Algorithms for the calculation of exact displacements ... · Elastic dislocation theory is widely...

Computers & Geosciences 33 (2007) 1064–1075

Algorithms for the calculation of exact displacements,strains, and stresses for triangular dislocationelements in a uniform elastic half space$

Brendan J. Meade�

Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138, USA

Received 5 August 2006; received in revised form 6 December 2006; accepted 10 December 2006

Abstract

We present algorithms for analytically calculating the displacements, strains, and stresses associated with slip on a

triangular dislocation element (TDE) in a homogeneous elastic half space. Following previous efforts, the solution is

constructed as a dislocation loop where the deformation fields for each of the three triangle legs are calculated by the

superposition of two angular dislocations. In addition to the displacements at the surface we derive the displacements and

strains at arbitrary depth. We give explicit formulas for the strains due to slip on an angular dislocation, the calculation of

angular dislocation slip components, a method for identifying observation coordinates affected by a solid body translation,

and rules for internally consistent vertex ordering allowing for the superposition of multiple TDEs. Examples of surface

displacements and internal stresses are given and compared with rectangular representations of geometrically complex

fault surfaces.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Elastic dislocation; Faulting; Seismotectonics

1. Introduction

Elastic dislocation theory is widely used forcalculating the displacements, strains, and stressesassociated with faulting from earthquake rupture tointerseismic time scales (e.g., Burgmann et al., 2002;McGuire and Segall, 2003; Savage and Burford,1973). Fault surfaces are often parameterized as acollection of point or rectangular sources due to theavailability of analytic solutions for the deformation

due to slip on dislocation elements of these shapes(e.g., Okada, 1985, 1992). However, some faults,such as those at subduction zones, may havegeometries with substantial variations in both strikeand dip. Rectangular parameterizations of complexfault surfaces exhibit geometric gaps due to the factthat it is often impossible to mesh non-planarsurfaces exactly using rectangular elements. Theeffect of these gaps is to complicate the accuratecalculation of displacements and stresses due tothe effects of strain singularities at the edge ofevery dislocation element. These effects vanish,except exactly at element edges, with gap-freeparameterizations of fault surfaces. With gap-freerepresentations of fault surfaces these effects vanish

ARTICLE IN PRESS

www.elsevier.com/locate/cageo

0098-3004/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cageo.2006.12.003

$Code available from server at http://www.iamg.org/

CGEditor/index.htm�Tel.: +1 617 495 8921; fax: +1 617 495 8839.

E-mail address: [email protected].

everywhere, except along dislocation edges, becauseeach singular boundary is matched exactly withanother. Gap-free tessellation of observationallyconstrained fault surface geometry can be achievedusing triangular meshes (e.g., Plesch et al., 2003)and we develop a solution for the static elasticresponse to slip on an arbitrary set of triangulardislocation elements (TDEs). These algorithmsfacilitate the accurate modeling of geometricallycomplex of fault zones and may be useful forCoulomb failure stress calculations, modeling geo-detic measurements, quasi-dynamic rupture simula-tions, and the assessment of the geometriccontinuity of active faults.

The concept of a TDE in a homogeneous elastichalf space is not a new one. Several studies over thelast 30 years have contributed to the development ofthe techniques that might be used to constructTDEs from primitive Green’s functions. Brown(1975) discussed the superposition of angular dis-locations to construct the sides of a polygon. Thisconstruction was enabled by the calculation ofGreen’s functions for an angular dislocation in auniform elastic half space given by Comninou(1973) and Comninou and Dunders (1975). BothComninou (1973) and Jeyakumaran et al. (1992)outlined the rotation and translation needed tomap an arbitrary dislocation leg into the angulardislocation coordinate system formalized by Com-ninou and Dunders (1975). Thomas (1993) extendedJeyakumaran et al.’s (1992) boundary elementformulation to arbitrary polygonal dislocationelements and noted typographical errors in theformulas for the displacements due to slip on anangular dislocation (Comninou and Dunders,1975). These algorithms have been used to studyfault interactions (Griffith and Cooke, 2004; Olsonand Cooke, 2005) and as the basis for the inversionof geodetic measurements for coseismic slip dis-tributions (Maerten et al., 2005). Despite thesecontributions the application of TDEs has not beenwidespread due to the fact that their constructionhas remained opaque.

