Angle-independent optimal adhesion in plane peeling of thin elastic films at large surface roughnesses

Figure 2 shows the geometry of our wavy peeling problem. Let $\mathbb{E}$ be a three-dimensional, oriented, Hilbert space, and let the Euclidean point space $\mathcal{E}$ be $\mathbb{E}$’s principle homogeneous space. The elements of $\mathbb{E}$ have units of length. The physical objects in our peeling problem are contained in $\mathcal{E}$. The origin of $\mathcal{E}$, which we denote as $O$, is marked in Figure 2. The vector set $\left(\hat{\mathbscr{e}}_{i}\right)_{i\in\mathcal{I}}$, where the index set $\mathcal{I}:=(1,2,3)$, is an orthonormal set in $\mathbb{E}$. That is, $\hat{\mathbscr{e}}_{i}\cdot\hat{\mathbscr{e}}_{j}=\delta_{ij}$, where $i$,$j\in\mathcal{I}$, the symbol $\cdot$ denotes the inner product operator between the elements of $\mathbb{E}$, and $\delta_{ij}$ is the Kronecker-delta symbol, which equals unity if $i=j$ and naught otherwise. A typical point $X\in\mathcal{E}$ is identified by its coordinates $\left(x_{i}\right)_{i\in\mathcal{I}}\in\mathbb{R}^{3}$¹¹1In our work the manner in which the information about a physical quantity’s units is stored is different from how that is usually done. We model different physical quantities as vectors belonging to different vector spaces. We store the information about a physical quantity’s units in the basis vectors we choose for that quantity’s vector space. For example, we may take $\hat{\mathbf{e}}_{1}$ to represent a motion of $1\,\mathrm{meters}$ (or $1\,\mathrm{micron}$) in a certain fixed direction in $\mathbb{E}$. In that case $\mathbf{e}_{1}$, as a consequence of its definition, would denote a motion of $\lambda\,\mathrm{meters}$ (resp. $\lambda\,\mathrm{microns}$) in the same direction as $\hat{\mathbf{e}}_{1}$. Thus, in our work the components of a physical quantity with respect to the basis vectors chosen for its vector space will always be non-dimensional. For example the components of $X$ with respect to $(\mathbf{e}_{i})$, namely $\left(x_{i}\right)_{i\in\mathcal{I}}$, are dimensionless. that are defined such that $X=O+\sum_{i\in\mathcal{I}}x_{i}\mathbscr{e}_{i}$. The vector $\sum_{i\in\mathcal{I}}x_{i}\mathbscr{e}_{i}\in\mathbb{E}$ is called $X$’s position vector.

Note that the vector set $(\mathbscr{e}_{i})_{i\in\mathcal{I}}$ is different from $(\hat{\mathbscr{e}}_{i})_{i\in\mathcal{I}}$; it is defined as $\mathbscr{e}_{i}:=\lambda\hat{\mathbscr{e}}_{i}$, where $\lambda>0$, for $i=1,~{}2$, and $\mathbscr{e}_{3}:=\hat{\mathbscr{e}}_{3}$. The set $\left(\mathbscr{e}_{i}\right)_{i\in\mathcal{I}}$ is an orthogonal, but not an orthonormal, basis for $\mathbb{E}$. We will be following the Einstein summation convention in this paper²²2We follow the Einstein summation convention in this paper. If a symbol has an italicized, light-faced latin character that appears as its subscript/superscript then that subscript/superscript denotes an index and that symbol along with that subscript/superscript denotes a component of a linear mapping. An index that appears only once in a term is called a “free” index. A free index in a term denotes that the term in fact represents the tuple of terms created by varying the free index in the original term over $\mathcal{I}$. An index that appears twice in a term is called a “repeated index”. A repeated index in a term denotes a sum of the terms that are created by varying the free index in the original term over $\mathcal{I}$., and write expressions such as $\left(\mathbscr{e}_{i}\right)_{i\in\mathcal{I}}$ and $\sum_{i\in\mathcal{I}}x_{i}\mathbscr{e}_{i}$ simply as $\left(\mathbscr{e}_{i}\right)$ and $x_{i}\mathbscr{e}_{i}$, respectively, and lists such as $\mathbscr{e}_{1},~{}\mathbscr{e}_{2}$, and $\mathbscr{e}_{3}$ simply as $\mathbscr{e}_{i}$.

