Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

Abstract

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

Appendix: Classes of Functions

For the case of the smooth contour $L$ with two tips $c=a$, $c=b$, the classes of functions considered in the present paper are defined as, see [58]:

• $H$ class

The function $\varphi (t )$ is said to belong to $H$ class on $L$, if for any two points $t_{1},\,t_{2}\in L$ it satisfies, for some $0<\alpha$, the following Hölder condition:

$$ \bigl\vert \varphi (t_{2} )-\varphi (t_{1} ) \bigr\vert \leq A \vert t_{2}-t_{1} \vert ^{\alpha } $$

(91)

in which $A$ is an arbitrary positive constant. More specifically, it could be said that the function $\varphi (t )$ belongs to $H (\alpha )$ class.

• $H^{*}$ class

The function $\varphi (t )$ is said to belong to $H^{*}$ class on $L$, if it satisfies Hölder condition with some $\alpha$ on every closed part of $L$ that does not contain its tips and near the tips $c=a$, $c=b$

$$ \varphi (t )= \frac{\varphi ^{*} (t )}{ (t-c )^{\beta }}, \quad 0 \leq \beta < 1 $$

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where $\varphi ^{*} (t )\in H$ class.

• $h_{0}$ and $h_{2}$ classes

These classes are particular cases of more general class $h (c_{1},\ldots ,c_{q} )$ introduced in [58] for the consideration of the Hilbert problems and singular integral equations for a union of smooth contours, see §79, 81, 88 in [58]. The definition of the latter class is more involved and require introduction of the material that is beyond the scope of the present paper. So, instead of giving formal definition of class $h (c_{1},\ldots ,c_{q} )$, we give its interpretation for the application related to our problem, see also the discussion on page 251 of [58].

In our case, the condition that the function (that belongs to $H$ class at any internal point of $L$) belongs to $h_{2}$ class simply means that this function is bounded near both tips of $L$, while the condition that the function belongs to $h_{0}$ class means that no such restrictions are required on the behavior of the function at any of two tips.

Finally, we reproduce the following result of §20 in [58] that was used in Sect. 5.1: if the function $\varphi (t )$ belongs to $H (\alpha )$ class on a smooth contour $L$, then

$$ \varPhi (t_{0} )=\frac{1}{2\pi i} \int \limits _{L} \frac{\varphi (t )dt}{t-t_{0}} $$

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also satisfies everywhere on $L$, except in the neighborhood of those of its tips at which $\varphi (t )\neq 0$, the $H (\alpha )$ condition for $\alpha <1$ and $H (1-\varepsilon )$ condition for $\alpha =1$, where $\varepsilon$ is an arbitrary small parameter.