eshelby’s solutions for ellipsoidal inclusions and ...xy742fz8741/rfp_2012_meng... · eshelby’s...

Eshelby’s solutions for ellipsoidal inclusions and heterogeneities with applications tolocalized volume deformation

Chunfang Meng, David D. Pollard

Department of Geological and Environmental Sciences, Stanford University, Stanford, CA94305-2115, the U.S., email: [email protected].

AbstractWe have developed a MatlabTMcode to evaluates Es-

helby’s solution for an arbitrary ellipsoidal inclusion orheterogeneity embedded in an elastic, isotropic and infi-nite body. The code evaluates the elastic fields (i.e. strain,stress and displacement) inside and outside an ellipsoidalinclusion/heterogeneity. With this code, we review some ofthe research on localized volumetric deformation in earth’scrust that used some special cases of Eshelby’s solution,e.g. a heterogeneity modeled as a 2D ellipse or an ellip-soid that has two equal axes known as a spheroid. Wediscuss how the arbitrary ellipsoidal geometry would im-prove the results. One application of the code is to evaluatethe stress fields about the tip-line of a compaction band orshear enhanced compaction band modeled as a flat ellip-soidal geometry to understand the propagation mechanism.Unlike numerical methods, e.g. finite element and bound-ary element methods, the accuracy is not affected by thedistance-to-tip at which the stress is evaluated. This featureenables us to accurately evaluate the stress field in an arbi-trary near-tip region. The previous criterion for the in/off-plane growth of the deformation is restricted to a straighttip-line. With this code, we investigate how the varying tip-line curvature would affect the near-tip stress and furtherthe deformation growth.

Keyword:Eshelby, volume reduction, eigenstrain, shear-

enhanced compaction band

1. Introduction

We review Eshelby’s solution for inclu-sion/heterogeneity problems and their application inmodeling localized volume reduction in porous sedimen-tary rock to form compaction bands and shear-enhancedcompaction bands. Previous work (Katsman et al. (2006))on localized volume deformations are compared to the newresults.

Alternatively the compaction band problem is modeledusing bifurcation theory (Borja and Aydin (2004), but herewe on focus on the inclusion/heterogeneity models.

We use empirical relationships between porosity andelastic moduli along with measured porosities of com-paction bands to specify the contrast in the elastic proper-ties of the ellipsoidal heterogeneities. Remote stress fields

are based on the geologic and tectonic history of the AztecSandstone. Fig. 1 shows a compaction band set shot at Val-ley of Fire State Park by Sternlof et al. (2005). The closelyspaced sub-parallel compaction bands are about 60m inlength and about 15mm in width.

The eigenstrain for the heterogeneity is based on a lin-ear relationship with the porosity loss. The local stressperturbation is examined to understand the propensity forpropagation along the elliptical tip-line of the heterogene-ity and variations in the direction of propagation for pureand shear-enhanced compaction bands.

1.1. Eshelby’s solutions

Eshelby’s solutions enable one to evaluate the quasi-static elastic fields (stress, strain and displacement) insideand outside an arbitrary ellipsoidal inclusion (same elas-tic moduli as surrounding) that has experienced a pre-scribed uniform strain, or an arbitrary ellipsoidal hetero-geneity (different moduli than surroundings) subjected toa prescribed remote state of stress (Eshelby (1957), Es-helby (1959) and Eshelby (1961)). These solutions havebeen applied in studies of rock deformation associated withfractures in sedimentary strata (Reches (1998)), pebblesin a conglomerate (Eidelman and Reches (1992)), fault-ing (Rudnicki (1977)), localized volume reduction struc-tures such as compaction bands (Sternlof et al. (2005)),and alterations of regional stress by a reservoir with differ-ent elastic constants than the surrounding rock (Rudnicki(1999)). Since Eshelby’s solutions are rather complex andlack analytical form for an arbitrary ellipsoid, researchershave degenerated the geometry to some special shapes suchthat analytical solutions could be formulated. Typically inprevious research one of the ellipsoid’s semi-axes was setmuch shorter than the other two that were set equal (i.e. adisk shaped spheroid) to approximate geological structures.Reches (1998) and Healy (2009) developed computer codesto evaluate the solution for spheroids. Recently, we devel-oped a MatlabTM code for arbitrary 3D ellipsoidal geome-tries (Meng et al. (2011)).

The inclusion problem described by Eshelby (1957)considers an ellipsoidal region in an infinite homogeneous,isotropic and elastic material known as the matrix (Fig. 2A). When a inclusion undergoes a change in shape and size,its misfit with the surrounding matrix causes both regions toattain new strain/stress states. The definition of eigenstrain,

Stanford Rock Fracture Project Vol. 23, 2012 E-1

Figure 1: Planar and parallel compaction bands along the northeast flank of Silica Dome. A total of 16 tip-to-tip thickness profileswere measured using a steel tape and calipers (in Sternlof et al. (2005)).

ε∗, summarized by Mura (1987)(p. 1), can be regarded asa uniform strain state that the inclusion will enter if we re-move the constraint of the matrix. Eshelby (1957) referredto this as stress-free strain. Eshelby solved for the elas-tic fields inside and outside the ellipsoidal inclusion for agiven eigenstrain and zero remote stress.

We denote the sub-domain occupied by the inclusion Ω

in the infinite domain D, and the uniform eigenstrain ε∗i j isprescribed to be non-zero throughout Ω and zero in D − Ω.The elastic moduli in the inclusion Ω and the matrix D −Ω

are the same. The displacement, u j, strain, εi j and stress,σi j for both the inclusion and matrix are given by Mura(1987)(p. 11).

Eshelby (1957), Eshelby (1959) and Eshelby (1961)derived solutions for the interior and exterior elastic strainfields for the inclusion problem.

εi j =S i jklε∗kl, for x ∈ Ω,

εi j(x) =Di jkl(x)ε∗kl, for x ∈ D −Ω,(1)

where S i jkl is known as the Eshelby tensor, Mura (1987)(p. 77); Di jkl(x) in general involves some incomplete ellip-tical integrals. Meng et al. (2011) evaluated the elliptical in-tegrals numerically with great precision using Igor (2005).Because of this the method is referred to as quasi-analytical.

