Comput MechDOI 10.1007/s00466-013-0947-3

ORIGINAL PAPER

Determination of an image point on a surface based on a πplane-based algorithm

Sang Inn Woo · Rodrigo Salgado

Received: 20 April 2013 / Accepted: 13 November 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Constitutive models of particulate materials oftenrely on distances between the current stress state in stressspace and various surfaces. Examples of these surfacesinclude the bounding surface and the dilatancy surface. Thispaper proposes a rigorous method for determination of dis-tance to a surface in stress space. It starts by examining oper-ations on stress variables defined in the π plane. Algorithmsfor determination of an image point on a surface are then pre-sented as a function of the location of the current stress statewith respect to the surface. For points within the surface, thebisection method is used; otherwise, the secant method isused. The paper shows that implementation of the proposedalgorithm locates the image point on a surface in stress spacewith accuracy and rigor, providing an accurate measure ofthe distance to the surface that can be used in hardening orflow rules.

Keywords Particulate materials · Bounding surfaceplasticity models · Deviatoric plane · Loading path ·Image point

1 Motivation

1.1 Surfaces in bounding plasticity models

Phenomenological constitutive models generally rely onsurfaces in stress space to help simulate aspects of the

S. I. Woo (B) · R. SalgadoSchool of Civil Engineering, Purdue University, 550 Stadium Malldrive, West Lafayette, IN 47907-1284, USAe-mail: woo4@purdue.edu

R. Salgadoe-mail: rodrigo@ecn.purdue.edu

mechanical response of materials. The commonly used sur-faces are: (1) the yield surface, which marks the change inresponse from elastic to plastic; (2) the bounding surface [1–6,8,10,11], which determines the rate at which a materialhardens or softens upon loading; and (3) the dilatancy sur-face [1–3,6,8,10], which is the locus of the points at whichthe plastic volumetric response of soils or particulate materi-als subjected to monotonic plastic shear deformation changesfrom contractive to dilative.

While traditional models were constructed based on a sin-gle surface (the yield surface) separating the plastic and elas-tic zones in stress space, some recently developed plasticitymodels use the concept of the bounding surface for steel [4],concrete [5,11], and soil [1–3,6,8,10]. The bounding surfaceis a surface within which the yield surface is constrained tomove [4]. Some plasticity models, called two-surface models[1–3,6,8,10], allow a stress state to exist marginally outsidethe bounding surface by disregarding the consistency con-dition on it; however, as soon as the stress steps outside thebounding surface, these models force the stress state to moveback to the bounding surface by setting the hardening (plas-tic) modulus to be negative. This follows from the fact thatthese models use the bounding surface to set the sign andmagnitude of the hardening modulus: the material is work-hardening when the stress state is inside the bounding sur-face and work-softening when the stress state is outside thebounding surface. Whether the consistency condition is ornot enforced at the bounding surface, the bounding surfaceis the final destination of the stress state as deformation pro-gresses.

In the bounding surface models, the magnitude of the hard-ening modulus is a function of the distance to the boundingsurface; the farther a stress state is from the bounding sur-face, the greater the hardening modulus is. The distance fromthe current stress state to the bounding surface is generally

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defined as the Frobenius norm of the subtraction of the cur-rent deviatoric stress from its image (conjugate) stress, usu-ally defined as the intersection of the projected loading pathwith the bounding surface. Thus, the key factor in the cal-culation of the hardening modulus in these models is thedetermination of the image stress on the bounding surface.

Dilatancy, a plastic volumetric response caused by plasticshear deformation, is an important feature of the mechanicalresponse of a granular material, such as sand, powder or grain.To simulate this, some plasticity models [1–3,6,8,10] defineanother surface, the dilatancy surface. The dilatancy surfaceis the locus of phase transformation points in stress space.Phase transformation is the change from plastic contraction toplastic dilation upon plastic shearing. In monotonic shearingstarting from the origin of the π plane, the dilatancy rateis positive inside and negative outside the dilatancy surface.As was true for the bounding surface, the key factor in thedetermination of the dilatancy rate is to locate the imagestress on the dilatancy surface.

1.2 Representation of stress and loading paths in the π

plane

The deviatoric stress components of every stress state in prin-cipal stress space and of every point of the yield, bound-ing and dilatancy surfaces are frequently mapped onto theπ plane, a plane defined in principal stress space by σp1 +σp2 + σp3 = 0 (i.e., I1 = σkk = 0) [7]. Figure 1a showsthe π plane in principal stress (σp1, σp2, and σp3) space,with σp1, σp2, and σp3 not defined in any specific order.The π plane is orthogonal to the hydrostatic axis (definedby σp1 = σp2 = σp3). Recalling the definition of the devi-atoric stress tensor, si j = σi j − 1/3 σkkδi j , the fact thatI1 = σkk = 0 on the π plane means that the deviatoricprincipal stresses (sp1, sp2, and sp3) are three independentvariables that can be plotted in the π plane. Thus, as illus-trated in Fig. 1b, there are three axes corresponding to thesethree variables in the π plane. A “vector” corresponding toa given stress deviator tensor s can be defined in the π planeby joining the origin of the plane to the point with the eigen-values sp1, sp2, and sp3 of the tensor s. We will refer to thisas the deviatoric stress vector. The π plane has the symme-try shown in Fig. 1b, with deviatoric stresses s1 through s6

able to express the same deviatoric stress state dependingon which of the three principal deviatoric stresses the axessp1, sp2 and sp3 are taken to represent.

