Consider a sharp crack of length R propagating with constant velocity v in a uniform elastic material in 2D plane strain conditions, whereby the crack tips are located at ( v t − R , 0 ) and ( v t , 0 ) . In terms of coordinates ( ξ , y ) = ( x − v t , y ) in which the crack tips are fixed at ( − R , 0 ) and ( 0 , 0 ) , the elastic displacement u ( x , y , t ) can be expressed in terms of scalar and vector potentials ϕ ( ξ , y ) and ψ → = ( 0 , 0 , ψ ( ξ , y ) ) as:

where:

for constants α P = 1 − v 2 / v P 2 and α S = 1 − v 2 / v S 2 specified by v and the P and SV wave velocities v P and v S [1]. In turn, the potentials ϕ ( ξ , y ) and ψ ( ξ , y ) can be expressed in terms of analytic functions F ( ζ ) and G ( ζ ) as:

for ζ P = ξ + i α P y and ζ S = ξ + i α S y , whereby elastic displacement field and stress field components are given by the equations:

for Lame parameters λ and μ .

Now assume the crack is Mode I, in which a uniform vertical traction σ p acts on the crack faces while the shear traction applied to the crack faces is 0 , as illustrated in Figure 1-left. In this case, the − R < ξ < 0 crack face boundary conditions are:

Figure 1. Stationary Yoffe Mode 1 crack (left) and Rice Mode II crack (right).

and the potentials ϕ ( ξ , y ) and ψ ( ξ , y ) have the symmetries ϕ ( ξ , − y ) = ϕ ( ξ , y ) and ψ ( ξ , − y ) = − ψ ( ξ , y ) imply u x ( x , − y , t ) = u x ( x , y , t ) and u y ( x , − y , t ) = − u y ( x , y , t ) , in keeping with x-axis reflection symmetry of a Mode I crack displacement solution. Furthermore, the Cauchy-Riemann equations imply F ( ζ ¯ ) = F ( ζ ) ¯ and G ( ζ ¯ ) = G ( ζ ) ¯ .

The symmetries of F and G , together with Equations (10) and (12), imply the function:

is continuous across the crack line between ζ = − R and ζ = 0 , and is therefore entire. Given that the elastic velocity and stress fields are zero at points remote from the crack, Lioiuville’s theorem on bounded entire functions implies:

Equations (9) and (14) together with condition (11) imply:

where D = 4 α P α S − ( 1 + α S 2 ) 2 , and discontinuity of I m   F ″ ( ζ ) across the crack line occurs as necessary to account for discontinuity in elastic displacement. Assuming ζ ( ζ + R ) is defined as a single valued function on the complex plane with a branch cut between ζ = − R and ζ = 0 along y = 0 , the function:

is continuous across the crack line, and is therefore an entire function that approaches 0 as | ζ | → ∞ , implying:

and:

Equations (17) and (18) imply the normal stress distribution σ y y ( ξ , 0 ) on the crack line ahead of the advancing crack tip is:

Integration of Equations (17) and (18) gives equations:

and:

in which constants of integration have been selected so elastic displacements are 0 at remote points. Assuming ζ ( ζ + R ) is defined as a single valued function on the complex plane with a branch cut between ζ = − R and ζ = 0 along y = 0 , Equations (20) and (21) together with Equation (7) imply the crack opening displacement is:

Now assume the crack is Mode II, in that a uniform shear traction σ s is applied to the crack faces while the vertical traction applied to the crack faces is 0, as illustrated in Figure 1-right [2]. In this case the − R < ξ < 0 crack face boundary conditions are:

and the potentials ϕ ( ξ , y ) and ψ ( ξ , y ) have the symmetries ϕ ( ξ , − y ) = − ϕ ( ξ , y ) and ψ ( ξ , − y ) = ψ ( ξ , y ) so that:

and:

as necessary to satisfy the Cauchy-Riemann equations for analytic functions F ( ζ ) and G ( ζ ) .