1.2. Equivalent inclusion method

The other problem solved by Eshelby (Mura (1987))considers the ellipsoidal region to have different elasticmoduli, i.e. C∗i jkl in Ω and Ci jkl in D − Ω, and the bodyis subjected to remote uniform stress σ∞i j (Fig. 2B). This isreferred to as the heterogeneity problem. Eshelby (1957)concluded that the stress perturbation of a uniformly ap-plied stress due to the presence of an ellipsoidal hetero-geneity can be determined by an inclusion problem whena fictitious eigenstrain ε∗i j is chosen properly. This is knownas the equivalent inclusion method.

To solve for ε∗i j, Meng et al. (2011) rewrites Mura(1987) (22.13):

(∆Ci jklS klmn −Ci jmn

)ε∗mn = −∆Ci jklε

∞kl −C∗i jklε

Pkl, (2)

where ∆Ci jkl = Ci jkl − C∗i jkl; and from the relation σ∞i j =

Ci jklε∞kl , we have the remote strain ε∞i j . ε

pi j is an arbitrary

initial eigenstrain that may be prescribed for the ellipsoidalheterogeneity.

The interior and exterior stress and strain fields are as


σ

σ

C

C ,ijkl εij( ) Ω

D −ijkl

*

, ij

Ω

= 0

A

C ijkl, ij

C ,ijkl* εij

p( ) Ω

D −Ω

B

=\ 0

Figure 2: A, inclusion problem: uniform elastic moduli acrossthe boundary, zero remote stress, inclusion under eigenstrainε∗; B, heterogeneity problem: different elastic moduli acrossthe boundary, non-zero remote stress, initial eigenstrain ε p inΩ.

follows, Mura (1987) (22.8-22.13):

εi j =ε∞i j + S i jmnε∗mn,

σi j =σ∞i j + Ci jkl(S klmnε∗mn − ε

∗mn), for x ∈ Ω;

εi j(x) =ε∞i j + Di jmn(x)ε∗mn,

σi j(x) =σ∞i j + Ci jklDklmn(x)ε∗mn, for x ∈ D −Ω.

(3)

We model the localized volume reduction problemswith an ellipsoidal heterogeneity that has axial-ratio ax

az and ay az, which gives suitable geometries for com-paction bands (Mollema and Antonellini (1996), Sternlofet al. (2005), Eichhubl et al. (2010)). The infinite elasticbody is under a tri-axial stress state with greatest principalcompressive stress perpendicular to the (x, y-)plane of thehighly eccentric ellipsoidal structure. For shear-enhancedcompaction bands the greatest compressive stress is obliqueto the band plane. The inelastic volume reduction is pre-scribed as the initial eigenstrain, ε p

i j. Analogous to Kats-man et al. (2006), we can set ε p

i j to be uni-axial. In generalhowever, e.g. when shearing across the band is clearly in-dicated, the initial eigenstrain is not uni-axial. Determininga realistic initial eigenstrain is address in sections 2.3 and5.1.

The elastic moduli of the heterogeneity are related tothe physical changes, e.g. porosity loss and grain size re-duction, and in general they are different than the moduli ofthe matrix.

1.3. Localized volume reduction in tabular structureLocalized volume reduction in a tabular structure illus-

trated in Fig. 3 has been modeled in 2D by Katsman et al.(2006).

2H

2h

2c

ε ijCijkl

Cijkl

σij = 0

ΩD −

Ω ( , )*

,

Figure 3: Localized volume reduction modeled as a 2D rect-angular inclusion whose thickness reduced by 2(H − h).

The prescribed uni-axial strain is uniform throughouta rectangular region of initial size 2H × 2c such that ε∗zz =

(H − h)/H. The volume reduction is in absence of remotestresses and the tabular structure has the same elastic mod-uli as the matrix. This is analogous to Eshelby’s inclusion


solution illustrated in Fig. 2, but the shape of the inclusionis rectangular not ellipsoidal. Katsman et al. (2006) citesthe laboratory experiments by Haimson (2003) and Baudet al. (2004) and concludes: these test results “demonstratethat compaction bands would have rectangular geometries.”We find however, an elliptical approximation of the com-paction band cross-section produces a better fit to the fielddata (Sternlof et al. (2005), see 2.1).

In comparing the 3D ellipsoidal inclusion and hetero-geneity models to the 2D tabular model, we set the param-eters such that the 3D model emulates the 2D model: theuni-axial eigenstrain ε∗zz = −(H−h)/H = −0.136; the aspectratio of the rectangular inclusion H/c = 8 × 10−4; and theaxial-ratios of the ellipsoid ay/ax = 20, az/ax = 8 × 10−4.Strictly speaking, ay should be infinite to match the 2D case,but we find as ay/ax ≥ 10, the result does not vary signifi-cantly.

The rectangular and ellipsoidal inclusions have thesame elastic moduli as the matrix, i.e. Ci jkl = C∗i jkl, whilethe ellipsoidal heterogeneity has different moduli. The elas-tic moduli are sampled from porosity-modulus relations forgiven porosities (see 2.2). In the inclusion cases, we let bothinclusion and matrix have porosity 25%. In the heterogene-ity case (see field data by Sternlof et al. (2005)), we set theporosities to φi = 10% for the ellipsoid and φm = 25% forthe matrix. Similar to Sternlof et al. (2005) we set the re-mote stress σ∞ii = [−2 × 107,−2 × 107,−4 × 107]Pa, wherei = x, y, z.

Fig. 4 plots the normal stress σzz along the ax axis forthe 2D tabular model, elliptical inclusion model and ellipti-cal heterogeneity model. In both 2D tabular and ellipsoidal

−1.5 −1 −0.5 0 0.5 1 1.5−4

−2

0

2

4

6

8x 10

7

x/ax

σzz [

Pa

]

Katsman

Eshelby inclusion

Eshelby heterogeneity

Figure 4: Normal stress σzz along the x axis by Katsman et al.(2006), Eshelby’s solution for inclusion and Eshelby’s solutionfor heterogeneity.

inclusion cases the inside and outside stresses have oppositesigns, tensile inside and compressive outside. Even though

the prescribed plastic strain is stress free, the elastic ma-trix resists this uni-axial contraction and pulls back on theinclusion, inducing the tensile stress. The contraction ofthe inclusion causes a local contraction of the matrix distalfrom the tips, thereby inducing a compressive stress.

