Sensitivity Analysis of Articular Contact Mechanics

Abstract

Asymptotic models of articular contact developed in the previous chapters assume, in particular, that the cartilage layers are of uniform thickness and are bonded to rigid substrates shaped like elliptic paraboloids. In this final chapter, treating the term “sensitivity” in a broad sense, we study the effects of deviation of the substrate’s shape from the elliptic (Sect. 9.1) and of nonuniform thicknesses of the contacting incompressible layers (Sect. 9.2). It is shown that these effects in multibody dynamics simulations can be minimized if the geometric parameters in question (in particular, the layer thicknesses) are determined in a specific way to minimize the corresponding error in the force-displacement relationship.

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9.1 Non-elliptical Contact of Thin Incompressible Viscoelastic Layers: Perturbation Solution

In this section, a more general three-dimensional unilateral contact problem for thin incompressible transversely isotropic viscoelastic layers bonded to rigid substrates, whose shapes are close to those of elliptic paraboloids, is considered and approximately solved by the perturbation technique.

9.1.1 Formulation of the Contact Problem

Consider the frictionless unilateral contact between two thin linear incompressible transversely isotropic viscoelastic layers firmly attached to rigid substrates. Introducing the Cartesian coordinate system $(y_1,y_2,z)$, we write the equations of the layer surfaces ($n=1,2$) in the form $z=(-1)^n\varphi _\varepsilon ^{(n)}(\mathbf{y})$, where $\mathbf{y}=(y_1,y_2)$. In the undeformed state, the two layer/substrate systems occupy convex domains $z\le -\varphi _\varepsilon ^{(1)}(\mathbf{y})$ and $z\ge \varphi _\varepsilon ^{(2)}(\mathbf{y})$ in contact with the plane $z=0$ at a single point chosen as the coordinate origin. Let us assume that

$$\begin{aligned} \varphi _\varepsilon ^{(n)}(\mathbf{y})=\varphi _0^{(n)}(\mathbf{y})+\varepsilon \phi _n(\mathbf{y}), \end{aligned}$$

(9.1)

where $\varphi _0^{(n)}(\mathbf{y})$ is an elliptic paraboloid, $\varepsilon$ is a small positive dimensionless parameter, and the function $\varepsilon \phi _n(\mathbf{y})$ describes a small deviation of the nth substrate surface from the paraboloid shape ($n=1,2$).

We denote the normal approach of the substrates by $\delta _\varepsilon (t)$. The linearized unilateral contact condition that the surface points of the viscoelastic layers do not penetrate one into another can then be written as follows:

$$\begin{aligned} \delta _\varepsilon (t)-w_0^{(1)}(t,\mathbf{y})-w_0^{(2)}(t,\mathbf{y})\le \varphi _\varepsilon ^{(1)}(\mathbf{y})+\varphi _\varepsilon ^{(2)}(\mathbf{y}). \end{aligned}$$

(9.2)

Here, $w_0^{(n)}(t,\mathbf{y})$ is the local indentation (i.e., the normal displacement of the surface points) of the nth layer ($n=1,2$).

According to the perturbation analysis performed in Sect. 2.5 (see, in particular, formula (2.152)), the leading-order asymptotic solution for the local indentation of an incompressible viscoelastic layer of thickness $h_n$ is given by

$$\begin{aligned} w_0^{(n)}(t,\mathbf{y})=-\frac{h_n^3}{3}\int \limits _{0^-}^t J^{\prime (n)}(t-\tau ) \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau . \end{aligned}$$

(9.3)

Here, $J^{\prime (n)}(t)$ is the out-of-plane shear creep compliance of the nth layer ($n=1,2$), $p_\varepsilon (t,\mathbf{y})$ is the contact pressure, and $\varDelta _y=\partial ^2/\partial y_1^2+\partial ^2/\partial y_2^2$ is the Laplace operator.

The equality in relation (9.2) determines the contact region $\omega _\varepsilon (t)$. In other words, the following equation holds within the contact area

$$\begin{aligned} w_0^{(1)}(t,\mathbf{y})+w_0^{(2)}(t,\mathbf{y})=\delta _\varepsilon (t)-\varphi _\varepsilon (\mathbf{y}),\quad \mathbf{y}\in \omega _\varepsilon (t), \end{aligned}$$

(9.4)

where we have introduced the notation $\varphi _\varepsilon (\mathbf{y})=\varphi _\varepsilon ^{(1)}(\mathbf{y})+\varphi _\varepsilon ^{(2)}(\mathbf{y})$.

According to (9.1), we have

$$\begin{aligned} \varphi _\varepsilon (\mathbf{y})=\varphi _0(\mathbf{y})+\varepsilon \phi (\mathbf{y}), \end{aligned}$$

(9.5)

where $\varphi _0(\mathbf{y})=\varphi _0^{(1)}(\mathbf{y})+\varphi _0^{(2)}(\mathbf{y})$ and $\phi (\mathbf{y})=\phi _1(\mathbf{y})+\phi _2(\mathbf{y})$. The function $\varepsilon \phi (\mathbf{y})$ will be called the gap function variation.

Without any loss of generality we may assume that

$$\begin{aligned} \varphi _0(\mathbf{y})=\frac{y_1^2}{2R_1}+\frac{y_2^2}{2R_2}, \end{aligned}$$

(9.6)

where the parameters $R_1$ and $R_2$ are positive and can be related to the coefficients of the paraboloids $\varphi _0^{(1)}(\mathbf{y})$ and $\varphi _0^{(2)}(\mathbf{y})$ by known formulas (see Sect. 2.1.1).

Substituting the expressions for displacements $w_0^{(1)}(t,\mathbf{y})$ and $w_0^{(2)}(t,\mathbf{y})$ given by formula (9.3) into Eq. (9.4), we obtain the contact condition in the following form:

$$\begin{aligned} -\sum _{n=1}^2\frac{h_n^3}{3}\int \limits _{0^-}^t J^{\prime (n)}(t-\tau ) \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau =\delta _\varepsilon (t)-\varphi _\varepsilon (\mathbf{y})\fancyscript{H}(t). \end{aligned}$$

(9.7)

Here, we assume that $\mathbf{y}\in \omega _\varepsilon (t)$, and $\fancyscript{H}(t)$ is Heaviside’s function introduced in a standard way, namely $\fancyscript{H}(t)=0$ for $t<0$ and $\fancyscript{H}(t)=1$ for $t\ge 0$.

Let $G_0^{\prime (n)}=1/J^{\prime (n)}(0^+)$ be the instantaneous out-of-plane shear elastic modulus of the nth layer. Then, the normalized creep function $\varPhi ^{\prime (n)}(t)$ is introduced by

$$\begin{aligned} J^{\prime (n)}(t)=\frac{1}{G_0^{\prime (n)}}\varPhi ^{\prime (n)}(t). \end{aligned}$$

(9.8)

Let us rewrite Eq. (9.7) in the form corresponding to one layer by introducing the compound creep function, $\varPhi _\beta (t)$, and the equivalent instantaneous shear elastic modulus, $G_0^{\prime }$, as follows:

$$\begin{aligned} \varPhi _\beta (t)=\beta _1\varPhi ^{\prime (1)}(t)+\beta _2\varPhi ^{\prime (2)}(t), \end{aligned}$$

(9.9)

$$\begin{aligned} G_0^{\prime }=\frac{(h_1+h_2)^3 G_0^{\prime (1)}G_0^{\prime (2)}}{ h_1^3 G_0^{\prime (2)}+h_2^3 G_0^{\prime (1)}}, \end{aligned}$$

(9.10)

$$\begin{aligned} \beta _1=\frac{h_1^3 G_0^{\prime (2)}}{ h_1^3 G_0^{\prime (2)}+h_2^3 G_0^{\prime (1)}}, \quad \beta _2=\frac{h_2^3 G_0^{\prime (1)}}{ h_1^3 G_0^{\prime (2)}+h_2^3 G_0^{\prime (1)}}. \end{aligned}$$

(9.11)

Moreover, let us introduce an auxiliary notation

$$\begin{aligned} m=\frac{3G_0^{\prime }}{h^3}, \end{aligned}$$

(9.12)

where $h=h_1+h_2$ is the joint thickness. Recall that formula (9.10) determines the equivalent modulus in such a way that $\beta _1+\beta _2=1$ and thus, $\varPhi _\beta (0)=1$.

Thus, taking into account (9.8)–(9.12), we rewrite Eq. (9.7) as

$$\begin{aligned} \int \limits _{0^-}^t \varPhi _\beta (t-\tau ) \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau =m\bigl (\varphi _\varepsilon (\mathbf{y})\fancyscript{H}(t)-\delta _\varepsilon (t)\bigr ). \end{aligned}$$

(9.13)

Equation (9.13) will be used to find the contact pressure density $p_\varepsilon (t,\mathbf{y})$. The contour $\varGamma _\varepsilon (t)$ of the contact area $\omega _\varepsilon (t)$ is determined from the condition that the contact pressure is positive and vanishes at the contour of the contact area:

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})>0, \quad \mathbf{y}\in \omega _\varepsilon (t), \end{aligned}$$

(9.14)

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _\varepsilon (t). \end{aligned}$$

(9.15)

In the case of the contact problem for an incompressible layer (see, in particular, Sect. 2.7.3), we additionally assume a smooth transition of the surface normal stresses from the contact region $\mathbf{y}\in \omega _\varepsilon (t)$ to the outside region $\mathbf{y}\not \in \omega _\varepsilon (t)$. Hence, we impose the following zero-pressure-gradient boundary condition [9, 10, 13, 17]:

$$\begin{aligned} \frac{\partial p_\varepsilon }{\partial n}(t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _\varepsilon (t). \end{aligned}$$

(9.16)

Here, $\partial /\partial n$ is the normal derivative directed outward from $\omega _\varepsilon (t)$.

We assume that the density $p_\varepsilon (t,\mathbf{y})$ is defined on the entire plane such that

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})= 0, \quad \mathbf{y}\not \in \omega _\varepsilon (t). \end{aligned}$$

(9.17)

From the physical point of view, the contact pressure between the smooth surfaces should satisfy the regularity condition, i.e., in the case (9.5), the function $p_\varepsilon (t,\mathbf{y})$ is assumed to be analytical in the domain $\omega _\varepsilon (t)$.

The equilibrium equation for the whole system is

$$\begin{aligned} \iint \limits _{\omega _\varepsilon (t)} p_\varepsilon (t,\mathbf{y})\,d\mathbf{y}=F(t), \end{aligned}$$

(9.18)

where F(t) denotes the external load, which is assumed to be known a priori.

For non-decreasing loads, when $dF(t)/dt\ge 0$, the contact zone should increase. Thus, we assume that the following monotonicity condition holds:

$$\begin{aligned} \omega _\varepsilon (t_1)\subset \omega _\varepsilon (t_2),\quad t_1\le t_2. \end{aligned}$$

(9.19)

Following Argatov and Mishuris [8], we construct an asymptotic solution for the three-dimensional contact problem formulated by Eq. (9.13), under the monotonicity condition (9.19). We will first consider the problem in its general formulation transforming it to a set of equations more suitable for further analysis.

9.1.2 Equation for the Contact Approach

Integrating Eq. (9.13) over the contact domain $\omega _\varepsilon (t)$, we find

$$\begin{aligned} \iint \limits _{\omega _\varepsilon (t)} \int \limits _{0^-}^t \varPhi _\beta (t-\tau ) \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau d\mathbf{y} =m\iint \limits _{\omega _\varepsilon (t)} \bigl (\varphi _\varepsilon (\mathbf{y})\fancyscript{H}(t)-\delta _\varepsilon (t)\bigr )\,d\mathbf{y}. \end{aligned}$$

(9.20)

In light of (9.17) and (9.19), we have $\omega _\varepsilon (\tau )\subset \omega _\varepsilon (t)$ for $\tau \in (0,t)$, as well as $p_\varepsilon (\tau ,\mathbf{y})\equiv 0$ for $\mathbf{y}\not \in \omega _\varepsilon (\tau )$. Therefore, the integral on the left-hand side of (9.20), which is denoted by $\fancyscript{I}(t)$, can be transformed into

$$\begin{aligned} \fancyscript{I}(t)= & {} \iint \limits _{\omega _\varepsilon (t)} \int \limits _{0^-}^t \varPhi _\beta (t-\tau ) \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau d\mathbf{y} \nonumber \\ { }= & {} \int \limits _{0^-}^t\varPhi _\beta (t-\tau )\iint \limits _{\omega _\varepsilon (t)} \varDelta _y\frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\mathbf{y}d\tau \nonumber \\ { }= & {} \int \limits _{0^-}^t\varPhi _\beta (t-\tau )\frac{\partial }{\partial \tau } \iint \limits _{\omega _\varepsilon (t)} \varDelta _y p_\varepsilon (\tau ,\mathbf{y})\,d\mathbf{y}d\tau . \end{aligned}$$

(9.21)

We note that, as a consequence of (9.15)–(9.17), the density $p_\varepsilon (t,\mathbf{y})$ is a smooth function of the variables $y_1$ and $y_2$ on the entire plane.

Further, by employing the second Green’s formula

$$\begin{aligned} \iint \limits _{\omega } \bigl (u(\mathbf{y})\varDelta v(\mathbf{y})-v(\mathbf{y})\varDelta u(\mathbf{y})\bigr )\,d\mathbf{y} = \int \limits _{\varGamma }\biggl (u(\mathbf{y})\frac{\partial v}{\partial n}(\mathbf{y}) -v(\mathbf{y})\frac{\partial u}{\partial n}(\mathbf{y}) \biggr )ds, \end{aligned}$$

(9.22)

where ds is the element of the arc length, we obtain

$$\begin{aligned} \iint \limits _{\omega _\varepsilon (t)} \varDelta _y p_\varepsilon (\tau ,\mathbf{y})\,d\mathbf{y}= \int \limits _{\varGamma _\varepsilon (t)}\frac{\partial p_\varepsilon }{\partial n}(\tau ,\mathbf{y})\,ds. \end{aligned}$$

(9.23)

In light of the boundary condition (9.16) and the monotonicity condition (9.19), the right-hand side of Eq. (9.23) vanishes for $t>0$ and, therefore, Eq. (9.20) reduces to

$$ \iint \limits _{\omega _\varepsilon (t)} \bigl (\varphi _\varepsilon (\mathbf{y})\fancyscript{H}(t)-\delta _\varepsilon (t)\bigr )\,d\mathbf{y}=0. $$

From here it immediately follows that

$$\begin{aligned} \delta _\varepsilon (t)=\frac{\fancyscript{H}(t)}{A_\varepsilon (t)}\iint \limits _{\omega _\varepsilon (t)}\varphi _\varepsilon (\mathbf{y})\,d\mathbf{y}, \end{aligned}$$

(9.24)

where $A_\varepsilon (t)$ is the area of $\omega _\varepsilon (t)$ given by the integral

$$\begin{aligned} A_\varepsilon (t)=\iint \limits _{\omega _\varepsilon (t)}d\mathbf{y}. \end{aligned}$$

(9.25)

Equation (9.24) connects the unknown contact approach $\delta _\varepsilon (t)$ with some integral characteristics of the contact domain $\omega _\varepsilon (t)$.