The fact that tensors with as many as six distinct compo-nents are represented in π plane using only three variablesdetermines which operations are valid in it. For example,the double dot product (i.e., A:B = Ai j Bi j ) of two differ-ent deviatoric stress tensors, which has been used to get theprojection of a tensor onto another tensor [2,3,6,8,10], is

(a)

(b)

Fig. 1 The π plane a in principal stress space and b viewed in thedirection of the hydrostatic axis

not properly represented by the dot (inner) product betweenthe corresponding principal deviatoric stress vectors in theπ plane unless the two tensors have the same eigenvectors,in which case principal stress space (and the π plane) hasthe same meaning for both tensors. If the tensors do not havethe same eigenvectors, the eigenvalues do not mean the samething in space, and cannot be directly operated on. Thus, thedot product of two deviatoric stress vectors is not a validoperation in the π plane because it only produces the correcttensor components under very specific conditions.

To see this in greater detail, consider deviatoric stress ten-sors A and B, given below together with the correspond-ing principal deviatoric stress vectors ap and bp (where thesuperscript p designates a principal deviatoric stress) in theπ plane and principal deviatoric stress directions vp

aI and vpbI

(with I = 1, 2, 3):

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A =⎡⎣

1 0 00 −1 00 0 0

⎤⎦ , ap =

⎡⎣

1−10

⎤⎦ ,

vpa1 =

⎡⎣

100

⎤⎦ , vp

a2 =⎡⎣

010

⎤⎦ , vp

a3 =⎡⎣

001

⎤⎦ (1)

B =⎡⎣

0 1 01 0 00 0 0

⎤⎦ , bp =

⎡⎣

1−10

⎤⎦ ,

vpb1 =

√2

2

⎡⎣

110

⎤⎦ , vp

b2 =√

2

2

⎡⎣−110

⎤⎦ , vp

b3 =⎡⎣

001

⎤⎦ (2)

A and B have the same principal deviatoric stress vectorsap = bp in the π plane, but their principal deviatoric stressdirections (eigenvectors) vp

aI and vpbI are different. In the π

plane, both tensors are represented by the same point becausetheir principal deviatoric stress vectors are identical; their dotproduct as they appear in Eqs. (1) and (2) is 2. However, theeigenvectors of a tensor need not be specified in any specificorder; if we reverse the order of the first two eigenvectors ofB, we must accordingly switch the order of the components1 and −1 of the vector bp, which means the dot product ofap and bp is now −2. This shows that the dot product is notdefined in the π plane. Additionally, the double dot productbetween A and B is zero; consequently, if their deviatoricstress vectors in the π plane were valid representations ofthese tensors, they would be orthogonal, which they are not.

Consider now a third deviatoric stress tensor C:

C =⎡⎣

1 0 00 −0.5 00 0 −0.5

⎤⎦ , cp =

⎡⎣

1−0.5−0.5

⎤⎦ ,

vpc1 =

⎡⎣

100

⎤⎦ , vp

c2 =⎡⎣

010

⎤⎦ , vp

c3 =⎡⎣

001

⎤⎦ (3)

Tensor C has the same set of eigenvectors as tensor A. Thisallows us to take the dot product between their principal devi-atoric stress vectors ap and cp, which is 1.5. The double dotproduct between A and C is 1.5, which is identical to thedot product between ap and cp. This shows that, for the veryspecific case of tensors with the same eigenvectors, the dou-ble dot product can be represented by the dot product of thecorresponding eigenvalue vectors in the π plane.

Calculation of the distance between a current stress stateand a stress state (the image stress) on the bounding or dila-tancy surface is central to models relying on such surfaces todetermine hardening or dilatancy rates. In general, the dis-tance between two stress points cannot be calculated in the π

plane. To see this, consider the distance between two second-order tensors, which can be calculated as the Frobenius norm

of their subtraction. The distance LAB between the second-order tensors A and B discussed earlier is

L AB =√(

Ai j − Bi j) (

Ai j − Bi j) = √3 (4)

In contrast, the distance L pAB calculated in the π plane using

the eigenvalues of A and B is

L pAB =

√(ap1 − bp1

)2+ (ap2 − bp2

)2+ (ap3−bp3

)2 (5)

If the distance between two stress points could be calcu-lated in the π plane according to Eq. (5), L AB would be thesame as L p

AB . However, because the eigenvectors of A and Bare different, there is no correspondence between the eigen-values of A and B, and the possible values of L p

AB are 0,21/2, 51/2, 61/2, and 81/2 (depending on the order in whichwe take the eigenvalues in Eq. (5)), all of which are differentfrom L AB .

As for the dot product, for the specific case of two second-order tensors that have identical eigenvectors, the distancebetween two stress points can be calculated in the π plane.For example, the distance between the second order tensors Aand C can be calculated from their eigenvalues. In this case,both L AC and L p

AC are equal to 0.51/2. It is also possibleto calculate the distance between a hydrostatic vector andan arbitrary stress deviatoric tensor using the eigenvalues.All eigenvalues of a hydrostatic tensor are the same, thusevery direction in space is an eigenvector of the tensor, whichmeans that its eigenvectors coincide with the eigenvectors ofany tensor. Consequently, distances between any deviatorictensor s and a hydrostatic tensor is simply represented by theeigenvalues vector of the arbitrary deviatoric tensor (whichrepresents the distance of < sp1, sp2, sp3 > from the originof the π plane).