Equations (27) and (28) imply the function:

is continuous across the crack line, and therefore entire. Furthermore, vanishing of stress and velocity field displacements as | ζ | → ∞ imply E ( ζ ) is bounded at infinity, and thus identically 0 by Liouville’s theorem. Consequently:

which together with boundary condition (23) implies that along the crack line:

where D = 4 α P α S − ( 1 + α S 2 ) 2 , and discontinuity of R e   F ″ ( ζ ) across the crack line occurs as necessary to account for discontinuity in elastic displacement. Assuming ζ ( ζ + R ) is defined as a single valued function on the complex plane with a branch cut between ζ = − R and ζ = 0 along y = 0 , the function:

is continuous across the crack line, and is therefore an entire function that approaches 0 as | ζ | → ∞ , implying:

and:

Integration of Equations (33) and (34) gives equations:

and:

in which constants of integration have been selected so elastic displacements are 0 at remote points. Assuming ζ ( ζ + R ) is defined as a single valued function on the complex plane with a branch cut between ζ = − R and ζ = 0 along y = 0 , Equations (35) and (36) together with Equation (6) imply the crack opening displacement is:

For the purpose of calculating multi-crack 2D elastostatic displacements, cracks will be identified with branch cuts of the complex plane specifying sheets of Riemann surfaces. To explain this statement in detail, consider the equation:

relating complex variables Y and ζ with 2 g + 2 different complex values b i . For each pair of values ( b 2 i − 1 , b 2 i ) the expression:

can be a singled valued analytic function on the complex X -plane with a branch cut joining the points b 2 i − 1 and b 2 i , and this function is unique up to choice of sign. Iterating selection of a branch for square root for 1 ≤ i ≤ g + 1 , it the product:

can be defined as a single valued analytic function on the complex X -plane, up to choice of sign, away from a finite set of branch cuts joining branch points b 2 i − 1 and b 2 i in pairs. For this reason, the projection:

from points satisfying Equation (38) to the complex X -plane is a double cover of the complex plane away from the branch cuts.

The projective completion of non-compact hyperelliptic curve (38) is the compact hyperelliptic curve:

in ℂ P 2 . From this point of view, the projection:

from points on curve (42) to the Riemann sphere ℂ P 1 is a double cover with branch points at X / Z = b i . When | X / Z | ≫ max | b i | , Equation (42) implies:

so the point ( X : Z ) with Z ≠ 0 is covered by the two points ( X : ± X g + 1 / Z g : Z ) . Therefore, for g = 0 the point at infinity ( 1 : 0 ) is double covered by ( 1 : 1 : 0 ) and ( 1 : − 1 : 0 ) while for g ≥ 1 the point at infinity ( 1 : 0 ) is double covered by ( 0 : 1 : 0 ) . With resolution of the g ≥ 1 singularity at ( X : Y : Z ) = ( 0 , 1 , 0 ) , projection (43) can be drawn as a projection from a non-singular genus g Rieman surface to the Riemann sphere as shown in Figure 2. In this image, the inverse image of each branch cut is shown as a closed loop on the surface.

For a superposition of vertical/shear traction acting on the crack faces, the Mode I and Mode II single crack elastostatic displacement potentials ϕ and ψ independently solve the 2D Laplace equation:

away from the crack lines in the ( ξ , y P ) = ( ξ , α P y ) and ( ξ , y S ) = ( ξ , α S y ) planes. By Gauss’ and Stokes’ theorems, the line integrals:

Figure 2. Double cover of Riemann sphere ( ℂ + ∞ ) by genus 1 and genus 0 hyperelliptic curves with 4 and 2 branch points.

(47)

and:

(48)

each obtain a constant value for any closed contours C P and C S encircling the crack once counterclockwise in the ζ P and ζ S planes. Moreover, these constant values can be non-zero because the analytic functions F ( ζ ) and G ( ζ ) are multi-valued, as may be anticipated based on 1 / | ζ | dependence of elastostatic displacement at points remote from the cracks. This dependence can be extracted from singular behavior of differential forms F ′ ( ζ ) d ζ and G ′ ( ζ ) d ζ at infinity evaluated using the substitution ζ ¯ = 1 / ζ . Noting that:

for | ζ | ≫ R , and d ζ = − d ζ ¯ / ζ ¯ 2 , it is evident that both F ′ ( ζ ) d ζ and G ′ ( ζ ) d ζ have a simple pole at infinity. For this reason, the integral of the pullback of either form to a Riemann sphere around the closed contour shown in Figure 2 is non-zero, because the form has a simple pole on both the inside and outside of the contour.

