The driving forces of plate motion are generally classified into three types (e.g., Forsyth and Uyeda 1975): body forces (e.g., slab-pull and ridge-push; hereafter expressed as \(T_\mathrm{BD}\)), plate boundary forces (e.g., collision, suction, and shear forces between neighboring plates; \(T_\mathrm{PB}\)), and resultant resistance forces (e.g., mantle drag and slab resistance; \(T_\mathrm{RS}\)). Here, mantle drag (a component of \(T_\mathrm{RS}\)) is treated as a resistive force rather than a driving force of plate spin motion, because the toroidal components in mantle convection are negligible (Hager and O’Connell 1978) and are unlikely to excite the spin motion of plates. Of these, \(T_\mathrm{BD}\), in particular the slab pull force, is the main driving force of the global plate system (Forsyth and Uyeda 1975); \(T_\mathrm{BD}\) contributes primarily to straight plate motion, whereas \(T_\mathrm{PB}\) may cause spin motion when torque occurs around the center of the plate. However, for the slab pull force, spin motion can be excited. One such example is the Cocos plate. Gorbatov and Fukao (2005) have shown that the northwestern part of the slab was torn away from the deeper Farallon slab. It induces the heterogeneity of the slab pull forces, including a strong northward force from the eastern part of the slab, which can lead to the observed counterclockwise spin motion. Another example is the Philippine Sea plate that exhibits an active spin motion (Seno et al. 1993). Seno (2000) suggests that the spin motion results from the eastward mantle flow against the Philippine sea slab subducted beneath the SW Japan–Ryukyu arc, which is indicated by observations of electric conductivity (Handa 2005; Shimoizumi et al. 1997) and mantle anisotropy (Long and Hilst 2005). To exclude such complexities associated with slabs, we focused on plates without slabs, in which case the torque balance around the center of the plate can be described with relevant \(T_\mathrm{PB}\) and \(T_\mathrm{RS}\) forces, as discussed below. We can express \(T_\mathrm{PB}\) as the driving shear stress along a plate boundary \(\sigma _\mathrm{PB}\) and the area receiving the stress \(S_\mathrm{PB}\) as

$$\begin{aligned} T_\mathrm{PB}= \sigma _\mathrm{PB} S_\mathrm{PB} R =\sigma _\mathrm{PB} D_\mathrm{PB} L_\mathrm{PB} R=2 \pi \lambda \sigma _\mathrm{PB} D_\mathrm{PB} R^2 \end{aligned}$$

(3)

where *R* is the plate radius, \(D_\mathrm{PB}\) is the average depth of the plate boundary sustaining the shear stress, \(L_\mathrm{PB}\) is the length of the plate boundary along which the driving force is applied, and \(\lambda \) is the ratio of \(L_\mathrm{PB}\) to the total length of the plate boundary. In Eq. (3), we assume a planar plate for simplicity in the calculation of its radius and area (which does not affect the results significantly), and we can assume that the plate is circular rather than spherical shell because the difference in the result is not sufficiently significant, the length of the driving plate boundary is proportional to the plate size, and \(\lambda \) is constant. We also assume the presence of a low-viscosity layer (hereafter referred to as the asthenosphere) with constant Newtonian viscosity \(\mu \) and thickness \(D_\mathrm{RS}\). Then, \(T_\mathrm{RS}\) can be expressed as

$$\begin{aligned} T_\mathrm{RS} =\int \sigma _\mathrm{RS} r \mathrm{d} S = \int _{0}^{R}\mu \frac{r \omega _C}{D_\mathrm{RS}} 2 \pi r^2 \mathrm{d} r = \frac{\pi \mu \omega _C }{2 D_\mathrm{RS}} R^{4}. \end{aligned}$$

(4)

From the torque balance and Eqs. (3) and (4), we obtain

$$\begin{aligned} \sigma _\mathrm{PB} = \frac{\omega _C}{4 \lambda D_\mathrm{PB} D_\mathrm{RS}} \mu R^{2}, \end{aligned}$$

(5)

which indicates that \(\sigma _\mathrm{PB}\) is proportional to \(R^{2}\) and \(\omega _C\). In other words, Eq. (5) shows that (for a constant \(\omega _C\)) as plate size increases (left to right in Fig. 4), the driving shear stress increases such that it induces spin motion onto a larger plate.