The following are valid operations in the π plane:

(1) addition or subtraction (as in si j ±�si j );(2) calculation of the distance from the origin to the current

deviatoric stress state, as in

l = √si j si j =√

2J2 (6)

(3) calculation of Lode’s angle, defined by

cos 3θ = √6si j s jkski(√

spqspq)3 =

3√

3

2

J3

(J2)32

(7)

where Lode’s angle ranges from zero to sixty degrees due tothe symmetry of the π plane discussed earlier.

These operations are also valid in p−q−θ space (wherep = σkk/3 is the mean stress; q = (3J2)

1/2 is the Misesstress, corresponding in the π plane to the distance from theorigin to the current deviatoric stress state; and θ is Lode’s

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Fig. 2 Loading path and projected loading paths in the π plane

angle, calculated from Eq. (7)) and in stress space (definedusing six different axes, one for each stress tensor compo-nent). However, not all the operations valid in stress space arevalid in the π plane or p−q−θ space. In the formulation ofconstitutive models, it is often advantageous to write expres-sions in terms of the deviatoric stress, and proper definitionof which operations are valid in the π plane then becomesessential.

Hardening or flow rules are expressed in terms of the dis-tance from the current stress state to an image stress on thebounding or dilatancy surface. The image stress is a functionof current stress state and current incremental loading direc-tion, which is projected towards the corresponding surface todetermine the image stress. The image stress does not dependon the actual (global) loading path, which is at the currentstate unknown. Figure 2 shows a hypothetical loading pathand the image stresses s(1)

b and s(2)b obtained on the load-

ing surface for two current stress states s(1)c and s(2)

c duringthe loading process. The loading direction is defined as a unitsecond-order tensor (the Frobenius norm of which is equal toone) aligned with the second-order tensor resulting from thesubtraction of the current stress from the kinematic hardeningtensor. For the two stress states shown in Fig. 2, the incre-mental loading directions are n(1) and n(2). The projectedloading path is the loading path that would be followed if thecurrent incremental loading direction were to be maintainedfrom the current stress all the way to the surface where animage stress is to be determined. The deviatoric stress tensorss(1)

c and s(2)c have different projected loading paths because

they have different stress components and incremental load-

ing directions; as a result, they have different image pointss(1)

b and s(2)b . The aim of the present paper is to define the

correct projected loading path in the π plane and propose arigorous methodology to determine the correct image stresson the bounding or dilatancy surface in the π plane and thedistance to the image stress from the current stress.

2 Projected loading path

In the remainder of the paper, we use geomechanics stresssign conventions and terminology. In geomechanics, normalstress (σi j , i = j) is positive in compression and a shearstress (σi j , i �= j) is positive if it works in the direction ofopening up the initially 90-degree angle between the planesof an element with faces parallel to the xi and x j axes, inthat order. The surfaces in this paper are assumed to be con-vex; convexity of a surface guarantees that there will be onlytwo image stress states along any straight line crossing thesurface (Fig. 3). A convex bounding surface is required forbounding-surface plasticity models to be stable numerically[6]. The image point is defined as the stress state that wouldbe reached if loading proceeded from the current stress statealong the projected loading path. The projected loading pathis defined by the current stress and the current incremen-tal loading direction. In the π plane, it can be expressed assc + ηn, where sc is the current deviatoric stress, n is theloading tensor, and η is a scale parameter.

The present paper shows that the current stress states andthe current incremental loading direction may be identicalin the π plane and yet have projected loading paths and,therefore, image points that are different. Additionally, thepaper develops an algorithm to account for this and avoid anerror in calculation of image stress points for use in hardeningand flow rules. Consider two cases showing this:

(1) a specimen initially subjected to a stress state such thatσ11 > σ22 = σ33 and a nonzero shear stress in the x1−x2

plane (σ12 > 0, σ23 = σ31 = 0) that is then subjected tosimple shear unloading (dσ12 < 0) in the x1− x2 plane ;

(2) a specimen initially subjected to three different normalstresses (σ11 > σ33 > σ22) with no shear stress in thereference planes that is then subjected to loading alongone of the deviatoric stress axes (dσ11 < 0, dσ22 >

0, dσ33 = 0) of the π plane.

An example of the first case can be constructed by con-sidering a material element initially subjected to the currentstress tensor:

σ(1)c =

⎡⎣

200 100 0100 100 0

0 0 100

⎤⎦ (8)

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(a)

(b)

Fig. 3 Bounding or dilatancy surface with a concave shape and b con-vex shape

The corresponding current deviatoric stress tensor is:

s(1)c =

⎡⎣

200/3 100 0100 −100/3 00 0 −100/3

⎤⎦ (9)

Consider now the loading direction given by:

n(1) = 1√2

⎡⎣

0 −1 0−1 0 00 0 0

⎤⎦ (10)

The deviatoric stress tensor corresponding to the stressobtained after addition of a stress increment with the direc-tion n(1) given by Eq. (10) may be written as:

s(1) = s(1)c + ηn(1)

=⎡⎣

200/3 100− η/√

2 0100− η/

√2 −100/3 0

0 0 −100/3

⎤⎦ (11)

where η is a positive scalar.

Fig. 4 Projected loading paths for case (1) and case (2) in the π plane

J (1)2 is expressed as:

J (1)2 =

1002

3+

(100− η√

2

)2

(12)

which means that J (1)2 is a quadratic function of η; it initially

decreases until it reaches a minimum value at η = 100√

2,where the deviatoric stress is on the sp1 axes (σ11 > σ22 =σ33, σi j = 0(i �= j)) of the π plane, as shown in Fig. 4. Onthat axis, the deviatoric stress is:

s(1)a =

⎡⎣

200/3 0 00 −100/3 00 0 −100/3

⎤⎦ (13)

Beyond the minimum at s(1)a , J (1)

2 starts to increase as η

increases. Figure 4 shows that the projected loading pathfor case (1) is a line that approaches axis sp1 in a directionof decreasing J2 and then reflects back from the axis at thesame angle with respect to the axis as that it approaches it at.