For g ≥ 1 , g + 1 cracks in the ζ P or ζ S plane can be viewed as branch cuts for a double cover of the plane by a hyperelliptic curve of genus g , and multi-crack elastostatic displacement solutions for constant vertical or shear surface tractions are associated with meromorphic 1-forms on the curve. The Riemann-Roch theorem applies to calculate the dimension of the complex vector space V ω of meromorphic 1-forms ω on a non-singular Riemann surface of genus g with simple poles at 2 prescribed points as being g + 1 . More specifically, the theorem for curves states that if K is a canonical divisor on a non-singular Riemann surface of genus g , then:

where l ( D ) is the complex dimension of the vector space of meromorphic functions on the Riemann surface bounded by divisor D [3]. Consequently, if an elastostatic displacement vector field u in the multi-cracked ( x − v t , y ) plane is associated with meromorphic 1-forms F ′ ( ζ P ) d ζ P and G ′ ( ζ S ) d ζ S in the ζ P and ζ S planes via Helmholtz decomposition (1) and Equations (4) and (5), the pullback of either 1-form to a non-singular genus g Riemann surface can each be expressed as a linear combination of a basis ω 1 , ω 2 , ⋯ , ω g , ω ∞ for a vector space of mermomorphic 1-forms V ω on the respective curve, where each ω i is holomorphic and ω ∞ has simple poles at the points ∞ ± 1 . When v = 0 , the ζ P and ζ S planes coincide and both meromorphic 1-forms F ′ ( ζ P ) d ζ P and G ′ ( ζ S ) d ζ S pullback to 1-forms on the same Rieman surface. In this case, the complex dimension g + 1 is in agreement with the fact that there are 2 g + 2 real parameters σ P , i and σ S , i available to specify constant vertical and shear surface tractions on crack faces of the multi-crack system, since the elastostatic displacement fields u associated with sets of constant surface tractions form a real vector space.

Consider a triaxial compression test on a rock sample for which axial compressive strain is applied along principal axis 1 while compressive stresses in the direction of principal axes 2 and 3 are held constant. Assume the initial compressive stresses satisfy σ 1 > σ 2 > σ 3 , and as σ 1 is increased towards a maximum value σ 1 , max , plastic strain hardening of the rock sample occurs. Also assume that when ϵ 1 is increased beyond the strain ϵ 1 , max at which σ 1 = σ 1 , max , strain softening of the rock occurs with formation of a shear band along a plane where the rock ultimately undergoes shear failure. Technically, the constitutive model of the rock may be defined using a Mohr-Coulomb yield criterion with mobilized friction angle/cohesion and a non-associated plastic flow rule [5].

Physically accurate and well defined mathematical description of the rock strain softening regime requires addition of at least two features to the strain hardening non-associated plasticity model:

Strain rate softening of rock shear strength [6].Regularization of rock plastic flow introducing a non-zero shear band width [7].

If it is assumed that with consideration of these technical details, and definition of a finite element mesh, a global elasto-plastic tangent stiffness operator K T e p ( τ ) can be defined for the rock sample throughout triaxial compression at a constant strain rate at each time τ , Newton’s equation of motion dictates that the equation:

describes time evolution of a vector perturbation U ( t ) to the finite element nodal elasto-plastic displacements at time τ , where M is a lumped mass matrix, C ( τ ) is a damping matrix at time τ , and F ( t ) is a perturbation of the externally applied nodal force vector [8].

