To understand how Earth’s spin changes may influence explosive activity at subduction volcanoes, we propose a conceptual model which takes into account the effects of LOD changes on the crustal stress state in subduction zones (Fig. 3). Previous works measured the effects of orbital and rotational variations on Earth’s lithosphere local strain field changes (Milyukov et al. 2013) and on Jupiter and Saturn’s icy satellites (Hoppa et al. 1999; Nimmo et al. 2007). Here, we focus on the capability of LOD-related stress changes to lead to the failure of the crustal rocks surrounding magma storage zones and thus to act as trigger mechanism for volcanic eruptions. In this regard, we use the term “trigger” to denote a process able to promote a fracturing event of magma chamber wall rocks on the condition that it acts in combination with the intrinsic dynamics of volcanoes (i.e. magma crystallization and differentiation, volatile pressure build-up). We propose that the rupture of magma chamber wall rocks occurs when the differential stress generated by LOD oscillation reaches a critical threshold (see Mohr’s circles, Fig. 4). The state of stress in the crust determined by plate tectonics is modified due to rotation instabilities. In E-directed subduction zones, where volcanic arcs develop in compressive tectonic regime, Earth’s spin decelerations (i.e. LOD increase) induce an increase of the horizontal stress (*σ*
_{1}) due to a lower tension in the crust and thus a larger differential stress favouring wall rock failure. On the contrary, in W-directed subduction zones, where volcanism develops in back-arc extensional regime, Earth’s spin accelerations (LOD decrease) produce larger differential stresses by enhancing the *σ*
_{3} tensor (Fig. 4).

To evaluate the presence of any contingent stress changes in the crust, related to Earth’s spin variations, we propose a simple model able to support statistical results. Our intent here is not to solve the problem entirely. We evaluate whether the *Δ*LOD_{eff}-related stress can be comparable with the other computed stress values, associated with other physical process (e.g. decompression rates related to glacio-eustatism or tidal forces), capable to promote the triggering of volcanic eruptions. We consider a rotating spherical shell with a specific thickness *d* = *R*
_{2}−*R*
_{1} = 20 km, where *R*
_{2} = 6371 km is the external radius, i.e. the Earth’s radius, and *R*
_{1} is the internal radius (i.e. *R*
_{1} = 6351 km), subjected to centrifugal effects on its mass. At the equatorial plane, the spherical shell becomes a circular crown. Under this cylindrical approximation, we are going to evaluate the stress generated on this rotating thick ring, subjected to a radial pressure due to centrifugal effects, using equations derived by Hearn (1997).

Inertial forces do not affect the dynamics of the Earth’s interior, and, with a constant Earth’s rotation, the centrifugal force does not affect the gravitational-centrifugal force equilibrium on the Earth’s surface. However, the rotational instability implies relatively small variations of the centrifugal component, with a net effect on the balance between gravitational and centrifugal forces, providing a continuous, relatively small dynamic inconstancy. This yearly disequilibrium produces unsettled values of the tension applied on the thick ring (i.e. the crust), and, consequently, a variable circumferential stress, or hoop stress (*σ*
_{H}), is generated (Hearn 1997).

Here, we compute the tensional hoop stress in the Earth crust, assuming, for sake of simplicity, a planar model on the equatorial plane. This corresponds to evaluate the radial *σ*
_{
R
} and hoop *σ*
_{H} stresses of a rotating circular crown, or a thick ring, with a thickness of *d* = 20 km, as a function of the radial distance, *r*; the crust Poisson’s ratio, *ν*; the crust density, *ρ*
_{C}; the Earth angular velocity, *ω*; and the internal and external circular crown radii, *R*
_{1} and *R*
_{2} (see Appendix). To analyse the variations of the radial and hoop stresses in our ideal crust that depend on *Δ*LOD_{eff} changes, under the planar model approximation, we define here the effective radial *σ*
_{
R
} and hoop *σ*
_{H} stresses as the *σ*
_{
R
} and *σ*
_{H} obtained with Eqs. (2) and (3), derived by Hearn (1997), per unit of ms, respectively. Figure 5 shows the variations of the effective radial, *σ*
_{
R
}, and hoop, *σ*
_{H}, stresses, as a function of *r* in the crust. We notice that for this particular geometry approximation, the equations derived by Hearn (1997) for a thick ring, used here to compute radial and hoop stresses, result in quite different values, when considering our ideal crust at the equatorial plane. We obtain that the effective radial *σ*
_{
R
} is zero at the base and at the top of the crust (*R*
_{1} = 6351 and *R*
_{2} = 6371 km, respectively), with a maximum values of 2.40 Pa/ms in the middle crust (Fig. 5). On the contrary, we have higher values (i.e. five orders of magnitude) for the hoop stress in the whole ideal geometry, with a maximum value of 600 kPa/ms at the base of the crust (Fig. 5). We consider these results, obtained at the equator, as the maximum values that can be reached when modelling with the simple approximations of a thick ring geometry and an ideal crust. In fact, we may expect even different values for the hoop stress if modelling with a rotating spherical shell, knowing that the centrifugal potential is greatest at the equator (i.e. the hoop stress would decrease as a function of the latitude) and also taking into account the unelastic behaviour of the Earth, crust heterogeneities and interactions with the other spherical shells (e.g. the mantle).

In this perspective, variations of *Δ*LOD_{eff} (1850 to 2009 period) and of the corresponding quantity *Δω*
_{eff} =*ω*
_{
n
} − *ω*
_{
n−1}, being the difference of the Earth’s rotation rate, over two consequent years are reported in Additional file 1: Figures S11 and S12, respectively. Thus, following the model by Hearn (1997), we obtain the variation of the hoop stress in the crust, *Δσ*
_{Heff} (Additional file 1: Figure S12), by Eq. (3), as a function of the variation of the *Δω*
_{eff.
}, which has values in the range between ±400 kPa for LOD variations of 1 ms (i.e. corresponding to ±45 Pa/h for typical annual *Δ*LOD values). For comparison, decompression rates related to glacio-eustatism and tidal forces, considered among possible controlling factors of volcanic eruptions, range in the order of a few Pa per year (Kutterolf, et al. 2012) and 150 Pa/h (Sottili et al. 2007), respectively, whilst magma volatile pressure build-up is reported around 400–720 Pa/h (Johnston and Mauk 1972; Jentzsch et al. 2001).