In this context, \(\omega _C\) of a slab-free plate, which is indicated by the blue circles in Fig. 5, generally increases with decreasing plate size. One notable feature in Fig. 5 is the rapid change in \(\omega _C\) around the critical plate size of \({\sim }1000\) km: above the critical size, \(\omega _C\) values are generally less than the average rotation rate of the global lithosphere (i.e., the net lithospheric rotation; Ricard et al. 1991), of \({\sim }\)0.43\(^\circ \)/Myr, as based on a hotspot reference frame; below the critical size, however, except for two plates, i.e., the Panama plate and the Shetland plate, the motions of which are not well determined in PB2002 model, all \(\omega _C\) values exceed 0.43 \(^\circ \)/Myr.

It should be noted that the *R*-\(\omega _C\) variation is not significantly affected by the choice of different plate models as shown in Fig. 5 [i.e., Pb2002 (Bird 2003), NNR-MORVLE (Argus et al. 2011) and GSRM v2.1 (Kreemer et al. 2014) that include both no-net-rotation and hotspot reference frames for the data acquired by several methods representing different timescales.
See “Additional file 2: Table S1” for details]: i.e., almost all the large plates (plate size \({>}{\sim }1000\) km) show negligibly small \(\omega _C\) less than the net-rotation rate (0.43), and the rotation direction (clockwise or counterclockwise) may vary depending on the reference frame chosen (as indicated by large error bars with downward arrows in Fig. 5), whereas the large \(\omega _C\) is seen only for small plates irrespective of the chosen frame (Fig. 5).

In order to test such a possibility, we examine the relationship between *R* and \(\sigma _\mathrm{PB}\) (the plate boundary shear stress driving the spin motion) based on Eq. (5) and observed \(\omega _C\) as shown in Fig. 6, assuming \(\mu = 10^{20}\) Pa s, \(D_\mathrm{PB} = 40\) km, \(D_\mathrm{RS} = 60\) km, and \(\lambda = 0.25\). The uncertainties associated with these assumptions will be discussed later. It is worth noticing that \(\sigma _\mathrm{PB}\) for plates with low rotation rates (the large plates, in general) have extremely large uncertainties, indicated by the error bars with downward arrows in Fig. 6, and should be regarded as upper bounds.

The dashed line in Fig. 6 represents the expected driving stress induced by the motion of the Pacific plate (corresponding to the dashed line in Fig. 5), which limits the upper bounds of \(\sigma _\mathrm{PB}\) for the small plates. The small plates plot along the dashed line are located next to (or very close to) the fast-moving Pacific plate and are expected to have a high \(\sigma _\mathrm{PB}\) as in Fig. 6, whereas other small plates located next to plates with slower velocities are expected to have a lower \(\sigma _\mathrm{PB}\), which is also seen in Fig. 6. Within this context, in order to discuss the maximum stress that the plate boundary can sustain, the upper bound of \(\sigma _\mathrm{PB}\) and its variation with plate size are thought to be more important than the overall data distribution. For large plates, \(\sigma _\mathrm{PB}\) is appreciably lower than the prediction of the dashed line, indicating that rheological weakening or yielding operates on the boundaries of large plates, irrespective of the choice of reference frame (Fig. 6).

For small plates (the plate size \({<} {\sim } 1000\) km, i.e., \(R < 500\) km), the shear stress driving the spin motion increases with *R* (e.g., the dashed line in Fig. 6), and above a critical size \(R_\mathrm{c}\), the stress becomes too large to transmit the stress across the boundary, causing rheological weakening or yielding. This critical stress is regarded as the strength of the plate boundary. From Fig. 6, the critical stress is estimated to be \({\sim }10 - 20\) MPa for plate sizes between \({\sim }350{-}630\) km. Accordingly, the rotation rate of plates with \(R > R_\mathrm{c}\) is small as compared to the dashed line in Fig. 5.