Unlike Katsman et al. (2006), the elliptical inclusionhas uniform inside stress given by Eq. (3). Also the stressimmediately distal from the tips is not singular. The 2D tab-ular region with a uniform uni-axial strain is equivalent totwo edge dislocations (Pollard and Fletcher (2006) p. 305),where the near-tip normal stresses on either side of the dis-location line are concentrated as σi j ∝ r−1 and have oppo-site signs. In contrast, the stress at the tip of the ellipsoidalinclusion is finite due to the bluntness of the tip profile (see4.1) .

For the ellipsoidal heterogeneity the stress curve is in-creased by a constant value relative to that of the ellipsoidalinclusion. The normal stress inside is less than the remotestress because the plastic deformation induces stress relief,but it remains compressive. Unlike the ellipsoidal inclusionthe heterogeneity is more consistent with the concept thatthe stress inside the region of a volume reduction (e.g. acompaction band) should be compressive if not zero.

2. Localized volume reduction in ellipsoidal structures

Based on the field data on single bands that show sig-nificant loss of porosity (Sternlof et al. (2005)), we suggestthat the ellipsoidal heterogeneity is more suitable for mod-eling compaction bands than the tabular or ellipsoidal in-clusions. In this section we review field data that documentthe cross-sectional shape of compaction bands and justifyusing the ellipsoidal shape. Also, we relate the changes inelastic moduli to the porosity losses (Avseth et al. (2005))and justify using the Eshelby’s heterogeneity model.

2.1. Cross-sectional shape of compaction bands: field dataSternlof et al. (2005) report data on the measured thick-

ness distributions of sixteen compaction bands exposed tipto tip in outcrops of Astec Sandstone in the Valley of FireState Park, NV. More than 1700 measurements on bandsranging in length from 1m to 62m show that a typical com-paction band is thicker in the middle and tapers toward thetip. In Fig. 16, we apply elliptical and piece-wise linear fitsto the length-width relations for eight measured bands. Forsome of the compaction bands (left column in Fig. 16), theelliptical fit performs better, i.e. has higher correlation co-efficients, while for some bands (right column in Fig. 16)the linear fit performs better. The elliptical fit yields anaverage correlation coefficient of 0.89, and the linear fityields 0.86, meaning the elliptical fit performs slightly bet-ter. Also, the elliptical fit produces smooth band profilesthat are consistent with the field observation. Both ellip-tical and piece-wise linear fits perform better statisticallythan the tabular shapes (Katsman et al. (2006)), with thick-ness approximated as the mean measured values yieldingcorrelation coefficients of only about 0.15.


2.2. Elastic moduli of compaction bands and surroundings

The compaction bands from the Astec Sandstone thatwere examined petrographically (Sternlof et al. (2005))have mean porosity of 11.9 ± 3%, whereas the sandstoneout side the bands yields 24.5 ± 2.9%. This suggests thatthe material inside the bands could have different elasticmoduli than outside (Gassmann (1951)). The elastic mod-uli (bulk modulus K, Young’s modulus E) and Poisson’sratio ν as functions of porosity φ for dry (subscript d) andwet (subscript w) sandstones are (Avseth et al. (2005)):

Kd =2G(1 + νd)3(1 − νd)

, where G = G0(1 − φ/φ0),

Kw =Kd +1 − (Kd/K0)2

φ/Kl + (1 − φ)/K0 − Kd/K20

,

where K0 = Kd|φ=φ0 ,

Ed =3Kd(1 − 2νd), Ew = 3Kd(1 − 2νd),

νd =0.07, νw =z − 1

1 + 2z, where z =

3Kw

2G,

(4)

where φ0 = 0.4 is the maximum porosity of the sandstone,G0 = 44[GPa] is the shear modulus of solid quartz, andKl = 2.2[GPa] is the effective bulk modulus of water.

Fig. 5 plots the moduli and Poisson’s ratio of wet sand-stone as functions of porosity φ. Given these variations inelastic moduli and the measured reductions in porosity, acompaction band is more appropriately modeled as a het-erogeneity problem, using the equivalent inclusion method(see 1.2) than as an inclusion problem. For demonstrationpurpose, we choose the curves of wet sandstone to samplethe elastic moduli and Poisson’s ratios for given porosities.As a result, the porosity change has an impact on both elas-tic moduli and Poisson’s ratio.

In the following sections, we set the axial ratio, az/ax =

10−3 √axay (given the thickness-to-length ratios measuredby Sternlof et al. (2005)). The elastic moduli are sampledfrom Fig. 5 for porosities inside and outside the compactionband φh = 10% and φm = 25% respectively. The resultingelastic moduli are Em = 37.31[GPa], νm = 0.13 for thematrix and Eh = 71.52[GPa], νh = 0.084 for the hetero-geneity.

2.3. Plastic strain in compaction bands: heuristic esti-mates

The initial eigenstrain, ε pi j, is defined as the plastic

strain caused by the compaction revealed, e.g. by poros-ity losses, from the field samples (Sternlof et al. (2005)).Resolving this plastic strain from the porosity change ishowever not a straight forward matter. The three principalcomponents ε p

ii can be related to the porosity change by:

(1 + εpxx)(1 + ε

pyy)(1 + ε

pzz) = 1 − (φm − φh) (5)

To determine all the six components of ε pi j, we need five

equations in addition to Eq. (5).

0 0.1 0.2 0.3 0.40

20

40

60

80

100

φ [−]

[GP

a]

Ed

Ew

Kd

Kw

0 0.1 0.2 0.3 0.40.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

[−]

νd

νw

Figure 5: Bulk modulus Kd(w), Young’s modulus Ed(w) (upper)and Poisson’s ratio νd(w) (lower) as functions of porosity φ fordry and wet sandstones.


In 5.1, we discuss a general plastic strain with normaland shear components, by introducing a weakening shearmodulus. For now however, we focus on pure compactionbands, i.e. the ones formed without plastic shear deforma-tion, Eichhubl et al. (2010). The strain components areε

pi j = 0, for i , j, which reduces the unknowns to three.

