1 Introduction

Moment of Inertia (MOI), also known as ‘rotational inertia’ and ‘second moment of mass’ indicates distribution of mass in a body that tends to restrict its rotation, and expresses the ease or difficulty of that body to rotate. MOI is a well-established concept in statics (e.g., Das and Mukherjee 2012) and its expression for several regular geometric bodies has been deduced (Spiegel and Liu 1999). The MOI controls the state of rest or motion of a rotating body about its rotational axis. The greater the mass concentrated away from the axis of rotation, the greater the MOI. Thus, the MOI depends on the mass distribution inside a moving/deforming body (Batra 2016) that in turn is sensitive to any variations in density. Bykov (2014, 2015) used MOI in modeling seismicity related to rotation of fault blocks (also see Dahlen 1977). To be specific, Bykov (2014; 2015) presented a differential equation involving inertial rotation angle, displacement and the MOI that represented the rotation waves in the ‘elastic fragmented massif’. Dahlen (1977), on the other hand, presented an expression of Earth in mechanical equilibrium during its rotation involving its MOI, and demonstrated mathematically how far this equilibrium disturbs locally due to faulting. Other applications of MOI can be found in engineering geology (Hudson and Harrison 1992), tectonics (Bombolakis 1994) and planetary sciences (Margot et al. 2012). For example, Bombolakis (1994) refined deformation behavior of rocks in modeled critical wedges that involved MOI in the analysis. Specifically, in structural geology, the MOI for fault blocks has been utilized in modeling the genesis of the related gouge material (equations 5 and 7 in Guo and Morgan 2007; equations A2, A5 and A7 of Guo and Morgan 2008).

Fig. 1
figure 1

Top-to-left simple shear on crustal blocks/mineral grains produce book-shelf gliding. Antithetic top-to-right (down) shear between the two blocks. Compaction syncline produced at the top part of the slided blocks, in regional context. L: dimension of the block in the third perpendicular direction. T: thickness of the left block. CM: orientation of line CQ before shear. \(\angle \hbox {QCM}=\theta \). In \(\Delta \hbox {ABC}\), \(\hbox {AC} = \hbox {T}\cdot \hbox {tan} \uptheta \), \(\hbox {Area}~\Delta \hbox {ABC} = 0.5\), and \(\hbox {AB}\cdot \hbox {AC} = 0.5\cdot \hbox {T}^{2}\cdot \hbox {tan}\uptheta \). Volume of the space \(\hbox {V} = 0.5\cdot \hbox {L}\cdot \hbox {T}^{2}\cdot \hbox {tan}\uptheta \). By symmetry, the same volume is opened up at the top pat as well.

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In case of book-shelf sliding, fault blocks rotate and slip by simple shear along pre-existing planes (figure 1), and are more common in deltaic shelf deposits (Mandl 1984, 1987, 2000), and also at times found in rift zones (Green et al. 2013). Such a deformation mechanism can work also on a microscopic scale amongst mineral grains (figure 2.23 in Mukherjee 2015). Tectonic loading is cited as one of the main factors for mega-scale bookshelf faulting (Narteau 2002). This work does not discuss ductile bookshelf sliding as recently discussed by Zuza and Yin (2013). Crustal blocks undergo rigid-body rotation so that the interfaces between individual blocks/books act as normal faults that rotate antithetically sheared with respect to the synthetic rotation (or primary simple shear) of the complete set of blocks/books. Deformations with these kinematics are of great importance in petroleum geosciences since the wide gaps that can develop along inter-block fault planes can transport hydrocarbons. Sediments deposited on the irregular top of bookshelf can form compaction synclines in which hydrocarbons can be stored preferentially in the hinge zone (Tin 1997). The bookshelf gliding of crustal blocks is important to seismicity studies (e.g., Wetzel et al. 1993), and has been elaborated using 2D Mohr diagrams and the Cosserat theory of elasticity (de Figueiredo et al. 2004). Also, Zuza and Yin (2016) deduced the velocity field along/across the displaced and rotated blocks in book-shelf faulting. Bookshelf gliding of mineral grains, often of micas and feldspars, are usually found in mylonites under an optical microscope (Meschede et al. 1997). The basal level of bookshelf normal faults can merge into a detachment (Peacock 1997). This article deduces MOI for bookshelf displaced and rotated blocks with geologically plausible density distributions.