It can be confirmed that the spin rates of small plates are higher than those of large plates, which is the overall result, from Fig. 7, which shows that many small plates along the “sides” (i.e., along strike-slip boundaries) of large fast-moving plates have high spin rates with a rotation direction (clockwise or counterclockwise) consistent with the nearly straight motions of large plates that subduct along their margins. This mechanism has been suggested for several individual microplates (e.g., Schouten et al. 1993). Figure 7, for example, demonstrates that in the southwestern Pacific, the Pacific plate (PA) excites spin motion of the Niuafo’ou plate (NI) and that the Australia plate (AU) induces spin motion of the Tonga plate (TO) and the Kermadec plate (KE). Along the East Pacific Rise, the Pacific plate and the Nazca plate (NZ) drive motions of the Easter plate (EA) and the Juan Fernandez plate (JZ). Although some of the abovelisted small plates, based on PB2002 plate model, are located within deforming zones identified by Kreemer et al. (2014) and could be inappropriate to consider them as rigid plates to define the spin rate (e.g., KE), the overall configuration remains unchanged, including large spin motions of NI, EA, and JZ as shown in Fig. 7.

These features, represented in Fig. 7, suggest that although some regions (especially Southeast Asia) exhibit complex spin directions probably due to interactions among the small plates, large fast-moving plates with subducting slabs induce spin motion in adjacent small plates through interactions along plate boundaries, which supports the idea presented above that \(T_{PB}\) drives the plate spin motions.

In addition, from Eq. (5) and Fig. 6, we obtain a quantitative relationship between the viscosity of the asthenospheric mantle beneath the plates \(\mu _\mathrm{a}\) and the strength of the plate boundary \(\sigma _\mathrm{st}\). We substitute the constraints at the critical condition, as specified in Fig. 6 (i.e., \(\omega _{C} \approx 8 ^{\circ }\)/Myr, plate size \(\approx 630\) km and \(\lambda \approx 0.25\), corresponding to the South Bismarck plate attaining the maximum \(\sigma _\mathrm{PB}\)), and set \(D_\mathrm{PB} = 40\) km as the thickness of the plate boundary sustaining the shear stress (e.g., Kohlstedt et al. 1995). Then, we obtain the following equation,

$$\begin{aligned} \sigma _\mathrm{st} \mathrm {(MPa)} = \frac{1.1}{10^{17}}\frac{\mu _\mathrm{a}\mathrm {(Pa \ s)}}{D_\mathrm{RS}\mathrm {(km)}}, \end{aligned}$$

(6)

which is the basis for Fig. 8. There is an appreciable uncertainty with regard to the thickness of the asthenosphere beneath the oceanic plates, as it is dependant on the observational methods used for measurement (Karato 2012); the asthenosphere is observed as a zone of low seismic velocity [e.g., \({\sim }120\) km (Kawakatsu et al. 2009)], a high attenuation layer [e.g., \({\sim }140\) km (Dziewonski and Anderson 1981); \({\sim }60\) km (Yingjie et al. 2007)] with significant seismic anisotropy [e.g., \({\sim }120\) km (Beghein and Trampert 2004)] and a high electric conductivity layer [e.g., \({\sim }60\) km (Evans et al. 2005)]. From these observations, we estimate the thickness of asthenosphere under the oceanic plates (\(D_\mathrm{RS}\)) to be 60–140 km.

Equation (6) and Fig. 8 impose several constraints on the plate–mantle dynamics. Substituting \(\mu _\mathrm{a} = 10^{21}\) Pa s, based on the representative viscosity of the upper mantle (Peltier 1998), into Eq. (6), we obtain \(\sigma _\mathrm{st}=78\)–183 MPa for \(D_\mathrm{RS} \approx 60\)–140 km. Considering a more realistic case and assuming a low-viscosity asthenosphere, which is estimated from post-glacial rebound, seismic data, and laboratory measurements as \(10^{19}\)–\(10^{20}\) Pa s (e.g., Karato and Wu 1993; Simons and Hager 1997; Forte and Mitrovica 2001), we substitute \(\mu _\mathrm{a} \approx 10^{19}\)–\(10^{20}\) Pa s, which gives \(\sigma _\mathrm{st} \approx 0.78\)–18 MPa. This estimate only considers the force along the fractional length of the plate boundary \(\lambda \), as in Eq. (3). If we consider the resistive forces along the remainder of the plate boundary, with length \(1-\lambda \), we obtain

$$\begin{aligned} \sigma _\mathrm{PB} = \frac{\omega _C}{4 \lambda D_\mathrm{PB} D_\mathrm{RS}} \mu R^{2} + \frac{1-\lambda }{\lambda } \sigma _\mathrm{R}, \end{aligned}$$