Still, two additional equations are needed to resolve the tri-axial strain.

An approach in previous research, e.g. Katsman et al.(2006) and Sternlof et al. (2005), is to assume a uni-axialplastic strain that is non-zero in the compaction direction,e.g. ε p

zz , 0 and the other components are zero. Eq. (5) isthen reduced to:

1 + εpzz = 1 − (φm − φh). (6)

For the parameters given in 2.2, the resulting uni-axialstrain is ε p

zz = −0.15.If we assume an isotropic tri-axial strain such that ε p

xx =

εpyy = ε

pzz, Eq. (5) becomes:

(1 + εpzz)3 = 1 − (φm − φh). (7)

The resulting tri-axial strain is then εpxx = ε

pyy = ε

pzz =

−0.0527.The thin section photos of compaction bands (from

Sternlof et al. (2005) and Eichhubl et al. (2010)) suggestthat the plastic deformation is indeed tri-axial, but more no-ticeable in the compaction (z) direction, e.g. ε

pzz < ε

pxx ≈

εpyy < 0. Neither the uni-axial strain nor the isotropic tri-

axial strain can capture this plastic deformation thoroughly.We propose an iterative algorithm that takes the tip-

incipient stress and the plastic strain as input and output ina loop and seeks a convergent tri-axial strain. The rationaleis demonstrate in Fig. 6. As a compaction band forms andpropagates, the tip-incipient matrix will join the band byundergoing the same plastic strain. The tip-incipient stressis responsible for such a plastic change. Here we formulatesome heuristic relationships between the tip stress and theplastic strain.

One simple relationship is linear by which we scale themagnitudes of three principal strains ε p

ii as:

εpxx =ε

pzzσxx − σ

c

σzz − σc H(σxx − σc),

εpyy =ε

pzzσyy − σ

c

σzz − σc H(σxx − σc),

(8)

where σii are the average principal stresses of the bandtip. σc is the critical principal stress above which plasticcompaction deformations will happen, H(·) is a heavy-sidefunction that makes the strain state zero when σii < σ

c.We use average values of N discrete tip-incipient points

to approximate σi j by:

σi j =

∮σiids∮

ds≈

∑Nm σ

(m)i j

2πkmN∑N

m2π

kmN

=

∑Nm σ

(m)i j

1km∑N

m1

km

, (9)

εxxp

εyyp

σzz

σxx

σyy

σxx

σyy

σxx

σyy

σxx

εxxp

εzzp

tip tip’(m+1)

(m+1)

(m)

(m)

(m−1)

(m−1)θ

(m−1)

(m−1)

Figure 6: When the band tip-line advances, marked by thedashed lines, the tip-incipient areas under stress σm

i j will jointhe compaction band by undergoing a plastic strain ε p

i j.

where σ(m)i j and km are tip-line stress and curvature of mth

discrete point, see Fig. 6 (see Fig. 6).With Eq. (5), Eq. (8) and Eq. (9), the principal plastic

strains ε pii can be resolved. When the new plastic strain is

fed back to the model, a new tip stress is produced. Bysolving for ε p

ii and σi j iteratively we seek the convergent tipstress and plastic strain.

If we feed the new plastic strain directly to the iteration,the results will oscillate and never converge. Therefore, wecombine the old plastic strain and the new one with a damp-ing factor α ∈ [0, 1]. The flow diagram for such a algorithmis given in Fig. 7.

We use the uni-axial strain and tri-axial strain given byEq. (6) and Eq. (7) as the initial guesses. For the modelparameter given in 2.2 and damping factor α = 0.3, theconvergent curves of the three principal strains are givenin Fig. 8. The results suggest that no matter which initialcondition we take, the strains always converge to the samethree values. The resulting plastic strain is tri-axial with thenormal component εzz greater than the other two principalcomponents by a factor of 10 suggesting the uni-axial guessis closer to the final result.

We plot the three principal plastic strains resulting fromthis iterative algorithm for different axial ratios ay/ax inFig. 9. When ay/ax changes from 0.2 to 0.5, i.e. the ma-jor and minor semi-axes switch positions, ε p

xx and ε pyy switch

values. The variation of ε pzz (within 0.8%) is less than vari-

ations of ε pxx(yy) (within 50%). This suggests that it is not

unreasonable to approximate a pure compaction band using


εpiiinitial guess for

(n)ijσ

p,(n−1)ij=ε

p,(n)ijε −ε

p,(n−1)ijij

p,(n))+α(ε

end

solve for

solve forp,(n)ijε

converge?

yes

n=n+1

Figure 7: Iterative scheme resolving the tip-incipient stress σi j

and plastic strain ε pi j.

a uni-axial plastic strain.

3. Compaction bands under symmetric remote stress

Arrays of sub-parallel compaction bands crop out inthe upper half of the Aztec Sandstone in the Valley of FireState Park (Fig. 17 A by Sternlof et al. (2005)). Theirlengths vary up to more than 100m and spacings vary upto more than 10m. In some cases the bands are approxi-mately planar with a regular spacing from a few decime-ters to a few meters, while in other cases they form a wavypattern (Fig. 17 B). Band segments may curve in prox-imity to adjacent segments forming eye shaped structures(Fig. 17 C), or otherwise form closely spaced parallel traces(Fig. 17 D). Despite these local irregularities in many out-crops, the bands form a symmetric set with approximatelyparallel members. This motivates consideration of a modelbased on the Eshelby heterogeneity under symmetric re-mote stresses.

In the following examples we examine compactionband modeled as an ellipsoidal heterogeneity with az ax

and az ay subjected to a remote stress (Sternlof et al.(2005)) σ∞ii = [−2, −2, −4] × 107Pa, where i = x, y, z.

2 4 6 8 10 12 14 16 18 20−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

number of iterationstr

ain

ay/a

x=0.5, φ

h=0.1, φ

m=0.25, σ

ii

∞=−[2, 2, 4]×10

7[Pa], σ

c=−4×10

7[Pa]

εp

xx, uni

εp

zz, uni

εp

yy, uni

εp

xx, tri

εp

zz, tri

εp

yy, tri

Figure 8: Convergent behaviors of the three principal plasticstrains ε p

ii for different guesses, i.e. uni-axial and tri-axial.