In the second example, a material element is consideredinitially subjected to the stress tensor:

σ(2)c = 50

⎡⎣

3+√5 0 00 3−√5 00 0 2

⎤⎦ (14)

with zero components everywhere but the diagonal; the prin-cipal stresses of case (2) are exactly the same as those of thecurrent stress tensor of case (1). The corresponding deviatoricstress tensor is:

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s(2)c = 50

⎡⎣

1/3+√5 0 00 1/3−√5 00 0 −2/3

⎤⎦ (15)

Consider now the incremental loading direction given by:

n(2) = 1√2

⎡⎣−1 0 00 1 00 0 0

⎤⎦ (16)

This incremental loading direction tensor has the same eigen-values as the incremental loading direction tensor in case (1).Consequently, both the initial stress tensor and the incremen-tal loading direction in cases (1) and (2) are expressed by thesame initial point and same incremental loading directionvector in the π plane (see Fig. 4).

In the same manner as case (1), a positive scalar η allowsthe writing of the deviatoric stress for case (2) after applica-tion of an incremental stress as:

s(2) = s(2)c + ηn(2)

=⎡⎣

50/3+ 50√

5− η/√

2 0 00 50/3− 50

√5+ η/

√2 0

0 0 −100/3

⎤⎦

(17)

The stress deviator J (2)2 corresponding to s(2) is given by:

J (2)2 =

502

3+

(50√

5− η√2

)2

(18)

The deviatoric stress s(2) meets the axis corresponding toσ11 > σ22 = σ33 and σi j = 0(i �= j) when η = 50

√10 −

50√

2. The deviatoric stress s(2)a1 at this axis is identical to

the deviatoric stress s(1)a of case (1). In case (1), however, s(1)

a

corresponded to minimum J2. That is not so for case (2); J (2)2

achieves its minimum value only at η = 50√

10 (which doesnot lie on the sp1 axis), where the deviatoric stress is:

s(2)m =

⎡⎣

50/3 0 00 50/3 00 0 −100/3

⎤⎦ (19)

Note that the deviatoric stress s(2)m is orthogonal to the incre-

mental loading direction n(2). Beyond s(2)m , J (2)

2 starts toincrease, and the deviatoric stress crosses another axis, sp2,(with σ22 > σ11 = σ33, σi j = 0(i �= j)) of the π plane atη = 50

√10+ 50

√2 (see Fig. 4), where the deviatoric stress

s(2)a2 is:

s(2)a2 =

⎡⎣−100/3 0 0

0 200/3 00 0 −100/3

⎤⎦ (20)

Figure 4 shows the projected loading path of case (2) togetherwith that for case (1). The projected loading path for case (1)is reflected from axis sp1 at point s(1)

a , whereas the projected

loading path in case (2) is a straight line that crosses sp1

and sp2 before reaching its final stress value on the boundingsurface.

Figure 5 plots the projected loading paths for case (1) andcase (2) in the q–θ plane. In case (1) (Fig. 5a), both q and θ

decrease along a nonlinear curve to s(1)a , where q is minimum

and θ is zero. Beyond s(1)a , q and θ start to increase until the

path reaches image point s(1)b on the bounding surface. In case

(2) (Fig. 5b), the projected loading path from s(2)c to s(2)

a is thesame as observed for case 1. However, beyond s(2)

a , q con-tinuously decreases and θ starts to increases until s(2)

m , whereq is minimum and θ is sixty degrees, is reached. Beyonds(2)

m , q and θ move to the image point s(2)b along the exact

same path as traversed from s(2)c to s(2)

a . The complexities ofloading paths in the q–θ plane make it less tractable in termsof obtaining an image stress than the π plane. Moreover, theprojected loading path in the p−q−θ space is a set of dis-cretized nonlinear curves, whereas it is a set of discretizedline segments in the π plane.

Examples (1) and (2) show that the projected loading pathcannot always be represented by a single line (vector) in theπ plane and that two initial stress states plotted at the samepoint in the π plane may not have the same image point onthe bounding surface. A natural question at this point is howto determine the correct image point on a surface for a giveninitial stress state and incremental loading direction. We willaddress this question in the remainder of the paper.

3 Determination of image points on the boundingsurface

3.1 Bounding surface and image point

For the sake of simplicity, the discussion that follows willrefer specifically to the bounding surface, but it appliesequally to the dilatancy surface or any surface in stress spacedefined in a similar manner.

The distance M from the origin of the π plane to thebounding surface in the π plane can be expressed as a func-tion of Lode’s angle θ :

M = M (θ) (21)

If s is the current deviatoric stress, convexity of the surfaceenables the determination of whether the current stress stateis inside or outside the bounding surface through calculationof f :

f = M (θ)−√s:s (22)

A positive f in Eq. (22) implies that the current stress state isinside the bounding surface; f is negative when the currentstress state is beyond the bounding surface. If the current

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Fig. 5 Projected loading pathsin q–θ plane for a case (1) and bcase (2)

(a)

(b)

stress state is on the bounding surface exactly, f is zero. Forthe image stress sb, which is exactly on the bounding surface,the corresponding value of f is:

fb = M (θb)−√sb:sb = 0 (23)

Under monotonic loading conditions, bounding surface mod-els simulate work-hardening and softening by enforcing thatthe sign of the hardening modulus matches the sign of f inEq. (22). However, in cyclic loading of a sand, this conceptdoes not work properly. If the load is reversed just outside thebounding surface, f is marginally negative and the distanceto the closest point on the bounding surface is very small.This would be consistent with a very small hardening modu-lus. In experiments, we observe very stiff stress-strain curvesupon unloading, implying a large positive hardening modu-lus. This apparent contradiction results from allowing stressstates to exist marginally outside the bounding surface in thecourse of a numerical calculation. As will be seen later, thisgets resolved by setting the correct image point with respectto the current stress state and incremental loading direction.

