1 Correction to: Arch Appl Mech (2017) 87:1859–1869 https://doi.org/10.1007/s00419-017-1293-2

The following text should be changed in the abstract:

Using the theory of generalized functions, some basic parameters at the half-space boundary are defined in a finite form, and no computation of any integral is needed. Knowledge of some of the Green’s functions in finite form allows us to derive the governing integral equations for the normal contact and crack problems, as well as to establish certain relationship between the integrands in Fourier transforms of the kernels of the relevant integral equations.

Text after equation (17) up to and including equation (24) should be replaced by:

In order to better visualize (17) we may deduce that

$$\begin{aligned} \sum _{s=1}^{5}\mathcal{X}_{ks}(-1)^{n+s}D_{c}^{(n,s)}=D_{\mathrm{cnk}}, \end{aligned}$$
(18)

which is the determinant of the matrix \(\mathcal{C}\) (see 11–12) with the nth row replaced by \(\mathcal{X}_{ks}\). Now (17) can be rewritten as

$$\begin{aligned} u_{k}(x_{1},x_{2},0)=\frac{1}{4\pi ^{2}}\int _{0}^{2\pi }\int _{0}^{\infty }\sum _{n=1}^{5}\alpha _{n}\frac{D_{\mathrm{cnk}}(\vartheta )}{D_{c}(\vartheta )}\exp \left[ -i\rho (x_{1}\cos \vartheta +x_{2}\sin \vartheta )\right] \mathrm{d}\rho \mathrm{d}\vartheta . \end{aligned}$$
(19)

Since \(\rho \) enters only the exponential, we can use the theory of generalized functions to get

$$\begin{aligned} \int _{0}^{\infty }\exp \left[ -i\rho (x_{1}\cos \vartheta +x_{2}\sin \vartheta )\right] \mathrm{d}\rho =\pi \delta (x_{1}\cos \vartheta +x_{2}\sin \vartheta )- \mathcal{P}\left( \frac{i}{x_{1}\cos \vartheta +x_{2}\sin \vartheta }\right) . \end{aligned}$$
(20)

Here \(\delta (\cdot )\) is the Dirac delta-function and \(\mathcal{P}\) stands for the principal value. The following property of delta-functions can be found in (Krein 1972)

$$\begin{aligned} \delta [f(\vartheta )]=\sum _{n}\frac{\delta \left( \vartheta -\vartheta _{n}\right) }{|f^{^{\prime }}(\vartheta _{n})|}, \end{aligned}$$
(21)

where \(\vartheta _{n}\) are all the roots of equation \(f(\vartheta )=0\). In our case, we have 2 roots in the interval \(0\le \vartheta <2\pi \)

$$\begin{aligned} \vartheta _{1}=\pi -\tan ^{-1}(x_{1}/x_{2}),\quad \vartheta _{2}=2\pi -\tan ^{-1}(x_{1}/x_{2}) \end{aligned}$$
(22)

with

$$\begin{aligned} |f^{^{\prime }}(\vartheta _{1})|=|f^{^{\prime }}(\vartheta _{2})|=\sqrt{x_{1}^{2}+x_{2}^{2}}. \end{aligned}$$
(23)

Now the final result will take the form

$$\begin{aligned} u_{k}(x_{1},x_{2},0)= & {} \frac{1}{4\pi } \left[ \sum _{n=1}^{5}\alpha _{n} \left( \frac{D_{\mathrm{cnk}}(-x_{2},x_{1})}{D_{c}(-x_{2},x_{1})}+ \frac{D_{\mathrm{cnk}}(x_{2},-x_{1})}{D_{c}(x_{2},-x_{1})}\right) \right] \nonumber \\&-\,\frac{i}{4\pi ^{2}}\mathcal{P}\int _{0}^{2\pi }\sum _{n=1}^{5}\alpha _{n}\frac{D_{\mathrm{cnk}}(\vartheta )\mathrm{d}\vartheta }{D_{c}(\vartheta )(x_{1}\cos \vartheta +x_{2}\sin \vartheta )}. \end{aligned}$$
(24)

