3-1 Source Mechanisms – Theory of AE 3-2 Moment Tensor – Basics 3-3 Moment Tensor – SiGMA Analysis

3-1 Source Mechanisms -Theory of AE October 18, 2007 (1/3)  Generalized Theory  Moment Tensor  Spatial Derivatives of Green’s Functions  Dislocation Model  Radiation Pattern

Generalized theory of AE  u k (x,t)= ∫ F [G ki (x,y,t)*t i (y,t) - T ki (x,y,t)*u i (y,t)]dS Separation  AE wave due to force: u k (x,t)= ∫ F G ki (x,y,t)*t i (y,t)dS  AE wave due to crack: u k (x,t)= ∫ F T ki (x,y,t)*u i (y,t)dS

 u k (x,t)= ∫ F T ki (x,y,t)*u i (y,t)dS  u k ( x,t)= G kp,q ( x, y,t)*S(t) C pqij n j l i  V In micromechanics [Mura 1982], tensor n j l i  V is defined as the eigenstrain, which is explicitly equivalent to the damage tensor in damage mechanics [Kachanov 1980].

 C pqkl l k n l  V= M pq  the moment tensor, M pq, is defined by the product of the elastic constants C pqkl [N/m 2 ] and the crack volume  V[m 3 ], which leads to the moment of physical unit [Nm].  This is a reason why the tesnor M pq is named the moment tenso r

 C pqkl =  pq  kl +     +  pl  qk

 Mathematically, the moment tensor is comparable to a stress as a symmetric second-order tensor, because components are readily obtained from the product of the eigenstrain and elastic constants.

 The case l = (1, 0, 0) n = (0, 1, 0) and  V = b  F.  b  F is defined as the seismic moment  Off-diagonal components of the moment tensor

 u k (x,t)= G kp,q (x,y,t)*S(t) C pqij n j l i  V  u k (x,t) = G kp,q (x,y,t) M pq *S(t)  AE source is represented by the moment tensor M pq and the source-time function S(t).  Crack kinetics is represented by S(t), which is solved by the deconvolution analysis.  Crack kinematics are represented by the moment tensor. AE waves due to crack nucleation

 u k (x,t) = G kp,q (x,y,t) M pq *S(t)  Spatial derivatives, G kp,q (x,y,t), is necessary, not Green’s function G kp (x,y,t).  Consequently, empirical Green’s functions (EGF) by a pencil-lead break are no use for AE waves due to crack nucleation, unless the spatial derivatives are obtained experimentally.  So, application of EGF is definitely non-sense.

 G i1,1 (x, y 1, t) = lim [G i1 (x, y 1 +  y 1,t) – G i1 (x, y 1, t)]/  y 1 u i = G i1,1 m 11 + …….

 G i1,2 (x, y 2, t) = lim[G i1 (x, y 2 +  y 2, t) – G i1 (x, y 2, t)]/  y 2.  u i =G i1,2 m 12 +…….

 Tensile motion:  normal vector n = (0, 0, 1)  motion vector l = (0, 0, 1).  The moment tensor becomes

 Shear crack  crack normal n = (0, 0, 1)  motion vector l = (1, 0, 0).

 An application of the moment tensor analysis to AE waves was earlier reported on cracking mechanisms of glass due to indentation [Kim & Sachse 1984], where only diagonal components of the tensor were assumed.  It is realized that the presence of tensor components is not actually associated with the type of the crack, but closely related with the coordinate system.

 Although the crack orientations are often assumed as parallel to the coordinate system [Saito, Takemoto et al. 1998, Takemoto 2000], they are generally inclined to the coordinate system because cracks are nucleated in random orientations.  As a result, the presence of all the components is to be reasonably assumed even though the type of the crack is of either the tensile or of the shear.

 By taking into account only P wave motion of the far field (1/R term) of Green’s function in an infinite space of an isotropic-elastic body,

Tensile Crack Shear Crack

 All positive amplirudes  4 quadrant model in earthquakes

 The radiation pattern and the equivalent force models of the dipole and the double-couple are not essential to study the source characterization of AE waves.  The important result is the fact that kinetics of AE source is recovered by the deconvolution analysis, and kinematics can be represented by the moment tensor.  The force models and the radiation patterns just show us the principal components of the moment tensor.