9.1.3 Equation for the Integral Characteristics the Contact Area

Substituting the functions $u(\mathbf{y})=p_\varepsilon (\tau ,\mathbf{y})$ and $v(\mathbf{y})=(y_1^2+y_2^2)/4$ into Green’s formula (9.22) for the domain $\omega _\varepsilon (t)$, assuming that $\tau <t$, and taking into account the boundary conditions (9.15), (9.16) and the monotonicity condition (9.19), we obtain the relation

$$\begin{aligned} \frac{1}{4}\iint \limits _{\omega _\varepsilon (t)} \vert \mathbf{y}\vert ^2\varDelta p_\varepsilon (\tau ,\mathbf{y})\,d\mathbf{y}= \iint \limits _{\omega _\varepsilon (\tau )} p_\varepsilon (\tau ,\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.26)

Using formula (9.26), we can evaluate the contact load (9.18). Indeed, by multiplying both sides of (9.13) by $(y_1^2+y_2^2)/4$ and integrating the obtained equation over the contact domain $\omega _\varepsilon (t)$, we obtain

$$\begin{aligned} \int \limits _{0^-}^t\varPhi _\beta (t-\tau )\frac{\partial }{\partial \tau } \iint \limits _{\omega _\varepsilon (\tau )} p_\varepsilon (\tau ,\mathbf{y})\,d\mathbf{y}d\tau =\frac{m}{4}\iint \limits _{\omega _\varepsilon (t)}\vert \mathbf{y}\vert ^2 \bigl (\fancyscript{H}(t)\varphi _\varepsilon (\mathbf{y})-\delta _\varepsilon (t)\bigr )\,d\mathbf{y}. \end{aligned}$$

(9.27)

Taking into account the notation (9.18) for the contact force, we rewrite (9.27) as

$$\begin{aligned} \int \limits _{0^-}^t\varPhi _\beta (t-\tau )\dot{F}(\tau )\,d\tau = \frac{m}{4}\fancyscript{H}(t) \iint \limits _{\omega _\varepsilon (t)}\vert \mathbf{y}\vert ^2\varphi _\varepsilon (\mathbf{y})\, d\mathbf{y} -\delta _\varepsilon (t)\frac{m}{4} \iint \limits _{\omega _\varepsilon (t)}\vert \mathbf{y}\vert ^2 d\mathbf{y}, \end{aligned}$$

(9.28)

where the dot denotes the differentiation with respect to time, i.e., $\dot{F}(t)=dF(t)/dt$.

Then, excluding the quantity $\delta _\varepsilon (t)$ from (9.28) by means of Eq. (9.24), we arrive at the following equation:

$$\begin{aligned} \int \limits _{0^-}^t\varPhi _\beta (t-\tau )\dot{F}(\tau )\,d\tau = \frac{m}{4}\fancyscript{H}(t)\iint \limits _{\omega _\varepsilon (t)} \biggl ( \vert \mathbf{y}\vert ^2-\frac{I_\varepsilon (t)}{A_\varepsilon (t)}\biggr ) \varphi _\varepsilon (\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.29)

Here, $I_\varepsilon (t)$ is the polar moment of inertia of $\omega _\varepsilon (t)$ given by the integral

$$\begin{aligned} I_\varepsilon (t)=\iint \limits _{\omega _\varepsilon (t)}\vert \mathbf{y}\vert ^2 d\mathbf{y}. \end{aligned}$$

(9.30)

Equation (9.29) connects the known contact force F(t) with some integral characteristics of the unknown contact area $\omega _\varepsilon (t)$.

9.1.4 Equation for the Contact Pressure

Let us rewrite Eq. (9.13) in the form

$$\begin{aligned} \varDelta _y P_\varepsilon (t,\mathbf{y})= m\bigl (\varphi _\varepsilon (\mathbf{y})-\delta _\varepsilon (t)\bigr ), \quad \mathbf{y}\in \omega _\varepsilon (t), \end{aligned}$$

(9.31)

where we have introduced the notation

$$\begin{aligned} P_\varepsilon (t,\mathbf{y})=\int \limits _{0^-}^t \varPhi _\beta (t-\tau ) \frac{\partial p_\varepsilon }{\partial \tau }(\tau ,\mathbf{y})\,d\tau . \end{aligned}$$

(9.32)

By denoting the integral operator on the right-hand side of the previous equation by $\fancyscript{K}$, we have

$$\begin{aligned} \fancyscript{K}y(\tau )=\int \limits _{0^-}^t \varPhi _\beta (t-\tau )\dot{y}(\tau )\,d\tau , \end{aligned}$$

(9.33)

so that formula (9.32) can be represented as

$$\begin{aligned} P_\varepsilon (t,\mathbf{y})=\fancyscript{K}p_\varepsilon (\tau ,\mathbf{y}). \end{aligned}$$

(9.34)

The inverse operator to ${\fancyscript{K}}$ denoted by ${\fancyscript{K}}^{-1}$ is defined by the formula

$$\begin{aligned} {\fancyscript{K}}^{-1}Y(\tau )=\int \limits _{0^-}^t \varPsi _\beta (t-\tau )\dot{Y}(\tau )\,d\tau , \end{aligned}$$

(9.35)

where $\varPsi _\beta (t)$ is the compound relaxation function defined by its Laplace transform

$$\begin{aligned} \tilde{\varPsi }_\beta (s)=\frac{1}{s^2\tilde{\varPhi }_\beta (s)}. \end{aligned}$$

(9.36)

Note that since

$$ \tilde{\varPhi }_\beta (s)=\beta _1\tilde{\varPhi }^{\prime (1)}(s)+\beta _2\tilde{\varPhi }^{\prime (2)}(s), $$

and

$$ \tilde{\varPhi }^{\prime (n)}(s)=\frac{1}{s^2\tilde{\varPsi }^{\prime (n)}(s)}, $$

where $\tilde{\varPsi }^{\prime (n)}(s)$ is the Laplace transform of the relaxation function in out-of-plane shear for the nth layer, formula (9.36) can be reduced to the following:

$$\begin{aligned} \tilde{\varPsi }_\beta (s)=\frac{ \tilde{\varPsi }^{\prime (1)}(s)\tilde{\varPsi }^{\prime (2)}(s)}{ \beta _1\tilde{\varPsi }^{\prime (2)}(s)+\beta _2\tilde{\varPsi }^{\prime (1)}(s)}. \end{aligned}$$

(9.37)

Recall also that the coefficients $\beta _1$ and $\beta _2$ are introduced by formulas (9.11) in such a way that $\varPsi _\beta (0)=1$.

As a result of the boundary conditions (9.15) and (9.16), the function $P_\varepsilon (t,\mathbf{y})$ must satisfy the following boundary conditions:

$$\begin{aligned} P_\varepsilon (t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _\varepsilon (t), \end{aligned}$$

(9.38)

$$\begin{aligned} \frac{\partial P_\varepsilon }{\partial n}(t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _\varepsilon (t). \end{aligned}$$

(9.39)

Thus, in the case of monotonically increasing contact area $\omega _\varepsilon (t)$, the problem (9.31), (9.38), (9.39) allows us to determine the domain $\omega _\varepsilon (t)$ based on the positivity condition for the contact pressure (9.14). Then, from Eq. (9.24) we can determine the contact approach $\delta _\varepsilon (t)$. Finally, by applying the inverse operator (9.35), we obtain a complete solution to the problem.

9.1.5 Limiting Case Problem: Elliptical Contact Area

We now consider the problem for the limiting case $\varepsilon =0$, when the function $\varphi _0(\mathbf{y})$ represents an elliptic paraboloid. In a special case of the integral operator ${\fancyscript{K}}$, the solution to this problem has previously been presented in [6]. Here we adopt it in the necessary form for the further asymptotic analysis, in order to construct a more general solution to the problem with the slightly perturbed boundary of an arbitrary shape.

In this case the right-hand side of (9.31) takes the form $m\bigl (\varphi _0(\mathbf{y})-\delta _0(t)\bigr )$. This suggests that we assume the domain $\omega _0(t)$ to be elliptical and so we set

$$\begin{aligned} P_0(t,\mathbf{y})=Q_0(t)\biggl (1-\frac{y_1^2}{a(t)^2}-\frac{y_2^2}{b(t)^2}\biggr )^2. \end{aligned}$$

(9.40)

In other words, the contour $\varGamma _0(t)$ is an ellipse with the semi-axises a(t) and b(t). It is simple to verify that the function $P_0(t,\mathbf{y})$ satisfies the boundary conditions (9.38) and (9.39) exactly.

Substituting (9.40) into Eq. (9.31), we obtain after some algebra the following system of algebraic equations:

$$\begin{aligned} \delta _0(t) = \frac{4Q_0(t)}{m}\biggl (\frac{1}{a(t)^2}+\frac{1}{b(t)^2}\biggr ), \end{aligned}$$

(9.41)

$$\begin{aligned} \frac{1}{R_1} = \frac{8Q_0(t)}{m a(t)^2}\biggl (\frac{3}{a(t)^2}+\frac{1}{b(t)^2}\biggr ), \quad \frac{1}{R_2} = \frac{8Q_0(t)}{m b(t)^2}\biggl (\frac{1}{a(t)^2}+\frac{3}{b(t)^2}\biggr ). \end{aligned}$$

(9.42)

The form of the ellipse $\varGamma _0(t)$ can be characterized by its aspect ratio s defined as

$$\begin{aligned} s=\frac{b(t)}{a(t)}, \end{aligned}$$

(9.43)

and from (9.42), it immediately follows that s is constant with time and is determined as a positive root of the equation

$$\begin{aligned} \frac{R_2}{R_1}=\frac{s^2(3s^2+1)}{3+s^2}. \end{aligned}$$

(9.44)

In turn, Eq. (9.44) can be reduced to a quadratic equation for $s^2$, so that

$$\begin{aligned} s^2=\sqrt{\biggl (\frac{R_1-R_2}{6R_1}\biggr )^2+\frac{R_2}{R_1}} -\frac{(R_1-R_2)}{6R_1}. \end{aligned}$$

(9.45)

Recall that, along with Eqs. (9.41) and (9.42), we have Eqs. (9.24) and (9.29), which connect the contact approach $\delta _\varepsilon (t)$ and the known contact force F( t) with some integral characteristics of the contact domain $\omega _\varepsilon (t)$.

By taking into account (9.45), we transform Eq. (9.24) into

$$\begin{aligned} \delta _0(t)=\frac{1}{8}\biggl (\frac{1}{R_1}+\frac{s^2}{R_2}\biggr )a(t)^2. \end{aligned}$$

(9.46)

Then, by excluding the quantity $\delta _0(t)$ from Eqs. (9.41) and (9.46), we obtain

$$\begin{aligned} Q_0(t)=\frac{m}{32}\frac{s^2}{(s^2+1)}\biggl (\frac{1}{R_1}+\frac{s^2}{R_2}\biggr )a(t)^4. \end{aligned}$$

(9.47)

By application of the same method, Eq. (9.29) becomes

$$\begin{aligned} {\fancyscript{K}}F(\tau )= \frac{\pi m}{384} \biggl (\frac{3s-s^3}{R_1}+\frac{3s^5-s^3}{R_2}\biggr )a(t)^6. \end{aligned}$$

(9.48)

This allows us to determine the major semi-axis a(t) of the contact area $\omega _0(t)$ as a function of time t in the form

$$\begin{aligned} a(t)=\biggl [\frac{\pi m}{384} \biggl (\frac{3s-s^3}{R_1}+\frac{3s^5-s^3}{R_2}\biggr )\biggr ]^{-1/6} \big ({\fancyscript{K}}F(\tau ) \big )^{1/6}. \end{aligned}$$

(9.49)

As a consequence of (9.49), formulas (9.46) and (9.47) determine the quantities $\delta _0(t)$ and $Q_0(t)$, respectively.

We now turn to evaluating the contact pressure in the case of elliptical contact. In light of (9.33) and (9.40), we obtain the following operator equation for the contact pressure density $p_0(t,\mathbf{y})$:

$$\begin{aligned} \fancyscript{K}p_0(\tau ,\mathbf{y})=Q_0(t)\biggl (1-\frac{y_1^2}{a(t)^2}-\frac{y_2^2}{b(t)^2} \biggr )^2, \quad \mathbf{y}\in \omega _0(t). \end{aligned}$$

(9.50)

By inverting Eq. (9.50) with the help of (9.35), we obtain

$$\begin{aligned} p_0(t,\mathbf{y})=\fancyscript{K}^{-1}\Biggl \{ Q_0(\tau )\biggl (1-\frac{y_1^2}{a(\tau )^2}-\frac{y_2^2}{b(\tau )^2}\biggr )^2 \fancyscript{H}\biggl (1-\frac{y_1^2}{a(\tau )^2}-\frac{y_2^2}{b(\tau )^2}\biggr )\Biggr \}, \end{aligned}$$

(9.51)

where the Heaviside factor indicates the contact domain $\omega _0(\tau )$.

Following the notation (9.35), Eq. (9.51) can be transformed into

$$\begin{aligned} p_0(t,\mathbf{y})= & {} \int \limits _{0^-}^t \varPsi _\beta (t-\tau ) Q_0(\tau )\biggl (1-\frac{y_1^2}{a(\tau )^2}-\frac{y_2^2}{b(\tau )^2}\biggr )_+^2d\tau . \end{aligned}$$

(9.52)

Here the positive part function $(x)_+=(x+\vert x\vert )/2$ is used as an indicator of the current contact area $\omega _0(\tau )$.