Here each parameter with the arguments \((-x_{2},x_{1})\) is understood as similar parameter in the article with \(\xi _{1}\) formally replaced by \(-x_{2}\) and \(\xi _{2}\) is replaced by \(x_{1}\). Numerical computations show that

$$\begin{aligned} Re\left[ D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )\right]= & {} Re\left[ D_{\mathrm{cnk}}(\vartheta +\pi )/D_{c}(\vartheta +\pi )\right] , Im\left[ D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )\right] \nonumber \\= & {} -Im\left[ D_{\mathrm{cnk}}(\vartheta +\pi )/D_{c}(\vartheta +\pi )\right] \end{aligned}$$
(24a)

and

$$\begin{aligned} Im[D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )+D_{\mathrm{ckn}}(\vartheta )/D_{c}(\vartheta )]=0,Im[D_{\mathrm{cnn}}(\vartheta )/D_{c}(\vartheta )]=0. \end{aligned}$$
(24b)

Here symbols Re and Im stand for the real and imaginary part, respectively. These properties allow us to conclude that the principal value of the integral of the \(Re[D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )]\) in the right-hand side of (24) is zero, while the principal value of the same integral of the \(Im[D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )]\) is zero in the case \(n=k\), so that final result in (24) is always real, as it should be. We can also conclude that the matrix \(\{D_{nk}\}=\{D_{\mathrm{cnk}}(\vartheta )/D_{c}(\vartheta )\}\) is Hermitian with \(D_{nk}\) equal to the complex conjugate of \(D_{kn}\) and the diagonal being real.

In the first paragraph of the section 4, the sentence “There is also no outside electrical or magnetic interference” should be replaced by:

“The outside electrical or magnetic parameters are chosen in such a way that the mechanical boundary conditions, described above, hold.”

Equations (52) and (53) should read:

$$\begin{aligned} K_{12}(x_{1},x_{2})= & {} \frac{1}{4\pi } \left( \frac{D_{12}(-x_{2},x_{1})}{D_{c}(-x_{2},x_{1})}+ \frac{D_{12}(x_{2},-x_{1})}{D_{c}(x_{2},-x_{1})}\right) -\frac{i}{4\pi ^{2}}\mathcal{P}\int _{0}^{2\pi }\frac{D_{12}(\vartheta )\mathrm{d}\vartheta }{D_{c}(\vartheta ) (x_{1}\cos \vartheta +x_{2}\sin \vartheta )}.\nonumber \\ \end{aligned}$$
(52)
$$\begin{aligned} K_{21}(x_{1},x_{2})= & {} \frac{1}{4\pi } \left( \frac{D_{21}(-x_{2},x_{1})}{D_{c}(-x_{2},x_{1})}+ \frac{D_{21}(x_{2},-x_{1})}{D_{c}(x_{2},-x_{1})}\right) - \frac{i}{4\pi ^{2}}\mathcal{P}\int _{0}^{2\pi }\frac{D_{21}(\vartheta )\mathrm{d}\vartheta }{D_{c}(\vartheta )(x_{1}\cos \vartheta +x_{2}\sin \vartheta )}.\nonumber \\ \end{aligned}$$
(53)

The last paragraph before section Discussion and unnumbered equation should be replaced by:

Numerical computations show the same properties hold for the ratios \(D_{nk}^{0}/D_{\mathrm{crt,}}\) as in (24a–24b). Taking into consideration that \(D_{nk}^{0}\) in (63–66) are homogeneous with respect to \(\xi _{1}\) and \(\xi _{2}\) of the order 4 and \(D_{\mathrm{crt}}\) is homogeneous of the order 3, we may conclude that the integrals (63–66) are divergent. They can be regularized and computed, as it was done in (36–38), except that now for the kernels \(K_{nk}\) with \(n\ne k\) we shall need to compute the principal value of the integral, similar to (52–53).