The substrate is a rigid solid whose points’ position vectors belong to the set

where $\alpha:=A/\lambda$, $A\geq 0$, $\varrho:\mathbb{R}\to[-1,1]$ is a surjective, 1-periodic function, and $\mathbb{R}$ is the set of real numbers. By 1-periodic we mean that $\varrho(x_{1}+1)=\varrho(x_{1})$ for all $x_{1}\in\mathbb{R}$. Without loss of generality we take $\int_{0}^{1}\varrho(x_{1})\,dx_{1}=0$. We also assume that $\varrho$ and its first and second derivatives, which we denote as $\dot{\varrho}$ and $\ddot{\varrho}$, respectively, are non-constant, continuous functions, i.e., $\varrho\in C^{2}$. We call $\alpha$, $A$, $\lambda$, and $\varrho$ the substrate surface’s aspect ratio, amplitude, periodicity, and profile, respectively.

We take the thin film to be of width $\mathscr{b}$, of thickness $\mathscr{h}$, and to be perfectly adhered to the substrate’s surface in its initial, or reference, configuration. Here $\mathscr{b}$ and $\mathscr{h}$ are physical parameters and have units of length. Say the units of $\mathbscr{e}_{i}$ and $\hat{\mathbscr{e}}_{i}$ is some length $\mathscr{m}$, which, for example, may stand for meters, then we would say that the magnitude of $\mathscr{b}$ (resp. $\mathscr{h}$) is $b$ (resp. $h$) iff $\mathscr{b}=b\,\mathscr{m}$ (resp. $\mathscr{h}=h\,\mathscr{m}$). Thus, initially the configuration of the thin film can be described as

where $\mathcal{D}:=[0,+\infty)$³³3As is standard in continuum mechanics, we will be referring to a particular thin film material particle using the coordinates of the the spatial point that it occupied in the initial configuration. That is, when we say a material particle $(x_{1},x_{2},x_{3})$, we in fact are referring to the material particle that in the initial configuration occupied the spatial point with coordinates $(x_{1},x_{2},x_{3})$. For the sake of brevity, we will be referring to a material particle $(x_{1},x_{2},x_{3})$ simply as $(x_{1},x_{3})$, since the second co-ordinate of any spatial point that was occupied by a thin film material particle in the initial configuration is fully determined by the point’s first co-ordinate (cf. (3.2)). Finally, when we say “the (thin-film) material particle $x_{1}$”, where $x_{1}\in\mathcal{D}$, we in fact mean the group of material particles $(x_{1},x_{3})$, where $\lvert x_{3}\rvert\leq b/2$. .

We model the peeling process by assuming that any deformed configuration $\bm{\kappa}$ of the thin film can be described as⁴⁴4In standard continuum mechanics the reference and deformed configurations are taken to belong to different spaces. In contrast, here we take both the reference and deformed configurations, $\mathbb{B}_{0}$ and $\mathbb{B}$, to belong to the space space, namely $\mathbb{E}$.

$u_{1},u_{2}:\mathcal{D}\to\mathbb{R}$ and their derivatives, denoted as $\dot{u}_{1}$ and $\dot{u}_{2}$, are continuous. We further assume that $u_{1}\left(x_{1}\right)=u_{2}\left(x_{1}\right)=0$ for all $x_{1}\geq a$, for some $a\in\mathcal{D}$, and $\dot{u}_{1}\left(x_{1}\right)=\dot{u}_{2}\left(x_{1}\right)=0$ for all $x_{1}>a$. We refer to $a$ as the de-adhered length. We call $P:=\left\{O+a\mathbscr{e}_{1}+\rho(a)\mathbscr{e}_{2}+x_{3}\mathbscr{e}_{3}\in\mathcal{E}\;|\;\lvert x_{3}\rvert\leq b/2\right\}$ the peeling front and $\Gamma_{a}=\{x_{1}\in\mathcal{D}~{}|~{}x_{1}>a\}$ the adhered region.