9.1.6 Slightly Perturbed Elliptical Contact Area

We now consider the gap function $\varphi _\varepsilon (\mathbf{y})$, given in a general form (9.5) with small $\varepsilon >0$. The solution corresponding to the limiting case $\varepsilon =0$ was defined above and is denoted by $p_0(t,\mathbf{y})$ and $\delta _0(t)$, where the contact domain $\omega _0(t)$ is bounded by an ellipse $\varGamma _0(t)$. Then, the function $P_0(t,\mathbf{y})=\fancyscript{K}p_0(\tau ,\mathbf{y})$ satisfies the problem

$$\begin{aligned} \varDelta _y p_0(t,\mathbf{y})=m\bigl (\varphi _0(\mathbf{y})-\delta _0(t)\bigr ), \quad \mathbf{y}\in \omega _0(t), \end{aligned}$$

(9.53)

$$\begin{aligned} p_0(t,\mathbf{y})=0, \quad \frac{\partial p_0}{\partial n}(t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.54)

Recall also that in accordance with (9.24), the contact approach $\delta _0(t)$ is given by

$$\begin{aligned} \delta _0(t)=\frac{1}{A_0(t)}\iint \limits _{\omega _0(t)}\varphi _0(\mathbf{y})\,d\mathbf{y}, \end{aligned}$$

(9.55)

where $A_0(t)$ is the area of $\omega _0(t)$. Moreover, in light of (9.29), we have

$$\begin{aligned} {\fancyscript{K}}F(\tau ) = \frac{m}{4}\iint \limits _{\omega _0(t)} B_0(t,\mathbf{y}) \varphi _0(\mathbf{y})\,d\mathbf{y}, \end{aligned}$$

(9.56)

where, with $I_0(t)$ being the polar moment of inertia of $\omega _0(t)$,

$$\begin{aligned} B_0(t,\mathbf{y})=\vert \mathbf{y}\vert ^2-\frac{I_0(t)}{A_0(t)}. \end{aligned}$$

(9.57)

We represent the solution to the perturbed auxiliary contact problem (9.31), (9.38), (9.39) as

$$\begin{aligned} P_\varepsilon (t,\mathbf{y})=P_0(t,\mathbf{y})+\varepsilon P_1(t,\mathbf{y})+O(\varepsilon ^2), \end{aligned}$$

(9.58)

$$\begin{aligned} \delta _\varepsilon (t)=\delta _0(t)+\varepsilon \delta _1(t)+O(\varepsilon ^2), \end{aligned}$$

(9.59)

and recall that the contact load F(t) is assumed to be specified, while the contact approach $\delta _\varepsilon (t)$ is unknown a priori.

By substituting (9.58) and (9.59) into Eq. (9.31), we arrive at the equation

$$\begin{aligned} \varDelta _y P_1(t,\mathbf{y})=m\bigl (\varphi (\mathbf{y})-\delta _1(t)\bigr ), \quad \mathbf{y}\in \omega _0(t). \end{aligned}$$

(9.60)

Let us assume that the unknown boundary $\varGamma _\varepsilon (t)$ of the contact area $\omega _\varepsilon (t)$ (see Fig. 9.1) is described by the equation

$$\begin{aligned} n=h_\varepsilon (t,\sigma ),\quad s\in \varGamma _0(t). \end{aligned}$$

(9.61)

Here, $\sigma$ is the arc length along $\varGamma _0(t)$, and n is the distance (taking the sign into account) measured along the outward (with respect to the domain $\omega _0(t)$) normal to the curve $\varGamma _0(t)$. The function $h_\varepsilon (t,\sigma )$ describes the variation of the contact area and should be determined by considering the boundary conditions for Eq. (9.60).

Fig. 9.1

Schematic representation of the contact domain $\omega _\varepsilon (t)$ with the boundary $\varGamma _\varepsilon (t)$, and the limit domain $\omega _0(t)$ with the boundary $\varGamma _0(t)$

In light of (9.58) and (9.59), we set

$$\begin{aligned} h_\varepsilon (t,\sigma )=\varepsilon h(t,\sigma ), \end{aligned}$$

(9.62)

where the function $h(t,\sigma )$ is assumed to be independent of $\varepsilon$.

Applying the perturbation technique (see, for example, [20]), we have

$$\begin{aligned} P_\varepsilon \bigr \vert _{\varGamma }=P_\varepsilon \bigr \vert _{\varGamma _0} +h_\varepsilon \frac{\partial P_\varepsilon }{\partial n}\Bigr \vert _{\varGamma _0} +O(\varepsilon ^2). \end{aligned}$$

(9.63)

Let $\mathbf{n}_\varepsilon$ be the unit outward normal vector to the curve $\varGamma _\varepsilon (t)$. Then, the following formula holds:

$$\begin{aligned} \mathbf{n}_\varepsilon =\frac{\bigl (1-\kappa (t,\sigma )h_\varepsilon (t,\sigma )\bigr )\mathbf{n}_0 -h_\varepsilon ^\prime (t,\sigma )\mathbf{t}_0}{ \sqrt{\bigl (1-\kappa (t,\sigma )h_\varepsilon (t,\sigma )\bigr )^2 +h_\varepsilon ^\prime (t,\sigma )^2}}. \end{aligned}$$

(9.64)

Here, $\mathbf{t}_0$ and $\mathbf{n}_0$ are the unit tangential and outward normal vectors to the curve $\varGamma _0(t)$, $\kappa (t,\sigma )$ is the curvature of $\varGamma _0(t)$, and the prime denotes the derivative with respect to the arc length $\sigma$.

Taking into account formula (9.64), we obtain

$$\begin{aligned} \frac{\partial p_\varepsilon }{\partial n}\Bigr \vert _{\varGamma }=\frac{\partial p_\varepsilon }{\partial n}\Bigr \vert _{\varGamma _0} -h_\varepsilon ^\prime \frac{\partial p_\varepsilon }{\partial \sigma }\Bigr \vert _{\varGamma _0} +O(\varepsilon ^2), \end{aligned}$$

(9.65)

and by substituting the expansion (9.58) into Eqs. (9.63), (9.65) and taking into account the boundary conditions (9.54) for the function $P_0(t,\mathbf{y})$, we derive the following boundary conditions at the unperturbed contact boundary:

$$\begin{aligned} P_1(t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.66)

$$\begin{aligned} \frac{\partial P_1}{\partial n}(t,\mathbf{y})=-h(t,\sigma ) \frac{\partial ^2 P_0}{\partial n^2}(t,\mathbf{y}), \quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.67)

Here, in accordance with Eq. (9.53), we have

$$\begin{aligned} \frac{\partial ^2 P_0}{\partial n^2}(t,\mathbf{y})= m\bigl (\varphi _0(\mathbf{y})-\delta _0(t)\bigr ), \quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.68)

Note that the right-hand side of (9.68) is strictly positive. This can be verified by employing the explicit formula obtained in Sect. 9.1.5, or proved by using the maximum principle for harmonic functions.

Finally, Eqs. (9.24) and (9.29) yield

$$\begin{aligned} A_0(t)\delta _1(t)=\iint \limits _{\omega _0(t)}\phi (\mathbf{y})\,d\mathbf{y} +\int \limits _{\varGamma _0(t)}\varphi _0(\mathbf{y})h(t,\sigma )\,d\sigma _y -A_1(t)\delta _0(t), \end{aligned}$$

(9.69)

from which it follows that, for $B_0(t,\mathbf{y})$ defined by (9.56),

$$\begin{aligned} 0= & {} \iint \limits _{\omega _0(t)}B_0(t,\mathbf{y})\phi (\mathbf{y})\,d\mathbf{y} -\delta _0(t)\biggl (I_1(t)-\frac{I_0(t)}{A_0(t)}A_1(t)\biggr ) \nonumber \\ { }{} & {} {}+\int \limits _{\varGamma _0(t)}B_0(t,\mathbf{y}) \varphi _0(\mathbf{y})h(t,\sigma )\,d\sigma _y. \end{aligned}$$

(9.70)

Here, $A_1(t)$ and $I_1(t)$ are the first-order perturbation coefficients of $A_\varepsilon (t)$ and $I_\varepsilon (t)$, respectively, given by

$$\begin{aligned} A_1(t)=\int \limits _{\varGamma _0(t)}h(t,\sigma )\,d\sigma _y, \end{aligned}$$

(9.71)

$$\begin{aligned} I_1(t)=\int \limits _{\varGamma _0(t)}\vert \mathbf{y}\vert ^2 h(t,\sigma )\,d\sigma _y. \end{aligned}$$

(9.72)

Equations (9.60), (9.66), (9.67), (9.69) and (9.70) constitute the first-order perturbation problem. By employing the relations (9.68) and (9.71), it is not hard to check that Eq. (9.69) coincides with the solvability condition (see, for instance, (9.23)) of the boundary-value problem (9.60), (9.66), (9.67).

9.1.7 Determination of the Contour of the Contact Area

First, let us express the solution to Eq. (9.60) in the form

$$\begin{aligned} P_1(t,\mathbf{y})=m\bigl (P_1^{[0]}(t,\mathbf{y})+P_1^{[1]}(t,\mathbf{y})\bigr ), \end{aligned}$$

(9.73)

where we have introduced the notation

$$\begin{aligned} P_1^{[0]}(t,\mathbf{y})=Y_\phi ^{[0]}(\mathbf{y})-\delta _1(t)Y_1^{[0]}(\mathbf{y}), \end{aligned}$$

(9.74)

$$\begin{aligned} Y_\phi ^{[0]}(\mathbf{y})=\frac{1}{2\pi }\iint \limits _{\omega _0(t)} \phi (\mathbf{y})\ln \vert \mathbf{y}-\mathbf{y}\vert \,d\mathbf{y}, \quad Y_1^{[0]}(\mathbf{y})=\frac{1}{2\pi }\iint \limits _{\omega _0(t)} \ln \vert \mathbf{y}-\mathbf{y}\vert \,d\mathbf{y}. \end{aligned}$$

(9.75)

Substituting the form (9.73) into Eqs. (9.60), (9.66) and (9.67), we obtain the following boundary value problem for the function $P_1^{[1]}(t,\mathbf{y})$:

$$\begin{aligned} \varDelta _y P_1^{[1]}(t,\mathbf{y})=0, \quad \mathbf{y}\in \omega _0(t), \end{aligned}$$

(9.76)

$$\begin{aligned} P_1^{[1]}(t,\mathbf{y})=-Y_\phi ^{[0]}(\mathbf{y})+\delta _1(t)Y_1^{[0]}(\mathbf{y}), \quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.77)

$$\begin{aligned} \frac{\partial P_1^{[1]}}{\partial n}(t,\mathbf{y})= & {} -h(t,\sigma )\bigl (\varphi _0(\mathbf{y})-\delta _0(t)\bigr ) \nonumber \\ { }{} & {} {}-\frac{\partial Y_\phi ^{[0]}}{\partial n}(t,\mathbf{y})+ \delta _1(t)\frac{\partial Y_1^{[0]}}{\partial n}(t,\mathbf{y}),\quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.78)

Here we have used relations (9.68) and (9.74).

Let us denote the first term on the right-hand side of (9.78) by $-\hat{h}(t,\sigma )$, so that

$$\begin{aligned} h(t,\sigma )=\frac{\hat{h}(t,\sigma )}{\varphi _0(\mathbf{y})-\delta _0(t)},\quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.79)

As a result of (9.71), (9.72) and (9.79), Eqs. (9.69), (9.70) and (9.78) respectively take the forms

$$\begin{aligned} \delta _1(t)=\frac{1}{A_0(t)}\iint \limits _{\omega _0(t)}\phi (\mathbf{y})\,d\mathbf{y} +\frac{1}{A_0(t)}\int \limits _{\varGamma _0(t)}\hat{h}(t,\sigma )\,d\sigma _y, \end{aligned}$$

(9.80)

$$\begin{aligned} 0 = \iint \limits _{\omega _0(t)}B_0(t,\mathbf{y})\phi (\mathbf{y})\,d\mathbf{y} +\int \limits _{\varGamma _0(t)}B_0(t,\mathbf{y})\hat{h}(t,\sigma )\,d\sigma _y. \end{aligned}$$

(9.81)

$$\begin{aligned} \frac{\partial P_1^{[1]}}{\partial n}(t,\mathbf{y})=-\hat{h}(t,\sigma ) -\frac{\partial Y_\phi ^{[0]}}{\partial n}(t,\mathbf{y})+ \delta _1(t)\frac{\partial Y_1^{[0]}}{\partial n}(t,\mathbf{y}),\quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.82)

Second, to proceed further with the computation of the contour of the contact area, we now define the Steklov—Poincaré (Dirichlet-to-Neumann) operator $\mathfrak {S}:H^{1/2}(\varGamma _0(t))\rightarrow H^{-1/2}(\varGamma _0(t))$ by

$$\begin{aligned} \bigl (\mathfrak {S}g\bigr )(\mathbf{y})=\frac{\partial w}{\partial n}(\mathbf{y}),\quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.83)

where $w(\mathbf{y})$ is the unique solution of the Dirichlet problem

$$\begin{aligned} \varDelta _y w(\mathbf{y})=0, \quad \mathbf{y}\in \omega _0(t); \qquad w(\mathbf{y})=g(\mathbf{y}), \quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.84)

The operator $\mathfrak {S}$ for a circular domain is well known, and we will use the above form later in Sect. 9.1.9. To the authors’ best knowledge there is no closed form representation for $\mathfrak {S}$ in the case of an elliptic domain. However, the finite element Steklov—Poincaré operator can be computed by standard FEM packages (see for details [15]). In [8], an alternative approach for constructing the operator numerically in terms of conformal mappings was presented.