Here, we present explicit algorithms to analyti-cally calculate the displacements and strains asso-ciated with slip on a TDE beginning with a reviewof the construction of a dislocation loop by meansof the superposition of angular dislocations. Wedetail the geometric transformations necessary tocalculate the angular dislocation geometries fromthe triangle vertices, and project the triangularelement slip vector into the slip vectors for each

angular dislocation element. Exact expressions forthe strains due to an angular dislocation in anelastic half space are presented (Appendix A). Thestrains and stresses at depth for a TDE are given bythe rotation and superposition of strains from eachangular dislocation. We develop an algorithm foridentifying observation coordinates below the sur-face of a TDE affected by a solid body offset. Thedeformation fields associated with multiple TDEsare readily constructed by superposition using avertex ordering scheme that is consistent fromelement to element. Comparisons between thedisplacement and stress fields for triangular andrectangular representations of non-planar faultsurfaces demonstrate the effects of alternate geo-metric parameterizations of fault surfaces.

2. Construction of TDEs

To calculate the displacements, u, at a point r, dueto slip, bi, on a dislocation element with a surface S,we can integrate the product of the displacementdiscontinuity and the stress produced at a point, r0,over the surface

uðrÞ ¼

ZS

biCijklnjqlukðr0 � rÞdS; (1)

where nj is a unit vector in the j direction, bi is theslip vector in the i direction, and umk is the tensorGreen’s function for the elastic displacements (e.g.,Hirth and Lothe, 1968). Assuming a homogeneouselastic body with Lame parameters, m and l, theelastic moduli tensor can be written as Cijkl ¼

mðdikdjl þ dildjkÞ þ ldijdkl (dij is Kronecker’s delta)(e.g., Malvern, 1969) and, along with Stokestheorem, Burgers (1939) demonstrated that (1) canbe rewritten in terms of contour integrals as

uðrÞ ¼bO4pþ

1

4p

IC

b� nl

jjr0 � rjjdl

þ1

4pð1� nÞr

IC

ðb� ðr0 � rÞÞ � nl

jjr0 � rjjdl; ð2Þ

where n ¼ m/2(l+m) is Poisson’s ratio. The firstterm in Eq. (2) gives the displacement discontinuity(solid angle offset) across the surface of thedislocation element, while the second and thirdterms are continuous and give the elastic effects asline integrals around the boundary of the disloca-tion surface serving as the foundation for theconcept of a dislocation loop (e.g., Hirth and Lothe,

ARTICLE IN PRESSB.J. Meade / Computers & Geosciences 33 (2007) 1064–1075 1065

1968). Thus, to calculate the total deformationassociated with uniform slip on a single TDE we cancalculate the appropriate line integrals around theelement edges (legs) so that the total displacement isgiven by the sum of the displacements from each ofthe three legs (Fig. 1). The basic concept is to useangular dislocations to construct a solid prism, withthe TDE defining its top surface, which is displacedin the same direction as the TDE slip vector (Fig. 1).Internally consistent vertex ordering causes theelastic effects due to prism edges below the TDEsurface to vanish leaving only the elastic effectsassociated with the boundary of the TDE. Thisapproach is identical to that outlined in previousstudies of the construction of TDEs (Brown, 1975;Comninou, 1973; Jeyakumaran et al., 1992;Thomas, 1993).

A TDE is geometrically defined by three vertices(p1,p2,p3) (Fig. 1) and for the purpose of thisapplication we define a vertex ordering system thatensures that the vertical legs of the angulardislocation elements cancel exactly. The vertexordering scheme we use here enforces a clockwisesense of circulation for triangle vertices whenviewed from above (Fig. 2). Triangle vertices aredefined as having a clockwise orientation if the z

component of the unit vector normal to the triangleface is positive, ððp2 � p1Þ � ðp3 � p1ÞÞz40. If thisquantity is negative then any two triangle verticescan be swapped and the triangle normal vector willthen be positive. With the vertices for all trianglesordered in a clockwise fashion the sense of slip andorientation of the unit vectors will be consistent

from one element to the next regardless of itsorientation. Using this vertex ordering system it isstraightforward to construct consistent representa-tions of geometrically complex surfaces using TDEsby superposition of individual elements.

The total deformation field for a TDE can becalculated by summing the deformation fields fromeach of the dislocation legs that defines theboundary of the element. Each of the three legscan be constructed by the superposition of twoangular dislocations between each set of trianglevertices (Fig. 3). We calculate the displacementfrom each leg, uk, by the superposition of deforma-tion fields from two angular dislocations (i.e., uk ¼

uð1Þk � u

ð2Þk ):

u ¼XN

j¼1

X3k¼1

uð1Þkj� uð2Þkj, (3)

where k and j are indices over the three legs of eachtriangle and each TDE respectively, and N is thenumber of TDEs (Comninou, 1973; Jeyakumaranet al., 1992). In constructing TDEs from angulardislocation legs there are two major tasks to enablethe summation in Eq. (3). The first is the calculationof the angular dislocation geometry in terms of the

ARTICLE IN PRESS

Fig. 2. Vertex ordering. (a) Clockwise vertex ordering where the

unit vector normal to the face has a positive (+) z component. (b)

Counter-clockwise vertex ordering where the unit vector normal

to the face has a negative (�) z-component.