Previous models of finding MOI involved constant density assumption for the faulted blocks. As stated in this work, in reality, density can vary either linearly or exponentially. Having considered that, the present work is an improved model of MOI for bookshelf glided blocks. Further, this work links the overall density of the rock with those of the pore fluid and the solid matrix, density gradients along different directions and the rock porosity. Finally, such an expression of density is linked with the MOI. With this model, therefore, one can test how MOI changes by changing its fundamental controlling factors. Testing the MOI in this way was not available in previous existing models.

2 Background and derivations

2.1 Case I

Density usually increases linearly vertically downward, especially for most oceanic crust (Reid 1987), sediments in basins (Motavalli-Anbaran et al. 2013), or even for the entire lithosphere (e.g., Xu et al. 2016) including the deep crust (Zhang and Chen 1992). However, the reverse can also happen (Ebbing et al. 2007 and its review) with depth, especially if evaporitic rocks are involved (Romer and Neugebauer 1991). An average density gradient can be \(0.32 \hbox { Mg m}^{-3} \hbox { km}^{-1}\) (Carlson and Herrick 1990), or \(13\pm 2 \hbox { kg m}^{-3} \hbox { km}^{-1}\) (Tenzer et al. 2012). Metamorphism can induce gradual density gradients in any orientation in some cases (Zhou 2009). Sedimentary facies variation during marine to non-marine transitions would alter density progressively. Densities varying laterally in rocks can occur over distances of at least \(\sim \! 20 \hbox { km}\) (Wu and Mereu 1990).

Referring to figure 2, say at point O(0, 0, 0) of a rectangular parallelepiped with dimensions \(\hbox {x}_{1}\), \(\hbox {y}_{1}\) and \(\hbox {z}_{1}\) along the axes X, Y and Z, the density is \(\rho _{0}\), and the linear density gradient in three perpendicular directions are \(k_{i}\) (\(i =x,\, y,\, z)\). Therefore, density variations along X-, Y- and Z-axes are:

$$\begin{aligned} \rho \left( {x,0,0} \right)= & {} \rho _0 +k_x x, \end{aligned}$$
(1)
$$\begin{aligned} \rho \left( {0,y_0 ,0} \right)= & {} \rho _0 +k_y y, \end{aligned}$$
(2)
$$\begin{aligned} \rho \left( {0,0,z} \right)= & {} \rho _0 +k_z z. \end{aligned}$$
(3)

Therefore, for any coordinate (x, y, z), the density would be given by

$$\begin{aligned} \rho \left( {x,y,z} \right) =\rho _0 +k_x x+k_y y+k_z z. \end{aligned}$$
(4)
Fig. 2
figure 2

Co-ordinate system for the crustal block.

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The lateral variation of density (along X- and Y-directions) can be caused by excess pore fluid pressure in sediments (Buryakovsky et al. 1995). Note for \({y} = {z} = 0\), \({x} = {y} = 0\) and \({z} = {x} = 0\), equation (4) satisfies equations (1), (2) and (3), respectively. Secondly, obviously \({k}_{{x}}= {k}_{{y}}= {k}_{{z} }= 0\) would indicate a homogeneous block with constant density ‘\(\rho _{0}\)’. The MOI about the Y-axis is given by, as per Das and Mukherjee (2012)

$$\begin{aligned} I_y = \mathop {\int }\nolimits _{0}^{x_1} \mathop {\int }\nolimits _{0}^{y_{1}} \mathop {\int }\nolimits _{0}^{z_1} \rho \left( {x,y,z} \right) \left( {z^{2}+x^{2}} \right) dx \,dy\,dz.\nonumber \\ \end{aligned}$$
(5)

Note that the volume of the block is

$$\begin{aligned} V=x_1 y_1 z_1. \end{aligned}$$
(6)