In terms of the Steklov—Poincaré operator, Eqs. (9.77) and (9.82) yield

$$\begin{aligned} \hat{h}(t,\sigma )= & {} \bigl (\mathfrak {S}Y_\phi ^{[0]}\bigr )(\mathbf{y}) -\frac{\partial Y_\phi ^{[0]}}{\partial n}(t,\mathbf{y}) \nonumber \\ { }{} & {} {}-\delta _1(t)\biggl (\bigl (\mathfrak {S}Y_1^{[0]}\bigr )(\mathbf{y}) -\frac{\partial Y_1^{[0]}}{\partial n}(t,\mathbf{y})\biggr ),\quad \mathbf{y}\in \varGamma _0(t). \end{aligned}$$

(9.85)

Note that the substitution of (9.85) into (9.80) results in an identity, which we check by verifying the following properties:

$$ \int \limits _{\varGamma _0(t)}\bigl (\mathfrak {S}g\bigr )(\mathbf{y})\,d\sigma _y=0,\quad \forall g\in H^{1/2}(\varGamma _0(t)), $$

$$ \iint \limits _{\omega _0(t)}\phi (\mathbf{y})\,d\mathbf{y}= \int \limits _{\varGamma _0(t)}\frac{\partial Y_\phi ^{[0]}}{\partial n}(t,\mathbf{y})\,d\sigma _y, \quad \forall \varphi \in L^2(\omega _0(t)). $$

Now, excluding the variable $\delta _1(t)$ from (9.85) by means of (9.80), we obtain

$$\begin{aligned} \hat{h}(t,\sigma )= & {} - \Biggl (\frac{1}{A_0(t)}\iint \limits _{\omega _0(t)}\phi (\mathbf{y})\,d\mathbf{y}+\hat{H}_0(t)\Biggr ) \biggl (\bigl (\mathfrak {S}Y_1^{[0]}\bigr )(\mathbf{y})- \frac{\partial Y_1^{[0]}}{\partial n}(t,\mathbf{y})\biggr ) \nonumber \\ {}{} & {} {}+\bigl (\mathfrak {S}Y_\phi ^{[0]}\bigr )(\mathbf{y}) -\frac{\partial Y_\phi ^{[0]}}{\partial n}(t,\mathbf{y}) ,\quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.86)

where $\hat{H}_0(t)$ is the relative weighted increment of the contact area defined as

$$\begin{aligned} \hat{H}_0(t)= \frac{1}{A_0(t)}\int \limits _{\varGamma _0(t)}\hat{h}(t,\sigma )\,d\sigma . \end{aligned}$$

(9.87)

At this point, the function $\hat{h}(t,\sigma )$ is determined by Eq. (9.86) with an accuracy up to its integral characteristics $\hat{H}_0(t)$.

By substitution of the expression (9.86) into Eq. (9.81), we arrive at the following simple equation to determine $\hat{H}_0(t)$:

$$\begin{aligned} \varXi _1^{[0]}(t)\hat{H}_0(t) = \varXi _\phi ^{[0]}(t)+\iint \limits _{\omega _0(t)}\biggl ( B_0(t,\mathbf{y})-\frac{\varXi _1^{[0]}(t)}{A_0(t)}\biggr )\phi (\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.88)

Here, both functions $\varXi _1^{[0]}(t)$ nd $\varXi _\phi ^{[0]}(t)$ are determined by the formula

$$\begin{aligned} \varXi _{\phi ;1}^{[0]}(t)=\int \limits _{\varGamma _0(t)}B_0(t,\mathbf{y}) \biggl ( \bigl (\mathfrak {S}Y_{\phi ;1}^{[0]}\bigr )(\mathbf{y}) -\frac{\partial Y_{\phi ;1}^{[0]}}{\partial n}(t,\mathbf{y})\biggr )\,d\mathbf{y}. \end{aligned}$$

(9.89)

It is clear that the solvability of Eq. (9.86) depends crucially on the property of having fixed sign for $\varXi _1^{[0]}(t)$. In [8], it was proven that $\varXi _1^{[0]}(t)<0$.

Thus, determining the function $\hat{H}_0(t)$ by dividing both sides of (9.88) by $\varXi _1^{[0]}(t)$ and substituting the obtained result into Eq. (9.86), we find $\hat{h}(t,\sigma )$ and, as a consequence of (9.79), uniquely determine the function $h(t,\sigma )$, which describes the variation of the contact domain $\omega _0(t)$.

9.1.8 Asymptotics of the Contact Pressure

In accordance with (9.34), the contact pressure is given by

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})=\fancyscript{K}^{-1} \bigl (P_\varepsilon (\mathbf{y},\tau )\fancyscript{I}_{\omega _\varepsilon (\tau )}(\mathbf{y})\bigr ), \end{aligned}$$

(9.90)

where the integral operator $\fancyscript{K}^{-1}$ is defined by formula (9.35), and $\fancyscript{I}_{\omega _\varepsilon (t)}(\mathbf{y})$ is the indicator function of the domain $\omega _\varepsilon (t)$ defined by

$$ \fancyscript{I}_{\omega _\varepsilon (t)}(\mathbf{y})=\left\{ {1, \ \ \mathbf{y}\in \omega _\varepsilon (t), \atop 0, \ \ \mathbf{y}\not \in \omega _\varepsilon (t). } \right. $$

In the interior of the contact area $\omega _\varepsilon (t)$, Eqs. (9.58) and (9.90) yield the asymptotic representation

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})=p_0(t,\mathbf{y})+\varepsilon p_1(t,\mathbf{y})+O(\varepsilon ^2), \end{aligned}$$

(9.91)

where

$$\begin{aligned} p_i(t,\mathbf{y})=\fancyscript{K}^{-1}\bigl (P_i(t,\mathbf{y})\fancyscript{I}_{\omega _0(t)}(\mathbf{y})\bigr ), \quad i=0,1. \end{aligned}$$

(9.92)

In the boundary-layer region near the contour $\varGamma _\varepsilon (t)$, the so-called outer asymptotic representation (9.91) does not work, and the so-called inner asymptotic representation should be constructed. Here we employ the terminology from the method of matched asymptotic expansions [20].

The inner asymptotic representation

$$\begin{aligned} p_\varepsilon (t,\mathbf{y})=\varepsilon ^2\fancyscript{K}^{-1}\fancyscript{P}(\tau ,s,\nu )+O(\varepsilon ^3) \end{aligned}$$

(9.93)

will be constructed by making use of the stretched coordinate

$$\begin{aligned} \nu =\varepsilon ^{-1}n. \end{aligned}$$

(9.94)

In light of (9.73)–(9.75), the function $P_1(t,\mathbf{y})$ will be determined completely as soon as we know the function $P_1^{[1]}(t,\mathbf{y})$ which satisfies the following Dirichlet problem (see (9.76) and (9.77)):

$$ \varDelta P_1^{[1]}(t,\mathbf{y})=0, \quad \mathbf{y}\in \omega _0(t);\quad P_1^{[1]}(t,\mathbf{y})=g_\phi ^{[0]}(t,\mathbf{y}), \quad \mathbf{y}\in \varGamma _0(t). $$

Here we have introduced the following notation (see Eqs. (9.80), (9.87) and (9.88)):

$$\begin{aligned} g_\phi ^{[0]}(t,\mathbf{y})=-Y_\phi ^{[0]}(\mathbf{y})+\delta _1(t)Y_1^{[0]}(\mathbf{y}), \end{aligned}$$

(9.95)

and as a consequence of said relations, we have

$$\begin{aligned} \delta _1(t)=\frac{\varXi _\phi ^{[0]}(t)}{\varXi _1^{[0]}(t)} +\frac{1}{\varXi _1^{[0]}(t)}\iint \limits _{\omega _0(t)}B_0(t,\mathbf{y})\phi (\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.96)

Near the boundary of the domain $\omega _0(t)$, we may use the Taylor expansions

$$\begin{aligned} P_0(t,\mathbf{y})=\frac{m}{2}\bigl [\varphi _0(\mathbf{y}_0(\sigma ))-\delta _0(t)\bigr ]n^2+O(n^3), \end{aligned}$$

(9.97)

$$\begin{aligned} P_1(t,\mathbf{y})=\frac{m}{2}\bigl [\phi (\mathbf{y}_0(\sigma ))-\delta _1(t)\bigr ]n^2 -m\hat{h}(t,\sigma )n+O(n^3), \end{aligned}$$

(9.98)

where $\mathbf{y}_0(\sigma )$ is the point of the curve $\varGamma _0(t)$ with the natural coordinate $\sigma$.

Applying the perturbation method developed by Nazarov [18], we construct the auxiliary function of the inner asymptotic representation (9.93) in the form

$$\begin{aligned} \fancyscript{P}(t,\sigma ,\nu )=\frac{m}{2}\bigl [\varphi _0(\mathbf{y}_0(\sigma )) -\delta _0(t)\bigr ](\nu -h(t,\sigma ))^2. \end{aligned}$$

(9.99)

The function (9.99) exactly satisfies the relations (9.15), while the boundary condition (9.16) is satisfied asymptotically. We note that the normals $\mathbf{n}_0$ and $\mathbf{n}_\varepsilon$ to the contours $\varGamma _0(t)$ and $\varGamma _\varepsilon (t)$ are, generally speaking, different (see formula (9.64)).

Finally, taking account of the relations (9.79), (9.94), (9.97)–(9.99), it is not hard to verify verify that the matching asymptotic condition for the outer (9.91) and inner (9.93) asymptotic representations is fulfilled.

9.1.9 Slightly Perturbed Circular Contact Area

Let us assume that a circular domain can be taken as a zero approximation in the form (9.5), where the function $\varphi (\mathbf{y})$ defining the perturbed boundary is given by the polynomials

$$\begin{aligned} \varphi _0(\mathbf{y})=\frac{1}{2R}(y_1^2+y_2^2), \quad \phi (\mathbf{y})=\sum _{n=0}^N\sum _{j=0}^n c_{nj}y_1^j y_2^{n-j}, \end{aligned}$$

(9.100)

where $c_{nj}$ are given dimensional coefficients.

In this case, the limit ($\varepsilon =0$) auxiliary contact problem (9.31), (9.38), (9.39) has the following solution:

$$ P_0(t,\mathbf{y})=Q_0(t)\biggl (1-\frac{y_1^2+y_2^2}{a_0(t)^2}\biggr )^2. $$

Correspondingly, Eqs. (9.46), (9.47) and (9.49) take the form

$$\begin{aligned} \delta _0(t)=\frac{a_0(t)^2}{4R},\quad Q_0(t)=\frac{m}{32}\frac{a_0(t)^4}{R}, \end{aligned}$$

(9.101)

$$\begin{aligned} a_0(t)=\Bigl (\frac{m\pi }{96R}\Bigr )^{-1/6} \Biggl (\ \, \int \limits _{0^-}^t \varPhi _\beta (t-\tau )\dot{F}(\tau )\,d\tau \Biggr )^{1/6}. \end{aligned}$$

(9.102)

The first-oder perturbation problem (9.60), (9.66), (9.67), (9.69) and (9.70) now can be written as follows:

$$\begin{aligned} \varDelta _y P_1(t,\mathbf{y})=m\biggl (\ \sum _{n=0}^N\sum _{j=0}^n c_{nj}y_1^j y_2^{n-j} -\delta _1(t)\biggr ), \quad \mathbf{y}\in \omega _0(t), \end{aligned}$$

(9.103)

$$\begin{aligned} P_1(t,\mathbf{y})=0, \quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.104)

$$\begin{aligned} \frac{\partial P_1}{\partial n}(t,\mathbf{y})=-mh(t,\sigma )\frac{a_0(t)^2}{4R}, \quad \mathbf{y}\in \varGamma _0(t), \end{aligned}$$

(9.105)

$$\begin{aligned} \pi a_0(t)^2\delta _1(t)=\sum _{n=0}^N\frac{a_0(t)^{n+2}}{n+2} \sum _{j=0}^n c_{nj}K_{nj}+\frac{a_0(t)^3}{4R}H_0(t), \end{aligned}$$

(9.106)

$$\begin{aligned} 0 = \frac{a_0(t)^5}{8R}H_0(t)+\sum _{n=0}^N\frac{n a_0(t)^{n+4}}{2(n+2)(n+4)} \sum _{j=0}^n c_{nj}K_{nj}. \end{aligned}$$

(9.107)

Here, $\omega _0(t)$ and $\varGamma _0(t)$ are the disc and the circle of the radius $a_0(t)$, $K_{nj}=0$ for odd n and j, while $K_{nj}=2B\bigl ((n+1-j)/2,(j+1)/2\bigr )$ for even values of n and j, and $B(\zeta ,\xi )$ is the Beta function defined by

$$ B(\zeta ,\xi )=\int \limits _0^1 t^{\zeta -1}(1-t)^{\xi -1}dt. $$

Moreover, $H_0(t)$ is an integral characteristics of the contour variation $h(t,\sigma )$ which can be described by the contour variation in polar coordinates as

$$ H_0(t)=\int \limits _0^{2\pi }h(\theta ,t)\,d\theta . $$

From (9.106) and (9.107), it immediately follows that

$$ H_0(t)=-\sum _{n=0}^N\frac{4R a_0(t)^{n-1}}{(n+2)(n+4)}\sum _{j=0}^n c_{nj}K_{nj}, $$

$$\begin{aligned} \delta _1(t)=\sum _{n=0}^N\frac{(n+3)a_0(t)^n}{\pi (n+2)(n+4)}\sum _{j=0}^n c_{nj}K_{nj}. \end{aligned}$$

(9.108)

Using Green’s function $\fancyscript{G}(t,\mathbf{y},\mathbf{y}^\prime )$ of the Dirichlet problem for the domain $\omega _0(t)$, we express the solution to the Dirichlet problem (9.103), (9.104) in the form

$$\begin{aligned} P_1(t,\mathbf{y})=m\iint \limits _{\omega _0(t)}\phi (\mathbf{y}^\prime ) \fancyscript{G}(t,\mathbf{y},\mathbf{y}^\prime )\,d\mathbf{y}^\prime -\frac{m}{4}\delta _1(t)(\vert \mathbf{y}\vert ^2-a_0(t)^2). \end{aligned}$$

(9.109)