Fig. 3. Construction of a single dislocation leg by means of

summation of two angular dislocations. Inclination of each

angular dislocation is given by a common inclination b while

vertices are buried to depths a1 and a2 respectively. Common,

dipping segment of each angular dislocation cancels after

summation.

Fig. 1. Triangular dislocation element and triangle leg geometry.

Arrows indicate sense of orientation of the slip components. Note

that vertical segments of each dislocation leg cancel exactly with

those of adjacent legs.

B.J. Meade / Computers & Geosciences 33 (2007) 1064–10751066

TDE vertices and the second is the calculation of theappropriate slip vectors on each angular disloca-tion. The displacements scale linearly with themagnitude of slip.

Green’s functions for the surface and internaldisplacements due to slip on a buried angulardislocation, uk in Eq. (3), are known for both awhole space (Yoffe, 1960) and, most relevant togeophysical applications, a half space (Comninou,1973; Comninou and Dunders, 1975). These studiesgive exact expressions (not repeated here) for thesurface and internal displacements due to strike-,dip-, and tensile-slip on a buried angular dislocationin the x�z plane with a vertex located at the originin the x�y plane. The geometry of an angulardislocation is defined by an area of uniform slipalong a vertical surface in the plane bounded by twosemi-infinite line segments which meet at a pointbelow the free surface at a depth, a, where the caretdenotes the variable’s association with a particularangular dislocation. The geometry of an angulardislocation is further defined by the inclination, b,which can be written in terms of the leg dip, d, as

b ¼p=2� d if dX0;

�p=2� d if do0;

((4)

where the leg dip is given by tan d ¼Dpz=ðDpx cos aþ Dpy sin aÞ. Note that the differencein position, Dp ¼ pi+1�pi, of the leg endpoints (twoTDE vertices) depends on the vertex ordering.Additionally, to maintain a consistent sense of theorientation for the vertical segment of each angulardislocation we ensure that 0pbpp=2. If the firstangular dislocation lies in the x�z plane with avertex at a depth a and is inclined at an angle b(inclination) a second dislocation element with avertex along the non-vertical line segment (at adepth of a) and slip opposite to that of the firstangular dislocation will cancel all deformationexcept that due to the polygonal slip patch belowthe leg of the triangular element (Fig. 3). Note thatthere is no deformation for the b ¼ 0 case.

Transforming the TDE vertices to the Comninouand Dunders (1975) coordinate system for each ofthe angular dislocations requires that the coordi-nates for each leg are mapped as ðp1; p2Þ ! ðp1; p2Þ,ðp2; p3Þ ! ðp1; p2Þ, ðp3; p1Þ ! ðp1; p2Þ where thedepth, a, to each vertex is defined as positive andthe coordinates ðp1; p2Þ define the rotated leg end-points for each of the TDE legs. The transformbetween the two coordinate systems is given by a

rotation of coordinates, p ¼ xp, where the rotationmatrix, x, is a function of the strike, tan a ¼ Dy=Dx,of each dislocation leg,

x ¼

cos a sin a 0

�sin a cos a 0

0 0 1

0B@

1CA. (5)

The observation coordinates can also be trans-lated and rotated into the Comninou and Dunders(1975) coordinate system so that each angulardislocation lies in the x� z plane.

s ¼ xðs� pÞ (6)

Using these relations and Green’s functions fromComninou and Dunders (1975) we can calculateboth the internal and surface displacement fields foran angular dislocation. As an example, consider theangular dislocation shown in Fig. 4 with a vertex at(x ¼ �25; y ¼ 0; z ¼ a ¼ 0) aligned along the x-axis(a ¼ 0) and dipping at 451 so that b ¼ 45� with unitstrike-slip motion. The internal displacements areshown in the x�y plane (Figs. 4a, d). Note therelatively uniform slip over the dislocation surfacethat is parallel to the strike of the dislocationelement and not a function of its inclination, b.Observation coordinates are offset slightly in thepositive y direction to highlight the elastic effects.This gives rise to non-zero displacements outside ofthe angular dislocation patch because the observa-tion coordinates are not coincident with the x�z

nodal plane. The surface displacements show thebasic four-quadrant pattern of deformation asso-ciated with right-lateral strike-slip motion on avertical dislocation (Fig. 4d). A second angulardislocation in the same plane and with the sameinclination as the previous one can be defined with avertex at (x ¼ 25; y ¼ 0; z ¼ a ¼ 50). With unitright-lateral strike-slip the displacements are clearlyrelated to the faulting mechanism in the cross-sectional view (Fig. 4b), while the surface displace-ments show a rather more diffuse pattern ofdeformation due to the fact that the angulardislocation source is buried (Fig. 4e). The differencebetween the displacement fields from these twoangular dislocations gives the displacements asso-ciated with a dislocation leg, ul ¼ uð1Þ � uð2Þ, be-tween the two vertices. The cross-sectional view(Fig. 4c) shows resulting dislocation surface andthat the displacements between nearly vanishoutside of the boundary of these two surfaces


four-quadrant pattern of deformation results at thesurface (Fig. 4f).

