{R}\boldsymbol{Y}^{*,\alpha},\quad\alpha=1,2,$
(2.49)

and at $\boldsymbol{Y}_{0}$,
$\boldsymbol{Y}_{,\alpha}=\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*}_{,\alpha},$
$\boldsymbol{Y}^{,\alpha}=\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*,\alpha}$,
for $\alpha=1,2$.

Let $\boldsymbol{\chi}:\mathbb{E}^{3}\rightarrow\mathbb{E}^{3}$ be a smooth
invertible deformation of Euclidean space. Since a superimposed rigid motion
does not affect the value of the surface energy, we will assume without loss
of generality that

$\displaystyle\boldsymbol{\chi}(\boldsymbol{Y}_{0})=\boldsymbol{Y}_{0}\mbox{
and at $\boldsymbol{Y}_{0}$,
}\boldsymbol{\mathsf{F}}:T_{\boldsymbol{Y}_{0}}\mathcal{S}\rightarrow
T_{\boldsymbol{Y}_{0}}\mathcal{S}.$ (2.50)

Following [17] we will impose the stronger requirement that

$\displaystyle\boldsymbol{\chi}(\boldsymbol{Y}_{0})=\boldsymbol{Y}_{0}\mbox{
and at $\boldsymbol{Y}_{0}$,
}\boldsymbol{F}=\tensor{\mathsf{F}}{{}^{\alpha}_{\beta}}\boldsymbol{Y}_{,\alpha}\otimes\boldsymbol{Y}^{,\beta}+\boldsymbol{N}\otimes\boldsymbol{N}.$
(2.51)

In particular, it follows that at $\boldsymbol{Y}_{0}$,
$\boldsymbol{\mathsf{F}}=\tensor{\mathsf{F}}{{}^{\alpha}_{\beta}}\boldsymbol{Y}_{,\alpha}\otimes\boldsymbol{Y}^{,\beta}$
and $\boldsymbol{n}=\boldsymbol{N}=\boldsymbol{F}\boldsymbol{N}$. The
deformation
$\boldsymbol{\chi}^{*}(\cdot):=\boldsymbol{\chi}(\boldsymbol{\lambda}(\cdot))$
when restricted to $\mathcal{S}^{*}$ has the same image as
$\boldsymbol{\chi}|_{\mathcal{S}}$, and in particular, the convected tangent
vector fields are the same,

$\displaystyle\boldsymbol{y}^{*}_{,\alpha}:=\frac{\partial}{\partial\theta^{\alpha}}\boldsymbol{\chi}^{*}(\boldsymbol{Y}^{*})=\frac{\partial}{\partial\theta^{\alpha}}\boldsymbol{\chi}(\boldsymbol{Y})=\boldsymbol{y}_{,\alpha}.$
(2.52)

Then the surface deformation gradients of $\boldsymbol{\chi}$ and
$\boldsymbol{\chi}^{*}$ at $\boldsymbol{Y}_{0}$ are related by

$\displaystyle\boldsymbol{\mathsf{F}}=\boldsymbol{y}_{\alpha}\otimes\boldsymbol{Y}^{,\alpha}=\boldsymbol{y}^{*}_{\alpha}\otimes\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*,\alpha}=\boldsymbol{\mathsf{F}}^{*}\boldsymbol{\mathsf{R}}^{T}\implies\boldsymbol{\mathsf{C}}=\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{C}}^{*}\boldsymbol{\mathsf{R}}^{T}.$
(2.53)

The components of the first and second fundamental forms associated to
$\mathcal{S}$ and $\mathcal{S}^{*}$ satisfy

$\displaystyle\mathsf{G}_{\alpha\beta}=\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{Y}_{,\beta}=\boldsymbol{R}\boldsymbol{Y}^{*}_{,\alpha}\cdot\boldsymbol{R}\boldsymbol{Y}^{*}_{\beta}=\boldsymbol{Y}^{*}_{,\alpha}\cdot\boldsymbol{Y}^{*}_{\beta}=\mathsf{G}^{*}_{\alpha\beta}\implies\boldsymbol{\mathsf{G}}=\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{G}}^{*}\boldsymbol{\mathsf{R}}^{T},$
(2.54)
$\displaystyle\mathsf{B}_{\alpha\beta}=\boldsymbol{Y}_{,\alpha\beta}\cdot\boldsymbol{N}=\boldsymbol{R}\boldsymbol{Y}^{*}_{,\alpha\beta}\cdot\boldsymbol{R}\boldsymbol{N}^{*}=\boldsymbol{Y}^{*}_{,\alpha\beta}\cdot\boldsymbol{N}^{*}=\mathsf{B}^{*}_{\alpha\beta}\implies\boldsymbol{\mathsf{B}}=\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{B}}^{*}\boldsymbol{\mathsf{R}}^{T},$
(2.55)

and thus,
$\boldsymbol{\mathsf{E}}=\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{E}}^{*}\boldsymbol{\mathsf{R}}^{T}$.
Since
$\boldsymbol{\mathsf{b}}=\mathsf{b}_{\alpha\beta}\boldsymbol{y}^{,\alpha}\otimes\boldsymbol{y}^{\beta}$
is the same for both deformations, we conclude that the relative curvature
tensors $\boldsymbol{\mathsf{K}}$ and $\boldsymbol{\mathsf{K}}^{*}$ associated
to $\boldsymbol{\chi}$ and $\boldsymbol{\chi}^{*}$ satisfy at
$\boldsymbol{Y}_{0}$,

$\displaystyle\boldsymbol{\mathsf{K}}=\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{K}}^{*}\boldsymbol{\mathsf{R}}^{T}.$
(2.56)

Finally, since the components of the first fundamental forms associated to
$\boldsymbol{\chi}(\mathcal{S})$ and $\boldsymbol{\chi}^{*}(\mathcal{S}^{*})$
satisfy

$\displaystyle\mathsf{g}_{\alpha\beta}=\boldsymbol{y}_{\alpha}\cdot\boldsymbol{y}_{\beta}=\boldsymbol{y}^{*}_{\alpha}\cdot\boldsymbol{y}^{*}_{\beta}=:\mathsf{g}^{*}_{\alpha\beta},$
(2.57)

we conclude that $\nabla\boldsymbol{\mathsf{E}}$ and
$\nabla^{*}\boldsymbol{\mathsf{E}}^{*}$ at $\boldsymbol{Y}_{0}$ satisfy

$\displaystyle\nabla\boldsymbol{\mathsf{E}}$
$\displaystyle=\frac{1}{2}\nabla_{\delta}\mathsf{g}_{\alpha\beta}\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}\otimes\boldsymbol{Y}^{,\delta}$
(2.58)
$\displaystyle=\frac{1}{2}\nabla^{*}_{\delta}\mathsf{g}_{\alpha\beta}^{*}\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*,\alpha}\otimes\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*,\beta}\otimes\boldsymbol{\mathsf{R}}\boldsymbol{Y}^{*,\delta}$
(2.59)
$\displaystyle=\boldsymbol{\mathsf{R}}[(\nabla^{*}\boldsymbol{\mathsf{E}}^{*})^{T}\boldsymbol{\mathsf{R}}^{T}]^{T}\boldsymbol{\mathsf{R}}^{T}.$
(2.60)

We denote the surface energy per unit reference area relative to
$\mathcal{S}^{*}$ by $U^{*}$. For the surface energy per unit mass to be
independent of the reference surface used, we must have

$\displaystyle
U^{*}(\boldsymbol{\mathsf{E}}^{*},\boldsymbol{\mathsf{K}}^{*},\nabla^{*}\boldsymbol{\mathsf{E}}^{*})=U(\boldsymbol{\mathsf{E}},\boldsymbol{\mathsf{H}},\nabla\boldsymbol{\mathsf{E}})$
(2.61)
$\displaystyle=U\Bigl{(}\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{E}}^{*}\boldsymbol{\mathsf{R}}^{T},\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{K}}^{*}\boldsymbol{\mathsf{R}}^{T},\boldsymbol{\mathsf{R}}\bigl{[}(\nabla^{*}\boldsymbol{\mathsf{E}}^{*})^{T}\boldsymbol{\mathsf{R}}^{T}\bigr{]}^{T}\boldsymbol{\mathsf{R}}^{T}\Bigr{)}.$
(2.62)

We now view $\boldsymbol{\chi}$ also as a deformation of $\mathcal{S}^{*}$,
but we denote its values by $\bar{\boldsymbol{\chi}}$,

$\displaystyle\bar{\boldsymbol{\chi}}(\boldsymbol{X})={\boldsymbol{\chi}}(\boldsymbol{X}),\quad\boldsymbol{X}\in\mathbb{E}^{3},$
(2.63)

and the associated kinematic variables relative to $\mathcal{S}^{*}$ are
denoted with an over-bar. We will now derive relationships between
$\bar{\boldsymbol{\mathsf{E}}}$, $\bar{\boldsymbol{\mathsf{K}}}$,
$\nabla^{*}\bar{\boldsymbol{\mathsf{E}}}$ and $\boldsymbol{\mathsf{E}}$,
$\boldsymbol{\mathsf{K}}$, $\nabla\boldsymbol{\mathsf{E}}$.