Recall that for a circular domain $\omega _0(t)$ of radius $a_0(t)$, Green’s function is

$$ \fancyscript{G}(t,\mathbf{y},\mathbf{y}^\prime )=\frac{1}{2\pi } \ln \frac{ a_0(t)\vert \mathbf{y}^\prime \vert \vert \mathbf{y}-\mathbf{y}^\prime \vert }{ \bigl \vert \,\vert \mathbf{y}^\prime \vert \mathbf{y}-a_0(t)^2\mathbf{x}^\prime \bigr \vert }, $$

whereas, in polar coordinates, we have

$$ \fancyscript{G}(t,\mathbf{y},\mathbf{y}^\prime )=\frac{1}{4\pi } \ln \frac{a_0(t)^2\bigl (r^2+r^{\prime 2}-2r^\prime r\cos (\theta -\theta ^\prime )\bigr ) }{a_0(t)^4+r^{\prime 2}r^2-2r^\prime r a_0(t)^2\cos (\theta -\theta ^\prime )}. $$

Calculating the normal derivative of the function (9.109), we obtain

$$\begin{aligned} \frac{\partial P_1}{\partial n}(t,\mathbf{y})=m\iint \limits _{\omega _0(t)} \phi (\mathbf{y}^\prime )\frac{\partial \fancyscript{G}}{\partial n} (t,\mathbf{y},\mathbf{y}^\prime )\,d\mathbf{y}^\prime - \frac{m}{2}\delta _1(t)\vert \mathbf{y}\vert , \end{aligned}$$

(9.110)

where

$$ \frac{\partial }{\partial n} \fancyscript{G}(t,\mathbf{y},\mathbf{y}^\prime )=\frac{1}{2\pi } \frac{a_0(t)^2-r^{\prime 2}}{ a_0(t)\bigl (a_0(t)^2+r^{\prime 2}-2r^\prime a_0(t)\cos (\theta -\theta ^\prime )\bigr )}. $$

By substitution of the expression (9.110) into the boundary condition (9.105) and taking into account (9.108), we obtain the function $h(t,\theta )$ describing variation of the contact domain in the following form:

$$\begin{aligned} h(t,\theta )= & {} \frac{2R}{\pi a_0(t)} \sum _{n=0}^N\frac{(n+3)a_0(t)^n}{(n+2)(n+4)}\sum _{j=0}^n c_{nj}K_{nj} \nonumber \\ { }{} & {} {}-\frac{2R}{\pi a_0(t)^3}\int \limits _0^{2\pi }d\theta ^\prime \int \limits _0^{a_0(t)} \biggl ( \sum _{n=0}^N r^{\prime n}\sum _{j=0}^n c_{nj}\cos ^j\theta ^\prime \sin ^{n-j}\theta ^\prime \biggr ) \nonumber \\ {}&\times \frac{a_0(t)^2-r^{\prime 2}}{ a_0(t)^2+r^{\prime 2}-2r^\prime a_0(t)\cos (\theta -\theta ^\prime )}\,r^\prime \,dr^\prime . \end{aligned}$$

(9.111)

Note that the integral with respect to $\theta ^\prime$ on the right-hand side of (9.111) can be evaluated with the help of the following relation (see, e.g., [14]):

$$ \int \limits _0^{\pi } \frac{\cos nxdx}{1-2\rho \cos x+\rho ^2}=\frac{\pi \rho ^n}{1-\rho ^2}, \quad \rho ^2<1. $$

It was shown [8] that formula (9.111) asymptotically coincides with the exact solution for the ellipse presented in Sect. 9.1.5 in the case of small eccentricity.

9.2 Contact of Two Bonded Thin Transversely Isotropic Elastic Layers with Variable Thicknesses

In this section, a three-dimensional unilateral contact problem for a thin transversely isotropic elastic layer with variable thickness bonded to a rigid substrate is considered. Two cases are studied sequentially: (a) the layer material is compressible; (b) the layer material is incompressible. It is well known that the asymptotic solution for a thin isotropic elastic layer undergoes a dramatic change in the limit as Poisson’s ratio $\nu$ tends to 0.5, so that the formulas obtained in the case (a) are not applicable when the layer material approaches the incompressible limit. After developing a refined asymptotic model for the deformation of one elastic layer of variable thickness, we apply sensitivity analysis to determine how “sensitive” the mathematical model for contact interactions of two thin uniform layers of thicknesses $h_1$ and $h_2$ is to variations in the layer thicknesses. We will consider the term “sensitivity” in a broad sense by allowing variable layer thicknesses $H_1(\mathbf{y})$ and $H_2(\mathbf{y})$, whereas the original model deals with the scalar parameters $h_1$ and $h_2$.

9.2.1 Unperturbed Asymptotic Model

As a rule, analytical models of articular contact assume rigid bones and represent cartilage as a thin elastic layer of constant thickness resisting deformation like a Winkler foundation consisting of a series of discrete springs with constant length and stiffness [12]. However, a subject-specific approach to articular contact mechanics requires developing patient-specific models for accurate predictions. A sensitivity analysis of finite element models of hip cartilage mechanics with respect to varying degrees of simplified geometry was performed in [2].

Based on the asymptotic analysis of the frictionless contact problem for a thin elastic layer bonded to a rigid substrate in the thin-layer limit (see Chap. 2), the following asymptotic model for contact interaction of two thin incompressible transversely isotropic layers was established:

$$\begin{aligned} -\biggl (\frac{h_1^3}{3G^\prime _1}+\frac{h_2^3}{3G^\prime _2}\biggr ) \varDelta _y p(\mathbf{y})=\delta _0-\varphi (\mathbf{y}), \quad \mathbf{y}\in \omega , \end{aligned}$$

(9.112)

$$\begin{aligned} p(\mathbf{y})=0,\quad \frac{\partial p}{\partial n}(\mathbf{y})=0,\quad \mathbf{y}\in \varGamma . \end{aligned}$$

(9.113)

Here, $p(\mathbf{y})$ is the contact pressure density, $h_n$ and $G^\prime _n$ are the thickness and out-of-plane shear modulus of the nth layer material, respectively, $n=1,2$, $\varDelta _y=\partial ^2/\partial y_1^2+\partial ^2/\partial y_2^2$ is the Laplace differential operator, $\delta _0$ is the vertical approach of the rigid substrates to which the layers are bonded, $\varphi (\mathbf{y})$ is the initial gap function defined as the distance between the layer surfaces in the vertical direction, $\omega$ is the contact area, and $\varGamma$ is the contour of $\omega$, $\partial /\partial n$ is the normal derivative.

In the isotropic case, the asymptotic model (9.112), (9.113) was developed in [4, 10]. It was shown [6, 9, 11] that this model describes the instantaneous response of thin biphasic layers to dynamic and impact loading. In [7], the isotropic elastic model (9.112), (9.113) was generalized for a general viscoelastic case.

With respect to articular contact, a case of special interest is when the subchondral bones are approximated by elliptic paraboloids, so that the gap function is given by

$$\begin{aligned} \varphi (\mathbf{y})=\frac{y_1^2}{2R_1}+\frac{y_2^2}{2R_2} \end{aligned}$$

(9.114)

with positive curvature radii $R_1$ and $R_2$.

In the case (9.114), the exact solution to the problem (9.112), (9.113) has the following form [6, 10]:

$$\begin{aligned} p(\mathbf{y})=p_0\biggl (1-\frac{y_1^2}{a_1^2}-\frac{y_2^2}{a_2^2}\biggr )^2. \end{aligned}$$

(9.115)

Integration of the contact pressure distribution (9.115) over the elliptical contact area $\omega$ with the semi-axes $a_1$ and $a_2$ results in the following force-displacement relationship [4]:

$$\begin{aligned} F=\frac{2\pi }{3}c_F(s)m R_1 R_2\delta _0^3. \end{aligned}$$

(9.116)

Here, $c_F(s)$ is a dimensionless factor depending on the aspect ratio $s=a_2/a_1$ (see Sect. 4.5.6), and the coefficient m is given by

$$\begin{aligned} m=3\biggl (\frac{h_1^3}{G^\prime _1}+\frac{h_2^3}{G^\prime _2}\biggr )^{-1}. \end{aligned}$$

(9.117)

The asymptotic model (9.112)–(9.114) assumes that the cartilage layers have constant thicknesses, whereas it is well known [1] that articular cartilage has a variable thickness and that the surface of subchondral bone deviates from the ellipsoid shape [19]. A sensitivity analysis of the model (9.112), (9.113) with respect to small perturbations of the gap function (9.114) was performed in [8]. In particular, it has been shown [4] that the influence of the gap function variation on the force-displacement relationship will be negligible if the effective geometrical characteristics $R_1$ and $R_2$ are determined by a least square method.

The two-dimensional contact problem for a thin isotropic elastic strip of variable thickness was solved by Vorovich and Peninin [21] using an asymptotic method, under the assumption that the Poisson’s ratio of the strip material is not very close to 0.5. A three-dimensional unilateral contact problem for a thin isotropic elastic layer of variable thickness bonded to a rigid substrate was studied in [5]. Here, the results obtained in [5] are generalized for the transversely isotropic case.

9.2.2 Contact Problem for a Thin Transversely Isotropic Elastic Layer with Variable Thickness

We consider (see Fig. 9.2) a homogeneous, isotropic, linearly elastic layer with a planar contact interface, $x_3=0$, and a variable thickness, $H(x_1,x_2)$, firmly attached to an uneven rigid surface

$$\begin{aligned} x_3=H(x_1,x_2). \end{aligned}$$

(9.118)

In the absence of body forces, the equilibrium equations and the strain-displacement relations governing small deformations of the elastic layer are

$$\begin{aligned} \frac{\partial \sigma _{1j}}{\partial x_1}+\frac{\partial \sigma _{2j}}{\partial x_2}+ \frac{\partial \sigma _{3j}}{\partial x_3}=0,\quad j=1,2,3, \end{aligned}$$

(9.119)

$$\begin{aligned} \varepsilon _{ij}=\frac{1}{2}\biggl (\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}\biggr ),\quad i,j=1,2,3, \end{aligned}$$

(9.120)

where $\sigma _{ij}$ is the Cauchy stress tensor, $\varepsilon _{ij}$ is the infinitesimal strain tensor, and $u_j$ is the displacement component along the $x_j$-axis. Here the Cartesian coordinate system $(y_1,y_2,z)$ will be used such that $y_1=x_1$, $y_2=x_2$, and $z=x_3$.

Fig. 9.2

Elastic layer with a variable thickness

Moreover, a transversely isotropic elastic body is characterized by the following stress-strain relationships:

$$\begin{aligned} \left( \begin{array}{c} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sigma _{23} \\ \sigma _{13} \\ \sigma _{12} \end{array} \right) = \left[ \begin{array}{cccccc} A_{11} &{} A_{12} &{} A_{13} &{} 0 &{} 0 &{} 0 \\ A_{12} &{} A_{11} &{} A_{13} &{} 0 &{} 0 &{} 0 \\ A_{13} &{} A_{13} &{} A_{33} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2A_{44} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 2A_{44} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2A_{66} \end{array} \right] \left( \begin{array}{c} \varepsilon _{11} \\ \varepsilon _{22} \\ \varepsilon _{33} \\ \varepsilon _{23} \\ \varepsilon _{13} \\ \varepsilon _{12} \end{array} \right) . \end{aligned}$$

(9.121)

Recall that $2A_{66}=A_{11}-A_{12}$, while $A_{11}$, $A_{12}$, $A_{13}$, $A_{33}$, and $A_{44}$ are five independent elastic constants.

We assume that the elastic layer is indented by a smooth rigid punch in the form of an elliptic paraboloid

$$ z=-\varphi (y_1,y_2), $$

where

$$\begin{aligned} \varphi (\mathbf{y})=\frac{y_1^2}{2R_1}+\frac{y_2^2}{2R_2}. \end{aligned}$$

(9.122)

Under the assumption of frictionless contact, we have

$$\begin{aligned} \sigma _{31}(\mathbf{y},0)=0,\quad \sigma _{32}(\mathbf{y},0)=0, \quad \mathbf{y}=(y_1,y_2)\in \mathbb {R}^2. \end{aligned}$$

(9.123)

Denoting by $\delta _0$ the indenter’s displacement, we formulate the boundary condition on the contact interface as

$$\begin{aligned} \begin{array}{c} u_3(\mathbf{y},0)\ge \delta _0-\varphi (\mathbf{y}),\quad \sigma _{33}(\mathbf{y},0)\le 0, \\ \bigl (u_3(\mathbf{y},0)-\delta _0+\varphi (\mathbf{y})\bigr )\sigma _{33}(\mathbf{y},0)=0,\quad \mathbf{y}\in \mathbb {R}^2, \end{array} \end{aligned}$$

(9.124)

while on the rigid substrate surface defined by Eq. (9.118) we have

$$\begin{aligned} u_j(\mathbf{y},H(\mathbf{y}))=0,\quad \mathbf{y}\in \mathbb {R}^2 \quad (j=1,2,3). \end{aligned}$$

(9.125)

Assuming that the layer is relatively thin in comparison to the characteristic dimensions of $\omega$, we introduce a small dimensionless parameter $\varepsilon$ and require that

$$\begin{aligned} \delta _0=\varepsilon \delta _0^*,\quad R_1=\varepsilon ^{-1}R_1^*,\quad R_2=\varepsilon ^{-1}R_2^*, \end{aligned}$$

(9.126)

$$\begin{aligned} H(\mathbf{y})=\varepsilon h_*(1+\varepsilon \psi _*(\mathbf{y})), \end{aligned}$$

(9.127)

where $\delta _0^*$, $R_1^*$, and $R_2^*$ are assumed to be comparable with $h_*$. Without loss of generality we can assume that $\vert \psi _*(\mathbf{y})\vert \le h_*$ for any $\mathbf{y}\in \mathbb {R}^2$.

The problem is to calculate the contact pressure distribution

$$\begin{aligned} p(x_1,x_2)=-\sigma _{33}(x_1,x_2,0),\quad (x_1,x_2)\in \omega , \end{aligned}$$

(9.128)

and the contact force required to indent the punch into the elastic layer

$$\begin{aligned} F=\iint \limits _\omega p(\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.129)

We introduce the notation

$$\begin{aligned} h=\varepsilon h_*,\quad \tilde{H}(\mathbf{y})=\varepsilon ^2 \tilde{H}_*(\mathbf{y}), \end{aligned}$$

(9.130)

where

$$\begin{aligned} \tilde{H}_*(\mathbf{y})=h_*\psi _*(\mathbf{y}). \end{aligned}$$

(9.131)

Hence, the following relation is evident:

$$\begin{aligned} H(\mathbf{y})=h+\tilde{H}(\mathbf{y}), \end{aligned}$$

(9.132)

where h is an average thickness, and $\tilde{H}(\mathbf{y})$ is a small variation such that $\tilde{H}(\mathbf{y})\ll h$.