The initial plastic strain (ε pi j) within the heterogeneity is de-

termined by the iterative algorithm in Fig. 7 with all shear(off-diagonal) components zero. The elastic constants aredetermined as in 2.2 for the heterogeneity and matrix.

We evaluate the elastic fields in two different observa-tion grids illustrated in Fig. 10. On the cross-section grid(e.g. in (x−o−y) plane) we illustrate how the stress changesalong the tip-line, and on the near-tip grid we focus on stressperturbation around the tip.

3.1. Three-dimensionality

In previous research, Healy (2009) and Reches (1998)evaluated the special case of the spheroidal inhomo-geneities, where two of the three semi-axes are equal, ax =

ay , az. Under symmetric remote stress states (princi-pal stresses parallel to the semi-axes of the ellipsoid), theresulting stress is uniformly distributed along the circulartip line. The three-dimensionality of the ellipsoidal hetero-geneity allows one to investigate the stress variation alongthe tip when ay/ax , 1 (Meng et al. (2011)).

We plot the normalized major (compressive) principalstress σ3/σ

∞3 for different axial-ratio ay/ax = [0.5 1 2 20]

about the tip (at longitude θ = 0, i.e. an end of the semi-axisax) on the cross-section plane y = 0 (first row in Fig. 18).As the ellipsoid stretches in the ay direction, the near-tip(θ = 0) stress increases, while the near-flank stress de-creases. This indicates that the regions near a band tip aremore likely to form new bands. Also, a band is more likelyto propagate from the tip with a shorter axis (e.g. ax in thefourth column of Fig. 18). Furthermore, new parallel bandsare more likely to form adjacent to a band with less axialratio (e.g. first column Fig. 21).


0 2 4 6 8 10−0.016

−0.015

−0.014

−0.013

−0.012

−0.011

−0.01

ay/a

x

str

ain

φh=0.1, φ

m=0.25, σ

ii

∞=−[2, 2, 4]×10

7[Pa], σ

c=−4×10

7[Pa]

0 2 4 6 8 10−0.127

−0.1268

−0.1266

−0.1264

−0.1262

−0.126

−0.1258

−0.1256

εp

xx

εp

yy

εp

zz

Figure 9: Principal plastic strain εpii for different axial-ratio

ay/ax.

We plot the normalized principal stress about the tipat θ = 0 on the circle (o′, r = 10−2 √axay, λ) (first rowin Fig. 19). The bi-modal distributed stress is symmetricin λ for a symmetric remote stress, but shifts somewhatwhile retaining the bimodal character for asymmetric re-mote stresses. The stress concentration is greater at theends of the minor semi-axis than at the ends of the majorsemi-axis. Note as ay/ax increases from 0.5 to above 1, ax

becomes the minor semi-axis from the major semi-axis.For a fixed axial-ratio ay/ax = 0.5 we plot the nor-

malized principal stress and shear stress σ3−σ1σ∞3 −σ

∞1

on the bandplane z = 0 (first row in Fig. 20) The thicker stress contoursin ay direction indicates a greater stress concentration. Asa result, one would expect an elliptical compaction band topropagate into less eccentric geometries that tend towardcircular, i.e. ax = ay.

C σ ij( ijkl, )

nt

nn

σrr

σλλ

σrλ

y

x

*ijkl,( )εij

pC

tip−line

θ

z

xo

nr

o’

A

B

z

y

Figure 10: A, Cartesian coordinate on cross-section planes ofthe ellipsoidal heterogeneity.B, polar coordinate (λ, r) aboutthe ellipsoid tip-line at longitude angle θ.

3.2. The influence region of a pure compaction band

Because of the stress relieved by the plastic deforma-tion (Fig. 18), the rock on the flanks of a pure compactionband is under less compressive stress than the remote field,whereas in front of the tips it has a greater compression.The concentration provides the stress required for propaga-tion (see 2.3), but here we focus on the diminution.

For another compaction band to form at some distanceaway from the flank of an existing one, the compressivestress there must exceed the critical value σc. The closespacing of the sub-parallel compaction bands observed inthe field (Fig. 17 by Sternlof et al. (2005)) indicate that thestress diminution on a band flank must be overcome easilyby an increasing tectonic stress.

We use a remote stress that is increasing in the nor-mal direction to simulate the tectonic process that wouldcreate a compaction band set. The normal principal stressσzz along z-axis under the increasing σ∞zz is plotted inFig. 11. When the normal remote stress increases slightly,


0 0.2 0.4 0.6 0.8 1−4.3

−4.2

−4.1

−4

−3.9

−3.8

−3.7

−3.6

−3.5x 10

7

z/(ax⋅ a

y)1/2

σzz [

Pa

]

ay/a

x=0.5, σ

∞

yy=σ

∞

xx=−20×10

6[Pa], x=y=0, z∈[0 1]⋅(a

x⋅ a

y)1/2

σ∞

zz=−40×10

6[Pa]

−41×106

−42×106

−43×106

−44×106

σc

Figure 11: Normal stress σzz evaluated along z-axis when theremote normal stress σ∞zz increases by up to 10%

e.g. within 10%, the compaction condition σzz > σc comes

dramatically closer to the band flank. This indicates thatwhile the accommodation capability of a single compactionband is quickly consumed in a tectonic process, so a closelyspaced parallel band (or band set) will form to further ac-commodate the remote stress build up.

3.3. Propagation of a pure compaction band

We investigate propagation of the compaction bandunder symmetric loading based on the near-tip principalstresses (first rows of Fig. 18 and Fig. 19). Such a near-tip stress distribution is similar to that of a mode I crackWilliams (1957), but of opposite sign (compressive insteadof tensile).

The bimodal distribution raises a question: why docompaction bands choose not to branch following the twomaximum compression planes, but rather propagate in be-tween these planes, e.g. λ = 0, along a strait path? This canbe explained by asserting that a compaction band can onlypropagate from its tip. This assertion has been employedby Erdogan and Sih (1963) in their study of mixed-mode III crack propagation.