On knowing the de-adhered length $a$ the length of the film peeled from the substrate can be computed as $\lambda l(a\,;\rho)\,\mathscr{m}$, where $l(\cdot\,;\rho):\mathcal{D}\to\mathcal{D}$ is defined by the equation

We refer to $l(a\,;\rho)$ as the peeled length, and will often abbreviate it as $l$.

In postulating the evolution law for de-adherence, we ignore all dynamic effects, such as inertial forces, kinetic energy, and viscoelastic behavior in the thin film. As we mentioned in §1, we do use the notion of time, but only so that we can speak about the sequence of events that take place in our peeling experiment.

In our peeling experiment the thin film de-adheres due to the application of the force $\bm{\mathfrak{f}}$ to its free end, which initially is at $O$, the origin of $\mathcal{E}$. We think of force as a linear map from $\mathbb{E}$ into a one dimensional vector space whose elements have units of energy. The set of all forces can, of course, be made into a vector space, which we will denote as $\mathbb{F}$. We use the set $\left(\bm{\mathfrak{f}}_{i}\right)$ as a basis for $\mathbb{F}$, where $\bm{\mathfrak{f}}_{i}$ are defined such that $\bm{\mathfrak{f}}_{i}\left(\mathbscr{e}_{j}\right)=\mathbio{E}_{c}\,\delta_{ij}$. The symbol $\mathbio{E}_{c}$, which appears in the last expression, is defined to be equal to the energy $\lambda{\mathpzc{E}}\mathscr{b}\mathscr{h}\mathscr{m}$, where ${\mathpzc{E}}$ is the thin film’s Young’s modulus. We express $\bm{\mathfrak{f}}$ as $F\hat{\bm{\mathfrak{f}}}$, where $F\geq 0$ and $\hat{\bm{\mathfrak{f}}}:=-\cos(\theta)\bm{\mathfrak{f}}_{1}+\sin(\theta)\bm{\mathfrak{f}}_{2}$. The angle $\theta$ is shown marked in Figure 3. It is prescribed as part of the problem’s definition. We call it the nominal peeling angle, or simply peeling angle for short. We denote the position vector of the free end of the thin film, on which $\bm{\mathfrak{f}}$ acts, as $\mathbscr{u}\in\mathbb{E}$, and will often refer to this quantity as simply the force-position-vector. Our experiment is force controlled and quasi-static. By force-controlled we mean that while the system’s configuration is changing, $\bm{\mathfrak{f}}$ is held fixed. If our experiment were a general force controlled experiment, then we would be allowed to vary $\bm{\mathfrak{f}}$ once the adhered region had stopped evolving. However, in our experiment we are only allowed to vary $F$ after the adhered region has stopped evolving. By quasi-static we mean that each variation in $F$ is of infinitesimal magnitude and is applied all at once at a particular instance in time.

Consider an experimentally observed configuration of the thin film $\bm{\kappa}_{0}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$, in which the de-adhered length is $a-\Delta a$, the peeled length is $l-\Delta l$, the force position vector is $\mathbscr{u}-\Delta\mathbscr{u}$, and the force is $(F-\Delta F)\hat{\bm{\mathfrak{f}}}$. Imagine that a force variation of $\Delta F\hat{\bm{\mathfrak{f}}}$ is then abruptly added to the force that was previously acting on the thin film so that the new force is $F\hat{\bm{\mathfrak{f}}}$. As a result, the thin film’s configuration will evolve to a new configuration $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$, in which the de-adhered length is $a$, the peeled length is $l$, the force-position-vector is $\mathbscr{u}$, and, of course, the force is $F\hat{\bm{\mathfrak{f}}}$. We postulate that the new configuration $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ is the one that locally minimizes the system’s total potential energy and is closest to $\bm{\kappa}_{0}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$.

