Correction: Thermoelastic instability on a frictional surface and its implication for size effect in friction experiments

Correction: Earth Planets Space (2023) 75: 71 https://doi.org/10.1186/s40623-023-01820-9

Following publication of the original article (Noda 2023), the author reported errors in Eqs. (12), (28) and (30).

In the published PDF, letter “c” at the beginning of Eqs. (12) and (28) should be removed.

The correct Eq. (12) should read:

$$\begin{array}{c}\Delta {\sigma }_{\text{n}}\left(x,t\right)=\left[{\int }_{-\infty }^{t}\text{d}{t}^{\prime}{\int }_{-\infty }^{\infty }\text{d}{x}^{\prime}\right]q\left({x}^{\prime},{t}^{\prime}\right)G\left(x-{x}^{\prime},t-{t}^{\prime}\right).\end{array}$$

(12)

The correct Eq. (28) should read:

$$\begin{array}{c}\widetilde{\eta }\left(s\right)\approx {\sum\limits_{i=1}^{n}}{a}_{i}\left[{\int }_{0}^{s}2\widetilde{\omega }\left({s}^{^{\prime}}\right){\mathrm{e}}^{-\left(s-{s}^{^{\prime}}\right)/{s}_{i}}\mathrm{d}{s}^{^{\prime}}\right]={\sum\limits_{i=1}^{n}}{a}_{i}{\phi }_{i} , \end{array}$$

(28)

The “dp” in Eq. (30) was mistakenly moved to the second line.

The correct Eq. (30) should read:

$$\begin{array}{c}\mathrm{erfc}\left(\sqrt{s}\right)={\int }_{0}^{\infty }\frac{H\left(p-1\right)}{\pi p\sqrt{p-1}}{\mathrm{e}}^{-sp}\,\,\mathrm{d}p \\ \quad\quad \quad\quad\quad\quad\quad\quad\quad={\int }_{0}^{1}\frac{1}{\pi }{s}_{\mathrm{c}}^{-1/2}{\left(1-{s}_{\mathrm{c}}\right)}^{-1/2}{\mathrm{e}}^{-s/{s}_{\mathrm{c}}}\mathrm{d}{s}_{\mathrm{c}} , \end{array}$$

(30)

The original article (Noda 2023) has been updated.