The effective density of this block is

$$\begin{aligned} \rho _{e}= & {} \{\rho _m -\left( {\rho _m -\rho _f } \right) \varPhi _0 \}+0.5\,\varSigma k_q q_{1} \nonumber \\&\left( {q=x,y,z} \right) \left( {\hbox {Mukherjee}~2017} \right) . \end{aligned}$$
(7)

Putting \(\rho (x, y, z)\) of equation (4) and ‘V’ of equation (6) into equation (5), integrating, and putting \(\rho _{e}\) of equation (10) in place of \(\rho _{0}\), and simplifying

$$\begin{aligned} I_y= & {} 0.33*V\cdot \rho _e \left( {z_1 ^{2}+x_1 ^{2}} \right) \nonumber \\&+\,k_x \cdot x_1 ^{2}\left( {0.11*z_1 ^{2}+0.25 *x_1 } \right) \nonumber \\&+\,0.17 *k_y \cdot y_1 \left( {x_1 ^{2}+z_1 ^{2}} \right) \nonumber \\&+\,0.5*k_z z_1 \left( {0.5*z_1 ^{2}+0.33*x_1 ^{2}} \right) . \end{aligned}$$
(8)

2.2 Case II

2.2.1 Exponential variation of porosity with depth

An exponential depth-density relation can exist in sediments (Goteti et al. 2012), especially for argillaceous sediments presumably compacted to shallow depths (Rieke and Chilingarian 1974). These workers referred to the following relationship amongst bulk wet density of sediments (\(\rho _{bw}\)), matrix density (\(\rho _{m}\)), fluid density (\(\rho _{f}\)) and porosity

(9)

On the other hand, Athy (1930) presented the following relation amongst surface porosity , porosity at depth z and compaction constant \(\lambda =b^{-1}\)

(10)

Combining equations (9 and 10),

(11)

Considering linear density variations along two horizontal perpendicular directions Y and Z as per equation (2) of Case I and equation (11), we get the following, in place of equation (4)

(12)

Using equations (5 and 12)

(13)

Recall, as per equation (6), here \(V=x_{1}y_{1}z_{1}\)

For \({k}_{{x}}= {k}_{{y}}= 0\), the expression simplifies to

(14)

For the case of depth independent density, i.e.

$$\begin{aligned} {b} = 0 \end{aligned}$$
(15)

or

(16)

One needs first to rewrite equation (4) as:

(17)

and use that in equation (5) for integration.

2.2.2 Linear variation of porosity with depth

A deviation from a linear relation between density and depth is noted in overpressure zones (figure 2.7 of Telford et al. 1990). Unlike equation (10) in Case II, porosity can also decrease linearly with depth (Lerche and O’Brien 1987)

$$\begin{aligned} \emptyset _z =\emptyset _0 -cz \end{aligned}$$
(18)

‘c’ is a constant. Therefore, equation (11) alters to

$$\begin{aligned} \rho _{bw} \left( {0,0,z} \right)&=\rho _{bwz}\nonumber \\&=\rho _m -\left( {\rho _m -\rho _f } \right) (\emptyset _0 -cz). \end{aligned}$$
(19)

This linear relation between \(\rho _{bw}\) and z can also be expressed as equation (3) of Case I with \({k}_{{z} }= (\rho _{{m}}-\rho _{{f}}\)).

Note: (i) for constant magnitudes of \(\rho _{m}\), \(\rho _{f}\), \(\varPhi _{0}\), \(x_{1}\), \(z_{1}\), \(k_{i}\, (i = x, y, z\)) in Case I and \(k_{i}\,(i = x,y)\) in Case II, \({I}_{{y}}\) is not proportional to \({y}_{1}\). This is because \(I_{y}\) is either in the form of \(I_{y}= Ay_{1}^{2}+By_{1}\) or \(I_{y}= Ay_{1}^{2}+By_{1} +C\). (ii) For the Case I, \(k_{i}= 0\) would indicate a homogeneous block with spatially \(\hbox {constant density} = \{\rho _{{m}}- (\rho _{{m}} -\rho _{{f}}) \varPhi _{0}\}\). In that case

$$\begin{aligned} I_y= & {} 0.33*{V}\{\rho _m -\left( {\rho _m -\rho _f } \right) \varPhi _0 \}\left( {z_1 ^{2}+x_1 ^{2}} \right) \nonumber \\= & {} 0.33*{M}\left( {z_1 ^{2}+x_1 ^{2}} \right) \end{aligned}$$
(20)

where M is the mass of the block.