For each of the principal (compressive) stresses σ3evaluated on the polar coordinates (o′, r, λ), there is aplane L passing the evaluation point and perpendicular toσ3. Among these planes there is a special one in betweenthe two planes associated with the two peak compressivestresses that contains the origin o′, i.e. |o′o′′| = 0, whereo′′ is the perpendicular projection of o′ on the plane L. Bythe forgoing assertion, this special plane contains the nextincrement of the propagating band originated from o′. For

symmetric loadings the special plane aligns with the planeof the band, e.g. the x − o − y plane.

4. Compaction bands under asymmetric remotestresses

Under symmetric remote stress, a pure compactionband, once formed, will tend to propagate in plane, as de-scribed in section 3 or with a wavy pattern about this plane.By observations however, some bands offset the sedimen-tary layering or older bands (Fig. 12) indicating they ac-commodated some shearing Eichhubl et al. (2010). This

Figure 12: Compaction bands penetrating sedimentary layers(upper); compaction bands intersect each other (lower).

suggests that in the process of a compaction band formationand propagation, the major principal (compressive) stressplane may not align well with the band plane. We modelthis miss alignment by rotating the remote stress σ∞i j aroundthe y-axis by an angle ηy.

We assume that a compaction band is already formedwithin the major principal stress plane, and its plastic strainε

pi j will not change with the remote stress rotations. Thus,

we let the band have a fixed plastic strain generated underthe symmetric remote stresses (produced by the iteration inFig. 7). For this plastic strain and different rotation angle ηy,


we evaluate the near-tip stress fields and try to relate themwith the observed band geometries.

In addition to the model parameters given in 2.2, weinvestigate the axial ratio ay/ax = 0.5 case in more detail.The remote stress rotation around the y-axis would createthe local stress at [ax 0 0] analogous to a crack tip undermixed mode I II loading (Lawn and Wilshaw (1975), chap-ter 3).

4.1. Three-dimensionality

Similarly to the symmetric case, We plot the near-tipnormalized principal stresses under different asymmetricloadings, i.e. for the rotation angel ηy = [15, 30, 45]o,on the Cartesian and circular coordinates (second throughfourth rows in Fig. 18 and Fig. 19). The bi-modal dis-tributions tilt following the remote stress rotation, andthe magnitudes slightly decrease. The resulting in-bandstresses also decrease. This Indicates that under asymmet-ric stresses a compaction band tends to propagate off-plane,however the tendency is slightly less than under a symmet-ric loading with the same magnitude.

We plot the normalized principal stress and shear stresson the band plane z = for a fixed axial-ratio ay/ax = 0.5(second through fourth rows in Fig. 19). The stress contoursshrink with the remote stress rotation, and the magnitudecontrast in ax and ay directions becomes less. This suggeststhat the asymmetric remote stress state may not favor thein-plane band propagation.

For ay/ax = 0.5, we zoom in to the tip at θ = 0 andplot the normalized principal stress σ3/σ

∞3 (first column

in Fig. 21). To have a quantitative indication of the off-plane growth tendency we plot the normalized circumfer-ential stress σλλ/σ∞3 with respected to the tip tangent vec-tor nt = [0 0 1] (second row in Fig. 20). The stress contourtilts following the rotation. This would be consistent withan off-plane growth tendency which is better indicated bythe maximum traces of the circumferential stress σλλ suchthat the off-plane angle is roughly equal to ηy.

4.2. Propagation

If we analyze the near-tip stress analogous to a mode III crack (see Erdogan and Sih (1963)), the band, with cer-tain conditions met, will propagate in the maximum circum-ferential stress direction. For example the tip (second andthird rows in Fig. 21) would kink along the maximum traceof σλλ.

Field observations by Sternlof et al. (2005) and Eich-hubl et al. (2010) document that compaction bands can havewavy or chevron geometries with decimeter wave lengths(Fig. 12). It is unlikely however, that the remote stress couldrotate this frequently. The fact that the tip-incipient ma-terial undergoes a plastic strain and joins the compactionband, and that the stress along the tip-line is finite moti-vate a propagation criterion different than the classic frac-ture mechanics criterion for a crack.

Eichhubl et al. (2010) suggest that shear stress mayenhance the compaction band formation. In Fig. 22, weplot the near-tip maximum shear stress and circumferen-tial shear stress normalized by the remote maximum shearstress on the near-tip grid (at longitude θ = 0). The maxi-mum shear stress has distributions similar to the major prin-cipal stress in Fig. 21. Comparing the magnitude of thesestresses (see columns) suggests that the band tip causesgreater concentration of the shear stress than of the prin-cipal stress.

When a compaction band forms, the band plane maynot be aligned with the plane normal to the greatest com-pression precisely, that is because the Aztec Sandstone isanisotropic due to cross bed layering and heterogeneitiesdue to sand dune boundaries. As a result, the stress σλλ(σrλ) will not be symmetric (anti-symmetric), e.g. about ax

in Fig. 21 (Fig. 22). One of the two anti symmetric wingsof σrλ shrinks or becomes weaker. This, combined withkinked σλλ, could enhance the compaction about the other(stronger) wing.

If the shear stress indeed enhances the compaction de-formation, the direction in which the band tip would ad-vance off-plane is likely to be in between the maximumtrace of σλλ and the stronger wing of σrλ. This leads to akink angle that overshoots the off-set angle, e.g. ηy. Whenthe new tip grows, the remote stress will correct this over-shoot by turning the tip back to its former orientation. Inthis way, the compaction band would swing back and forthforming a wavy shape.

The stress fields calculated using Eshelby’s hetero-geneity combined with some propagation criteria can beused to predict the incipient direction of out-of-plane propa-gation. Unfortunately because of its geometrical limitation,if the compaction band develops into some non-ellipsoidalgeometry, Eshelby’s solution will cease to apply.

5. Plastic shear strain

Plastic shear deformations are observed by Eichhublet al. (2010) along some planar bands in Valley of Fire. Inthis section we discuss how to model a failure-like shear de-formation. Also, by the field observations of Sternlof et al.(2005) and Eichhubl et al. (2010), we assume a fixed shear-normal plastic strain ratio, and investigate the shear-normalstress combinations at the tip that may in turn favor thisshear-enhanced compaction band.