We note that as a consequence of (9.128), the unilateral contact condition (9.124) can be rewritten as follows:

$$ p(\mathbf{y})\ge 0,\quad \mathbf{y}\in \mathbb {R}^2, $$

$$ p(\mathbf{y})>0 \ \ \Rightarrow \ \ u_3(\mathbf{y},0)=\delta _0-\varphi (\mathbf{y}), $$

$$ p(\mathbf{y})=0 \ \ \Rightarrow \ \ u_3(\mathbf{y},0)\ge \delta _0-\varphi (\mathbf{y}). $$

We pose the following frictionless unilateral contact problem: for a given value of the punch displacement $\delta _0$, find the contact pressure $p(\mathbf{y})$ such that the Signorini boundary condition (9.124 ) is satisfied both inside the contact area $\omega$ and outside (that is $p(\mathbf{y})=0$ for $\mathbf{y}\in \mathbb {R}^2\setminus \bar{\omega }$).

9.2.3 Perturbation Solution

First, we introduce the so-called stretched coordinate

$$\begin{aligned} \zeta =\varepsilon ^{-1}z. \end{aligned}$$

(9.133)

Substituting (9.121) and (9.120) into Eqs. (9.119) and taking into account the variable transformation (9.133), we arrive at the following Lamé system for the displacement vector $\mathbf{u}=(\mathbf{v},w)$:

$$\begin{aligned} \varepsilon ^{-2}A_{44}\frac{\partial ^2\mathbf{v}}{\partial \zeta ^2} +&\varepsilon ^{-1}(A_{13}+A_{44})\nabla _y\frac{\partial w}{\partial \zeta } \nonumber \\ { } { } {}&+A_{66}\varDelta _y\mathbf{v}+(A_{11}-A_{66})\nabla _y\nabla _y\!\cdot \mathbf{v}=\mathbf{0}, \end{aligned}$$

(9.134)

$$\begin{aligned} \varepsilon ^{-2}A_{33}\frac{\partial ^2 w}{\partial \zeta ^2} +\varepsilon ^{-1}(A_{13}+A_{44})\nabla _y\!\cdot \frac{\partial \mathbf{v}}{\partial \zeta } +A_{44}\varDelta _y w=0. \end{aligned}$$

(9.135)

Here, $\nabla _y=(\partial /\partial y_1,\partial /\partial y_2)$ is the Hamilton differential operator, and the dot denotes the scalar product, so that $\nabla _y\!\cdot \nabla _y=\varDelta _y$ is the Laplace operator.

Correspondingly, the boundary condition (9.123) takes the form

$$\begin{aligned} \varepsilon ^{-1}\frac{\partial \mathbf{v}}{\partial \zeta }+ \nabla _y w\biggr \vert _{\zeta =0}=\mathbf{0}. \end{aligned}$$

(9.136)

In light of (9.124) and (9.126), we have

$$\begin{aligned} w(\mathbf{y},0)=\varepsilon (\delta _0^*-\varphi ^*(\mathbf{y})),\quad \mathbf{y}\in \omega , \end{aligned}$$

(9.137)

where we have introduced the notation (see Eqs. (9.122) and (9.126))

$$\begin{aligned} \varphi ^*(\mathbf{y})=\frac{y_1^2}{2R_1^*}+\frac{y_2^2}{2R_2^*}. \end{aligned}$$

(9.138)

Furthermore, by stretching the normal coordinate, Eq. (9.128) is reduced to

$$\begin{aligned} -p(\mathbf{y})=\varepsilon ^{-1}A_{33}\frac{\partial w}{\partial \zeta } +A_{13}\nabla _y\!\cdot \mathbf{v}\biggr \vert _{\zeta =0},\quad \mathbf{y}\in \omega . \end{aligned}$$

(9.139)

The boundary conditions (9.125) on the substrate surface (see Eqs. (9.118) and (9.127))

$$\begin{aligned} \zeta =h_*(1+\varepsilon \psi _*(\mathbf{y})) \end{aligned}$$

(9.140)

take the following form:

$$\begin{aligned} \mathbf{v}\bigl (\mathbf{y},h_*+\varepsilon h_*\psi _*(\mathbf{y})\bigr )=\mathbf{0},\quad w\bigl (\mathbf{y},h_*+\varepsilon h_*\psi _*(\mathbf{y})\bigr )=0. \end{aligned}$$

(9.141)

Observe that among Eqs. (9.134)–(9.137), (9.139) and (9.141), there are only two inhomogeneous equations, namely (9.137) and (9.139). The form of (9.137) suggests the asymptotic expansion

$$\begin{aligned} w(\mathbf{y},\zeta )=\varepsilon w^0(\mathbf{y},\zeta )+\varepsilon ^2 w^1(\mathbf{y},\zeta )+\cdots \,. \end{aligned}$$

(9.142)

In light of (9.139) and (9.142), we suggest that $p(\mathbf{y})=O(1)$ as $\varepsilon \rightarrow 0$. Taking into account the homogeneous conditions (9.136), (9.141), we set

$$\begin{aligned} \mathbf{v}(\mathbf{y},\zeta )=\varepsilon \mathbf{v}^0(\mathbf{y},\zeta )+\varepsilon ^2 \mathbf{v}^1(\mathbf{y},\zeta )+\cdots \,. \end{aligned}$$

(9.143)

We emphasize that the asymptotic ansatz (9.142), (9.143) is valid only inside the contact region $\omega$. In other words, a plane boundary layer should be constructed near the edge of the contact area. We refer to [3, 13] for more details.

9.2.4 Derivation of Asymptotic Expansions

Substituting (9.142) and (9.143) into Eqs. (9.134) and (9.135), we obtain

$$\begin{aligned} \varepsilon ^{-2}A_{44}\frac{\partial ^2\mathbf{v}^0}{\partial \zeta ^2}+ & {} \varepsilon ^{-1}\biggl ((A_{13}+A_{44})\nabla _y\frac{\partial w^0}{\partial \zeta }+ A_{44}\frac{\partial ^2\mathbf{v}^1}{\partial \zeta ^2}\biggr ) \nonumber \\ { }+ & {} \varepsilon ^0\biggl \{ A_{66}\varDelta _y\mathbf{v}^0+(A_{11}+A_{66})\nabla _y\nabla _y\!\cdot \mathbf{v}^0 \nonumber \\ { }+ & {} (A_{13}+A_{44})\nabla _y\frac{\partial w^1}{\partial \zeta }+ A_{44}\frac{\partial ^2\mathbf{v}^2}{\partial \zeta ^2}\biggr \}+\cdots =\mathbf{0}, \end{aligned}$$

(9.144)

$$\begin{aligned} \varepsilon ^{-2}A_{33}\frac{\partial ^2 w^0}{\partial \zeta ^2}+ & {} \varepsilon ^{-1}\biggl (A_{33}\frac{\partial ^2 w^1}{\partial \zeta ^2}+ (A_{13}+A_{44})\nabla _y\!\cdot \frac{\partial \mathbf{v}^0}{\partial \zeta }\biggr ) +\varepsilon ^0\biggl \{A_{33}\frac{\partial ^2 w^2}{\partial \zeta ^2}\nonumber \\ { }+ & {} (A_{13}+A_{44})\nabla _y\!\cdot \frac{\partial \mathbf{v}^1}{\partial \zeta } +A_{44}\varDelta _y w^0\biggr \}+\cdots =0. \end{aligned}$$

(9.145)

Further, by substituting (9.142) and (9.143) into the boundary conditions (9.136) and (9.139) at the contact region, we find

$$\begin{aligned} \varepsilon ^{-1}\frac{\partial \mathbf{v}^0}{\partial \zeta }+ \varepsilon ^0\biggl (\nabla _y w^0+\frac{\partial \mathbf{v}^1}{\partial \zeta }\biggr ) +\varepsilon \biggl (\nabla _y w^1+\frac{\partial \mathbf{v}^2}{\partial \zeta }\biggr )+\cdots \ \biggr \vert _{\zeta =0}=\mathbf{0}, \end{aligned}$$

(9.146)

$$\begin{aligned} A_{33}\frac{\partial w^0}{\partial \zeta }+ & {} \varepsilon \biggl (A_{13}\nabla _y\!\cdot \mathbf{v}^0 +A_{33}\frac{\partial w^1}{\partial \zeta }\biggr )\nonumber \\ { }+ & {} \varepsilon ^2\biggl (A_{13}\nabla _y\!\cdot \mathbf{v}^1 +A_{33}\frac{\partial w^2}{\partial \zeta }\biggr )+\cdots \ \biggr \vert _{\zeta =0}=-p(\mathbf{y}). \end{aligned}$$

(9.147)

The substitution of (9.142), (9.143) into Eqs. (9.141) then yields

$$\begin{aligned} \mathbf{v}^0+\varepsilon \biggl (\mathbf{v}^1+\tilde{H}_*\frac{\partial \mathbf{v}^0}{\partial \zeta }\biggr ) +\varepsilon ^2\biggl (\mathbf{v}^2+\tilde{H}_*\frac{\partial \mathbf{v}^1}{\partial \zeta } +\frac{\tilde{H}_*^2}{2}\frac{\partial ^2\mathbf{v}^0}{\partial \zeta ^2}\biggr ) +\cdots \ \biggr \vert _{\zeta =h_*}=\mathbf{0}, \end{aligned}$$

(9.148)

$$\begin{aligned} w^0+\varepsilon \biggl (w^1+\tilde{H}_*\frac{\partial w^0}{\partial \zeta }\biggr ) +\varepsilon ^2\biggl (w^2+\tilde{H}_*\frac{\partial w^1}{\partial \zeta } +\frac{\tilde{H}_*^2}{2}\frac{\partial ^2 w^0}{\partial \zeta ^2}\biggr ) +\cdots \ \biggr \vert _{\zeta =h_*}=0, \end{aligned}$$

(9.149)

where we have used notation (9.131).

Thus, on the basis of Eqs. (9.144)–(9.149), we have arrived at a recurrence system of boundary-value problems for the functions $\mathbf{v}^k$ and $w^k$ ($k=0,1,\ldots$). Let us construct the first several terms of the asymptotic series (9.142) and (9.143).

9.2.5 Asymptotic Solution for a Thin Compressible Layer

According to (9.144)–(9.149), the first-order problem takes the form

$$\begin{aligned} \frac{\partial ^2 w^0}{\partial \zeta ^2}=0, \quad \zeta \in (0,h_*),\quad A_{33}\frac{\partial w^0}{\partial \zeta }\biggr \vert _{\zeta =0}=-p(\mathbf{y}),\quad w^0\bigr \vert _{\zeta =h_*}=0; \end{aligned}$$

(9.150)

$$\begin{aligned} \frac{\partial ^2\mathbf{v}^0}{\partial \zeta ^2}=\mathbf{0}, \quad \zeta \in (0,h_*),\quad \frac{\partial \mathbf{v}^0}{\partial \zeta }\biggr \vert _{\zeta =0}=\mathbf{0},\quad \mathbf{v}^0\bigr \vert _{\zeta =h_*}=\mathbf{0}. \end{aligned}$$

(9.151)

From (9.150) and (9.151), it immediately follows that

$$\begin{aligned} w^0(\mathbf{y},\zeta )=\frac{p(\mathbf{y})}{A_{33}}(h_*-\zeta ), \end{aligned}$$

(9.152)

$$\begin{aligned} \mathbf{v}^0(\mathbf{y},\zeta )\equiv \mathbf{0}. \end{aligned}$$

(9.153)

As a consequence of (9.153), the second-order problem, derived from (9.144)–(9.149) is

$$ \frac{\partial ^2 w^1}{\partial \zeta ^2}=0, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} \frac{\partial w^1}{\partial \zeta }\biggr \vert _{\zeta =0}=0,\quad w^1\bigr \vert _{\zeta =h_*}=-\tilde{H}_*(\mathbf{y})\frac{\partial w^0}{\partial \zeta }\biggr \vert _{\zeta =h_*}; \end{aligned}$$

(9.154)

$$ A_{44}\frac{\partial ^2\mathbf{v}^1}{\partial \zeta ^2}=-(A_{13}+A_{44})\nabla _y\frac{\partial w^0}{\partial \zeta }, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} \frac{\partial \mathbf{v}^1}{\partial \zeta }\biggr \vert _{\zeta =0}=-\nabla _y w^0\bigr \vert _{\zeta =0},\quad \mathbf{v}^1\bigr \vert _{\zeta =h_*}=\mathbf{0}. \end{aligned}$$

(9.155)

It is readily seen that the solution of the problem (9.154) is given by

$$\begin{aligned} w^1(\mathbf{y},\zeta )=\frac{p(\mathbf{y})}{A_{33}}\tilde{H}_*(\mathbf{y}). \end{aligned}$$

(9.156)

On the other hand, the solution of the problem (9.155) is

$$\begin{aligned} \mathbf{v}^1(\mathbf{y},\zeta )=\varPsi (\zeta )\nabla _y p(\mathbf{y}), \end{aligned}$$

(9.157)

where we have introduced the notation

$$\begin{aligned} \varPsi (\zeta )=\frac{(A_{13}+A_{44})}{2A_{33}A_{44}}(h_*-\zeta )^2 -\frac{h_*A_{13}}{A_{33}A_{44}}(h_*-\zeta ). \end{aligned}$$

(9.158)

Collecting Eqs. (9.142), (9.152) and (9.156), we arrive at the two-term asymptotic approximation for the normal displacement

$$\begin{aligned} w(\mathbf{y},\zeta )\simeq \varepsilon \frac{p(\mathbf{y})}{A_{33}}(h_*-\zeta ) +\varepsilon ^2\frac{p(\mathbf{y})}{A_{33}}\tilde{H}_*(\mathbf{y}). \end{aligned}$$

(9.159)

By taking into account the scaling relations (9.130), we rewrite (9.159) in the form

$$\begin{aligned} u_3(\mathbf{y},z)\simeq \frac{p(\mathbf{y})}{A_{33}} \bigl (h+\tilde{H}(\mathbf{y})-z\bigr ). \end{aligned}$$

(9.160)

By substituting the expression (9.160) into the contact condition

$$\begin{aligned} u_3(\mathbf{y},0)=\delta _0-\varphi (\mathbf{y}),\quad \mathbf{y}\in \omega , \end{aligned}$$

(9.161)

we derive the following equation for the contact pressure density:

$$\begin{aligned} \frac{h+\tilde{H}(\mathbf{y})}{A_{33}}p(\mathbf{y})=\delta _0-\varphi (\mathbf{y}),\quad \mathbf{y}\in \omega . \end{aligned}$$

(9.162)

In light of the condition $p(\mathbf{y})>0$ for $\mathbf{y}\in \omega$, we obtain

$$\begin{aligned} p(\mathbf{y})=\frac{A_{33}}{h+\tilde{H}(\mathbf{y})} \bigl (\delta _0-\varphi (\mathbf{y})\bigr )_+, \end{aligned}$$

(9.163)

where $(x)_+=\max \{x,0\}$ is the positive-part function.