We first note that since ${\boldsymbol{C}}=\bar{\boldsymbol{C}}$, we
immediately conclude that

$\displaystyle\boldsymbol{\mathsf{E}}$
$\displaystyle=\frac{1}{2}[(\boldsymbol{C}-\boldsymbol{I})\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{Y}_{,\beta}]\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}$
(2.64)
$\displaystyle=\frac{1}{2}[(\bar{\boldsymbol{C}}-\boldsymbol{I})\boldsymbol{Y}_{,\alpha}^{*}\cdot\boldsymbol{Y}_{,\beta}^{*}]\boldsymbol{Y}^{*,\alpha}\otimes\boldsymbol{Y}^{*,\beta}=\bar{\boldsymbol{\mathsf{E}}}.$
(2.65)

Let

$\displaystyle\mbox{Grad}\,{\boldsymbol{C}}=\frac{\partial C_{ij}}{\partial
X^{a}}\boldsymbol{e}^{i}\otimes\boldsymbol{e}^{j}\otimes\boldsymbol{e}^{a},$
(2.66)

a third order tensor on $\mathbb{R}^{3}$. The identity
$\mathsf{g}_{\alpha\beta}=\boldsymbol{C}\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{Y}_{,\beta}$,
the Gauss equations
$\boldsymbol{Y}_{,\alpha\delta}=\tensor{\Gamma}{{}^{\mu}}{{}_{\alpha\delta}}\boldsymbol{Y}_{,\mu}+\mathsf{B}_{\alpha\delta}\boldsymbol{N}$
on $\mathcal{S}$, and the symmetry of $\boldsymbol{C}$ imply that

$\displaystyle\mathsf{g}_{\alpha\beta,\delta}$
$\displaystyle=\mbox{Grad}\,\boldsymbol{C}[\boldsymbol{Y}_{,\alpha}\otimes\boldsymbol{Y}_{,\beta}\otimes\boldsymbol{Y}_{,\delta}]+\boldsymbol{C}\boldsymbol{Y}_{,\alpha\delta}\cdot\boldsymbol{Y}_{,\beta}+\boldsymbol{C}\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{Y}_{,\beta\delta}$
(2.67)
$\displaystyle=\mbox{Grad}\,\boldsymbol{C}[\boldsymbol{Y}_{,\alpha}\otimes\boldsymbol{Y}_{,\beta}\otimes\boldsymbol{Y}_{,\delta}]+\tensor{\Gamma}{{}^{\mu}}{{}_{\alpha\delta}}\mathsf{g}_{\mu\beta}+\mathsf{B}_{\alpha\delta}\boldsymbol{C}\boldsymbol{N}\cdot\boldsymbol{Y}_{,\beta}$
(2.68)
$\displaystyle\quad+\tensor{\Gamma}{{}^{\mu}}{{}_{\beta\delta}}\mathsf{g}_{\mu\alpha}+\mathsf{B}_{\beta\delta}\boldsymbol{C}\boldsymbol{N}\cdot\boldsymbol{Y}_{,\alpha}.$
(2.69)

By (2.51), we have $\boldsymbol{C}\boldsymbol{N}=\boldsymbol{N}$, and thus, at
$\boldsymbol{Y}_{0}$,

$\displaystyle\nabla\boldsymbol{\mathsf{E}}$
$\displaystyle=\frac{1}{2}\mbox{Grad}\,\boldsymbol{C}[\boldsymbol{Y}_{,\alpha}\otimes\boldsymbol{Y}_{,\beta}\otimes\boldsymbol{Y}_{,\delta}]\boldsymbol{Y}^{,\alpha}\otimes\boldsymbol{Y}^{,\beta}\otimes\boldsymbol{Y}^{,\delta}.$
(2.70)

In particular, since $\bar{\boldsymbol{C}}=\boldsymbol{C}$ we conclude that

$\displaystyle\nabla^{*}\bar{\boldsymbol{\mathsf{E}}}=\nabla\boldsymbol{\mathsf{E}}.$
(2.71)

Finally, as shown in Section 6 of [23], we have

$\displaystyle\bar{\boldsymbol{\mathsf{K}}}=\boldsymbol{\mathsf{K}}.$ (2.72)

Inspired by Murdoch and Cohen’s extension of Noll’s theory of material
symmetry, we say that $\boldsymbol{Y}_{0}\in\mathcal{S}$ is symmetry related
to $\boldsymbol{Y}_{0}\in\mathcal{S}^{*}$ if the mechanical responses to the
arbitrary deformation $\boldsymbol{\chi}$ are identical, i.e.,

$\displaystyle
U(\boldsymbol{\mathsf{E}},\boldsymbol{\mathsf{K}},\nabla\boldsymbol{\mathsf{E}})=U^{*}(\bar{\boldsymbol{\mathsf{E}}},\bar{\boldsymbol{\mathsf{K}}},\nabla^{*}\bar{\boldsymbol{\mathsf{E}}}).$
(2.73)

By (2.62), (2.65), (2.71), (2.72), this requirement leads to the following
definition: the rotation $\boldsymbol{\mathsf{R}}$ is in the _symmetry set_ of
$\boldsymbol{Y}_{0}$ relative to $\mathcal{S}$ if for every smooth, invertible
deformation $\boldsymbol{\chi}:\mathbb{E}^{3}\rightarrow\mathbb{E}^{3}$
satisfying (2.51), we have

$\displaystyle
U(\boldsymbol{\mathsf{E}},\boldsymbol{\mathsf{K}},\nabla\boldsymbol{\mathsf{E}})=U\Bigl{(}\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{E}}\boldsymbol{\mathsf{R}}^{T},\boldsymbol{\mathsf{R}}\boldsymbol{\mathsf{K}}\boldsymbol{\mathsf{R}}^{T},\boldsymbol{\mathsf{R}}\bigl{[}\nabla\boldsymbol{\mathsf{E}}^{T}\boldsymbol{\mathsf{R}}^{T}\bigr{]}^{T}\boldsymbol{\mathsf{R}}^{T}\Bigr{)}.$
(2.74)

As in the case of the standard theory of Noll [18], one can verify that the
symmetry set of $\boldsymbol{Y}_{0}$ relative to $\mathcal{S}$ is a subgroup
of the group of proper rotations of $T_{\boldsymbol{Y}_{0}}\mathcal{S}$. In
the case that the symmetry set of $\boldsymbol{Y}_{0}$ relative to
$\mathcal{S}$ equals the group of proper rotations of
$T_{\boldsymbol{Y}_{0}}\mathcal{S}$, we say that the surface energy density
$U$ is _hemitropic_ at $\boldsymbol{Y}_{0}$.

###  2.3. Field equations

The field equations for the body $\mathcal{B}$ with strain-gradient elastic
surface $\mathcal{S}\subseteq\partial\mathcal{B}$ are defined to be the Euler-
Lagrange equations for the Lagrangian energy functional

$\displaystyle\mathcal{A}[\boldsymbol{\chi}]=\Phi[\boldsymbol{\chi}]+V[\boldsymbol{\chi}]$
(2.75)

where $V[\boldsymbol{\chi}]$ is the potential energy associated to the applied
forces. In this work, we assume that the Gâteaux derivative of the load
potential takes the form

$\displaystyle\dot{V}=-\int_{\mathcal{B}}\boldsymbol{f}\cdot\boldsymbol{u}\,dV-\int_{\mathcal{S}}\boldsymbol{t}\cdot\boldsymbol{u}\,dA,$
(2.76)

where $\boldsymbol{f}$ is a prescribed external body force on $\mathcal{B}$
and $\boldsymbol{t}$ is a prescribed boundary traction on $\mathcal{S}$.

Let $\boldsymbol{\chi}(\cdot;\epsilon)$ be a one-parameter family of
deformations of $\mathcal{B}$ such that
$\boldsymbol{\chi}(\cdot;\epsilon)|_{\partial\mathcal{B}\backslash\mathcal{S}}=\boldsymbol{\chi}_{0}(\cdot)$,
and denote

$\displaystyle\dot{}:=\frac{d}{d\epsilon}\Big{|}_{\epsilon=0},\quad\boldsymbol{u}:=\dot{\boldsymbol{\chi}}(\cdot;\epsilon).$
(2.77)

Then $\boldsymbol{u}|_{\partial\mathcal{B}}$ vanishes to first order on
$\partial\mathcal{B}\backslash\mathcal{S}$. Using the chain rule and
integration by parts we have the classical identity

$\displaystyle\int_{\mathcal{B}}\dot{W}\,dV=\int_{\partial\mathcal{B}}\boldsymbol{P}\boldsymbol{N}\cdot\boldsymbol{u}\,dA-\int_{\mathcal{B}}\mbox{Div}\,\boldsymbol{P}\cdot\boldsymbol{u}\,dV$
(2.78)

where
$\boldsymbol{P}=\tensor{P}{{}_{i}^{a}}\boldsymbol{e}^{i}\otimes\boldsymbol{e}_{a}$
is the Piola stress with $\tensor{P}{{}_{i}^{a}}=\frac{\partial
W}{\partial\tensor{F}{{}^{i}_{a}}}$ and
$\mbox{Div}\,\boldsymbol{P}=\bigl{(}\partial_{X^{a}}\tensor{P}{{}_{i}^{a}}\big{)}\boldsymbol{e}^{i}$.
Using the chain rule we have that

$\displaystyle\int_{\mathcal{S}}\dot{U}\,dA=\int_{\mathcal{S}}\Bigl{(}\boldsymbol{\mathsf{T}}^{\alpha}\cdot\boldsymbol{u}_{,\alpha}+\boldsymbol{\mathsf{M}}^{\alpha\beta}\cdot\boldsymbol{u}_{,\alpha\beta}\Bigr{)}dA,$
(2.79) $\displaystyle\boldsymbol{\mathsf{T}}^{\alpha}:=\frac{\partial
U}{\partial
y^{k}_{\alpha}}\boldsymbol{e}^{k},\quad\boldsymbol{\mathsf{M}}^{\alpha\beta}:=\frac{\partial
U}{\partial y^{k}_{,\alpha\beta}}\boldsymbol{e}^{k}.$ (2.80)

We define surface stress vectors $\boldsymbol{\mathsf{P}}^{\alpha}$ by

$\displaystyle\quad\boldsymbol{\mathsf{P}}^{\alpha}:=\boldsymbol{\mathsf{T}}^{\alpha}-G^{-1/2}(G^{1/2}\boldsymbol{\mathsf{M}}^{\alpha\beta})_{,\beta},$
(2.81)

and use (2.78), (2.79) and integration by parts to obtain the Euler-Lagrange
equations associated to the Lagrangian energy functional (2.75),

$\displaystyle\begin{split}&\mbox{Div}\,\boldsymbol{P}+\boldsymbol{f}=\boldsymbol{0},\quad\mbox{on
}\mathcal{B},\\\
&\boldsymbol{P}\boldsymbol{N}=G^{-1/2}(G^{1/2}\boldsymbol{\mathsf{P}}^{\alpha})_{,\alpha}+\boldsymbol{t},\quad\mbox{on
}\mathcal{S},\\\
&\boldsymbol{\chi}(\boldsymbol{X})=\boldsymbol{\chi}_{0}(\boldsymbol{X}),\quad\mbox{on
}\partial\mathcal{B}\backslash\mathcal{S}.\end{split}$ (2.82)

##  3\. Small strain models

Our principle motivation for modeling an elastic solid with strain-gradient
elastic boundary surface is the study of brittle fracture. In this setting (to
be discussed more in the following section), the surface $\mathcal{S}$
possessing strain-gradient surface elasticity will be the crack front and
strains will be linearized, motivating the introduction of a quadratic surface
energy density. In this section, we present a model uniform, hemitropic,
quadratic surface energy density that requires the same material constants
(with the same physical interpretations) as found in the narrower Steigmann-
Ogden theory. In contradistinction, the surface energy incorporates the
surface’s resistance to geodesic distortion and satisfies the strong
ellipticity condition. Moreover, the surface energy density may be viewed as a
geometric generalization of that introduced and advocated for by Hilgers and
Pipkin in [7, 8].