Typically, plastic deformation of rock material occurs with cracking (i.e. damage). In seismology, earthquake faults are observed to be surrounded by damage zones, and detecting fault damage zone formation by monitoring real time variation of local earthquake coda wave signals on seismograms has been identified as a possible method of predicting earthquakes [9]. According to this method, in a frequency band centered at angular frequency ω , time variation of the local earthquake coda Q ( ω ) value at a particular seismic station can be attributed to time variation of the anelastic and/or scattering Q values Q i ( ω ) and Q s ( ω ) according to the equation:

and decrease of Q s ( ω ) in time may reflect earthquake precursor dynamics in the lithosphere. More specifically, based on a single backscattering model of coda wave signals, Q s scales with signal frequency according to the power law:

and may therefore contribute to real time variation of Q ( ω ) if the Hurst exponent H changes in time [10]. Fundamentally, the Hurst exponent H is related to the power spectrum of random S-wave velocity model inhomogeneities in the lithosphere W 3 D ( k ) , where k is wave number, via the relation [11]:

These random inhomogeneities can be attributed to a fracture network superimposed on a fixed seismic velocity model background, such as a 1D layered velocity model, with fracture size statistics characterized by Equation (54).

Suppose a scalar crack phase field influences Lame elastic parameters λ ( x , y , z , τ ) and μ ( x , y , z , τ ) of a 3D laboratory rock sample or lithospheric region via the equations:

where λ 0 ( x , y , z ) and μ 0 ( x , y , z ) are the Lame parameters of uncracked rock at location ( x , y , z ) , ϕ ( x , y , z , τ ) is the crack phase field with value 0 for uncracked material and 1 for maximally damaged material, and λ r and μ r are constants [12]. If the crack phase field can be directly measured at a particular 3D location P at regular time intervals, the sequence of phase field values ϕ [ i ] = ϕ ( P , τ 0 + ( i − 1 ) Δ τ ) can be used to write a trajectory matrix X ( ϕ [ i ] ) :

whose singular value decomposition can be used to identify dominant signal components in the discrete time series ϕ [ i ] [13]. More generally, the scalar time series ϕ [ i ] can be replaced with a vector time series of phase field values at different 3D locations, and multichannel singular spectrum analysis can be applied to identify dominant vector signal components in the time series.

Realistically, crack phase field values are not measured at individual points in a laboratory sample or the lithosphere, but may be inferred from changes in local earthquake coda wave signals. Therefore, it is noted that single/multi-channel singular spectrum analysis allows for replacement of the scalar/vector time series of crack phase field measurements ϕ [ i ] with a scalar/vector time series Φ [ i ] of experimentally observed data, such as relative changes in average S-wave velocity between seismic station pairs measured by monitoring coda wave signals [14]. It is conjectured that each Φ [ i ] is the measurement of an observable on a nonlinear dynamical system with a finite dimensional phase space ℳ , whereby the rows of the trajectory matrix X ( Φ [ i ] ) provide an embedding of the phase space trajectory in ℳ into ℝ n . Based on this conjecture, an estimate of the dimension of the phase space trajectory in ℳ is provided by computing the correlation dimension of the phase space trajectory embedded in ℝ n .

The conjectured existence of a finite dimensional nonlinear dynamical system describing time evolution of the fault damage zone is a generalization of a previous claim by seismologists that the average damage along an earthquake fault preceding its rupture evolves according to a 1 dimensional nonlinear dynamical system [15]. From this point of view, if the damage field can be decomposed into a linear superposition of finitely many time independent damage modes ϕ i ( x , y , z ) with time dependent coefficients a i ( τ ) :

a nonlinear dynamical system:

specifies time evolution of the phase field and ℳ is contained in ℝ m . So that each damage mode is associated with localization of strain in the lithosphere, it is further conjectured that each damage modes corresponds to a nodal displacement eigenvector of an elasto-plastic stiffness matrix K T e p ( τ ) with complex eigenvalue β i ( τ ) satisfying | β i ( τ ) | ≪ 1 . Whether or not there is good reason to identify the eigenvalues β 1 ( τ ) , β 2 ( τ ) , ⋯ , β m ( τ ) as branch points of a cover of the complex plane by a τ -dependent Riemannn surface remains unclear.