5.1. Increasing shear strain

We could scale the shear components of ε pi j to the

amount that the average tip shear stresses σi j exceeds a crit-ical shear stress σc

sh similarly to Eq. (8).Unlike the principal plastic strains however, the shear

strains do not affect the volume. There is no analogous re-lation as Eq. (5) to provide enough equations that wouldallow one to resolve the shear strains. To overcome this,we simulate the shear deformation as a failure-like process


with a weakening shear modulus. This allows one to relatethe shear strain and stress explicitly.

The plastic shear deformation process is demonstratedby a stress-strain cure in Fig. 13. In this process, the tip-

σxy

εxy

Gm

Gp

εxyp

Gh

Figure 13: Shear deformation stress-strain curve: elasticshear for the near-tip material with shear modulus Gm; plasticshear, with effective shear modulus Gp, upon the compactionband formation; elastic shear, with modulus Gh, of the in-bandmaterial

incipient material would undergo a failure like shear defor-mation which can be simulated by a stress-strain relationwith a modulus much lower than the shear moduli (Gh andGm) of the heterogeneity and matrix.

We scale the plastic shear strains of the band by:

εpi j = Gp(σi j − σ

csh)H(σi j − σ

csh), for i , j, (10)

where, σcsh is a critical shear stress above which plastic

shears will happen. The effective band shear modulus isscaled to the average shear modulus Gp = c(Gh +Gm)/2 fora decreasing c ∈ (0, 1).

Adding Eq. (10) to the iteration in Fig. 6, we can re-solve all six components of the plastic strain ε p

i j.If we set the critical shear stress σc

sh as the maximumremote shear stress (σ∞3 − σ

∞1 )/2, the resulting shear com-

ponents of ε pi j are all zeros. If we consider that the existence

of the band tip would cause the incipient area to shear plas-tically at a stress that is lower than the maximum remoteshear stress, the resulting plastic shear strain ε

pxy could be

non zero.Analogous to Fig. 8, we plot the convergence curves of

the full plastic strain in Fig. 14, with σcsh set to be 80% of

the remote maximum shear stress (while σc = σ∞3 ). Underthe rotated remote stress that creates an analogous modeI II shear (σxy) at the ends of ax, the only non-zero shearcomponent of the plastic strain is ε p

xy.

0 5 10 15 20 25 30

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

number of iteration

str

ain

ax/a

y=.5, σ

ii

∞=−[2 2 4]×10

7[Pa] rotates about y−axis by π/4,

Gp=.125(Gh+G

m)/2, σ

c=−4×10

7[Pa], σ

sh=8×10

6[Pa]

εp

xx

εp

yy

εp

zz

εp

xy

εp

xz

εp

yz

εp

xz>0

Figure 14: For Gp = 18 (Gm + Gh)/2, all six components of ε p

i jevolving along the iteration.

We plot the plastic strains (ε pi j) and tip stresses (σi jby

Eq. 9) during the shear deformation (i.e. for a decreasingGp) in Fig. 23. As the effective shear modulus Gp de-creases, both the non zero shear strain ε

pxy and stress σxy

increase dramatically, while other shear components stayzero. σxy not only stays above the critical shear stress(σc

sh = 8MPa) but also stays above the maximum remoteshear stress ((σ∞3 − σ

∞1 )/2 = 10MPa).

This suggests that once the plastic shear deformationhappens, for a critical shear stress σh

sh < σ∞3 − σ∞1 )/2, the

resulting tip stress σxy will easily penetrate σ∞3 − σ∞1 )/2

and in return amplify the shear strain, e.g. by Eq. 10, untilthe end of the plastic deformation (see Fig. 13). This selfenhancing process could result in what referred to (in Eich-hubl et al. (2010)) as shear enhanced compaction bands.

The shear deformation has barely any affect on the prin-cipal stresses and strains in the ellipsoidal heterogeneity.

5.2. Fixed shear strain

Eichhubl et al. (2010) estimate that the plastic shearstrain is roughly equal to the normal strain that is about10%. The shear-enhanced compaction bands are mostlyplanar, which means only the in-plane tip-incipient areaswould join the band as the tip advances. The band plane isin about 45o to the greatest compression plane.

We set the asymmetric remote stress with ηy = 45o,which would cause non-zero ε p

xz, mentioned in 5.1. We im-pose a linear relation ε

pxz = cε p

zz where the shear-normalstrain ratio c ∈ [0.1, 1.5], with other shear componentsof ε p

i j zero when we solve for the convergent strains, as in2.3. For different c we investigate what is the average shear-normal stress ratio σxz/σzz along the tip.

For a disk shaped compaction band, i.e. ax = ay (otherparameters set in 2.2), we plot the normalized stress com-


ponents of σi j and the shear-normal stress ratio as func-tions of ε p

xz/εpzz (Fig. 15). The shear stress σxz and the ratio

σxz/σzz vary linearly with the strain ratio. If, by Eichhublet al. (2010), ε p

xz/εpzz ≈ 1, the corresponding tip stress ratio

σxz/σzz ≈ 1.3. This means that the in-plane shear-enhancedcompaction band has formed under the shear-compressionratio about 1.3.

6. Conclusion and discussion

The Eshelby’s solution offers unique insight in model-ing the localized volume reductions, e.g. compaction andshear-enhanced compaction bands. By reviewing the previ-ous researches, we find the extra aspect ratio ay/ax enabledby the 3D model has significant impact on the near-tip stressdistribution.

The compaction bands are depicted as highly eccentricellipsoids which show more resemblance to the field obser-vations than the tabular structures in Katsman et al. (2006).Also, the model provides near-tip stress fields that are con-sistent with the band formation mechanism.

We formulate the truncated linear relationships be-tween the plastic strains and the tip-incipient stresses,Eq. (8) and Eq. (10). These heuristic relationships are sub-jected to improvements, as one explores more on the plasticstress-strain relationships.

The Eshelby’s heterogeneity model provides possibleexplanations for the wavy bands, and provides the shear-compression ratio under which shear-enhanced compactionbands are favored.

The three dimensionality, enabled by the axial ratioay/ax , 1 which makes the model more sophisticated thanprevious Eshelby evaluations, has significant impact on thetip stress distributions.