###  3.1. Hilgers-Pipkin surface energy

For the surface $\mathcal{S}$, we propose the uniform, quadratic, hemitropic
surface energy density

$\displaystyle U$
$\displaystyle=\frac{\lambda_{s}}{2}(\tensor{\mathsf{E}}{{}^{\alpha}_{\alpha}})^{2}+\mu_{s}\mathsf{E}_{\alpha\beta}\mathsf{E}^{\alpha\beta}+\frac{\zeta}{2}\Bigl{[}(\tensor{\mathsf{K}}{{}^{\alpha}_{\alpha}})^{2}+(\mathsf{g}^{-1})^{\mu\nu}\tensor{\mathsf{L}}{{}_{\mu}{}_{\alpha}^{\alpha}}\tensor{\mathsf{L}}{{}_{\nu}{}_{\beta}^{\beta}}\Bigr{]}$
(3.1)
$\displaystyle\quad+\eta\Bigl{[}\tensor{\mathsf{K}}{{}_{\alpha}{}_{\beta}}\mathsf{K}^{\alpha\beta}+(\mathsf{g}^{-1})^{\mu\nu}\tensor{\mathsf{L}}{{}_{\mu}{}_{\alpha}{}_{\beta}}\tensor{\mathsf{L}}{{}_{\nu}^{\alpha}{}^{\beta}}\Bigr{]}.$
(3.2)

Here $\lambda_{s}$, $\mu_{s}$, $\zeta$ and $\eta$ are positive numbers that
can be interpreted as the surface Lamé constants and pure bending moduli.
Indices are raised and lowered using the reference metric
$\boldsymbol{\mathsf{G}}$, but note that $\boldsymbol{y}^{,\alpha}$ are the
dual vector fields to $\boldsymbol{y}_{,\alpha}$ and are not given by
$\boldsymbol{y}_{,\beta}(\mathsf{G}^{-1})^{\alpha\beta}$. In the case that
$\mathcal{S}$ is contained in a plane with flat coordinates
$(\theta^{1},\theta^{2})$, we have

$\displaystyle\mathsf{E}_{\alpha\beta}=\frac{1}{2}(\mathsf{g}_{\alpha\beta}-\delta_{\alpha\beta}),\quad\mathsf{L}_{\mu\alpha\beta}=\boldsymbol{y}_{,\mu}\cdot\boldsymbol{y}_{,\alpha\beta},$
(3.3)
$\displaystyle\boldsymbol{y}_{,\alpha\beta}=\mathsf{L}_{\mu\alpha\beta}\boldsymbol{y}^{,\mu}+\mathsf{K}_{\alpha\beta}\boldsymbol{n},\quad\sum_{\alpha=1}^{2}\boldsymbol{y}_{,\alpha\alpha}=\tensor{\mathsf{L}}{{}_{\mu}{}_{\alpha}^{\alpha}}\boldsymbol{y}^{,\mu}+\tensor{\mathsf{K}}{{}^{\alpha}_{\alpha}}\boldsymbol{n},$
(3.4)
$\displaystyle\Bigl{|}\sum_{\alpha=1}^{2}\boldsymbol{y}_{,\alpha\alpha}\Bigr{|}^{2}=(\tensor{\mathsf{K}}{{}^{\alpha}_{\alpha}})^{2}+(\mathsf{g}^{-1})^{\mu\nu}\tensor{\mathsf{L}}{{}_{\mu}{}_{\alpha}^{\alpha}}\tensor{\mathsf{L}}{{}_{\nu}{}_{\beta}^{\beta}},$
(3.5)
$\displaystyle\sum_{\alpha,\beta=1}^{2}|\boldsymbol{y}_{,\alpha\beta}|^{2}=\tensor{\mathsf{K}}{{}_{\alpha}{}_{\beta}}\mathsf{K}^{\alpha\beta}+(\mathsf{g}^{-1})^{\mu\nu}\tensor{\mathsf{L}}{{}_{\mu}{}_{\alpha}{}_{\beta}}\tensor{\mathsf{L}}{{}_{\nu}^{\alpha}{}^{\beta}},$
(3.6)

and (3.2) becomes

$\displaystyle
U=\frac{\lambda_{s}}{2}(\tensor{\mathsf{E}}{{}^{\alpha}_{\alpha}})^{2}+\mu_{s}\mathsf{E}_{\alpha\beta}\mathsf{E}^{\alpha\beta}+\frac{\zeta}{2}\Bigl{|}\sum_{\alpha=1}^{2}\boldsymbol{y}_{,\alpha\alpha}\Bigr{|}^{2}+\eta\sum_{\alpha,\beta=1}^{2}|\boldsymbol{y}_{,\alpha\beta}|^{2}.$
(3.7)

Up to a choice of constants, the surface energy density (3.7) is precisely
that introduced by Hilgers and Pipkin in [7], and therefore, we refer to (3.2)
as a _quadratic Hilgers-Pipkin surface energy_. In [7, 8], Hilgers and Pipkin
advocated for the use of (3.7) over the classical surface energy

$\displaystyle U$
$\displaystyle=\frac{\lambda_{s}}{2}(\tensor{\mathsf{E}}{{}^{\alpha}_{\alpha}})^{2}+\mu_{s}\mathsf{E}_{\alpha\beta}\mathsf{E}^{\alpha\beta}+\frac{\zeta}{2}\tensor{\mathsf{K}}{{}^{\alpha}_{\alpha}}+\eta\mathsf{K}_{\alpha\beta}\mathsf{K}^{\alpha\beta}$
(3.8)
$\displaystyle=\frac{\lambda_{s}}{2}(\tensor{\mathsf{E}}{{}^{\alpha}_{\alpha}})^{2}+\mu_{s}\mathsf{E}_{\alpha\beta}\mathsf{E}^{\alpha\beta}+\frac{\zeta}{2}\Bigl{|}\sum_{\alpha=1}^{2}(\boldsymbol{n}\cdot\boldsymbol{y}_{,\alpha\alpha})\Bigr{|}^{2}+\eta\sum_{\alpha,\beta=1}^{2}(\boldsymbol{n}\cdot\boldsymbol{y}_{,\alpha\beta})^{2}$
(3.9)

on the basis of (3.7) being analytically simpler than (3.9). Indeed, with
little effort one sees that for (3.7),

$\displaystyle\boldsymbol{\mathsf{T}}^{\alpha}$
$\displaystyle=(\lambda_{s}\tensor{\mathsf{E}}{{}^{\gamma}_{\gamma}}\delta^{\alpha\beta}+2\mu_{s}\mathsf{E}^{\alpha\beta})\boldsymbol{y}_{,\beta},$
(3.10) $\displaystyle\boldsymbol{\mathsf{M}}^{\alpha\beta}$
$\displaystyle=\zeta\Bigl{(}\sum_{\gamma}\boldsymbol{y}_{,\gamma\gamma}\Bigr{)}\delta^{\alpha\beta}+2\eta\boldsymbol{y}_{,\alpha\beta}.$
(3.11)

Moreover, it is simple to see that (3.7) satisfies the strong ellipticity
condition,

$\displaystyle\forall(\mathsf{a}_{1},\mathsf{a}_{2})\in\mathbb{R}^{2}\backslash\\{\boldsymbol{(}0,0)\\},\boldsymbol{b}\in\mathbb{R}^{3}\backslash\\{\boldsymbol{0}\\},\quad\mathsf{a}_{\alpha}\mathsf{a}_{\beta}\boldsymbol{b}\cdot\Bigl{(}\boldsymbol{C}^{\alpha\beta\delta\gamma}\mathsf{a}_{\delta}\mathsf{a}_{\gamma}\boldsymbol{b}\Bigr{)}>0,$
(3.12)
$\displaystyle\boldsymbol{C}^{\alpha\beta\gamma\delta}:=\frac{\partial^{2}U}{\partial
y^{i}_{,\alpha\beta}\partial
y^{j}_{,\delta\gamma}}\boldsymbol{e}^{i}\otimes\boldsymbol{e}^{j},$ (3.13)

while (3.9) does not (see (3.28) and (3.30) below). Physically, this may be
viewed as a consequence of the surface energy (3.2) incorporating the
surface’s resistance to geodesic distortion via also including dependence on
the tensor $\boldsymbol{\mathsf{L}}$.