 Moment Tensor Inversion  Theoretical Backgrounds  Far-Field Approximation  Sensor Calibration  SiGMA Procedure

 In seismology, linear inversion techniques were proposed to determine the moment tensor component in both time and frequency domains [Stump & Johnson 1977] and [Kanamori & Given 1981].  Although all the components of the moment tensor must be determined, the moment tensor inversion with constraints has been normally applied in seismology [Dziewonski & Woodhouse 1981].  This is partly because a fault motion of an earthquake is primarily associated with shear motion, corresponding to off-diagonal components in the moment tensor.

 Both tensile motion of diagonal components and shear motion of off-diagonal are definitely present in crack motions as an AE source.  Elsewhere, another procedure named the relative moment tensor inversion is proposed [Dahm 1996]. They have named the relative moment tensor analysis.

 Basic equation:  To inversely solve the above equation and to determine all components of the moment tensor, the spatial derivatives of Green's functions are inevitably required.  Accordingly, numerical solutions are obtained by the Finite Difference Method (FDM) [Enoki, Kishi et al. 1986] and by the Finite Element Method (FEM) [Hamstad, O’Gallagher et al. 1999].

 These solutions, however, need a vector processor for computation and are not readily applicable to processing a large amount of AE waves.  Consequently, based on the far-filed term of P wave, a simplified procedure was developed [Ohtsu, Okamoto et al. 1998], which is suitable for a PC- based processor and robust in computation.  The procedure is now implemented as a SiGMA (Simplified Green's functions for Moment tensor Analysis) code.

 Taking into account only P wave motion of the far field (1/R term) of Green’s function in an infinite space, the displacement U i (x,t) of P wave motion is obtained,  In the case that we are interested in motions of AE waves at the observation point, the first approximation could be an elastic wave in a half space.

Displacement motions detected at location A in a half space (solid curves in a) and b)), compared with solutions in an infinite space (broken curve in a)) and of the far-filed (broken curve in b)).

 In all the cases, it is observed that the amplitude of the first motion (P wave) in a half space is almost as twice as the amplitudes both of the infinite-space solution and the far-filed solution. The ratio of the amplitude in a half space to that of the infinite space is equivalent to the reflection coefficient Re(t,r),

 t is the orientation vector of sensor sensitivity and k = v p /v s and a is the scalar product of vector r and vector t.  In the case that P wave is incident vertically to the surface (a=1), Re(t,r) =2.  Consequently, the first motions of AE waves detected at the observation point can be approximated with the good accuracy as the product of the far-field solution and the reflection coefficient.

 Considering the effect of reflection at the surface, the amplitude of the first motion Ao(x,t) in the far-field due to an applied force f(t) is derived

 The relative calibration coefficient Cs of equivalent sensitivity is obtained for each sensor.

 In a few cases, absolutely calibrated sensors are available. In this respect, a moment tensor analysis to determine the relative tensor components is preferable in practical applications.

 Since the moment tensor is a symmetric tensor of the 2 nd rank, the number of independent components is six.  These components can be determined from the observation of the first motions at more than six sensor locations.

 To solve the equation and determine six components of the moment tesnor, the coefficient Cs, the reflection coefficient Re(t,r), the distance R, and its direction cosine vector r are necessary.  The determination of Cs and Re(t,r) is already discussed.  Other values can be obtained from the source (flaw) location analysis. Thus, the location analysis is essential to perform the moment tensor analysis.

 In the SiGMA analysis, two parameters of the arrival time (P1) and the amplitude of the first motion (P2) are visually determined from AE waveform,

SiGMA analysis  In the location procedure, the crack location y is determined from the arrival time differences t i between the observation x i and x i+1, solving equations, R i – R i+1 = | x i – y | - | x i+1 – y | = v p t i.  Then, the distance R and its direction vector r are determined. The amplitudes of the first motions at more than 6 channels are substituted, and the components of the moment tensor are determined from a series of algebraic equations.  Since the SiGMA code requires only relative values of the moment tensor components, the relative calibration coefficient Cs of AE sensors is sufficient enough.

