Z. Angew. Math. Phys. (2018) 69:4
2017 Springer International Publishing AG,
part of Springer Nature
published online December 9, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Interaction between a punch and an arbitrary crack or inclusion in a transversely
I. Fabrikant, E. Karapetian and S. V. Kalinin
Abstract. We consider the problem of an arbitrary shaped rigid punch pressed against the boundary of a transversely
isotropic half-space and interacting with an arbitrary ﬂat crack or inclusion, located in the plane parallel to the boundary.
The set of governing integral equations is derived for the most general conditions, namely the presence of both normal
and tangential stresses under the punch, as well as general loading of the crack faces. In order to verify correctness of
the derivations, two diﬀerent methods were used to obtain governing integral equations: generalized method of images and
utilization of the reciprocal theorem. Both methods gave the same results. Axisymmetric coaxial case of interaction between
a rigid inclusion and a ﬂat circular punch both centered along the z-axis is considered as an illustrative example. Most of
the ﬁnal results are presented in terms of elementary functions.
Mathematics Subject Classiﬁcation. Integral equations 45, Linear integral equations 45A05.
Mechanical characteristics of solids and surfaces are relevant in multiple areas of science and engineering
applications ranging from structural mechanics to bioengineering to corrosion. The need to probe mechan-
ical behavior of surfaces have spurred the development of multiple characterization techniques ranging
from micro- and nanoindentation  to scanning probe microscopies including atomic force acoustic mi-
croscopy [2–11] and frequency tracking  and band excitation [13–17] dynamic probes. Measured in
these methods are the tip-surface forces as a function of indentation depth (nanoindentation), or resonance
frequency shifts (AFAM) directly related to the tip-surface stiﬀness.
Interpretation of these data in terms of materials functionalities requires the known functional rela-
tionships between the force acting on the probe and measured displacement or resonant frequency shift,
i.e., relevant contact mechanics model.
Voluminous and signiﬁcant research has been published by the authors (with other co-authors) [2,18–
27], where the results of theoretical and experimental investigations were presented on validation of
Hertzian type solutions for the cases of nanoindentation and their practical applications to various types
of scanning probe microscopy and piezoresponse force microscopy. A variety of materials were studied
both inorganic and biological [18,19]. The theoretical basis for the research is given , where the exact
solution in terms of elementary functions was obtained for an arbitrary point force and point source acting
on the boundary of a piezoelectric transversely isotropic half-space. Nanoindentation of ﬂat, conical and
spherical indenters [24,26] was studied in the cases of normal as well as tangential (frictional) contact.
The more complicated case of ﬂat and non-ﬂat indenters of arbitrary planform is presented . The
investigation of the weak and strong indentations and their applications to piezoresponse force microscopy
can be found in .
However, these analyses are limited to the surfaces of uniform materials of various symmetries and
dissimilar piezoelectric or thermal properties and generally allow only for the certain deviations of surface
geometry from planar. The eﬀect of this topographic cross talk on SPM imaging is well explored .
At the same time, realistic materials can contain below-surface imperfections such as cracks, voids, and