3. Slip components on the angular dislocations

To calculate the deformation associated withgeneralized slip in an arbitrary direction on aTDE we must convert the TDE Burgers vector intostrike-, dip-, and tensile-slip components on eachangular dislocation. Comninou (1973) and Jeyaku-maran et al. (1992) outlined a method based on theconstruction of rotation matrices as functions ofunspecified angles. We construct our solution byrelating the geometry of the TDE to that of eachangular dislocation using orthogonal unit vectors todescribe their respective orientations, and projectingthe slip vectors from the TDE on to each angular

dislocation patch. Using this method with the vertexordering scheme described above ensures thatcoordinate transforms are consistent from leg toleg and triangle to triangle.

Given component tensile-, strike-, and dip-slipcomponents of slip, s, on a triangular fault patch wewould like relate these to a total slip vector, vs. Wedo this by first projecting the Burgers vector intox,y,z coordinates by defining unit vectors normal tothe TDE face, one parallel to the direction of strikeand one down-dip. A unit vector normal to the faceof the triangle is given by, nt ¼ ðp1 � p2Þ=jjðp1 � p2Þjj, while one parallel to the strike of thepatch is ns ¼ ð�sin g; cos g; 0Þ, where g ¼ tan�1ðn

ðyÞt =

nðxÞt Þ. A third direction vector can be defined parallel

to the TDE dip as, nd ¼ nt� ns. The fault slip vectoris then calculated by dotting the slip components

ARTICLE IN PRESS

Fig. 4. Displacement fields for strike-slip motion on angular dislocations. (a) Cross-sectional view of displacement for an angular

dislocation with a vertex at zero depth and an inclination of 451. Displacements are non-zero off of the displacement surface due to the fact

that observation plane has been offset slightly from nodal plane along y ¼ 0 to highlight elastic effects. (b) Same as (a) only for a

dislocation with a vertex at a depth of 50 and offset in the positive y direction by 50 units. Thus dipping segment of angular dislocation

aligns with that from (a). (c) Displacements from dislocation leg calculated by subtracting (b) from (a). (d–f) Surface displacement fields

corresponding to dislocations shown in panels (a–c).


into each unit vector:

vs ¼

vðxÞs

vðyÞs

vðzÞs

0B@

1CA ¼

nðxÞt nðxÞs n

ðxÞd

nðyÞt nðyÞs n

ðyÞd

nðzÞt nðzÞs n

ðzÞd

0BB@

1CCA

st

ss

sd

0B@

1CA. (7)

Note that neither vs nor the slip components oneach triangular patch are, in general, not the sameas the slip components on any angular dislocation.The slip vector must be projected into strike-, dip-,and tensile-slip components for each of the threedifferent dislocation legs. To do this we first find theorientation of unit normal vectors in the direction ofeach component of slip on each angular dislocation.The strike- and tensile-slip components are paralleland perpendicular to the strike of the dislocation legns ¼ ðcosðaÞ; sinðaÞ; 0Þ, nt ¼ ð�sinðaÞ; cosðaÞ; 0Þ, re-spectively, while the dip-slip unit vector is orthogo-nal to the plane defined by these two vectors,nd ¼ ns � nt. Due to the fact that ns and nt always liein the x� y plane the unit vector in the direction ofthe dip-slip on an angular dislocation is alwaysvertical, nd ¼ nðzÞ ¼ ð0; 0; 1Þ. For each angular dis-location we must resolve the slip vector into thecorresponding strike-, dip-, and tensile-slip compo-nents.

s ¼

st

ss

sd

0B@

1CA ¼

nðxÞt nðxÞs nðxÞd

nðyÞt nðyÞs nðyÞd

nðzÞt nðzÞs nðzÞd

0BB@

1CCA

vðxÞs

vðyÞs

vðzÞs

0B@

1CA, (8)

where the subscripts t, s, and d refer to the tensile-,strike-, and dip-slip components, respectively. Theprojection of the slip vector, vs, to slip componentson an angular dislocation depends only on the strikeof the angular dislocation, a, not on inclination ofthe angular dislocation, b. While the slip compo-nents on each angular dislocation are not a functionof the inclination, the displacements from each legare. With the slip components known for eachangular dislocation element we can calculate u

ð1Þ

and uð2Þ for each angular dislocation. These

displacement fields can then be rotated, uðiÞ ¼

x�1uðiÞ, and summed, as shown in Eq. (3), to givethe total displacement, u, associated with slip on aTDE.