Figure 6 illustrates three typical combinations of currentstress and incremental loading direction: in case (1), the cur-rent stress state sc1 is inside the bounding surface (the incre-mental loading direction may only point towards the bound-ing surface); in case (2), the current stress state sc2 is outsidethe bounding surface and the incremental loading directionn2 points away from the bounding surface; in case (3), the

Fig. 6 Definition of the image point on the bounding surface for thethree possible cases: (1) stress state inside the bounding surface, (2)stress outside the bounding surface and incremental loading directionpointing away from it and (3) stress outside the bounding surface andincremental loading direction pointing towards it

current stress state sc3 is outside the bounding surface and theincremental loading direction n3 points toward the boundingsurface. For simplicity, the bounding surface is assumed tobe a circle. Determination of the image stress state (sb1) incase (1) is straightforward: it is unique for a stress state (sc1)

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inside the bounding surface. There is no uniqueness in cases(2) and (3) because, numerically, the stress state is locatedoutside the bounding surface and, in each case, there are twocandidate image points.

The easiest way to determine the correct image point ineach case is to refer back to the conceptual underpinningof bounding surface models. In the original bounding sur-face models, stress states were not allowed to exist outsidethe bounding surface, a constraint enforced through the con-sistency condition. So a stress state marginally outside thebounding surface with a loading path pointing toward theoutside of the bounding surface (case (2)) should be viewed,conceptually, as a stress state on the bounding surface thatis its own image point. In two-surface plasticity, therefore,the correct image point would be the proximate image pointcandidate.

In case (3), the stress state is on the bounding surfacebut the orientation of the loading path reverses, so it nowpoints to the opposite side of the bounding surface, wherethe image point lies. In two-surface plasticity, this meansthat the image point of a stress state that exists marginallyoutside the bounding surface is the most distant of the twocandidates.

3.2 Numerical determination of the image point on thebounding surface

3.2.1 Algorithm 1: the bisection method

If the current deviatoric stress is inside the bounding surface,there is only one possible image point along the incrementalloading direction due to the convex shape of the bounding sur-face. Andrianopoulos et al. [1] first introduced the bisectionmethod for estimation of the image point. Their algorithmstarts with the step of determining the range within whichthe solution (image point) is located; a more efficient algo-rithm will skip this step, as the maximum size of the bound-ing surface, expressed through Mmax, is typically knownfrom experiments measuring shear strength as a function ofchanges of incremental loading direction, and so the rangeis in fact known at the start. In this study, the range withinwhich the image point exists is set as that extending from thecurrent stress state sc to the stress state sc+2Mmaxn to ensurethat the solution (image point) is inside the range. The bisec-tion method to determine the image point is schematicallyillustrated in Fig. 7, in which the subscripts 1 and 2 desig-nate the first and second bounds of the search range, and thesuperscript denotes the order of iteration. The input valuesin this method are the current stress state sc, the incrementalloading direction n, and parameters η1 (initially 0) and η2

(initially 2Mmax) corresponding to the bounds s1 = sc+η1nand s2 = sc + η2n defining the search range. The equationthat we are attempting to solve is f = 0 (since f = 0 at the

Fig. 7 Bisection method to get the correct image stress in the π plane

image point on the bounding surface), with f defined by Eq.(22). For the first and second bounds (s1 and s2), f1 and f2

are estimated using Eq. (22) after calculation of their Lode’sangles from Eq. (7).

The bisection algorithm aims to find ηb in sb = sc +ηbn that places sb on the bounding surface. Assumption oflinearity gives an approximation of ηb in terms of η1, η2, f1

and f2 as:

ηb = η1 − f1η1 − η2

f1 − f2(24)

The value fb of f that corresponds to ηb can be calcu-lated using Eqs. (7) and (22). If the absolute value of fb

is smaller than the convergence tolerance TOL (set as 10−5

here), sb(= sc + ηbn) is practically on the bounding surfaceand calculations are terminated, returning the image point sb.If the absolute value of fb is greater than the tolerance, thealgorithm needs to update either the first or second bound ofthe search range. The solution is located between η1 and ηb

if f1 fb is negative; if so, η2 and f2 are updated to ηb and fb.If the value of f2 fb is negative, the solution is in the ηb − η2

range, and η1 and f1 are updated to ηb and fb. After updat-ing the search range, the algorithm calculates a new linearapproximation for ηb. This process, which continues untila value of f within the tolerance limit is achieved, is illus-trated in Fig. 8, in which subscripts c, 1 and 2 denote currentstate, the first and second bounds of the search range, respec-tively, and superscripts denote the number of iterations. Theproposed bisection algorithm is summarized in Table 1.