In general, for (3.2) we have

$\displaystyle\boldsymbol{\mathsf{T}}^{\alpha}=\frac{\partial
U}{\partial\boldsymbol{y}_{,\alpha}}=\frac{\partial
U}{\partial\mathsf{E}_{\beta\gamma}}\frac{\partial\mathsf{E}_{\beta\gamma}}{\partial\boldsymbol{y}_{,\alpha}}+\frac{\partial
U}{\partial\mathsf{K}_{\delta\nu}}\frac{\partial\mathsf{K}_{\delta\nu}}{\partial\boldsymbol{y}_{,\alpha}}+\frac{\partial
U}{\partial\mathsf{L}_{\beta\gamma\delta}}\frac{\partial\mathsf{L}_{\beta\gamma\delta}}{\partial\boldsymbol{y}_{,\alpha}}.$
(3.14)

Using

$\displaystyle\frac{\partial\mathsf{E}_{\beta\gamma}}{\partial\boldsymbol{y}_{,\alpha}}=\frac{1}{2}\Bigl{(}\tensor{\delta}{{}^{\alpha}_{\beta}}\boldsymbol{y}_{,\gamma}+\tensor{\delta}{{}^{\alpha}_{\gamma}}\boldsymbol{y}_{,\beta}\Bigr{)},$
(3.15)
$\displaystyle\frac{\partial\mathsf{K}_{\mu\nu}}{\partial\boldsymbol{y}_{,\alpha}}=-\tensor{\gamma}{{}^{\alpha}_{\mu}{}_{\nu}}\boldsymbol{n},\quad\frac{\partial\mathsf{K}_{\mu\nu}}{\partial\boldsymbol{y}_{,\alpha\beta}}=\frac{1}{2}(\tensor{\delta}{{}^{\alpha}_{\mu}}\tensor{\delta}{{}^{\beta}_{\nu}}+\tensor{\delta}{{}^{\alpha}_{\nu}}\tensor{\delta}{{}^{\beta}_{\mu}})\boldsymbol{n},$
(3.16)
$\displaystyle\frac{\partial\mathsf{L}_{\beta\gamma\mu}}{\partial\boldsymbol{y}_{,\alpha}}=\tensor{\delta}{{}^{\alpha}_{\beta}}\boldsymbol{y}_{\gamma\mu}-(\tensor{\delta}{{}^{\alpha}_{\nu}}\boldsymbol{y}_{,\beta}+\tensor{\delta}{{}^{\alpha}_{\beta}}\boldsymbol{y}_{,\nu})\tensor{\Gamma}{{}^{\nu}_{\gamma}{}_{\mu}},$
(3.17)
$\displaystyle\frac{\partial\mathsf{L}_{\gamma\mu\sigma}}{\partial\boldsymbol{y}_{,\alpha\beta}}=\frac{1}{2}(\tensor{\delta}{{}^{\alpha}_{\mu}}\tensor{\delta}{{}^{\beta}_{\sigma}}+\tensor{\delta}{{}^{\alpha}_{\sigma}}\tensor{\delta}{{}^{\beta}_{\mu}})\boldsymbol{y}_{\gamma},$
(3.18)

we readily compute that

$\displaystyle\boldsymbol{\mathsf{T}}^{\alpha}$
$\displaystyle=\Bigl{[}\lambda_{s}\tensor{\mathsf{E}}{{}^{\mu}_{\mu}}(\mathsf{G}^{-1})^{\alpha\gamma}+2\mu_{s}\mathsf{E}^{\alpha\gamma}$
(3.19)
$\displaystyle\qquad-(\mathsf{g}^{-1})^{\mu\alpha}(\mathsf{g}^{-1})^{\nu\gamma}(\zeta\tensor{\mathsf{L}}{{}_{\mu}{}_{\delta}^{\delta}}\tensor{\mathsf{L}}{{}_{\nu}{}_{\sigma}^{\sigma}}+2\eta\tensor{\mathsf{L}}{{}_{\mu}{}_{\delta}{}_{\sigma}}\tensor{\mathsf{L}}{{}_{\nu}^{\delta}{}^{\sigma}})\Bigr{]}\boldsymbol{y}_{,\gamma},$
(3.20)
$\displaystyle\quad-\Bigl{[}\zeta\tensor{\mathsf{K}}{{}^{\mu}_{\mu}}(\mathsf{G}^{-1})^{\delta\nu}\tensor{\gamma}{{}^{\alpha}_{\delta}{}_{\nu}}+2\eta\mathsf{K}^{\delta\nu}\tensor{\gamma}{{}^{\alpha}_{\delta}{}_{\nu}}\Bigr{]}\boldsymbol{n},$
(3.21)
$\displaystyle\quad+\Bigl{[}\zeta(\mathsf{g}^{-1})^{\mu\alpha}\tensor{\mathsf{L}}{{}_{\mu}{}_{\nu}^{\nu}}(G^{-1})^{\gamma\delta}+2\eta(g^{-1})^{\mu\alpha}\tensor{\mathsf{L}}{{}_{\mu}^{\gamma}{}^{\delta}}\Bigr{]}\boldsymbol{y}_{,\gamma\delta}$
(3.22)
$\displaystyle\quad-\zeta\Bigl{[}(\mathsf{g}^{-1})^{\mu\alpha}\tensor{\mathsf{L}}{{}_{\mu}{}_{\nu}^{\nu}}\tensor{\Gamma}{{}^{\beta}_{\delta}{}^{\delta}}+(\mathsf{g}^{-1})^{\mu\beta}\tensor{\mathsf{L}}{{}_{\mu}{}_{\nu}^{\nu}}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}^{\delta}}\Bigr{]}\boldsymbol{y}_{,\beta}$
(3.23)
$\displaystyle\quad-2\eta\Bigl{[}(\mathsf{g}^{-1})^{\mu\alpha}\tensor{\mathsf{L}}{{}_{\mu}^{\delta}{}^{\sigma}}\tensor{\Gamma}{{}^{\beta}_{\delta}{}_{\sigma}}+(\mathsf{g}^{-1})^{\mu\beta}\tensor{\mathsf{L}}{{}_{\mu}^{\delta}{}^{\sigma}}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}_{\sigma}}\Bigr{]}\boldsymbol{y}_{,\beta}.$
(3.24)

and

$\displaystyle\boldsymbol{\mathsf{M}}^{\alpha\beta}$
$\displaystyle=\Bigl{[}\zeta\tensor{\mathsf{K}}{{}^{\mu}_{\mu}}(\mathsf{G}^{-1})^{\alpha\beta}+2\eta\mathsf{K}^{\alpha\beta}\Bigr{]}\boldsymbol{n}$
(3.25)
$\displaystyle\quad+\Bigl{[}\zeta(\mathsf{g}^{-1})^{\mu\gamma}\tensor{\mathsf{L}}{{}_{\mu}{}_{\nu}^{\nu}}(\mathsf{G}^{-1})^{\alpha\beta}+2\eta(\mathsf{g}^{-1})^{\mu\gamma}\tensor{\mathsf{L}}{{}_{\mu}^{\alpha}{}^{\beta}}\Bigr{]}\boldsymbol{y}_{,\gamma}.$
(3.26)

To see that the strong ellipticity condition is satisfied for (3.2), we
compute

$\displaystyle\boldsymbol{C}^{\alpha\beta\mu\nu}=\Bigl{(}\zeta(\mathsf{G}^{-1})^{\alpha\beta}(\mathsf{G}^{-1})^{\mu\nu}+2\eta[(\mathsf{G}^{-1})^{\alpha\mu}(\mathsf{G}^{-1})^{\beta\nu}+(\mathsf{G}^{-1})^{\alpha\nu}(\mathsf{G}^{-1})^{\beta\mu}]\Bigr{)}\boldsymbol{I},$
(3.27)

and thus, for all
$(\mathsf{a}_{1},\mathsf{a}_{2})\in\mathbb{R}^{2}\backslash\\{\boldsymbol{(}0,0)\\}$
and $\boldsymbol{b}\in\mathbb{R}^{3}\backslash\\{\boldsymbol{0}\\}$,

$\displaystyle\mathsf{a}_{\alpha}\mathsf{a}_{\beta}\boldsymbol{b}\cdot\Bigl{(}\boldsymbol{C}^{\alpha\beta\delta\gamma}\mathsf{a}_{\delta}\mathsf{a}_{\gamma}\boldsymbol{b}\Bigr{)}=(\zeta+2\eta)[(\mathsf{G}^{-1})^{\alpha\beta}\mathsf{a}_{\alpha}\mathsf{a}_{\beta}]^{2}|\boldsymbol{b}|^{2}>0.$
(3.28)

For the classical surface energy (3.9) of Steigmann-Ogden type, we have

$\displaystyle\boldsymbol{C}^{\alpha\beta\mu\nu}=\Bigl{(}\zeta(\mathsf{G}^{-1})^{\alpha\beta}(\mathsf{G}^{-1})^{\mu\nu}+2\eta[(\mathsf{G}^{-1})^{\alpha\mu}(\mathsf{G}^{-1})^{\beta\nu}+(\mathsf{G}^{-1})^{\alpha\nu}(\mathsf{G}^{-1})^{\beta\mu}]\Bigr{)}\boldsymbol{n}\otimes\boldsymbol{n},$
(3.29)

and thus, for all $(\mathsf{a}_{1},\mathsf{a}_{2})\in\mathbb{R}^{2}$ and
$\boldsymbol{b}\in\mathbb{R}^{3}$,

$\displaystyle\mathsf{a}_{\alpha}\mathsf{a}_{\beta}\boldsymbol{b}\cdot\Bigl{(}\boldsymbol{C}^{\alpha\beta\delta\gamma}\mathsf{a}_{\delta}\mathsf{a}_{\gamma}\boldsymbol{b}\Bigr{)}=(\zeta+2\eta)[(\mathsf{G}^{-1})^{\alpha\beta}\mathsf{a}_{\alpha}\mathsf{a}_{\beta}]^{2}(\boldsymbol{n}\cdot\boldsymbol{b})^{2}.$
(3.30)

In particular, (3.30) shows that the surface energy (3.9) satisfies the
Legendre-Hadamard condition but not the strong ellipticity condition since the
right side of (3.30) is 0 for $\boldsymbol{b}\neq\boldsymbol{0}$ and
orthogonal to $\boldsymbol{n}$.