Invoking the notation (9.132) for the variable thickness of the elastic layer, we rewrite formula (9.163) as

$$\begin{aligned} p(\mathbf{y})=\frac{A_{33}}{H(\mathbf{y})}\bigl (\delta _0-\varphi (\mathbf{y})\bigr )_+. \end{aligned}$$

(9.164)

Formula (9.164) shows that a thin compressible elastic layer deforms like a Winkler foundation with the variable foundation modulus

$$\begin{aligned} k(\mathbf{y})=\frac{A_{33}}{H(\mathbf{y})}. \end{aligned}$$

(9.165)

Let us recall that the elastic parameter $A_{33}$ is related to Young’s moduli, E and $E^\prime$, and Poisson’s ratios, $\nu$ and $\nu ^\prime$, by the formulas

$$\begin{aligned} A_{33}=\frac{E^\prime (1-\nu )}{\displaystyle 1-\nu -\frac{2E}{E^\prime }\,\nu ^{\prime 2}}, \end{aligned}$$

(9.166)

Finally, according to [16] (see also Sect. 2.4), the denominator of (9.166) would approach zero if the layer material became more incompressible. It is readily seen from (9.165) that $k(\mathbf{y})\rightarrow \infty$ in the incompressibility limit, implying that the case of an incompressible elastic layer requires special consideration.

9.2.6 Asymptotic Solution for a Thin Incompressible Layer

Let us continue the process of constructing terms of the asymptotic expansions (9.142) and (9.143). In light of (9.156), Eqs. (9.144)–(9.149) yield the following third-order problem:

$$ A_{33}\frac{\partial ^2 w^2}{\partial \zeta ^2}= -(A_{13}+A_{44})\nabla _y\!\cdot \frac{\partial \mathbf{v}^1}{\partial \zeta } -A_{44}\varDelta _y w^0, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} A_{33}\frac{\partial w^2}{\partial \zeta }\biggr \vert _{\zeta =0}= -A_{13}\nabla _y\!\cdot \mathbf{v}^1\bigr \vert _{\zeta =0},\quad w^2\bigr \vert _{\zeta =h_*}=0; \end{aligned}$$

(9.167)

$$ \frac{\partial ^2\mathbf{v}^2}{\partial \zeta ^2}=\mathbf{0}, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} \frac{\partial \mathbf{v}^2}{\partial \zeta }\biggr \vert _{\zeta =0}= -\nabla _y w^1\bigr \vert _{\zeta =0},\quad \mathbf{v}^2\bigr \vert _{\zeta =h_*}= -\tilde{H}_*(\mathbf{y})\frac{\partial \mathbf{v}^1}{\partial \zeta }\biggr \vert _{\zeta =h_*}. \end{aligned}$$

(9.168)

Substituting (9.152) and (9.157) into Eqs. (9.167), we derive the problem

$$ \frac{\partial ^2 w^2}{\partial \zeta ^2}=\frac{A_{13}\varDelta _y p(\mathbf{y})}{A_{33}^2 A_{44}} \bigl [(A_{13}+2A_{44})(h_*-\zeta )-(A_{13}+A_{44})h_*\bigr ], \quad \zeta \in (0,h_*), $$

$$\begin{aligned} \frac{\partial w^2}{\partial \zeta }\biggr \vert _{\zeta =0}= -\frac{A_{13}(A_{44}-A_{13})h_*^2}{2A_{33}^2 A_{44}}\varDelta _y p(\mathbf{y}),\quad w^2\bigr \vert _{\zeta =h_*}=0. \end{aligned}$$

(9.169)

It can be verified that the solution to (9.169) condenses into the form

$$\begin{aligned} w^2(\mathbf{y},\zeta )= & {} \frac{A_{13}\varDelta _y p(\mathbf{y})}{A_{33}^2}\biggl \{ \frac{A_{13}+2A_{44}}{6A_{44}}(h_*-\zeta )^3\nonumber \\ {}{} & {} {}-\frac{(A_{13}+A_{44})h_*}{2A_{44}}(h_*-\zeta )^2 +\frac{h_*^2}{2}(h_*-\zeta )\biggr \}. \end{aligned}$$

(9.170)

Further, in light of (9.156) and (9.157), the problem (9.168) takes the form

$$ \frac{\partial ^2\mathbf{v}^2}{\partial \zeta ^2}=\mathbf{0}, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} \frac{\partial \mathbf{v}^2}{\partial \zeta }\biggr \vert _{\zeta =0}= -\frac{\nabla _y (p\tilde{H}_*)}{A_{33}},\quad \mathbf{v}^2\bigr \vert _{\zeta =h_*}= -\frac{A_{13} h_*}{A_{33}A_{44}}\tilde{H}_*\nabla _y p, \end{aligned}$$

(9.171)

where we have omitted the arguments of functions $p(\mathbf{y})$ and $\tilde{H}_*(\mathbf{y})$ for clarity.

It can be easily verified that the solution to (9.171) has the form

$$\begin{aligned} \mathbf{v}^2(\mathbf{y},\zeta )= \frac{h_*-\zeta }{A_{33}}\nabla _y (p\tilde{H}_*) -\frac{A_{13} h_*}{A_{33}A_{44}}\tilde{H}_*\nabla _y p. \end{aligned}$$

(9.172)

We emphasize that in contrast to the two-term approximation (9.159), the third term (9.170) does not vanish at the contact interface in the limit as $\nu \rightarrow 0{.}5$. Indeed, formula (9.170) yields

$$\begin{aligned} w^2(\mathbf{y},0)=-\frac{h_*^3 A_{13}(A_{13}-A_{44})}{3A_{33}^2A_{44}}\varDelta _y p(\mathbf{y}). \end{aligned}$$

(9.173)

In the incompressibility limit, we have

$$\begin{aligned} \frac{A_{13}(A_{13}-A_{44})}{A_{33}^2A_{44}}\rightarrow \frac{1}{a_{44}}, \end{aligned}$$

(9.174)

where $a_{44}=A_{44}=G^\prime$ is the out-of-plane shear modulus.

In order to construct a correction for the leading asymptotic term (9.170), we consider the following problem:

$$ A_{33}\frac{\partial ^2 w^3}{\partial \zeta ^2}= -(A_{13}+A_{44})\nabla _y\!\cdot \frac{\partial \mathbf{v}^2}{\partial \zeta } -A_{44}\varDelta _y w^1, \quad \zeta \in (0,h_*), $$

$$\begin{aligned} A_{33}\frac{\partial w^3}{\partial \zeta }\biggr \vert _{\zeta =0}= -A_{13}\nabla _y\!\cdot \mathbf{v}^2\bigr \vert _{\zeta =0},\quad w^3\bigr \vert _{\zeta =h_*}= -\tilde{H}_*(\mathbf{y})\frac{\partial w^2}{\partial \zeta }\biggr \vert _{\zeta =h_*}. \end{aligned}$$

(9.175)

By substituting (9.156), (9.170) and (9.172) into Eqs. (9.175), we find

$$\begin{aligned} \frac{\partial ^2 w^3}{\partial \zeta ^2}= \frac{A_{13}}{A_{33}^2}\varDelta _y (p\tilde{H}_*), \quad \zeta \in (0,h_*), \end{aligned}$$

(9.176)

$$\begin{aligned} \frac{\partial w^3}{\partial \zeta }\biggr \vert _{\zeta =0}= \frac{A_{13} h_*}{A_{33}^2 A_{44}}\bigl ( A_{13}\nabla _y\!\cdot (\tilde{H}_*\nabla _y p)- A_{44} \varDelta _y (p\tilde{H}_*)\bigr ), \end{aligned}$$

(9.177)

$$\begin{aligned} w^3\bigr \vert _{\zeta =h_*}= \frac{A_{13} h_*^2}{2A_{33}^2}\tilde{H}_*\varDelta _y p. \end{aligned}$$

(9.178)

Integrating Eq. (9.176), we obtain

$$\begin{aligned} w^3(\mathbf{y},0)=\frac{A_{13}}{2A_{33}^2}\varDelta _y (p\tilde{H}_*)\zeta ^2+ C_1(\mathbf{y})\zeta +C_0(\mathbf{y}), \end{aligned}$$

(9.179)

where the integration functions $C_1(\mathbf{y})$ and $C_0(\mathbf{y})$ are determined by the boundary conditions (9.177) and (9.178). It can be checked that

$$\begin{aligned} C_1(\mathbf{y})=\frac{A_{13} h_*}{A_{33}^2 A_{44}}\bigl ( A_{13}\nabla _y\!\cdot (\tilde{H}_*\nabla _y p)- A_{44} \varDelta _y (p\tilde{H}_*)\bigr ), \end{aligned}$$

(9.180)

$$\begin{aligned} C_0(\mathbf{y})=\frac{A_{13} h_*^2}{2A_{33}^2} [\tilde{H}_*\varDelta _y p+\varDelta _y (p\tilde{H}_*)] -\frac{A_{13}^2 h_*^2}{A_{33}^2A_{44}}\nabla _y\!\cdot (\tilde{H}_*\nabla _y p). \end{aligned}$$

(9.181)

From (9.179), it immediately follows that

$$\begin{aligned} w^3(\mathbf{y},0)=C_0(\mathbf{y}), \end{aligned}$$

(9.182)

and it can be shown that in the incompressibility limit (when $A_{13}/A_{33}\rightarrow 1$ and $A_{44}/A_{33}\rightarrow 0$), we arrive at the following result:

$$\begin{aligned} C_0(\mathbf{y})=-\frac{h_*^2}{a_{44}}\nabla _y\!\cdot (\tilde{H}_*\nabla _y p). \end{aligned}$$

(9.183)

Thus, collecting Eqs. (9.142), (9.173), (9.182) and (9.183), we obtain the following two-term asymptotic approximation for the normal displacement at the contact interface in the case of the incompressible elastic layer:

$$\begin{aligned} w(\mathbf{y},0)\simeq -\varepsilon ^3\frac{h_*^3}{3a_{44}}\varDelta _y p(\mathbf{y}) -\varepsilon ^4\frac{h_*^2}{a_{44}}\nabla _y\!\cdot (\tilde{H}_*(\mathbf{y})\nabla _y p(\mathbf{y})). \end{aligned}$$

(9.184)

Recalling the scaling relations (9.130) and the notation $a_{44}=G^\prime$, we rewrite (9.184) in the form

$$\begin{aligned} u_3(\mathbf{y},0)\simeq -\frac{h^3}{3G^\prime }\varDelta _y p(\mathbf{y}) -\frac{h^2}{G^\prime }\nabla _y\!\cdot (\tilde{H}(\mathbf{y})\nabla _y p(\mathbf{y})). \end{aligned}$$

(9.185)

Now, substituting the expression (9.185) into the contact condition (9.161), we arrive at a partial differential equation in the domain $\omega$ with respect to the function $p(\mathbf{y})$. According to the asymptotic analysis [13] (see also Sect. 2.7.3), at the contour $\varGamma$ of $\omega$, we impose the following boundary conditions:

$$\begin{aligned} p(\mathbf{y})=0,\quad \frac{\partial p}{\partial n}(\mathbf{y})=0,\quad \mathbf{y}\in \varGamma . \end{aligned}$$

(9.186)

Here, $\partial /\partial n$ is the normal derivative. We underline that the location of the contour $\varGamma$ must be determined as part of the solution. For analytical evaluation of the contour $\varGamma$, a perturbation-based method was developed in [8].

9.2.7 Perturbation of the Contact Pressure in the Compressible Case

Collecting Eqs. (9.142), (9.152), (9.156), (9.170) and (9.179), we obtain

$$\begin{aligned} w(\mathbf{y},0)\simeq & {} \varepsilon \frac{h_*}{A_{33}}p(\mathbf{y}) +\varepsilon ^2\frac{\tilde{H}_*(\mathbf{y})}{A_{33}}p(\mathbf{y}) -\varepsilon ^3\frac{h_*^3 A_{13}(A_{13}-A_{44})}{3A_{33}^2 A_{44}}\varDelta _y p(\mathbf{y}) \nonumber \\ {}{} & {} {}+\varepsilon ^4\biggl \{\frac{h_*^2 A_{13}}{2A_{33}^2} \bigl [\tilde{H}_*(\mathbf{y})\varDelta _y p(\mathbf{y})+\varDelta _y \bigl (p(\mathbf{y})\tilde{H}_*(\mathbf{y})\bigr )\bigr ] \nonumber \\ {}{} & {} {}-\frac{h_*^2 A_{13}^2}{A_{33}^2 A_{44}}\nabla _y\!\cdot \bigl (\tilde{H}_*(\mathbf{y})\nabla _y p(\mathbf{y})\bigr )\biggr \}. \end{aligned}$$

(9.187)

By substitution of the asymptotic expansion (9.187) into the contact condition (9.137) and using the notation (9.131), we arrive at the equation

$$\begin{aligned} p(\mathbf{y})+ & {} \varepsilon \psi _*(\mathbf{y})p(\mathbf{y}) -\varepsilon ^2\frac{h_*^3A_{13}(A_{13}-A_{44})}{3A_{33}A_{44}}\varDelta _y p(\mathbf{y})\nonumber \\ {}+ & {} {}\varepsilon ^3\frac{A_{13} h_*^2}{2A_{33}A_{44}} \biggl \{ A_{44}\bigl [\psi _*(\mathbf{y})\varDelta _y p(\mathbf{y})+\varDelta _y \bigl (p(\mathbf{y})\psi _*(\mathbf{y})\bigr )\bigr ]\nonumber \\ {}- & {} 2A_{13}\nabla _y\!\cdot \bigl (\psi _*(\mathbf{y})\nabla _y p(\mathbf{y})\bigr )\biggr \} =\frac{A_{33}}{h_*}f^*(\mathbf{y}), \end{aligned}$$

(9.188)

where we have introduced the shorthand notation

$$\begin{aligned} f^*(\mathbf{y})=\delta _0^*-\varphi ^*(\mathbf{y}). \end{aligned}$$

(9.189)

It should be noted that Eq. (9.188) is applied in the case of compressible materials when its right-hand side makes sense.