3.2.2 Algorithm 2: the secant method

If the current deviatoric stress is outside the bounding sur-face, there are two possible image points along the incre-mental loading direction. As mentioned earlier, the nearestimage point is the correct image point when the incremen-

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Fig. 8 Bisection method to obtain the correct image stress

Table 1 Bisection algorithm to get the image point on the boundingsurface

Step Description

1 Input: sc, n, η1(= 0), and η2 (= 2Mmaxwhere Mmax is the maximum value of M(θ))

2 Calculate:

s1 = sc + η1n, θ1 = θ(s1), f1 = M(θ1)−√s1 : s1s2 = sc + η2n, θ2 = θ(s2), f2 = M(θ2)−√s2 : s2

3 Calculate:

ηb = η1 − f1η1−η2f1− f2

sb = sc + ηbn, θb = θ (sb) , fb = M (θb)−√sb : sb

4 If | fb| ≤ TOL = 10−5, then exit with sb, else go to step 5

5 If f1 fb < 0, then η2 ← ηb and f2 ← fb and go to step 3

If f1 fb > 0, then η1 ← ηb and f1 ← fb and go to step 3

tal loading direction points away from the bounding surface,whereas the farthest image point is the correct image point ifthe incremental loading direction points towards the bound-ing surface; the algorithm must distinguish between thesetwo possible image points.

In order to easily determine whether the incremental load-ing direction points towards or away from the bounding sur-face, a small negative number κ (= −10 TOL in this study)is used to compute the stress state sκ (= sc + κn). If fκcorresponding to sκ is greater than f at sc, the incrementalloading direction n points away from the bounding surface atthe current stress state, and the correct image point is the oneclosest to the current stress state, as shown in Fig. 9a. In thiscase, the secant algorithm starts from the current stress state(s1 = sc in the figure). If, on the other hand, fκ < f (sc),as illustrated in Fig. 9b, then the farthest image point is thecorrect image point; the secant algorithm in this second casestarts from a stress state on the other side of the bounding sur-face, s1 = sc + 3Mmaxn, which guarantees that the startingpoint is both outside and on the opposite side of the boundingsurface with respect to the current stress state sc.

The two initial points for starting off the secant algorithmare s1 and s2, corresponding to η1 (initially either 0 or 3Mmax)

(a)

(b)

Fig. 9 Secant method to obtain the correct image stress in the π planea when the incremental loading direction points outward from thebounding surface and b when the incremental loading direction pointstoward the bounding surface

and η2 (initially η1 + κ). After determining f1 and f2 corre-sponding to η1 and η2 using Eqs. (7) and (22), the algorithmfirst checks whether they have different signs. If so, the bisec-tion method is used with sc, n, η1 and η2. Otherwise, ηb isestimated using Eq. (24), and an estimate of sb is calculatedfrom ηb. If the absolute value of fb calculated at sb using Eq.(22) is sufficiently close to zero (less than the tolerance limitTOL=10−5), sb is taken as the image point and the algorithmis terminated. If | fb| is greater than the tolerance limit andfb f2 < 0, the solution corresponds to a value of η betweenη2 and ηb, and the bisection method is used with sc, n, η2,and ηb to yield the next estimate of ηb. If | fb| is greater thanthe tolerance and fb f2 > 0, η1, η2, f1, and f2 are updated toη2, ηb, f2, and fb, respectively, and ηb is recalculated usingEq. (24). This process is repeated until | fb| is less than TOL.

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(a)

(b)

Fig. 10 Secant method to get the correct image stress a when the incre-mental loading direction points outward from the bounding surface andb when the incremental loading direction points toward the boundingsurface

The secant algorithm is illustrated schematically in Fig. 10.Subscripts c, 1 and 2 denote current state, the first and sec-ond points used to kick off the algorithm, respectively, andthe superscript denotes the order of the iteration. Table 2 sum-marizes the secant method as used to determine the imagepoint.

3.2.3 Convergence of the secant method

In contrast to the bisection method, the secant method doesnot have guaranteed convergence. Figure 11 shows that thefunction f may be locally convex or concave with respect tothe scalar η. In Fig. 11a, local concavity of f causes the secantmethod to diverge. However, the secant method convergesto the correct image point if f is convex, as shown in Fig.11b. Thus, the secant algorithm requires the function f tobe convex throughout. A convex bounding surface leads to aconvex function f .

Figure 12 shows the change of f along the path from acurrent stress sc to its image stress sb for a concave bound-ing surface (Fig. 12a) and for a convex bounding surface(Fig. 12b). If the bounding surface is concave, the functionf does not increase monotonically from sc to sb; in Fig. 12a,function value f(1) at s(1) is less than function value fc at sc

even though s(1) is nearer to the image point sb than sc. Thisleads to concavity of the f versus η curve, and the secantmethod might diverge. In Fig. 12b, the bounding surface isconvex, and, consequently, f increases monotonically, with

Table 2 Secant algorithm to getthe image point on the boundingsurface

Step Description

1 Input: sc, n, κ (small negative number ≈− 10 TOL)

Calculate following

sκ = sc + κn, θκ = θ (sκ ) , fκ = M (θκ )−√sκ : sκ

If f < fκ , the image point is along the direction opposite to incremental loading direction

→ set η = 0

If f > fκ , the image point is along the direction same as incremental loading direction

→ set η = 3Mmax

2 Set η1 = η and η2 = η + κ

3 Calculate:

s1 = sc + η1n, θ1 = θ (s1) , f1 = M (θ1)−√s1 : s1

s2 = sc + η2n, θ2 = θ (s2) , f2 = M (θ2)−√s2 : s2

4 If f1 f2 < 0, run bisection algorithm with s, n, η1, and η2.

If f1 f2 > 0, go to step 5

5 Calculate:

ηb = η1 − f1η1−η2f1− f2

sb = sc + ηbn, θb = θ (sb) , fb = M (θb)−√sb : sb

6 If | fb| ≤ TOL = 10−5, then exit with sb, else go to step 7

7 If f2 fb < 0, run bisection algorithm with sc, n, η2, and ηb.

If f2 fb > 0, then η1 ← η2, η2 ← ηb, f1 ← f2, and f2 = fb and go to step 5

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(a)

(b)

Fig. 11 Shape of function f with respect to η and convergence of thesecant method: a concavity of f may induce divergence and b convexityof f ensures convergence

no inflection points, from sc to sb, where f is zero. As a result,f versus η is convex, and the secant method converges to theimage point sb. This holds for any surface defined, like thebounding surface, based on the concept that a certain quantityf varies monotonically with distance to the surface.