The surface displacements for three legs of adipping TDE with a uniform strike-slip displace-ment discontinuity and the total displacement fieldshow how the deformation from individual legs sumto give the total displacement field (Fig. 5). Vertex

coordinates, inclination angles, normal vectors andslip components are given in Table 1. For the legparallel to the direction of TDE slip (Fig. 5a) theangular dislocation slip components are strike-sliponly while they are tensile-slip only for the legoriented perpendicular to the slip direction (Fig. 5c)and mixed for the obliquely oriented leg (Fig. 5b).Note that while slip on individual TDE legs mayyield symmetric displacements (Fig. 5a) the overalleffect of TDE dip is to generate an asymmetricdisplacement field where displacements are larger onthe hanging wall side (Fig. 5d) due the effects of thedipping legs (Figs. 5b, c).

4. Displacements at depth

The solution by summation of angular disloca-tions involves the displacement of the entire prismbelow the TDE in the direction of the displacementvector. By summing the angular dislocations theelastic effects of down-going dislocation segmentscancel and we are left with only the elastic effectsassociated with the triangular element. However,the displacements directly below the TDE seecontributions from not only the elastic effects ofthe TDE but also from the translation of the entireprism. This effect can be modeled by subtracting thedisplacement vector for observation points belowthe TDE.

For those observation points whose x–y coordi-nates lie within the projection of the TDE into thex–y plane we can determine which are below theTDE. If a line segment between the observationcoordinates, s, and a point sharing the same x,ycoordinates at the free surface intersects with theplane defined by the vertices of the TDE then theequations for the two will be equivalent at the pointof intersection. Using a parametric representationof both the plane and line segment we can solve fora single parameter, t, whose sign determines whe-ther or not the observation point is below the TDE:

t ¼ det

1 1 1 1

pðxÞ1 p

ðxÞ2 p

ðxÞ3 sðxÞ

pðyÞ1 p

ðyÞ2 p

ðyÞ3 sðyÞ

pðzÞ1 p

ðzÞ2 p

ðzÞ3 sðzÞ

��

��Cdet

1 1 1 0

pðxÞ1 p

ðxÞ2 p

ðxÞ3 0

pðyÞ1 p

ðyÞ2 p

ðyÞ3 0

pðzÞ1 p

ðzÞ2 p

ðzÞ3 sðzÞ

��

��,

(9)

where (p1,p2,p3) are the triangle vertices and s arethe observation coordinates (e.g., Tremblay, 2004).For observation points that are located within theprism and below the TDE the total displacement


or's

pe

rson

al

copy

vector can be calculated by subtracting the slipvector vs:

u ¼u if t40;

u� vs if to0

((10)

assuming that the depth of the observation co-ordinate, s(z), is always positive. This correction

does not eliminate the singularities along the edgesof all angular dislocations which preclude displace-ment calculations directly below any TDE vertex.

5. Strains and stresses for a TDE

Similar to the displacements, the strains andstresses associated with a TDE can be calculated by

ARTICLE IN PRESS

Fig. 5. Surface displacement fields for individual dislocation legs and a complete triangular dislocation element. (a–c) Surface

displacement fields for three dislocation legs described in Table 1. (d) Total displacement field resulting from summation of displacement

fields from all three legs (a–c). Note that near-fault displacements are larger in hanging wall side than footwall side.


summing the stresses from six separate angulardislocations. The expressions for the stresses froman angular dislocation in an elastic half space can bederived by differentiating Green’s functions givenby Comninou and Dunders (1975) and applying thedefinition of strain, e ¼ 1

2ðruþ ðruÞTÞ, where ru is

the deformation gradient tensor. The derivatives ofthe displacements are somewhat lengthy and werecalculated with symbolic algebra computer softwarein an effort to avoid human error (see Appendix A).The strains from each dislocation leg, el ¼ eð1Þ � eð2Þ,must be rotated about the z-axis and summed get totalstrain tensor:

e ¼X3k¼1

x�1k ðeð1Þk � eð2Þk Þxk, (11)

where the rotation matrix is given by Eq. (5). Oncethe strains have been summed the stresses can becalculated using Hooke’s law for a linear elasticsolid, s ¼ l trðeÞIþ 2me, where tr(e) is the trace ofthe strain tensor and I is the identity matrix.Analytic expressions for the strains are somewhatlengthy and are given in Appendix A of supple-mentary data (see Appendix A). In addition,MATLAB implementations of the complete set ofalgorithms for both the displacements and strainsare given in Appendix B of supplementary data (seeAppendix A).