To make the postulate precise, consider a configuration $\widetilde{\bm{\kappa}}$ that is close to $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and has the same force as $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$, i.e., $F\hat{\bm{\mathfrak{f}}}$. The de-adhered and peeled lengths, and the force-position-vector in $\widetilde{\bm{\kappa}}$ are, however, different from those in $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$; we denote them, respectively, as $a+\delta a$, $l+\delta l$, and $\mathbscr{u}+\delta\mathbscr{u}$. Since $F\geq 0$, i.e. the peeled section of the film is in tension, the variations $\delta l$ and $\delta\mathbscr{u}$, in fact, depend on $\delta a$. Thus, these variations are to be interpreted as abbreviations for $\delta l(\delta a\,;a,F,\rho)$ and $\delta\mathbscr{u}(\delta a\,;a,F,\rho)$, respectively. Let the difference in the system’s potential energy between the configurations $\widetilde{\bm{\kappa}}$ and $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ be $\mathbio{E}_{c}\,\delta E$, where $\mathbio{E}_{c}$ is a constant and has unit of energy, and we refer to $\delta E\in\mathbb{R}$ as the non-dimensional potential energy variation. In our model of the wavy peeling experiment, we take $\delta E$ to be a sum of three different energy variations. These variations take place, respectively, in the energy stored in the interbody adhesive interactions between the thin film and the substrate, the elastic strain energy stored in the peeled part of the thin film, and, finally, the energy stored in the apparatus that maintains the constant force $F\hat{\bm{\mathfrak{f}}}$ between the configuration $\bm{\kappa}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and $\widetilde{\bm{\kappa}}$.

We model adhesive interactions between the thin film and the substrate using the JKR theory [20]. According to this theory, the formation of an interface region of area $\delta\mathscr{A}$ lowers the system’s total potential energy, irrespective of the shape of the interface region or any other details of the experiment, reversibly by $\mathscr{w}\delta\mathscr{A}$. Here, $\mathscr{w}$ is the Dupré work of adhesion [43, p. 30]. Thus, the variation in the potential energy stored in the interbody adhesive interactions is

where $w$ is defined such that ${\mathpzc{E}}\mathscr{h}w=\mathscr{w}$. That is, $w$ is the non-dimensional work of adhesion.

We assume that the peeled part of the film is in uniform, uniaxial tension. It follows from this assumption that the strain in the peeled part of the film is also uniform, and uniaxial, and that its magnitude is equal to $F$. It also follows that the variation in the elastic strain energy stored in the thin film is

Usually, the above term is supplemented by an additional term that corresponds to bending energy in the thin film [42]. We, however, ignore the film’s bending energy in our model. As we state in our closing remarks, we believe that ignoring the bending energy is unlikely to significantly affect our estimates for the film’s peel-off force in the type of the peeling experiments that we consider in this paper.

The variation in the potential energy of the apparatus maintaining the constant force $F\hat{\bm{\mathfrak{f}}}$ is $-\bm{\mathfrak{f}}(\delta\mathbscr{u})$. Say $\delta\mathbscr{u}=\delta u_{j}\mathbscr{e}_{j}$. This variation can therefore be written as

where $\bm{F}:=F\hat{\bm{F}}$, $\hat{\bm{F}}:=(-\cos(\theta),\sin(\theta),0)$, and $\delta\bm{u}:=\left(\delta u_{i}\right)$. It follows from the previous discussion that,

As is the case with $\delta l$ and $\delta\bm{u}$, the symbol $\delta E$ in (3.8) is, in fact, an abbreviation for the value $\delta E(\delta a;\,a,F,\rho)$.

Since $\delta l$ and $\delta\bm{u}$ depend on $\delta a$, the formula for $\delta E$ given by (3.8) needs to be further refined before it can be used for determining whether or not a particular de-adhered length $a$ is in equilibrium; and, if $a$ is in equilibrium, then for determining the stability of that equilibrium. The particulars of the requisite refinement depend on whether or not the peeled part of the film contacts the substrate. Therefore, we will be making separate refinements of (3.8) for the cases of no-contact and contact in §3.2 and §3.3, respectively. In both cases, however, the refinement process will involve determining the asymptotic dependence of $\delta l$ and $\delta\bm{u}$ on $\delta a$ as $\delta a\to 0$, and then using that information to determine the asymptotic dependence of $\delta E$ on $\delta a$.