For concave surfaces (yield surfaces, bounding surfaces ordilatancy surfaces), the secant algorithm may not convergebecause of the resulting non-convex shape of f with respectto η, and the bisection method may provide a wrong imagepoint because more than two intercepts between the bound-ing surface and the projected loading path can exist. Shenget al. [9] proposed the M2 Steffensen’s method to find theintercepts between a loading path and a yield surface. Thismethod could be used if lack of convexity were an issue; how-ever, in the models in which bounding and dilatancy surfaceshave been used, it is not.

4 Implementation of algorithm in soil bounding surfacemodels

In particulate materials (soils, grains, or powder), shearstrength is generally a function of mean effective stress p′,which is the mean total stress p(= σkk/3) less the pore waterpressure u. In soil mechanics, bounding surface models havebeen successfully used to simulate aspects of the mechanicalbehavior of soil [1–3,6,8,10]. The bounding surface in thesemodels is typically cone-shaped, expanding in the directionof increasing p′. Thus, soil bounding surface models are usu-ally expressed in terms of a normalized π plane defined by

(a)

(b)

Fig. 12 Evolution of f along the incremental loading direction n: afor a concave bounding surface and b for a convex bounding surface

normalized principal deviatoric stresses rp(= sp/p′), plot-ted in the three axes in the plane. The distance from the originto the stress state in the normalized π plane is defined as:

l = √ri j ri j =√

si j si j

p′= 3√

2J2

I ′1(25)

where ri j is the normalized deviatoric stress tensor (= si j/p′)and the prime symbol denotes that the stress is an effectivestress (σ ′i j = σi j − 1/3δi j u, where u is the pore water pres-sure). The radius of the bounding surface can be expressedin terms of the image point:

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M = √rb : rb (26)

According to Loukidis and Salgado [6], the bounding surfacecan be expressed as:

M = 2μc[(1+ c1/μ

)− (1− c1/μ

)cos (3θ)

]μ Mmax (27)

where c is Mmin/Mmax; Mmin and Mmax are the minimumand maximum radius of the bounding surface; θ is Lode’sangle; and μ is a parameter that sets the convexity of thebounding surface. Equation (27) implies that the image pointrepresented by rb that appears in Eq. (26) depends on theincremental loading direction. Figure 13 illustrates the typi-cal shape of a soil bounding surface.

The bounding surface has the maximum radius in triaxialcompression (dσ ′p1 > dσ ′p2 = dσ ′p3, σ ′p1 �= σ ′p2 = σ ′p3,and θ = 0◦) and the minimum radius in triaxial extension(dσ ′p1 < dσ ′p2 = dσ ′p3, σ

′p1 �= σ ′p2 = σ ′p3, and θ = 60◦).

Although the size of the bounding surface changes withrespect to factors such as the density of the soil and meaneffective stress, the present paper assumes the size of thebounding surface to be fixed to simplify calculations. In soilbounding surface models [1–3,6,8,10], the hardening mod-ulus is a function of the distance db from the current stressstate to the bounding surface. In the absence of a precisedefinition of the image point, bounding surface models havetraditionally estimated db using:

db = M (θ)− ri j ni j (28)

in which Lode’s angle θ is calculated using the incremen-tal loading direction ni j ; it is fixed unless the incrementalloading direction changes.

In the present paper, the distance is defined as the Frobe-nius norm of the subtraction between the current stress stateand the image point:

db = ±‖r − rb‖F = ±‖ηbn‖F = ηb (29)

in which the sign of the distance is positive if the image pointis along the incremental loading direction (sb1 and s′b3 in Fig.6) and is negative when the image point lies in the directionopposite to the incremental loading direction (sb2 in Fig. 6).Equation (29) shows that db is equal to ηb. The magnitudeof db is the same as that of ηb because the Frobenius normof the loading tensor n is 1. If the line from the stress stateto the image point has the same orientation as the loadingpath, ηb is positive (as shown in Figs. 7 and 9b), and so is db;otherwise, ηb is negative (as shown in Fig. 9a); so db and ηb

have again the same sign.Figure 13 also shows a simple shear unloading path (s1 →

s2 → s3) for a sand specimen with initial stress state definedby σ ′11 > σ ′22 = σ ′33, σ ′12 = τ , and σ ′23 = σ ′31 = 0 (i.e.,with shear stress only in the x1 − x2 plane). Two samplecases are examined. In case A, σ ′11 and σ ′22 are 200 and 100

Fig. 13 Typical soil bounding surface and the loading path examined

Fig. 14 Loading path and distances to the bounding surface in case A

throughout loading, while σ ′11 and σ ′22 are 300 and 100 duringloading in case B. During loading, the shear stress componentτ changes from 100 to −100 in both cases. The boundingsurface parameters Mmax, c, and μ in Eq. (27) are assumedto have values 1.0, 0.7 and 0.35, respectively.