6. Examples of displacement and stress fields

A few simple comparisons can be made betweenthe well-known solutions for the deformation fieldssurrounding a rectangular dislocation element(e.g., Okada, 1985) and those from two TDEscombined to form a rectangular dislocation element

(Fig. 6). Upon summation the common diagonallegs, lc, from each triangle cancel leaving only thecontributions from the perimeter legs (l1, l2, l3, l4). Ifwe construct a rectangular dislocation element fromtwo TDEs and compare the surface displacementsfor unit strike-slip on a vertical fault 20 km2, the twodisplacement fields agree to one part in 1015 which ison the order of double precision for values nearunity. This is in agreement with Jeyakumaranet al.’s (1992) finding that the case of two verticalTDEs (used to form a composite rectangle) yield asolution that is algebraically identical to thatpresented by Chinnery (1961) for a vertical strike-slip fault.

A second reference case is that developed byThomas’ (1993) consisting of a unit dip-slip on asingle 2� 2 km rectangular dislocation elementburied with its centroid at a depth of 2 km andinclined at 451. Similar to the construction shown inFig. 6 we can also calculate the deformationassociated with slip on this same dislocation patchby the superposition of two TDEs which divide thereference rectangle into two across a diagonal oflength 4=

ffiffiffi2p

km. For this problem we find that thedifference between surface displacement fields fromthe rectangular solution (e.g., Okada, 1985) and theTDE solution presented here differs by a maximumof one part in 1015 while Thomas (1993) reported aagreement at the level of one part in 104. Jeyaku-maran et al. (1992) give a maximum discrepancy ofone part in 103 for an unspecified dipping faultmodel.

Thus far we have considered TDEs dominatedby right and 451 angles. The displacements andstresses resulting from slip on an obliquely ori-ented, dipping and partially buried TDE highlightthe generality of the algorithms presented aboveand the complexity of the resulting deformationfields. Consider a TDE with vertices located at

ARTICLE IN PRESS

Table 1

Parameters for a simple triangular dislocation element

Quantity Leg 1 Leg 2 Leg 3

p1 (40,50,0) (60,50,0) (40,30,20)

p2 (60,50,0) (40,30,20) (40,50,0)

b 90 �tan�1ffiffiffi2p

45

a 180 45 �90

ns (�1,0,0) ðffiffiffi2p

=2;ffiffiffi2p

=2; 0Þ (0,�1,0)

nd (0,0,1) (0,0,1) (0,0,1)

nt (0,�1,0) ð�ffiffiffi2p

=2;ffiffiffi2p

=2; 0Þ (1,0,0)

ss �1ffiffiffi2p

=2 0

sd 0 0 0

st 0 �ffiffiffi2p

=2 1

Fig. 6. Representation of a rectangular dislocation element by

summation of two triangular elements. Common leg, lc, cancels if

vertex ordering is same for both triangles. This construction is

used for two of the examples in discussion.

B.J. Meade / Computers & Geosciences 33 (2007) 1064–1075 1071

Autho

r's

pers

c

p1 ¼ ð20; 50; 5Þ, p2 ¼ ð80; 80; 0Þ, p3 ¼ ð50; 30; 20Þ(Fig. 7). The surface displacements for strike-, dip-, and tensile-slip are shown in Figs. 7a–c. Note againthe asymmetry across the strike of the dislocationplane where displacements are larger above thehanging wall than in the footwall. Contours of thesecond invariant of the stress tensor, I2 ¼

12ðr : r

�trðrÞ2Þ, on a horizontal plane cutting through theTDE at a depth of 10 km for the same three cases(Figs. 7d–f) show both the characteristic four-quadrant pattern of deformation and the commonlocations of the nodal surfaces emanating from theTDE edges.

As an example of the use of multiple TDEs weconsider the sub-vertical Whittier fault at theeastern edge of the Los Angeles basin. Plesch et al.(2003) constrained the location of the fault surfaceusing well logs, earthquake relocations, and high-

resolution seismic reflection data and parameterizedthis surface using a gap free triangular mesh of 2068elements (Fig. 8) and a six element rectangularrepresentation. The rectangular representation ofthe fault surface includes six elements with acharacteristic length scale of �10 km, somewhatsmaller than those typically used to study inter-seismic deformation at intermediate scales, butlarger than those commonly used to infer coseismicslip distributions. By assuming unit strike-slip on alldislocation elements we can calculate the deforma-tion fields exactly and determine the sensitivity tofault geometry parameterizations using triangulardislocations as a tool to calculate displacements andstresses from the triangular mesh. The surfacedisplacement fields from both the coarse rectangularand fine triangular parameterizations are character-ized by a macro-scale four-quadrant pattern of

ARTICLE IN PRESS

Fig. 7. Displacement and stress fields resulting from slip on an obliquely oriented, dipping, and buried triangular dislocation element.