This matches with standard derivations available in statics texts for rectangular parallelepipeds. However, note that where we choose the Y-axis, whether inside, outside or at some other margin of the block, obviously modifies the expression for the MOI (derivations 11.2 in Spiegel and Liu 1999). Equation (20) for homogeneous block shows (recalling \(V = x_{1}y_{1}z_{1}\)), unlike the non-homogeneous case, \({I}_{{y}}\propto {y}_{1}\). This shows obviously that the presently considered MOI for the density-distributed blocks differ significantly from that of the homogeneous block case. Another point, if the density of the rock varies temporally, such as due to increased tectonic loading at its top, the present work would require a refinement in terms of time-dependent density.

2.2.3 Product of inertia

One can further deduce the product of inertia (POI) in the context of the tectonics and structural geology considered in this study. The POI has also been studied in the context of landscape pattern and geomorphological features (Zhang et al. 2006). In this case, the POI with respect to X- and Y-axes

$$\begin{aligned} I_{XY }= \mathop {\int }\nolimits _{0}^{x_{1}} \mathop {\int }\nolimits _{0}^{y_{1}} \rho (x, y, z) xy \,dx \,dy. \end{aligned}$$
(21)

For Case I, substituting \(\rho (x,y,z)\) from equation (4), and after performing definite integral

$$\begin{aligned} I_{XY }= & {} 0.5*x_{1}^{2}y_{1}^{2 }{[0.5(\rho }_{0}+k _{z})\nonumber \\&+\,0.33(k_{x}x_{1}+ k_{y}y_{1})]. \end{aligned}$$
(22)

Similarly,

$$\begin{aligned} I_{YZ }= & {} 0.5*y_{1}^{2}z_{1}^{2 }{ [0.5(\rho }_{0}+k_{x})\nonumber \\&+\,0.33(k_{x}y_{1}+ k_{y}z_{1})] \end{aligned}$$
(23)

and

$$\begin{aligned} I_{ZX }= & {} 0.5*x_{1}^{2}z_{1}^{2 }{ [0.5(\rho }_{0}+k_{y})\nonumber \\&+\,0.33(k_{z}y_{1}+ k_{x}z_{1})]. \end{aligned}$$
(24)

We firstly note that the POI increases non-linearly with increases in both the dimension of the parallelepiped (i.e., \(x_{1}\), \(y_{1}\) and \(z_{1})\) and its density gradient (\({k}_{{i}}\)). Secondly, \(I_{XY}\), \(I_{YZ}\), \(I_{ZX }\ne 0\). This means that neither of XY, YZ and XZ are the planes of symmetry, which as per figure 2 is obvious and matches our intuition. In other words, X-, Y- and Z-directions are not the principal axes at the corner point O. Thirdly, if we consider the X, Y and the Z axes passing through the point (\(0.5{x}_{1}\), \(0.5{y}_{1}\), \(0.5{z}_{1})\) and that this point is taken as the origin (0, 0, 0), the POI becomes

$$\begin{aligned} I_{XY }= & {} \mathop {\int }\nolimits _{0.5x_{1}}^{0.5x_{1 }} \mathop {\int }\nolimits _{-0.5y_{1}}^{0.5y_{1}} \rho (x,y,z) xy \,dx \,dy \nonumber \\= & {} I_{YZ}= I_{ZX} = 0. \end{aligned}$$
(25)

This is for both the Cases I and II of \(\rho (x,y,z)\) as per equations (4) and (12), respectively. This is as expected (Gere and Goodno 2009) since in this case the X-, Y- and Z-axes are the symmetry axes passing through the geometric centroid of the parallelepiped.