Figure 14 shows the selected points along the loading pathand their corresponding distance db to the bounding surfacecalculated by both the proposed method and Eq. (28) for caseA. At points I and VI, located outside the bounding surface,the secant method was used to determine the correspondingimage points, whereas the bisection method was used forpoints II, III, and IV, which are inside the bounding surface.The figure shows that db is zero at the image point V when theproposed method is used, whereas Eq. (28) slightly overes-timates db for the entire loading path in this case. Therefore,the hardening modulus that follows from Eq. (28) is slightly

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Fig. 15 Loading path and distances to the bounding surface in case B

greater than that resulting from the proposed method; thestress increment is slightly overestimated when Eq. (28) isused. During shearing, this small difference, which appearsfor every increment, accumulates and may lead to differencein results that may be significant.

The loading path and db values calculated using both theproposed method and Eq. (28) are illustrated in Fig. 15 forcase B. In case B, the specimen is unloaded in simple shear inthe x1–x2 plane from a highly anisotropic stress state with alarger J2 than observed in case A. Figure 15 shows that thereis a significant difference in db calculated by the proposedalgorithm and Eq. (28). At the image point denoted by IVin Fig. 15, the proposed method produces zero db, whereasEq. (28) gives db approximately equal to 0.6. At point V,which is clearly outside the bounding surface and where theincremental loading direction points away from the bound-ing surface, Eq. (28) still yields positive db, which meansthe plasticity model generates a work-hardening responsewhen a work-softening response is required. The db result-ing from the proposed algorithm has negative sign at pointV and restores the stress state back to the bounding surfaceby setting the hardening modulus as negative. The proposedmethod guarantees that a stress state eventually reaches thebounding surface, whereas Eq. (28) does not. This differencebetween the two approaches may cause a significant differ-ence in the simulation of mechanical response of sand ormaterials like it. This method of calculating the distance tothe bounding surface may be used in any bounding surfacemodel.

5 Summary and conclusions

Valid operations in the π plane include addition, subtrac-tion, calculation of distance from the origin (i.e., (2J2)

1/2)

and calculation of Lode’s angle. The proposed methodol-ogy to calculate the image point on the bounding surfacecorresponding to an arbitrary stress state is based on theseoperations. The dot product of two different principal stressvectors and double dot product of two different deviatorictensors are not, in general, valid operations in the π plane,although bounding surface plasticity models have used it intheir formulation.

This paper showed that it is possible for stress states withthe same principal deviatoric stresses and principal incremen-tal loading direction to have different projected loading pathsin the π plane. This results from the fact that the π plane hasonly three independent variables but the incremental loadingdirection has the form of a second-order symmetric tensor,having six, not three independent variables. Consequently,projected loading paths must be plotted in the π plane withreference to their tensorial representation. After expressionof the projected loading path in this manner, the image pointon the bounding surface corresponding to the initial stresstensor was defined as the intersection of the surface and theprojected loading path.

The algorithm to estimate the image point is proposedbased on the bisection method and the secant method. Thebisection method is used for the cases where current stressstate is inside the bounding surface. It guarantees conver-gence for a convex surface, as there is only one image pointalong the incremental loading direction from the stress stateinside the bounding surface. If the current stress state is out-side the bounding surface, there are two possible image pointsalong the incremental loading direction. The nearest imagepoint applies if the incremental loading direction points awayfrom the bounding surface, while the farthest image pointapplies if the incremental loading direction points towardsthe bounding surface. To estimate the correct image pointin this case, the secant algorithm was used. As the secantmethod cannot guarantee convergence, the bisection methodis applied as soon as the solution existence is confirmed dur-ing iterations of the secant method.

References

1. Andrianopoulos KI, Papadimitriou AG, Bouckovalas GD (2010)Explicit integration of bounding surface model for the analysis ofearthquake soil liquefaction. Int J Numer Anal Methods Geomech34:1586–1614. doi:10.1002/Nag.875

2. Chakraborty T, Salgado R, Loukidis D (2013) A two-surface plas-ticity model for clay. Comput Geotech 49:170–190. doi:10.1016/j.compgeo.2012.10.011

3. Dafalias YF, Papadimitriou AG, Li XS (2004) Sand plasticitymodel accounting for inherent fabric anisotropy. J Eng Mech130:1319

4. Dafalias YF, Popov EP (1975) A model of nonlinearly hardeningmaterials for complex loading. Acta Mech 21:173–192. doi:10.1007/BF01181053

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5. El-tawil S, Deierlein GG (2001) Nonlinear analysis of mixed steel-concrete frames. I: element formulation. J Struct Eng 127:647–655

6. Loukidis D, Salgado R (2009) Modeling sand response usingtwo-surface plasticity. Comput Geotech 36:166–186, 2008. doi:10.1016/J.Compgeo.02.009

7. Lubliner J (1990) Plasticity theory. Macmillan, New York8. Manzari MT, Dafalias YF (1997) A critical state two-surface plas-

ticity model for sands. Géotechnique 47:255–272

9. Sheng D, Augarde CE, Abbo AJ (2011) A fast algorithm for find-ing the first intersection with a non-convex yield surface. ComputGeotech 38:465–471. doi:10.1016/j.compgeo.2011.02.008

10. Taiebat M, Dafalias YF (2008) SANISAND: simple anisotropicsand plasticity model. Int J Numer Anal Methods Geomech32:915–948. doi:10.1002/Nag.651

11. Yang B-L, Dafalias YF, Herrmann LR (1985) A bounding surfaceplasticity model for concrete. J Eng Mech 111:359–380

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