Let $\bm{T}\left(x_{1}\right)$ be $(\mathbscr{e}_{1}\cdot\dot{\bm{\mathbscr{x}}}(x_{1}),\mathbscr{e}_{2}\cdot\dot{\bm{\mathbscr{x}}}(x_{1}),0)\in\mathbb{R}^{3}$, where $\mathbscr{x}(x_{1})$ is defined in (3.3b). Because a material particle $x_{1}$ is de-adhered from or adhered to the substrate (depending on whether $x_{1}$ is less than or greater than $a$), the vector $\bm{T}(x_{1})$ will often be discontinuous at $x_{1}=a$. However, since we have assumed $\dot{\varrho}$, $\dot{u}_{1}$, and $\dot{u}_{2}$ to be continuous, the following right and left hand limits of $\bm{T}\left(x_{1}\right)$ as $x_{1}\to a$ are well defined:

The vector $\bm{T}\left(a^{+}\right)$ is essentially the tangent to the substrate’s surface profile at $x_{1}=a$, and the vector $-\bm{T}\left(a^{-}\right)$ points in the direction the detached part of the thin film leaves the substrate’s surface at the peeling front. (See Figure 3.) We call the angle between $\bm{T}\left(a^{-}\right)$ and $\bm{T}\left(a^{+}\right)$ the true peeling angle $\psi(a)$. For the thin film to not go through the substrate immediately after de-adhering from it, it is necessary that

We refer to (3.9) as the local compatibility condition.

It can be shown that once the local compatibility condition is satisfied, the peeled part of the thin film will not contact the substrate anywhere, irrespective of the location of the peeling front iff

We refer to (3.10) as the global compatibility condition.

Since the peeled part of the thin film is not in contact with the substrate, it follows that

where $\varepsilon$ is the uniform, uni-axial strain in the peeled part of the thin film, and $\delta\rho:={\rho}({a}+\delta{a})-{\rho}({a})$. It follows from (3.4) that

where

Substituting the asymptotic expansions for $\delta\rho$ and $\delta l$ as $\delta a\to 0$ into (3.11), and then substituting the resulting asymptotic expansion for $\delta\bm{u}$ into (3.7), we find that

The symbol $o$, that appears in (3.14) and elsewhere, is the Bachmann-Landau “Small-Oh” symbol. Its primary property of relevance is that $o\left(\left(\delta a\right)^{n}\right)/(\delta a)^{n}\to 0$, where $n=0,~{}1,\ldots$, as $\delta a\to 0$.

In the current case of no contact it also follows that the true peeling angle is given by

By recognizing from (3.15) that

we can rewrite (3.14) as

By substituting (3.17) into (3.8), and noting that on account of our non-dimensionalization scheme $\varepsilon=F$, we get that

Since we take $\rho\in C^{2}$, it follows that $\dot{\rho}$ and $\dot{l}$ are continuous, and, as a consequence, that $\delta E_{1}(\cdot;\rho,F)$ is continuous. It then follows that for $a$ to be an equilibrium de-adhered length it is necessary that $\delta E_{1}(a\,;\rho,F)=0$. For a given $F$ and $\rho$ there can, however, be more than one de-adhered length that is in equilibrium. We characterize all those lengths by saying that they belong to the set

We plot the set of points

for various $F$ values, for the example surfaces shown in Figures 4a and c in Figures 4b and d, respectively. As can be seen, the peeling force for peeling on wavy surface is not constant, but varies with the same periodicity of the wavy surfaces. It can be shown that the point set $\mathcal{D}^{\circ}(F,\rho)\mathbin{\leavevmode\hbox to6.46pt{\vbox to6.46pt{\pgfpicture\makeatletter\hbox{\hskip 3.22914pt\lower-3.22914pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}\pgfsys@setlinewidth{0.43056pt}\pgfsys@invoke{ }{}{{}}{} {}{}{}{{}}{} {}{}{}\pgfsys@moveto{-3.01387pt}{-3.01387pt}\pgfsys@lineto{3.01387pt}{3.01387pt}\pgfsys@moveto{-3.01387pt}{3.01387pt}\pgfsys@lineto{3.01387pt}{-3.01387pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\{F\}$, for any admissible $F$, falls on the graph of the periodic function