Acknowledgments: We thank Atilla Aydin for intro-ducing us to compaction bands in the field; Peter Eichhublfor the concept of shear-enhanced compaction bands; KurtSternlof and John Rudnicki for field and theoretical guid-ance with respect to compaction bands. This research wassupported by grant DE-FG02-04ER15588 from the Depart-ment of Energy, Basic Energy Sciences Program and theStanford University Rock Fracture Project.

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D. Healy, Short note: Elastic field in 3d due to a spheroidal inclusion-matlab code for eshelby’s solution, Comput. Geosci. 35 (2009) 2170–2173.

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0 0.5 1 1.50

2

4

6

8

10

12

14

εp

xz/ε

p

zz

σij/σ

3∞

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

εp

xz/ε

p

zz

σxz/σ

zz

σxx

/σ3

∞

σyy

/σ3

∞

σzz

/σ3

∞

σxy

/σ3

∞

σxz

/σ3

∞

σyz

/σ3

∞

l

l

l

_

_

_

_

_

_

Figure 15: Normalized tip stresses σi j/σ∞3 (upper) and stress

ratio σxz/σzz (lower) as functions of the shear-compaction ra-tio ε p

xz/εpzz.


−1 −0.5 0 0.5 10

1

2

3

ce=0.9533, c

l=0.8517

x/ax

z [m

m]

−1 −0.5 0 0.5 10

5

10

15

ce=0.8701, c

l=0.8740

−1 −0.5 0 0.5 10

5

10

15

ce=0.9353, c

l=0.8703

−1 −0.5 0 0.5 10

5

10

ce=0.9503, c

l=0.9525

−1 −0.5 0 0.5 10

5

10

15

ce=0.9410, c

l=0.9041

−1 −0.5 0 0.5 10

5

10

15

ce=0.9292, c

l=0.9653

−1 −0.5 0 0.5 10

5

10

ce=0.9618, c

l=0.9433

−1 −0.5 0 0.5 10

5

10

ce=0.9182, c

l=0.9419

Figure 16: Compaction band thickness z as a function of normalized length x/ax; elliptical and piece-wise linear fits for thewidth-length relation and associated correlation coefficients, ce and cl.


Figure 17: A, pure compaction bands; B, pure compaction bands in wavy pattern; C, two band tips forming an eye shapedstructure resulted from band interaction; D, closely spaced parallel compaction bands.


−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

x/ax

ay/a

x=.5

z/a

x

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

−1 0 1

−1

0

1

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

2

−1 0 1

−1

0

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

20

−1 0 1

−1

0

1

−1 0 1

−1

0

1

−1 0 1

−1

0

1

0.5

1

1.5

2

15o

45o

30o

σ3/σ

3

∞

ηy=0

Figure 18: Normalized principal stress σ3/σ∞3 evaluated the cross-section plane y = 0 for axial-ratio ay/ax = [0.5 1 2 20] (column),

under a pure tensile remote stress (ηy = 0) and under the stress rotated around y-axis by ηy = [15 30 45]o (row)


−100 0 100

1

1.5

2

λ[o]

σ3/σ

3∞

r=10−2

(axa

y)1/2

, ay/a

x=.5

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

1

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

20

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

−100 0 100

1

1.5

2

ηy=0

15o

30o

45o

Figure 19: Normalized principal stress σ3/σ∞3 evaluated on a circle (r = 10−2 √axay, λ) around the tip at longitude θ = 0 for

axial-ratio ay/ax = [0.5 1 2 20] (column), under a pure tensile remote stress (ηy = 0) and under the stress rotated around y-axis byηy = [15 30 45]o (row)


−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

σ3/σ

3

∞

x/ax

y/a

x

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

(σ3−σ

1)/(σ

∞

3−σ

∞

1)

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

σ3/σ

3

∞, η

y=15

o

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

(σ3−σ

1)/(σ

∞

3−σ

∞

1), η

y=15

o

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

σ3/σ

3

∞, η

y=45

o

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

(σ3−σ

1)/(σ

∞

3−σ

∞

1), η

y=45

o

1

1.05

1.1

1.15

1.2

1.25

1.3

1

1.05

1.1

1.15

1.2

1.25

1.3

Figure 20: Normalized major principal stress σ3/σ∞3 and maximum shear stress σ3−σ1

σ∞3 −σ∞1

evaluated on the cross-section plane z = 0under remote stress σ∞ii = −[20 20 40]MPa and under this stress rotated around y-axial by angle ηy = [15o 45o].


Figure 21: Normalized major principal stress σ3/σ∞3 and circumferential stress σλλ/σ

∞3 evaluated on the near-tip grid about

[ax 0 0] (i.e. an end of the long semi axis) under remote stress σ∞ii = −[20 20 40]MPa and under this stress rotated round y-axis byangle ηy = [15o 45o].


Figure 22: Near-tip maximum shear stresses (left) and circumferential shear stresses (right) normalized by remote maximumshear stress under remote stress σp

ii = −[20 20 40]MPa and under this stress rotated about y-axis for angle ηy = [15o 45o].


0.1 0.2 0.3 0.4 0.5−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

2Gp/(Gh+G

m)

εp ii

ay/a

x=.5, σ

∞

ii=−[2 2 4]×10

7[Pa] rotates about y−axis by π/4

0.1 0.2 0.3 0.4 0.5−3

−2.5

−2

−1.5

−1

−0.5

0x 10

8

2Gp/(Gh+G

m)

σtip

ii [

Pa

]

0.1 0.2 0.3 0.4 0.5−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−3

2Gp/(Gh+G

m)

εp ij

0.1 0.2 0.3 0.4 0.5

0

5

10

15

20x 10

6

2Gp/(Gh+G

m)

σijtip [

Pa

]

εp

xx

εp

zz

εp

zz

εp

xz

εp

xy

εp

yz

σxz

σxy

σyz

σc

sh

σxx

σzz

σyy

σc

shear

shear

_

_

_

_

_

_

Figure 23: For shear modulus Gp(Gm+Gh)/2 ∈ [1/8, 1/2], plastic strain ε p

i j (left) and tip stress σi j (right).


eshelby’s solutions for ellipsoidal inclusions and ...xy742fz8741/rfp_2012_meng... · eshelby’s...

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