By employing a perturbation method, the third-order asymptotic solution to Eq. (9.188) is expressed in the form

$$\begin{aligned} p(\mathbf{y})\simeq \frac{A_{33}}{h_*}\bigl ( \sigma _0(\mathbf{y})+\varepsilon \sigma _1(\mathbf{y})+\varepsilon ^2\sigma _2(\mathbf{y})+\varepsilon ^3\sigma _3(\mathbf{y}) \bigr ). \end{aligned}$$

(9.190)

Upon substitution of (9.195) into (9.188), we straightforwardly obtain

$$\begin{aligned} \sigma _0=f^*,\quad \sigma _1=-\psi _* f^*,\quad \sigma _2=\psi _*^2 f^* +\frac{h_*^2A_{13}(A_{13}-A_{44})}{3A_{33}A_{44}}\varDelta _y f^*, \end{aligned}$$

(9.191)

$$\begin{aligned} \sigma _3= & {} -\psi _*^3 f^* -\frac{h_*^2A_{13}(A_{13}-A_{44})}{3A_{33}A_{44}} \bigl (\psi _*\varDelta _y f^*+\varDelta _y(\psi _* f^*)\bigr ) \nonumber \\ {}{} & {} {}-\frac{h_*^2A_{13}}{2A_{33}A_{44}} \bigl \{A_{44} \bigl [\psi _*\varDelta _y f^*+\varDelta _y (f^*\psi _*)\bigr ] -2A_{13}\nabla _y\!\cdot (\psi _*\nabla _y f^*)\bigr \}, \qquad \qquad \end{aligned}$$

(9.192)

where, for the sake of brevity, the argument $\mathbf{y}$ is omitted.

Further, by making use of the differential identities

$$ \nabla \cdot (\psi \nabla f)=\nabla \psi \cdot \nabla f+\psi \varDelta f, $$

$$ \varDelta (f\psi )=\psi \varDelta f+f\varDelta \psi +2\nabla f\cdot \nabla \psi , $$

we simplify formula (9.192) as follows:

$$\begin{aligned} \sigma _3= & {} -\psi _*^3 f^* -\frac{h_*^2A_{13}(2A_{13}+A_{44})}{6A_{33}A_{44}} f^*\varDelta _y \psi _* \nonumber \\ {}{} & {} {}+\frac{h_*^2A_{13}(A_{13}-A_{44})}{3A_{33}A_{44}} \bigl (\nabla _y f^*\cdot \nabla _y \psi _*+\psi _*\varDelta _y f^*\bigr ). \end{aligned}$$

(9.193)

Observe also that formula (9.181) can be transformed to

$$\begin{aligned} C_0(\mathbf{y})=-\frac{h_*^2A_{13}(A_{13}-A_{44})}{A_{33}^2A_{44}} \bigl [\nabla _y \tilde{H}_*\cdot \nabla _y p+\tilde{H}_*\varDelta _y p\bigr ] +\frac{h_*^2A_{13}}{2A_{33}^2}p\varDelta _y\tilde{H}_*, \end{aligned}$$

(9.194)

where the expression in the brackets in (9.194) is equal to $\nabla _y\!\cdot (\tilde{H}_* \nabla _y p)$.

In the isotropic case, Eqs. (9.190)–(9.193) reduce to the following [5]:

$$\begin{aligned} p(\mathbf{y})\simeq \frac{2\mu +\lambda }{h_*}\bigl ( \sigma _0(\mathbf{y})+\varepsilon \sigma _1(\mathbf{y})+\varepsilon ^2\sigma _2(\mathbf{y})+\varepsilon ^3\sigma _3(\mathbf{y}) \bigr ), \end{aligned}$$

(9.195)

$$\begin{aligned} \sigma _0=f^*,\quad \sigma _1=-\psi _* f^*,\quad \sigma _2=\psi _*^2 f^* +\frac{h_*^2\lambda (\lambda -\mu )}{3\mu (2\mu +\lambda )}\varDelta _y f^*, \end{aligned}$$

(9.196)

$$\begin{aligned} \sigma _3= & {} -\psi _*^3 f^* -\frac{h_*^2\lambda (2\lambda +\mu )}{6\mu (2\mu +\lambda )} f^*\varDelta _y \psi _* \nonumber \\ {}{} & {} {}+\frac{h_*^2\lambda (\lambda -\mu )}{3\mu (2\mu +\lambda )} \bigl (\nabla _y f^*\cdot \nabla _y \psi _*+\psi _*\varDelta _y f^*\bigr ). \end{aligned}$$

(9.197)

It can be easily checked that the four-term asymptotic expansion (9.195), with the coefficients given by (9.196) and (9.197) in the 2D case, recovers the corresponding solution obtained in [21], where the next asymptotic term for the contact pressure in (9.195) was given explicitly.

9.2.8 Application to Sensitivity Analysis of the Contact Interaction Between Two Thin Incompressible Layers

According to (9.185) and (9.186), the refined asymptotic model for contact interaction of thin incompressible layers bonded to rigid substrates takes the form

$$\begin{aligned} -m^{-1}\varDelta _y p(\mathbf{y}) -\sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\nabla _y\!\cdot (\tilde{H}_n(\mathbf{y})\nabla _y p(\mathbf{y})) =\delta _0-\varphi (\mathbf{y}),\quad \mathbf{y}\in \tilde{\omega }, \end{aligned}$$

(9.198)

$$\begin{aligned} p(\mathbf{y})=0,\quad \frac{\partial p}{\partial n}(\mathbf{y})=0,\quad \mathbf{y}\in \tilde{\varGamma }, \end{aligned}$$

(9.199)

where $\tilde{\varGamma }$ is the contour of the contact region $\tilde{\omega }$, and m is given in (9.117).

Let us define

$$\begin{aligned} p(\mathbf{y})=\bar{p}(\mathbf{y})+\tilde{p}(\mathbf{y}), \end{aligned}$$

(9.200)

where $\bar{p}(\mathbf{y})$ is the solution to the original asymptotic model (9.112), (9.113), and $\tilde{p}(\mathbf{y})$ represents a perturbation due to the variability of the layer thickness.

Then, under the assumption that the thickness variation functions $\tilde{H}_1(\mathbf{y})$ and $\tilde{H}_2(\mathbf{y})$ introduce a small variation into the elliptical contact region $\omega$ corresponding to the density $\bar{p}(\mathbf{y})$, we derive from (9.198)–(9.200) the following limit problem for the variation of the contact pressure density:

$$\begin{aligned} -m^{-1}\varDelta _y \tilde{p}(\mathbf{y})= \sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\nabla _y\!\cdot (\tilde{H}_n(\mathbf{y})\nabla _y \bar{p}(\mathbf{y})),\quad \mathbf{y}\in \omega , \end{aligned}$$

(9.201)

$$\begin{aligned} \tilde{p}(\mathbf{y})=0,\quad \mathbf{y}\in \varGamma . \end{aligned}$$

(9.202)

Here, $\varGamma$ is the contour corresponding to the contact pressure (9.115).

Moreover, the thickness variation functions $\tilde{H}_1(\mathbf{y})$ and $\tilde{H}_2(\mathbf{y})$ will not greatly influence the resulting force-displacement relationship, if

$$\begin{aligned} \iint \limits _\omega \tilde{p}(\mathbf{y})\,d\mathbf{y}=0. \end{aligned}$$

(9.203)

In this case, the contact force P is related to the displacement $\delta _0$ by the same relation as that derived in framework of the original asymptotic model (9.112), (9.113).

Let us derive the conditions for $\tilde{H}_1(\mathbf{y})$ and $\tilde{H}_2(\mathbf{y})$ under which the equality (9.203) holds true. With this aim we consider an auxiliary problem

$$\begin{aligned} \varDelta _y \varTheta (\mathbf{y})=1,\quad \mathbf{y}\in \omega ,\quad \varTheta (\mathbf{y})=0,\quad \mathbf{y}\in \varGamma \end{aligned}$$

(9.204)

with the solution

$$ \varTheta (\mathbf{y})=-\frac{a_1^2 a_2^2}{2(a_1^2+a_2^2)}\theta (\mathbf{y}), $$

where

$$\begin{aligned} \theta (\mathbf{y})=1-\frac{y_1^2}{a_1^2}-\frac{y_2^2}{a_2^2}. \end{aligned}$$

(9.205)

As a result of (9.204), we rewrite Eq. (9.203) as

$$\begin{aligned} \iint \limits _\omega \tilde{p}(\mathbf{y})\varDelta _y \varTheta (\mathbf{y})\,d\mathbf{y}=0. \end{aligned}$$

(9.206)

Applying the second Green’s formula and taking into account Eqs. (9.201), (9.202) and (9.204), we reduce Eq. (9.206) to

$$\begin{aligned} \iint \limits _\omega \theta (\mathbf{y})\sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\nabla _y\!\cdot (\tilde{H}_n(\mathbf{y})\nabla _y \bar{p}(\mathbf{y}))\,d\mathbf{y}=0. \end{aligned}$$

(9.207)

After rewriting Eq. (9.207) in the form

$$ \iint \limits _\omega \theta (\mathbf{y})\sum _{k=1}^2\frac{\partial }{\partial y_k} \biggl (\frac{\partial \bar{p}}{\partial y_k}(\mathbf{y}) \sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\tilde{H}_n(\mathbf{y})\biggr )d\mathbf{y}=0 $$

and integrating by parts with (9.205) taken into account, we find

$$\begin{aligned} {}{} & {} -\iint \limits _\omega \sum _{k=1}^2 \frac{2y_k}{a_k^2} \frac{\partial \bar{p}}{\partial y_k}(\mathbf{y}) \sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\tilde{H}_n(\mathbf{y})\,d\mathbf{y} \nonumber \\ {}{} & {} {} +\int \limits _\varGamma \theta (\mathbf{y})\sum _{k=1}^2\cos (n,y_k) \frac{\partial \bar{p}}{\partial y_k}(\mathbf{y}) \sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\tilde{H}_n(\mathbf{y})\,ds_y=0. \end{aligned}$$

(9.208)

It is clear that the line integral in (9.208) vanishes due to the boundary condition (9.205). Hence, taking into account the exact expression (9.115) for $\bar{p}(\mathbf{y})$, we finally transform Eq. (9.208) into

$$\begin{aligned} \sum _{n=1}^2\frac{h_n^2}{G^\prime _n}\iint \limits _\omega \tilde{H}_n(\mathbf{y})\rho (\mathbf{y})\,d\mathbf{y}=0, \end{aligned}$$

(9.209)

where we have introduced the notation

$$\begin{aligned} \rho (\mathbf{y})=\biggl (\frac{s y_1^2}{a_1^2}+\frac{y_2^2}{s a_2^2}\biggr ) \biggl (1-\frac{y_1^2}{a_1^2}-\frac{y_2^2}{a_2^2}\biggr ). \end{aligned}$$

(9.210)

Based on the derived Eq. (9.209), we suggest the following optimization criterion for determining the average thicknesses $h_1$ and $h_2$:

$$\begin{aligned} \min _{h_n}\iint \limits _{\omega _*} (H_n(\mathbf{y})-h_n)^2\rho _*(\mathbf{y})\,d\mathbf{y}. \end{aligned}$$

(9.211)

Here, $\omega _*$ is a characteristic elliptic domain with semi-axes $a_1^*$ and $a_2^*$. In particular, in the capacity of $\omega _*$ we can take the average elliptic contact area for a class of admissible contact loadings, while $\rho _*(\mathbf{y})$ is given by

$$\begin{aligned} \rho _*(\mathbf{y})=\biggl (\frac{s^* y_1^2}{{a_1^*}^2}+\frac{y_2^2}{s^* {a_2^*}^2}\biggr ) \biggl (1-\frac{y_1^2}{{a_1^*}^2}-\frac{y_2^2}{{a_2^*}^2}\biggr ), \end{aligned}$$

(9.212)

where $s^*=a_2^*/a_1^*$ is the aspect ratio of $\omega _*$.

It is clear that the necessary optimality condition for (9.211) has the form

$$\begin{aligned} \iint \limits _{\omega _*} (H_n(\mathbf{y})-h_n)\rho _*(\mathbf{y})\,d\mathbf{y}=0, \end{aligned}$$

(9.213)

from which it follows that

$$\begin{aligned} h_n=\frac{1}{R_*} \iint \limits _{\omega _*} H_n(\mathbf{y})\rho _*(\mathbf{y})\,d\mathbf{y}, \end{aligned}$$

(9.214)

where

$$ R_*=\iint \limits _{\omega _*}\rho _*(\mathbf{y})\,d\mathbf{y}=\frac{\pi }{12}\bigl ( a_1^{*2}+a_2^{*2}\bigr ). $$

It remains to show that Eq. (9.209) follows from (9.213) if $\omega _*$ coincides with $\omega$. Indeed, as a consequence of (9.132), Eq. (9.213) is equivalent to the following:

$$\begin{aligned} \iint \limits _{\omega _*} \tilde{H}_n(\mathbf{y})\rho _*(\mathbf{y})\,d\mathbf{y}=0,\quad n=1,2. \end{aligned}$$

(9.215)

By adding the two equations above, multiplied by $h_n^2/G^\prime _n$, $n=1,2$, respectively, we arrive at Eq. (9.209).

It is interesting to observe that Eq. (9.214) indicates that, in order to obtain the optimal average thickness $h_n$, the corresponding variable thickness $H_n(\mathbf{y})$ has been averaged with the weight function $\rho _*(\mathbf{y})$ given by (9.212).

To conclude, we note that in the case of compressible layers, the optimal value of the average thickness $h_n$ coincides with the simple average of $H_n(\mathbf{y})$.