We only consider cases in which the work of adhesion $w$ is non-negative. It therefore follows from (3.20b) that $\mathpzc{F}(\psi)$ is always non-negative. The graph of $\mathcal{F}$ for different $w$ values is shown in Figure 5. As can be seen, $\mathcal{F}$ is a strictly decreasing function whose value at any admissible $\psi$ increases with $w$.

Kendall analyzed the peeling of a thin film on a flat smooth surface [19]. By setting $\rho(x_{1})=0$ for a flat surface, which leads to $\psi(a)=\theta$ from (3.15), Kendall’s result can be immediately recovered from (3.20) which gives the peeling force as

which is a constant for a given nominal peeling angle $\theta$.

For a general peeling process we define the supremum and infimum equilibrium force values, denoted as $F^{+}$ and $F^{-}$, respectively, as

It can be shown that the maximum and minimum values of the function $\mathpzc{F}$ are $(2w)^{1/2}$ and $(4+2w)^{1/2}-2$, respectively. This implies that $F^{+}$ is bounded above by $(2w)^{1/2}$ and $F^{-}$ is bounded below by $(4+2w)^{1/2}-2$.

We denote the maximum and minimum values of the true peeling angle during peeling as $\psi^{+}$ and $\psi^{-}$, respectively. In the current case of peeling with no contact it follows from the fact that $\mathcal{F}$ is a monotonically decreasing function that

Since $\mathpzc{F}(\psi)$ is always non-negative it follows from (3.23a) that $F^{\pm}\geq 0$.

The inequality (3.28) implies that the derivative of the total energy w.r.t $a$ will be negative when $F$ is greater than $F^{+}$ irrespective of the value of $a$, $w$, or the nature of $\rho$ (see, e.g., Figures 6a and c). Thus, if $F>F^{+}$ the de-adhered length will grow without bound. Realistically, however, the de-adhered length will keep growing until the film completely detaches from the substrate. For that reason we call $F^{+}$ the peel-off force.

The result that (3.31) holds in the case where $0\leq F<F^{-}$ implies that the derivative of the total energy with respect to the de-adhered length in that case is positive irrespective of any other details in the problem (see, for example, Figures 6b and d). Therefore, if $0\leq F<F^{-}$ and the de-adhered length is initially naught, then the de-adhered length will not grow or if it is initially non-zero then it will keep decreasing, i.e., the peel front will keep receding until the entire thin film is adhered to the substrate. For this reason, we call $F^{-}$ the peel-initiation force.

We study the stability of the local equilibria by examining the sign of the second variation of the total potential energy. Specifically, a configuration with the de-adhered length $a$ is a stable equilibrium state iff $a$ belongs to the set

It is a neutral equilibrium state iff $a$ belongs to the set

and is an unstable equilibrium state if $a$ belongs to the set $\mathcal{D}^{\otimes}(F,\rho)$, which is a set of all de-adhered lengths that belong to $\mathcal{D}^{\circ}(F,\rho)$ but not to $\mathcal{D}^{\leavevmode\hbox to1.9pt{\vbox to1.9pt{\pgfpicture\makeatletter\hbox{\hskip 0.95pt\lower-0.95pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.75pt}{0.0pt}\pgfsys@curveto{0.75pt}{0.41422pt}{0.41422pt}{0.75pt}{0.0pt}{0.75pt}\pgfsys@curveto{-0.41422pt}{0.75pt}{-0.75pt}{0.41422pt}{-0.75pt}{0.0pt}\pgfsys@curveto{-0.75pt}{-0.41422pt}{-0.41422pt}{-0.75pt}{0.0pt}{-0.75pt}\pgfsys@curveto{0.41422pt}{-0.75pt}{0.75pt}{-0.41422pt}{0.75pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}(F,\rho)$ or $\mathcal{D}^{\odot}(F,\rho)$.