###  3.2. Linearized equations

We now compute the linearization of (4.4) about the reference configuration.
For the bulk solid, we adopt a classical quadratic, isotropic energy density

$\displaystyle W=\frac{\lambda}{2}(\tensor{E}{{}^{i}_{i}})^{2}+\mu
E_{ij}E^{ij}.$ (3.31)

Here indices are raised using the flat metric on $\mathbb{R}^{3}$, and
$\lambda$ and $\mu$ are the Lamé constants for the bulk solid. For the surface
$\mathcal{S}$, the surface energy density is given by (3.2).

Let $\boldsymbol{u}:\mathcal{B}\rightarrow\mathbb{R}^{3}$ be a displacement
field such that
$\boldsymbol{u}|_{\partial\mathcal{B}\backslash\mathcal{S}}=\boldsymbol{0}$
and

$\displaystyle\sup_{\boldsymbol{X}\in\mathcal{B}}\Bigl{[}|\boldsymbol{u}(X)|+|\mbox{Grad}\,\boldsymbol{u}(\boldsymbol{X})|\Bigr{]}+\sum_{\alpha,\beta=1}^{2}\sup_{\boldsymbol{Y}\in\mathcal{S}}|\boldsymbol{u}_{,\alpha\beta}(\boldsymbol{Y})|\leq\delta_{0}.$
(3.32)

Assume that the body force $\boldsymbol{f}$, boundary traction
$\boldsymbol{t}$, and Dirichlet condition $\boldsymbol{\chi}_{0}$ satisfy

$\displaystyle|\boldsymbol{f}|=O(\delta_{0}),\quad|\boldsymbol{t}|=O(\delta_{0}),\quad|\boldsymbol{\chi}_{0}-\mbox{Id}|=O(\delta_{0}).$
(3.33)

If
$\boldsymbol{\chi}(\boldsymbol{X})=\boldsymbol{X}+\boldsymbol{u}(\boldsymbol{X}),$
then $E_{ij}=\varepsilon_{ij}+O(\delta_{0}^{2})$,
$\mathsf{E}_{\alpha\beta}=\epsilon_{\alpha\beta}+O(\delta_{0}^{2})$, and
$\mathsf{K}_{\alpha\beta}=\mathsf{k}_{\alpha\beta}+O(\delta_{0}^{2})$ where

$\displaystyle\varepsilon_{ij}=\frac{1}{2}\Bigl{(}\boldsymbol{e}_{i}\cdot\frac{\partial\boldsymbol{u}}{\partial
X^{j}}+\boldsymbol{e}_{j}\cdot\frac{\partial\boldsymbol{u}}{\partial
X^{i}}\Bigr{)}=O(\delta_{0}),$ (3.34)

and on $\mathcal{S}$,

$\displaystyle\epsilon_{\alpha\beta}=\frac{1}{2}\bigl{(}\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{,\beta}+\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{,\beta}\bigr{)}=O(\delta_{0}),\quad\mathsf{k}_{\alpha\beta}=\boldsymbol{N}\cdot\boldsymbol{u}_{;\alpha\beta}=O(\delta_{0}),$
(3.35)

see (3.12) in [23]. Now we observe that
$\mathsf{L}_{\alpha\beta\delta}=\mathsf{l}_{\alpha\beta\delta}+O(\delta_{0}^{2}),$
with

$\displaystyle\mathsf{l}_{\alpha\beta\delta}$
$\displaystyle=\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{,\beta\delta}+\boldsymbol{Y}_{,\beta\delta}\cdot\boldsymbol{u}_{,\alpha}-\tensor{\Gamma}{{}^{\mu}_{\beta}{}_{\delta}}\bigl{(}\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{,\mu}+\boldsymbol{Y}_{,\mu}\cdot\boldsymbol{u}_{,\alpha}\bigr{)}$
(3.36)
$\displaystyle=\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{;\beta\delta}+\boldsymbol{Y}_{;\beta\delta}\cdot\boldsymbol{u}_{,\alpha}$
(3.37)
$\displaystyle=\boldsymbol{Y}_{,\alpha}\cdot\boldsymbol{u}_{;\beta\delta}+(\boldsymbol{N}\cdot\boldsymbol{u}_{,\alpha})\mathsf{B}_{\beta\delta}=O(\delta_{0}).$
(3.38)

Then
$\boldsymbol{\mathsf{T}}^{\alpha}=\boldsymbol{\mathsf{t}}^{\alpha}+O(\delta_{0}^{2})$
and
$\boldsymbol{\mathsf{M}}^{\alpha\beta}=\boldsymbol{\mathsf{m}}^{\alpha\beta}+O(\delta_{0}^{2})$
where

$\displaystyle\boldsymbol{\mathsf{t}}^{\alpha}$
$\displaystyle:=\Bigl{[}\lambda_{s}\tensor{\epsilon}{{}^{\mu}_{\mu}}(\mathsf{G}^{-1})^{\alpha\gamma}+2\mu_{s}\epsilon^{\alpha\gamma}\Bigr{]}\boldsymbol{Y}_{,\gamma}-\Bigl{[}\zeta\tensor{\mathsf{k}}{{}^{\mu}_{\mu}}\tensor{\Gamma}{{}^{\alpha}_{\nu}{}^{\nu}}+2\eta\mathsf{k}^{\delta\nu}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}_{\nu}}\Bigr{]}\boldsymbol{N},$
(3.39)
$\displaystyle\quad+\Bigl{[}\zeta\tensor{\mathsf{l}}{{}^{\alpha}_{\nu}{}^{\nu}}\tensor{\mathsf{B}}{{}^{\delta}_{\delta}}+2\eta\mathsf{l}^{\alpha\delta\sigma}\mathsf{B}_{\delta\sigma}\Bigr{]}\boldsymbol{N}-\Bigl{[}\zeta\tensor{\mathsf{l}}{{}^{\beta}_{\nu}{}^{\nu}}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}^{\delta}}+2\eta\mathsf{l}^{\beta\delta\sigma}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}_{\sigma}}\Bigr{]}\boldsymbol{Y}_{,\beta},$
(3.40) $\displaystyle\boldsymbol{\mathsf{m}}^{\alpha\beta}$
$\displaystyle:=\Bigl{[}\zeta\tensor{\mathsf{k}}{{}^{\mu}_{\mu}}(\mathsf{G}^{-1})^{\alpha\beta}+2\eta\mathsf{k}^{\alpha\beta}\Bigr{]}\boldsymbol{N}+\Bigl{[}\zeta\tensor{\mathsf{l}}{{}^{\gamma}_{\nu}{}^{\nu}}(\mathsf{G}^{-1})^{\alpha\beta}+2\eta\tensor{\mathsf{l}}{{}^{\gamma}{}^{\alpha}{}^{\beta}}\Bigr{]}\boldsymbol{Y}_{,\gamma}.$
(3.41)

The linearization of (2.82) about the reference configuration is obtained by
omitting the $O(\delta_{0}^{2})$ terms from $\boldsymbol{P}$,
$\boldsymbol{\mathsf{T}}^{\alpha}$ and
$\boldsymbol{\mathsf{M}}^{\alpha\beta}$, yielding

$\displaystyle\begin{split}&\mbox{Div}\,\boldsymbol{\sigma}+\boldsymbol{f}=\boldsymbol{0},\quad\mbox{on
}\mathcal{B},\\\
&\boldsymbol{\sigma}\boldsymbol{N}=G^{-1/2}(G^{1/2}\boldsymbol{\mathsf{p}}^{\alpha})_{,\alpha}+\boldsymbol{t},\quad\mbox{on
}\mathcal{S},\\\ &\boldsymbol{u}=\boldsymbol{0},\quad\mbox{on
}\partial\mathcal{B}\backslash\mathcal{S},\end{split}$ (3.42)

where
$\boldsymbol{\sigma}=\lambda(\textrm{tr}\boldsymbol{\varepsilon})\boldsymbol{I}+2\mu\boldsymbol{\varepsilon}$
and
$\boldsymbol{\mathsf{p}}^{\alpha}=\boldsymbol{\mathsf{t}}^{\alpha}-G^{-1/2}(G^{1/2}\boldsymbol{\mathsf{m}}^{\alpha\beta})_{,\beta}$.
We observe that solutions to the linearized equations (3.42) are critical
points of the energy functional

$\displaystyle\mathcal{A}_{L}[\boldsymbol{u}]$
$\displaystyle=\int_{\mathcal{B}}\Bigl{[}\frac{\lambda}{2}(\tensor{\varepsilon}{{}^{i}_{i}})^{2}+\mu\varepsilon_{ij}\varepsilon^{ij}\Bigr{]}dV-\int_{\mathcal{B}}\boldsymbol{f}\cdot\boldsymbol{u}\,dV+\int_{\mathcal{S}}\Bigl{[}\frac{\lambda_{s}}{2}(\tensor{\epsilon}{{}^{\alpha}_{\alpha}})^{2}+\mu_{s}\epsilon_{\alpha\beta}\epsilon^{\alpha\beta}\Bigr{]}dS$
(3.43)
$\displaystyle\quad+\int_{\mathcal{S}}\Bigl{[}\frac{\zeta}{2}\Bigl{(}(\tensor{\mathsf{k}}{{}^{\alpha}_{\alpha}})^{2}+\tensor{\mathsf{l}}{{}_{\mu}{}_{\alpha}^{\alpha}}\tensor{\mathsf{l}}{{}^{\mu}_{\beta}{}^{\beta}}\Bigr{)}+\eta\Bigl{(}\tensor{\mathsf{k}}{{}_{\alpha}{}_{\beta}}\mathsf{k}^{\alpha\beta}+\tensor{\mathsf{l}}{{}_{\mu}{}_{\alpha}{}_{\beta}}\tensor{\mathsf{l}}{{}^{\mu}{}^{\alpha}{}^{\beta}}\Bigr{)}\Bigr{]}dA-\int_{\mathcal{S}}\boldsymbol{t}\cdot\boldsymbol{u}\,dA$
(3.44)

over the set of $\boldsymbol{u}$ satisfying
$\boldsymbol{u}|_{\partial\mathcal{B}\backslash\mathcal{S}}=\boldsymbol{0}$.