 Eigenvalue Analysis  Unified Decomposition of Eigenvalues  Crack Orientation  Two-Dimensional (2-D) Treatment

 In the SiGMA code, classification of a crack is performed by the eigenvalue analysis of the moment tensor [Ohtsu, 1991]  This is because that the presence of tensor components is not actually associated with the type of the crack, but closely related with the coordinate system.  Although the crack orientations are often assumed as parallel to the coordinate system, they are generally inclined to the coordinate system.

 The moment tensor for a shear crack: From the eigenvalue analysis, three eigenvalues are obtained as  V, 0, and -  V. Setting the ratio of the maximum shear contribution as X, three eigenvalues for the shear crack are represented as X, 0, -X.

 The matrix is already diagonalized, and diagonal components are identical to three eigenvalues  V,  V,  V, which are decomposed as,

 The components can be decomposed into the deviatoric (non-volumetric) components [1 st term] and the isotropic components [2 nd term].  Setting the ratio of the maximum deviatoric tensile component as Y and the isotropic tensile as Z, three eigenvalues are denoted as, -Y/2 + Z, -Y/2 + Z and Y + Z.

1.0 = X + Y + Z,  the intermediate eigenvalue/the maximum eigenvalue = 0 - Y/2 + Z,  the minimum eigenvalue/the maximum eigenvalue = -X - Y/2 + Z.

 The ratios X, Y, and Z are mathematically determined in an isotropic solid. Setting the angle, c, between crack vector l and normal vector to the crack surface n, as cos c = (l,n)  X = [(1 - 2n) - (1 - 2n)cos c]/[(1 - 2n) + cos c],  Y = 4(1 - 2n)cos c/[3(1 - 2n) + 3cos c],  Z = 2(1 + n)cos c/[3(1 - 2n) + 3cos c],

 In the SiGMA code, AE sources of which the shear ratios are less than 40% are classified into tensile cracks.  The sources of X > 60% are classified into shear cracks.  In between 40% and 60%, cracks are referred to as mixed mode.

 From the eigenvalue analysis of the moment tensor, three eigenvectors e1, e2, e3 are also obtained. These are presented by the two vectors l and n, e1 = l + n e2 = l x n e3 = l – n.

 In the first version of SiGMA [Ohtsu 1991], the orientations of tensile cracks are determined from the vector e1, and those of shear cracks are presented by two vectors l and n, which are usually perpendicular.

 Deformations of the plate are classified into two motions. One is in-plane motion where a crack surface is generated as the normal vector to the crack plane is vertical to the x 3 -axis and AE waves are detected at the edge of the plate.  The other is out-of-plane motion where the crack surface is created parallel to the x 1 -x 2 plane.

 In in-plane motions, the x 3 -components of both the vector l and n are equal to zero. Still, SiGMA is available.  In the case of the out-of-plane observation, only the case that a tensile crack is generated parallel to the x 1 -x 2 plane can be treated. Otherwise, no information can be recovered.

 the moment tensor in an isotropic solid, In the case that AE sensors are attached at the edge of the plate, the components of the tensor are readily defined except m 33 component, because no motion in the x 3 -direction is detected.

 The m 33 component is actually determined from [Shigeishi & Ohtsu 1999],  m 33 = l k n k = (m 11 +m 22 )/(2 +2  ) = (m 11 +m 22 ).

 The component of the moment tensor: m 11, m 12, m 22 are determined, solving the following equation, Since the m 33 component is obtained, the unified decomposition of the eigenvalues and the orientation analysis by the eigenvectors are readily performed in the same manner as those of the 3-D problems.

 To locate AE sources, 5-channel system is at least necessary for three-dimensional (3-D) analysis. Since 6-channnel system is the minimum requirement for the SiGMA analysis, more than 6-channel system is in demand.  In contrast, although 3-channel system is available for the SiGMA-2D analysis, the location analysis requires 4-channel system.