The following note can be made regarding equation (15) and later onwards in the main text that involves the following definite integral

$$\begin{aligned}&\mathop {\int }\nolimits _{0}^{z_{1}}(z^{2}e^{-bz})dz = b ^{-1}[2*b^{-2}\nonumber \\&\quad - e^{-bz}{1}(z_{1}^{2}+2*z_{1}b^{-1}+2*b^{-2}). \end{aligned}$$
(26)

One needs to consider the case of \({b} = 0\) before the definite integral operation, such as

$$\begin{aligned} \mathop {\int }\nolimits _{0}^{z_{1}} {z}^{2} \,dz = 0.33*z_{1}^{3}. \end{aligned}$$
(27)

Instead, if \(b = 0\) is put on the right hand side of equation (26), the result is invalid since ‘b’ occurs also in the denominator.

3 Discussions

This article uses a few simple cases. In reality, both \(\rho _{m}\) and \(\rho _{f}\) can be depth (pressure and temperature) dependent (Djomani et al. 2001), while \(\rho _{f}\) can increase with depth (Patwardhan 2012). Even though the rock/sedimentary body remains ‘the same’ during bookshelf gliding, a change in ‘\(\rho _{f}^{{\prime }}\) that could be due to circulation of fluid along the fault plane would change the magnitude of MOI at any point inside the block. More refined deductions of MOI can be attempted by optimists if required for tectonic modeling. In any case, we would require (at least empirical) equations (e.g., equations 1, 2 and  3 of Case I and equation 10 of Case II) of density variation within a rock/sediment column to find its MOI. Therefore, the present approach may not work as it is for book-shelf faulted blocks in metamorphic rocks, which are likely to have either locally non-specific (Reynolds 2011) or unknown density distributions (Gorbatsevich et al. 2017). Also note, since rotation rates that can only be constrained on faults \(>\!\!1 \hbox { km}\) long in tectonic/geological cases are very slow, e.g., \(3^{\circ } \hbox { Ma}^{-1}\) (Price and Scott 1994), \(1\!-\!2^{\circ } \hbox { Ma}^{-1}\) (Kreemer et al. 2009) or \(0.25 \,\upmu \hbox {rad yr}^{-1}\) (Sigmundsson 2006) with rotations of \(\sim \, 22^{\circ }\) (Tapponier et al. 1990) would take place over a long geological time with a total significant temporal change in \(\rho _{f}\) alone in the rotating block can alter the MOI.

Bookshelf gliding (figure 1) and rotational faulting are the two well known cases of deformation involving rigid body rotation in structural geology and tectonics. Of these two types of faults, the former affects rectangular parallelepiped crustal blocks en mass, so that analyzing their MOI for geologically realistic density distribution within blocks on scales of kms becomes relatively easy; it is much more difficult to constrain the slip rates of small scale bookcase faults. For faulted blocks with irregular geometric shapes, such as for rotationally faulted blocks (Mukherjee and Khonsari 2017), the analysis would become difficult. MOI of irregular objects can be deduced either experimentally (Koyama et al. 2010) or using computer models such as AMINERTIA (Internet reference).

4 Conclusions

The moment of inertia is deduced for book-shelf faulting of crustal blocks having geologically realistic density distributions. The deductions involve porosity, densities of rock matrix and that of the pore fluid. This work does not describe the evolution/genesis of such faulting, but estimates MOI when such a deformation takes place. Depending on their occurrence, such crustal blocks are potentially important in our understanding of ocean floor kinematics (Sigmundsson 2006), and in petroleum geosciences, their kinematic analyses become important, especially in analyzing basins formed above book-shelf faulted blocks (Veeken 2007; such as in the Afar region: Tapponnier et al. 1990). Considering the effect of porosity in the expression of MOI for bookshelf faulted blocks is important since such arrays are common in sedimentary environments, deltaic deposits in particular (Mandl 2000) where the density of each block is likely to be depth-controlled (along the Z-direction as per the present work). The MOI for the density-distributed blocks differ much from the case of the homogeneous block case, for example only for the latter case, the MOI about the Y-axis is proportional to the dimension of the parallelepiped along the Y-direction. The presented model is capable of handling MOI and POI for time-dependent density of the parallelepiped as well. The POI depends non-linearly on the length, width and height of the parallelepiped and its density gradient.