In the case of the classical Hilgers-Pipkin surface energy (3.7), we see from
(3.10) and (3.11) that

$\displaystyle\boldsymbol{\mathsf{t}}^{\alpha}=(\lambda_{s}\tensor{\epsilon}{{}^{\gamma}_{\gamma}}\delta^{\alpha\beta}+2\mu_{s}\epsilon^{\alpha\beta})\boldsymbol{Y}_{,\beta},\quad\boldsymbol{\mathsf{m}}^{\alpha\beta}=\zeta\delta^{\alpha\beta}\tensor{\boldsymbol{u}}{{}_{,\gamma}^{\gamma}}+2\eta\boldsymbol{u}^{,\alpha\beta}.$
(3.45)

Writing
$\boldsymbol{u}=\boldsymbol{\mathsf{u}}+\mathsf{u}^{3}\boldsymbol{N}=\mathsf{u}^{\gamma}\boldsymbol{Y}_{\gamma}+\mathsf{u}^{3}\boldsymbol{N}$,
it follows that (3.42) becomes

$\displaystyle\begin{split}&\mbox{Div}\,\boldsymbol{\sigma}+\boldsymbol{f}=\boldsymbol{0},\quad\mbox{on
}\mathcal{B},\\\
&\boldsymbol{\sigma}\boldsymbol{N}=\mu_{s}\tensor{\boldsymbol{\mathsf{u}}}{{}_{,\alpha}^{\alpha}}+(\lambda_{s}+\mu_{s})\tensor{\mathsf{u}}{{}^{\gamma}_{,\alpha\gamma}}\boldsymbol{Y}^{\alpha}-(\zeta+2\eta)\partial_{\alpha}\partial_{\beta}\tensor{\boldsymbol{u}}{{}^{,\alpha\beta}}+\boldsymbol{t},\quad\mbox{on
}\mathcal{S},\\\ &\boldsymbol{u}=\boldsymbol{0},\quad\mbox{on
}\partial\mathcal{B}\backslash\mathcal{S}.\end{split}$ (3.46)

##  4\. Mode-III Fracture Problem

In this section, we apply the linearized theory (3.46) to the problem of a
brittle infinite plate, with a straight crack $\mathcal{C}$ of length $2\ell$,
under far-field anti-plane shear loading $\sigma$. As discussed in the
Introduction and in contrast to ascribing either a quadratic Gurtin-Murdoch or
Steigmann-Ogden surface energy (3.9) to the crack fronts, the use of (3.2)
yields a model that predicts bounded strains and stresses up to the crack tips
(see Theorem 4.4).

###  4.1. Formulation and governing equations

We consider a brittle, infinite plate under anti-plane shear loading,
$\lim_{x^{2}\rightarrow\pm\infty}\sigma_{13}=0$ and
$\lim_{x^{2}\rightarrow\pm\infty}\sigma_{23}=\sigma$, with a straight crack
$\mathcal{C}=\\{(x^{1},0,x^{3})\mid x^{1}\in[-\ell,\ell]\\}$ of length $2\ell$
(see Figure 2). For anti-plane shear, the displacement field takes the form

$\displaystyle\boldsymbol{u}(x^{1},x^{2},x^{3})=u(x^{1},x^{2})\boldsymbol{e}_{3},$
(4.1)

Then the only nonzero components of the stress are

$\displaystyle\sigma_{13}=\mu u_{,1},\quad\sigma_{23}=\mu u_{,2}$ (4.2)

By the symmetry of the problem, $u$ can be taken to be even in $x^{1}$ and odd
in $x^{2}$, so we will focus only on the strain and stress fields for
$x^{2}\geq 0$. The governing field equations are (3.46) on
$\mathcal{B}=\\{(x^{1},x^{2},x^{3})\mid x^{2}\geq 0\\}$ with
$\mathcal{S}=\\{(x^{1},0,x^{3})\mid x^{1}\in[-\ell,\ell]\\}$,
$\boldsymbol{t}=\boldsymbol{0}$ and $\boldsymbol{f}=\boldsymbol{0}$.

Figure 2. Schematic of the mode-III problem with the crack $\mathcal{C}$
appearing in blue.

We define dimensionless variables

$\displaystyle x=\frac{x^{1}}{\ell},\quad y=\frac{x^{2}}{\ell},\quad
z=\frac{x^{3}}{\ell},\quad
w(x,y,z)=\frac{1}{\ell}\Bigl{(}u(x^{1},x^{2},x^{3})-\frac{\sigma}{\mu}x^{2}\Bigr{)}.$
(4.3)

Then the field equations take the dimensionless form

$\displaystyle\begin{split}&\Delta w(x,y)=0,\quad y>0,\\\ &-w_{y}(x,0)=\alpha
w_{xx}(x,0)-\beta w_{xxxx}(x,0)+\gamma,\quad x\in(-1,1),\\\
&w(x,0)=0,\quad|x|\geq 1,\\\ &w_{x}(\pm 1,0)=0,\end{split}$ (4.4)

with the decay condition $\lim_{y\rightarrow\infty}|\nabla w(x,y)|=0$. We note
that the boundary conditions $w_{x}(\pm 1,0)=0$ imply that the crack opening
is cusp shaped rather then blunted (see also Figure 3 and Figure 4). The
dimensionless parameters $\alpha,\beta$ and $\gamma$ are given by

$\displaystyle\alpha=\frac{\mu_{s}}{\mu\ell}>0,\quad\beta=\frac{\zeta+2\eta}{\mu\ell^{3}}>0,\quad\gamma=\frac{\sigma}{\mu},$
(4.5)

and in particular, we see from (4.5) that the behavior of the displacement $w$
depends on the length of the crack, $\ell$. For macro cracks satisfying
$\beta\ll\alpha\ll 1$, we expect $w(x,0)$ to be well-approximated by the
singular, rounded opening profile from the classical linear elastic fracture
mechanics except in small regions near the crack tips (boundary layers). See
Figure 4.

We remark that in using the Steigmann-Ogden surface energy (3.9) rather than
(3.2), the boundary conditions at $y=0$ are replaced by

$\displaystyle\begin{split}&-w_{y}(x,0)=\alpha w_{xx}(x,0)+\gamma,\quad
x\in(-1,1),\\\ &w(x,0)=0,\quad|x|\geq 1.\end{split}$ (4.6)

One may view this loss of higher order derivatives in the boundary conditions
as a consequence of the fact that the Steigmann-Ogden surface energy does not
satisfy the strong-ellipticity condition (3.12): for anti-plane shear,
$\boldsymbol{b}=\boldsymbol{u}(x^{1},0,x^{3})$ is orthogonal to the surface’s
normal $\boldsymbol{n}=-\boldsymbol{e}_{2}$ (see (3.30)). As discussed in [25,
9], the boundary conditions (4.6) _do not_ lead to a model predicting bounded
strains up to the crack tips $x=\pm 1$, i.e., the displacement field satisfies

$\displaystyle\sup_{y>0}|\nabla w(\pm 1,y)|=\infty.$ (4.7)

We see that (4.4) is the system of Euler-Lagrange equations for the energy
functional

$\displaystyle\mathcal{A}_{L}[w]$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\int_{-\infty}^{\infty}|\nabla
w(x,y)|^{2}dxdy+\int_{-\infty}^{\infty}\Bigl{(}\frac{\alpha}{2}|w_{x}(x,0)|^{2}+\frac{\beta}{2}|w_{xx}(x,0)|^{2}\Bigr{)}dx$
(4.8) $\displaystyle\quad-\gamma\int_{-\infty}^{\infty}w(x,0)dx$ (4.9)

defined for $w$ with ${\nabla}w\in L^{2}(\\{y>0\\})$, $w(\cdot,0)\in
H^{2}(\mathbb{R})$ and $w(x,0)=0$ for all $|x|\geq 1$. Motivated by this
observation, we define the Hilbert space $H$ to be the completion of
$C^{\infty}_{c}((-1,1))$ under the norm

$\displaystyle\|f\|^{2}_{H}:=\int_{-\infty}^{\infty}\bigl{(}\alpha|f^{\prime}(x)|^{2}+\beta|f^{\prime\prime}(x)|^{2}\bigr{)}dx.$
(4.10)

It is straightforward to verify the following facts using the fundamental
theorem of calculus and Cauchy-Schwarz inequality:

  * •

(Sobolev embedding) If $f\in H$ then $f\in C_{c}^{1,1/2-}(\mathbb{R})$ and
$f(x)=0$ for all $|x|\geq 1$, and for all $\delta\in[0,1/2)$, there exists a
constant $A>0$ depending on $\alpha,\beta$ and $\delta$, such that for all
$f\in H$,

$\displaystyle\|f\|_{C^{1,\gamma}(\mathbb{R})}\leq A\|f\|_{H}.$ (4.11)

  * •

$f\in H$ if and only if $f\in H^{2}(\mathbb{R})$ and $f(x)=0$ for all $|x|\geq
1$. Moreover, there exist $b,B>0$ depending on $\alpha$ and $\beta$ such that
for all $f\in H$,

$\displaystyle b\|f\|_{H}\leq\|f\|_{H^{2}(\mathbb{R})}\leq B\|f\|_{H}.$ (4.12)

The problem (4.4) can be reduced completely to a problem on the boundary by
using the Dirichlet-to-Neumann map $-w_{y}(x,0)=\mathcal{H}w_{x}(x,0)$ where
$\mathcal{H}$ is the Hilbert transform

$\displaystyle\mathcal{H}f(x)=\frac{1}{\pi}\,\mbox{p.v.}\,\int_{-\infty}^{\infty}\frac{f(s)}{x-s}ds,\quad
f\in H.$ (4.13)

Then finding $w$ with $\nabla w\in L^{2}(\\{y>0\\})$ and $w(\cdot,0)\in H$
satisfying (4.4) is equivalent to determining $w(\cdot,0)=:f\in H$
satisfying666Once $f$ is found, $w$ is determined on the upper half plane
using the standard Poisson kernel for the upper half plane.