(a–c) Surface displacements for strike-, dip, and tensile-slip. Arrows give horizontal displacements and contours show vertical

deformation. (d–f) Second invariant of stress tensor on a horizontal plane cutting through dislocation surface at a depth of 10 km. Note

that for every case a four-quadrant pattern of deformation is observed near intersection of observation plane and triangular dislocation

element.


's

pers

onal

co

py

ARTICLE IN PRESS

Fig. 8. Displacement and stress fields resulting from slip on different geometric representation of Whittier fault in Los Angeles Basin area

of southern California. Fault dips to north at its western end and progressively steepens to southeast. (a) Black arrows are horizontal

displacements from unit strike-slip motion on each of 2068 triangular elements shown in gray. (b) Horizontal displacements predicted by a

model of unit strike-slip motion across each of six rectangular dislocation elements representing fault surface. (c) Residual (triangular-

rectangular parameterizations) displacement field at five times scale of vectors in (a, b). Note that differential displacements are largest

near hanging wall side of fault where different geometric parameterizations are closest to free surface. (c–f) Same as (a–c) only showing

base 10 logarithm of absolute value of second invariant of deviatoric stress tensor on a horizontal surface cutting through fault plane at a

depth of 5 km. Dark and light areas denote high and low stresses, respectively. The multiple dark regions in rectangular case are due to

large stresses at gaps between rectangular dislocation elements. Note that these stress concentrations bulge outward away from fault and

have effect of propagating larger stresses farther away from main fault zone than gap-free triangular representation (d). Differential

stresses (f) show that largest mismatch between two parameterizations exists in hanging wall where fault surface is closest to observation

coordinates.

B.J. Meade / Computers & Geosciences 33 (2007) 1064–1075 1073

compression and extension (Figs. 8a, b). Thedifferential surface displacement field (Fig. 8c)highlights the residual displacements (up to �40%at the given observation coordinates) of the calcu-lated model displacements. These effects are largestin the hanging wall block where the fault dip isshallowest and contributions from dislocation ele-ments at depths greater than a few kilometerscontribute significantly to the predicted displace-ments. The differences in predicted surface displace-ment may be large enough to use geodeticmeasurements to constrain the continuity of activefaults at depth.

The second invariant of the stress tensor, I2, atdepth shows significant differences between the twogeometric parameterizations. Most significantly therectangular parameterization shows the presence ofhigh magnitude stress concentrations around eachof the gaps between major segments (Figs. 8a–c).These excess stresses have the effect of adding to theoverall magnitude of I2 away from the fault zoneand introducing localized stress concentrations nearthe fault. Thus, while a gap-free representation offault surface may be considered more complex itmay lead to a somewhat smoother stress field.Similar calculations of Coulomb failure stressesmight be used to predict the locations of near-faultaftershocks.

A simple example of how rectangular approxima-tions converge toward the exact triangular solutionscan be seen by approximating a single vertical TDEwith a variable number of thin, vertical, rectangulardislocation elements. We define a vertical right TDEin the x–z plane with vertices at p1 ¼ ð40; 50; 0Þ,p2 ¼ ð60; 50; 0Þ, p3 ¼ ð60; 50; 20Þ with unit strike-slipmotion and calculate the stresses along a linealigned parallel to the dipping leg but offset in thepositive y direction by 1 km. The sxx components ofthe stress tensor for the exact triangular solutionwith 10 and 100 element rectangular discretizationsare compared in Fig. 9. Differences between theexact and discretized approximations are minimizedas the number of rectangular elements is increased,however at the coarsest 10 element discretization thestresses are, in general, lower than the exactsolutions with complex behavior in between elementedges. The bumps in the 10 element discretizationare a result of the high stresses generated by thestrain singularities at the base of the triangularapproximation where the rectangular elements donot meet exactly. These comparisons demonstratethe utility of using TDEs to calculate high-resolu-

tion displacement fields rather than having tonumerically integrate point sources or computeapproximate rectangular discretizations.

7. Conclusions

TDEs allow for the exact calculation of displace-ment strain and stress fields with gap-free represen-tations of fault surfaces. We have detailed thealgorithms necessary to carry out these calculations,including the calculation of angular dislocation slipcomponents, accounting for the solid body displa-cement below the TDE, exact expressions for thestrains, and defining a triangle vertex orderingscheme that enables the calculation of deformationfields from multiple TDEs in an internally consis-tent framework. These algorithms may find applica-tion in the analysis of geodetic data over inter-, co-,and post-seismic time scales as well is in thecalculation of Coulomb failure stresses and slipevolution modeling.