(7)

where \(\sigma _\mathrm{R}\) is the average resistive stress along the plate boundary (which must be less than the strength of plate boundary \(\sigma _\mathrm{st}\)). As a result, a lower viscosity \(\mu _\mathrm{a}\) is required to reproduce the same \(\sigma _\mathrm{st}\), as compared with the results of equation (6) (broken lines, Fig. 8). Setting \(\sigma _\mathrm{R}\)(\(\le \sigma _\mathrm{st} \approx 0.78\)–18 MPa), \(\mu _\mathrm{a}\) = \(10^{19}\)–\(10^{20}\) Pa s, and \(D_\mathrm{RS} = 60{-}140\) km, and considering the uncertainty of plate size in plate model, we estimate \(\sigma _\mathrm{st} \approx 3\)–75 MPa.

In previous studies, the strength of a plate boundary was estimated based on seismic observations, particularly the spatial mapping of earthquake focal mechanisms and their corresponding temporal changes before and after a large earthquake; e.g., Hasegawa et al. (2012) used high-resolution data on the approximately 4000 earthquake focal mechanisms in northeast Japan between 2003 and 2011 and found that the 2011 Tohoku earthquake almost completely released the accumulated stress along the plate boundary, and they estimated the stress at release to be as small as 5–15 MPa, suggesting that the presence of water weakened the plate boundary fault. Hardebeck and Hauksson (2001) used the focal mechanism data of approximately 50,000 earthquakes along the San Andreas Fault, mainly between 1981 and 1999, including the 1992 Landers earthquake, and estimated the strength of the fault to be about \(20 \pm 10\) MPa, due in part to the low mechanical strength of smectite (Carpenter et al. 2011). In this study, although the estimated strength of 3–75 MPa may be regarded as a global average strength for a number of plate boundaries of various types, including convergent and transform boundaries, the estimated strength is consistent with a stress level deduced from high-resolution seismic observations of specific areas, as mentioned above.

To naturally reproduce plate-like structures and motions as part of a mantle convection process, including the case of rigid plates with soft plate boundaries, three-dimensional numerical simulations have been used to investigate critical conditions and requirements, especially those concerning rock rheology. Tackley (2000b) and Richards et al. (2001) estimated the required yield stress of a plate to reproduce Earth-like plate motion on the basis of surface velocity fields and obtained results of 17–170 and 50–150 MPa, respectively. In addition, Bercovici (1993, 1995, 2003) suggested that a pseudo-stick-slip rheology, in which the stress decreases with increasing strain rate after yielding, can reproduce plate-like motions, in particular a high toroidal/poloidal kinetic energy ratio of up to 0.8, which is consistent with ratios observed in previous studies (Bercovici and Wessel 1994; Hager and O’Connell 1978; O’Connell et al. 1991) as well as in this study (Fig. 1). In this context, after exceeding the yield stress of plates, the strain can be concentrated to form a plate boundary composed of a “damaged” weak zone (e.g., Bercovici and Ricard 2014), where the stress level is significantly reduced, possibly to the range estimated from seismology and this study, especially when water is present to weaken the plate boundary.

The exact rheology and the physical–chemical state of plate boundary is a vital problem that will help understand the mechanisms of plate tectonics (e.g., Bercovici and Ricard 2014; Gordon 1998, 2000). At present, it is difficult to constrain the exact rheology from the approach in this study; however, by combining with other approaches, such as seismic and geodetic observations on both rigid plates and deformation zones (Gordon 1998, 2000; Kreemer et al. 2014), field and laboratory studies on rock and fault rheology (e.g., Kohlstedt et al. 1995; Sibson 2003), and numerical simulation of combined plate motion and mantle convection (e.g., Bercovici 2003; Richards et al. 2001; Tackley 2000b), tighter constraints can be obtained to quantify the Earth’s dynamics with regard to plate tectonics.