$\displaystyle\beta f^{\prime\prime\prime\prime}(x)-\alpha
f^{\prime\prime}(x)+\mathcal{H}f^{\prime}(x)=\gamma,\quad x\in(-1,1).$ (4.14)

By using the Plancherel theorem, the Fourier representation of the Hilbert
transform (see (4.44)), and (4.12), we have for all $f\in H$,

$\displaystyle\|\mathcal{H}f^{\prime}\|_{H^{1}(\mathbb{R})}\leq\|f^{\prime}\|_{H^{1}(\mathbb{R})}\leq\|f\|_{H^{2}(\mathbb{R})}\leq
B\|f\|_{H}.$ (4.15)

######  Definition 4.1.

A function $f\in H$ is a _weak solution_ to the integro-differential equation
(4.14) if for all $g\in H$

$\displaystyle\int_{-\infty}^{\infty}[\beta
f^{\prime\prime}(x)g^{\prime\prime}(x)+\alpha
f^{\prime}(x)g^{\prime}(x)+\mathcal{H}f^{\prime}(x)g(x)\Bigr{]}dx=\int_{-\infty}^{\infty}\gamma
g(x)dx.$ (4.16)

We remark that since $f,g\in H$, the integrals appearing in (4.16) are in fact
over the interval $(-1,1)$.

A function $f\in H$ is a _classical solution_ to (4.14) if $f\in
C^{4}((-1,1))\cap H$ and $f$ satisfies (4.14) pointwise.

We note that by (4.15) and Cauchy-Schwarz, (4.16) is well-defined for each
$f,g\in H$.

###  4.2. Solution of the integro-differential equation

We now establish that there exists a unique classical solution to (4.14), and
the solution’s behavior is consistent with the linearization assumption
(3.32). We denote the following Green function

$\displaystyle
G(x,\tau)=\begin{cases}\frac{1}{24}(x-1)^{2}(\tau+1)^{2}(1+2x-2\tau-x\tau)&\quad\tau\in[-1,x],\\\
\frac{1}{24}(\tau-1)^{2}(x+1)^{2}(1+2\tau-2x-x\tau)&\quad\tau\in[x,1],\end{cases}$
(4.17)

satisfying $G_{xxxx}(x,\tau)=\delta(x-\tau)$, $G(\pm 1,\tau)=0$, $G_{x}(\pm
1,\tau)=0$. We note that $G(x,\tau)=G(\tau,x)$ for all $\tau,x\in[-1,1]$,
$G\in C^{2}([-1,1]\times[-1,1])$ and
$\int_{-1}^{1}G(x,\tau)d\tau=\frac{1}{24}(1-x^{2})^{2}$. In particular, we
have for all $f\in H$,

$\displaystyle\int_{-1}^{1}G_{\tau\tau}(x,\tau)f^{\prime\prime}(\tau)d\tau=f(x).$
(4.18)

Figure 3. Numerical solutions for the equivalent formulation of (4.14) as a
Fredholm problem (4.36). The parameters $(\beta,\alpha,\gamma)$ range over
$(1,1,1)$, $(5,1,5)$ and $(10,1,10)$. For $\gamma=\beta$ and $\beta\gg
1\simeq\alpha$, the opening profile is well approximated by the limiting
opening profile $f_{\infty}(x)=\frac{1}{24}(1-x^{2})^{2}$ on $[-1,1]$.

######  Lemma 4.2.

A function $f\in H$ is a weak solution to (4.14) if and only if $f$ satisfies

$\displaystyle\beta f(x)+\int_{-1}^{1}G(x,\tau)(-\alpha
f^{\prime\prime}(\tau)+\mathcal{H}f^{\prime}(\tau))d\tau=\frac{\gamma}{24}(1-x^{2})^{2},\quad
x\in[-1,1].$ (4.19)

###### Proof.

Let $h\in L^{2}(\mathbb{R})$ with $h(x)=0$ for all $|x|>1$, and set

$\displaystyle g(x)=\begin{cases}\int_{-1}^{1}G(x,\tau)h(\tau)d\tau&\mbox{ if
}|x|\leq 1,\\\ 0&\mbox{ if }|x|>1.\end{cases}$ (4.20)

Since $G\in C^{2}([-1,1]\times[-,1,1])$, $G(\pm 1,\tau)=0$, and $G_{x}(\pm
1,\tau)=0$, $g$ is twice continuously differentiable on
$\mathbb{R}\backslash\\{\pm 1\\}$ and continuously differentiable on
$\mathbb{R}$ with

$\displaystyle g^{\prime}(x)$ $\displaystyle=\chi_{\\{|x|\leq
1\\}}(x)\int_{-1}^{1}G_{x}(x,\tau)h(\tau)d\tau,$ (4.21) $\displaystyle
g^{\prime\prime}(x)$ $\displaystyle=\chi_{\\{|x|\leq
1\\}}(x)\int_{-1}^{1}G_{xx}(x,\tau)h(\tau)d\tau,$ (4.22)

where $\chi_{E}$ is the indicator function of a subset $E\subseteq\mathbb{R}$.
In particular, we conclude that $g\in H^{2}(\mathbb{R})$ and thus, $g\in H$.
Inserting $g$ into (4.16), integrating by parts in the second term and using
that $g(\pm 1)=0$ yield

$\displaystyle\beta\int_{-1}^{1}\int_{-1}^{1}G_{xx}(x,\tau)f^{\prime\prime}(x)h(\tau)d\tau
dx$ (4.23) $\displaystyle+\int_{-1}^{1}\int_{-1}^{1}G(x,\tau)(-\alpha
f^{\prime\prime}(x)+\mathcal{H}f^{\prime}(x))h(\tau)d\tau dx$ (4.24)
$\displaystyle=\gamma\int_{-1}^{1}\int_{-1}^{1}G(x,\tau)h(\tau)d\tau dx.$
(4.25)

Interchanging the order of integration, using the symmetry of $G$ and
relabeling the integration variables lead to

$\displaystyle\beta\int_{-1}^{1}\int_{-1}^{1}G_{\tau\tau}(x,\tau)f^{\prime\prime}(\tau)d\tau\,h(x)dx$
(4.26) $\displaystyle+\int_{-1}^{1}\int_{-1}^{1}G(x,\tau)(-\alpha
f^{\prime\prime}(\tau)+\mathcal{H}f^{\prime}(\tau))d\tau\,h(x)dx$ (4.27)
$\displaystyle=\int_{-1}^{1}\frac{\gamma}{24}(1-x^{2})^{2}h(x)dx.$ (4.28)

Finally, by (4.18) we conclude that

$\displaystyle\int_{-1}^{1}\Bigl{[}\beta f(x)+\int_{-1}^{1}G(x,\tau)(-\alpha
f^{\prime\prime}(\tau)+\mathcal{H}f^{\prime}(\tau))d\tau\Bigr{]}\,h(x)dx$
(4.29) $\displaystyle=\int_{-1}^{1}\frac{\gamma}{24}(1-x^{2})^{2}h(x)dx,$
(4.30)

for all $h(x)\in L^{2}(\mathbb{R})$ with $h(x)=0$ for all $|x|>1$, proving
(4.19).

Conversely, if $f\in H$ and (4.19) holds, then for all $g\in H$, we have

$\displaystyle\beta\int_{-\infty}^{\infty}f^{\prime\prime}(x)g^{\prime\prime}(x)dx$
$\displaystyle=\int_{-1}^{1}\int_{-1}^{1}G_{xx}(x,\tau)(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))g^{\prime\prime}(x)dx$
(4.31)
$\displaystyle\quad+\int_{-1}^{1}\frac{\gamma}{6}(3x^{2}-1)g^{\prime\prime}(x)dx.$
(4.32)

We again interchange the order of integration and use integration by parts and
$\int_{-1}^{1}G_{xx}(x,\tau)g^{\prime\prime}(x)dx=g(\tau)$ to conclude that

$\displaystyle\int_{-1}^{1}\int_{-1}^{1}$ $\displaystyle G_{xx}(x,\tau)(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))g^{\prime\prime}(x)dx+\int_{-1}^{1}\frac{\gamma}{6}(3x^{2}-1)g^{\prime\prime}(x)dx$
(4.33) $\displaystyle=\int_{-1}^{1}(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))g(\tau)d\tau+\int_{-1}^{1}\gamma
g(x)dx$ (4.34) $\displaystyle=-\int_{-1}^{1}(\alpha
f^{\prime}(x)g^{\prime}(x)+\mathcal{H}f^{\prime}(x)g(x))dx+\int_{-1}^{1}\gamma
g(x)dx$ (4.35)

This proves $f\in H$ satisfies (4.16) and concludes the proof of the lemma. ∎

Figure 4. Numerical solutions for the macro-crack regime $\beta\ll\alpha\ll
1$. The parameters $(\beta,\alpha,\gamma)$ range over $(10^{-1},10^{-1},1)$,
$(10^{-2},10^{-1},1)$, $(10^{-5},10^{-2},1)$ and $(10^{-6},10^{-3},1)$. For
$\beta\ll\alpha\ll 1$, we expect the crack opening $f(x)$ to be well-
approximated by the singular, rounded opening profile predicted by classical
linear elastic fracture mechanics away from the crack tips where
$f^{\prime}(\pm 1)=0$ (and the profile is cusped).

Via integration by parts and straightforward computations, we conclude from
Lemma 4.2 that the crack opening profile $f$ must satisfy the following
Fredholm integral equation of the second kind (4.36). We will show in Theorem
4.4 that this equation is uniquely solvable for _arbitrary_ $\alpha,\beta>0$
and $\gamma$. We remark that since the kernel extends to a continuous function
on $[-1,1]\times[-1,1]$ (the singularities are removable), the numerical
computation of solutions is relatively straightforward via the Nÿstrom method
with the trapezoidal rule to approximate the integral (see Figure 3 and Figure
4).