Acknowledgments

Discussions with Robert Simpson and WilliamStuart helped to focus this work on a cleardescription of the algorithmic details. Reviews byDavid Healy and an anonymous reviewer greatlyimproved the writing and content of this paper.

ARTICLE IN PRESS

Fig. 9. Stresses from a triangular dislocation element and

variable resolution rectangular approximations. Triangle lies in

y–z plane with a width and depth of 20 km. Stresses are calculated

along a set of observation coordinates parallel to dipping leg of a

triangular dislocation element and offset in y direction by 1 km.

Triangle surface is approximated using multiple rectangular

dislocations whose edges match exactly except along bottom

dipping leg of triangle. Stresses from rectangular approximations

of triangular dislocation converge as the number of rectangular

elements increases.


Funding was provided by the Southern CaliforniaEarthquake Center and Harvard University.

Appendix A. Supplementary material

Supplementary data associated with this articlecan be found in the online version at doi:10.1016/j.cageo.2006.12.003.

References

Brown, R.L., 1975. A dislocation approach to plate interaction,

Ph. D. Dissertation, Massachusetts Institute of Technology,

Cambridge, MA, 449pp.

Burgers, J.M., 1939. Some considerations on the fields of stress

connected with dislocations in a regular crystal lattice I In:

Proceedings of the Koninklijke Nederlandse Akademie Van

Wetenschappen, vol. 42, pp. 293–325.

Burgmann, R., Ergintav, S., Segall, P., Hearn, E.H., McClusky,

S., Reilinger, R.E., Woith, H., Zschau, J., 2002. Deformation

during the 12 November 1999 Duzce, Turkey, earthquake,

from GPS and InSAR data. Bulletin of the Seismological

Society of America 92, 161–171.

Chinnery, M., 1961. The deformation of the ground around

surface faults. Bulletin of the Seismological Society of

America 51, 355–372.

Comninou, M., 1973. Angular dislocation in a half space. Ph.D.

Dissertation, Northwestern University, Evanston, Ill, 45pp.

Comninou, M., Dunders, J., 1975. Angular dislocation in a half

space. Journal of Elasticity 5, 203–216.

Griffith, W.A., Cooke, M.L., 2004. Mechanical validation of the

three-dimensional intersection geometry between the Puente

Hills blind-thrust system and the Whittier fault, Los Angeles,

California. Bulletin of the Seismological Society of America

94, 493–505.

Hirth, J.P., Lothe, J., 1968. Theory of Dislocations. McGraw-

Hill, New York, NY, 780pp.

Jeyakumaran, M., Rudnicki, J.W., Keer, L.M., 1992. Modeling

slip zones with triangular dislocation elements. Bulletin of the

Seismological Society of America 82, 2153–2169.

Maerten, F., Resor, P., Pollard, D., Maerten, L., 2005. Inverting

for slip on three-dimensional fault surfaces using angular

dislocations. Bulletin of the Seismological Society of America

95, 1654–1665.

Malvern, L.E., 1969. Introduction to the Mechanics of a

Continuous Medium. Prentice Hall, Upper Saddle River,

NJ, 713pp.

McGuire, J.J., Segall, P., 2003. Imaging of aseismic fault slip

transients recorded by dense geodetic networks. Geophysical

Journal International 155, 778–788.

Okada, Y., 1985. Surface deformation due to shear and tensile

faults in a half-space. Bulletin of the Seismological Society of

America 75, 1135–1154.

Okada, Y., 1992. Internal deformation due to shear and tensile

faults in a half-space. Bulletin of the Seismological Society of

America 82, 1018–1040.

Olson, E.L., Cooke, M.L., 2005. Application of three fault

growth criteria to the Puente Hills thrust system, Los Angeles,

California, USA. Journal of Structural Geology 27,

1765–1777.

Plesch, A., Shaw, J.H., SCEC CFM Working Group, 2003.

SCEC CFM—AWWW accessible community fault model for

southern California. Eos. Transactions of the American

Geophysical Union, 84.

Savage, J.C., Burford, R.O., 1973. Geodetic determination of

relative plate motion in central California. Journal of

Geophysical Research 78, 832–845.

Thomas, A., 1993. POLY3D: A three-dimensional, polygonal

element, displacement discontinuity boundary element com-

puter program with applications to fractures, faults, and

cavities in the Earth’s crust. M.Sc. Thesis, Stanford Uni-

versity, 69pp.

Tremblay, C., 2004. Mathematics for Game Developers. Thomp-

son, Boston, MA, 627pp.

Yoffe, E.H., 1960. The angular dislocation. Philosophical

Magazine 5, 161–175.


Algorithms for the calculation of exact displacements ... · Elastic dislocation theory is widely...

Documents

Transcript of Algorithms for the calculation of exact displacements ... · Elastic dislocation theory is widely...