######  Corollary 4.3.

A function $f\in H$ is a weak solution to (4.14) if and only if $f$ satisfies
the Fredholm equation

$\displaystyle\beta
f(x)+\int_{-1}^{1}K(x,s)f(s)ds=\frac{\gamma}{24}(1-x^{2})^{2},\quad
x\in[-1,1],$ (4.36)

where $K(x,s)=-\alpha
G_{ss}(x,s)+\frac{1}{\pi}\mbox{p.v.}\,\int_{-1}^{1}\frac{G_{\tau}(x,\tau)}{s-\tau}d\tau$,

$\displaystyle G_{ss}(x,s)$
$\displaystyle=\begin{cases}-\frac{1}{4}(x-1)^{2}(2s+xs+1)&\quad
s\in[-1,x],\\\ -\frac{1}{4}(x+1)^{2}(-2s+xs+1)&\quad s\in[x,1],\end{cases}$
(4.37)

and

$\displaystyle\frac{1}{\pi}\mbox{p.v.}\,\int_{-1}^{1}\frac{G_{\tau}(x,\tau)}{s-\tau}d\tau$
$\displaystyle=\frac{1}{4\pi}(sx-1)(x^{2}-1)+\frac{1}{2\pi}(s-x)^{2}\log|x-s|$
(4.38) $\displaystyle\quad-\frac{1}{8\pi}(x-1)^{2}(-x+2s+sx)(1+s)\log(1+s)$
(4.39) $\displaystyle\quad-\frac{1}{8\pi}(x+1)^{2}(x-2s+sx)(1-s)\log(1-s).$
(4.40)

######  Theorem 4.4.

There exists $C>0$ depending on $\alpha$ and $\beta$ such that the following
hold. There exists a unique classical solution $f$ to (4.14), and $f$
satisfies

$\displaystyle\|f\|_{C^{4}([-1,1])}\leq C|\gamma|.$ (4.41)

Moreover, the displacement field
$w(x,y)=\int_{-\infty}^{\infty}P_{y}(x-s)f(s)ds$, where $P_{y}(\cdot)$ is the
Poisson kernel for the upper half plane, has bounded strains up to the crack
tips:

$\displaystyle\|w\|_{C^{1}(\\{y\geq 0\\})}\leq C|\gamma|.$ (4.42)

###### Proof.

In what follows, $C$ will denote a positive constant depending only on
$\alpha$ and $\beta$ that may change from line to line, and we denote the
Fourier transform and inverse Fourier transform by

$\displaystyle\hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi
ix\xi}dx,\quad\check{f}(x)=\int_{-\infty}^{\infty}f(\xi)e^{2\pi ix\xi}\,d\xi.$
(4.43)

We recall that

$\displaystyle\widehat{f^{\prime}}(\xi)=2\pi
i\xi\hat{f}(\xi),\quad\widehat{\mathcal{H}f}(\xi)=-i{\mathrm{sgn}}(\xi)\hat{f}(\xi),$
(4.44)

the latter relation following from the Fourier representation of $w$ on the
upper half plane (see (4.50)).

We define a bilinear form $B(\cdot,\cdot):H\times H\rightarrow\mathbb{R}$ by

$\displaystyle B(f,g)=\int_{-\infty}^{\infty}[\beta
f^{\prime\prime}(x)g^{\prime\prime}(x)+\alpha
f^{\prime}(x)g^{\prime}(x)+\mathcal{H}f^{\prime}(x)g(x)\Bigr{]}dx,\quad f,g\in
H.$ (4.45)

By Cauchy-Schwarz, (4.15), and (4.12) we conclude that for all $f,g\in H$,

$\displaystyle|B(f,g)|\leq(1+B^{2})\|f\|_{H}\|g\|_{H},$ (4.46)

so that $B(\cdot,\cdot)$ is a bounded bilinear form on $H$. Moreover, by the
Plancherel theorem and (4.44), we have for all $f\in H$,

$\displaystyle\int_{-\infty}^{\infty}\mathcal{H}f^{\prime}(x)f(x)dx=\int_{-\infty}^{\infty}2\pi|\xi||\hat{f}(\xi)|^{2}d\xi\geq
0.$ (4.47)

Thus, the bilinear form is coercive. Since for all $g\in H$,

$\displaystyle\Bigl{|}\int_{-\infty}^{\infty}\gamma
g(x)dx\Bigr{|}\leq|\gamma|\|g\|_{C([-1,1])}\leq|\gamma|A\|g\|_{H},$ (4.48)

the classical Lax-Milgram theorem implies that there exists a unique weak
solution $f\in H$ to (4.14), and moreover,

$\displaystyle\|f\|_{H}\leq A|\gamma|.$ (4.49)

To prove (4.42), we express $w$ via the Fourier transform,

$\displaystyle w(x,y)=\int_{-\infty}^{\infty}e^{-2\pi y|\xi|}e^{2\pi
ix\xi}\hat{f}(\xi)d\xi.$ (4.50)

Since $f\in H\subset H^{2}(\mathbb{R})$, we have

$\displaystyle\int(1+|\xi|)^{4}|\hat{f}(\xi)|^{2}d\xi\leq
C_{0}\|f\|_{H^{2}(\mathbb{R})}^{2}\leq C\|f\|_{H}^{2}.$ (4.51)

Thus, by Cauchy-Schwarz

$\displaystyle|w(x,y)$ $\displaystyle|+|\nabla w(x,y)|$ (4.52)
$\displaystyle\leq\int_{-\infty}^{\infty}(1+2\pi|\xi|\sqrt{2})|\hat{f}(\xi)|^{2}\,d\xi$
(4.53) $\displaystyle\leq
2\pi\sqrt{2}\Bigl{(}\int_{-\infty}^{\infty}|\xi|^{2}(1+|\xi|)^{-4}d\xi\Bigr{)}^{1/2}\Bigl{(}\int_{-\infty}^{\infty}(1+|\xi|)^{4}|\hat{f}(\xi)|^{2}d\xi\Bigr{)}^{1/2}$
(4.54) $\displaystyle\leq C\|f\|_{H}\leq C|\gamma|.$ (4.55)

We now show that the weak solution $f$ is a classical solution, and (4.41)
holds. By a density argument in $H$, we have that
$\int_{-1}^{1}G(\cdot,\tau)(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau\in H^{3}([-1,1])$
with

$\displaystyle\frac{d^{k}}{dx^{k}}\int_{-1}^{1}$ $\displaystyle
G(x,\tau)(\alpha f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau$
(4.56) $\displaystyle=\int_{-1}^{1}\partial_{x}^{k}G(x,\tau)(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau,\quad k=1,2,$ (4.57)
$\displaystyle\frac{d^{3}}{dx^{3}}\int_{-1}^{1}$ $\displaystyle
G(x,\tau)(\alpha f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau$
(4.58) $\displaystyle=\int_{-1}^{x}\frac{1}{4}(2-\tau)(\tau+1)^{2}(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau$ (4.59)
$\displaystyle\quad-\int_{x}^{1}\frac{1}{4}(2+\tau)(\tau-1)^{2}(\alpha
f^{\prime\prime}(\tau)-\mathcal{H}f^{\prime}(\tau))d\tau.$ (4.60)

By Lemma 4.2, we conclude that $f\in H^{3}([-1,1])$, and by (4.19) (4.60),
Cauchy-Schwarz, and (4.49) we have

$\displaystyle\|f^{\prime\prime\prime}\|_{L^{2}([-1,1])}$ $\displaystyle\leq
C(\|f^{\prime\prime}\|_{L^{2}(\mathbb{R})}+\|\mathcal{H}f^{\prime}\|_{L^{2}(\mathbb{R})}+|\gamma|)$
(4.61) $\displaystyle\leq C(\|f\|_{H}+|\gamma|)\leq C|\gamma|.$ (4.62)

Moreover, by (4.49), (4.62) and the fundamental theorem of calculus, $f\in
C^{2}([-1,1])$ with

$\displaystyle\|f\|_{C^{2}([-1,1])}\leq C|\gamma|.$ (4.63)

By (4.16) and integration by parts, it follows that for all $g\in
C^{\infty}_{c}((-1,1))\subset H$,

$\displaystyle\int_{-1}^{1}f^{\prime\prime\prime}(x)g^{\prime}(x)dx=\frac{1}{\beta}\int_{-1}^{1}[-\alpha
f^{\prime\prime}(x)+\mathcal{H}f^{\prime}(x)-\gamma]g(x)dx,$ (4.64)

and, thus, $f\in H^{4}([-1,1])$ and

$\displaystyle\beta f^{\prime\prime\prime\prime}(x)-\alpha
f^{\prime\prime}(x)+\mathcal{H}f^{\prime}(x)=\gamma,\quad\mbox{for a.e.
}x\in(-1,1).$ (4.65)

Then by (4.65), (4.63), and the fact that $\mathcal{H}f^{\prime}\in
H^{1}(\mathbb{R})\hookrightarrow C(\mathbb{R})$, we conclude that
$f^{\prime\prime\prime\prime}\in C([-1,1])$ and

$\displaystyle\|f^{\prime\prime\prime\prime}\|_{C([-1,1])}$ $\displaystyle\leq
C\Bigl{(}\|f\|_{C^{2}([-1,1])}+\|\mathcal{H}f^{\prime}\|_{C([-1,1])}+|\gamma|\Bigr{)}$
(4.66) $\displaystyle\leq
C\Bigl{(}\|\mathcal{H}f^{\prime}\|_{H^{1}(\mathbb{R})}+|\gamma|\Bigr{)}$
(4.67) $\displaystyle\leq C\Bigl{(}\|f\|_{H}+|\gamma|\Bigr{)}\leq C|\gamma|.$
(4.68)

By (4.62), (4.63) and (4.68), it follows that $f\in C^{4}([-1,1])$ is a
classical solution to (4.14) and (4.41) holds. ∎

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C. Rodriguez

Department of Mathematics, University of North Carolina

Chapel Hill, NC 27599, USA