subdifferential-based implicit return-mapping operators in

Zeitschrift fur Angewandte Mathematik und Mechanik, 14 June 2017

Subdifferential-based implicit return-mapping operators in Mohr-Coulomb plasticity

Stanislav Sysala1,∗, Martin Cermak2, and Tomas Ligursky1

1 Institute of Geonics, Czech Academy of Sciences, Ostrava, Czech Republic2 VSB–Technical University of Ostrava, Ostrava, Czech Republic

Key words Infinitesimal plasticity, Mohr-Coulomb yield surface, implicit return-mapping scheme, consistent tangentoperator, semismooth Newton method, incremental limit analysis, slope stability.MSC (2010) 35Q74, 74C05, 74S05, 90C25

The paper is devoted to constitutive solution, limit load analysis and Newton-like methods in elastoplastic problems con-taining the Mohr-Coulomb yield criterion. Within the constitutive problem, we introduce a self-contained derivation of theimplicit return-mapping solution scheme using a recent subdifferential-based treatment. Unlike conventional techniquesbased on Koiter’s rules, the presented scheme a priori detects a position of the unknown stress tensor on the yield surfaceeven if the constitutive solution cannot be found in a closed form. This eliminates blind guesswork from the scheme andenables to analyze properties of the constitutive operator. It also simplifies the construction of the consistent tangent oper-ator, which is important for the semismooth Newton method when applied to the incremental boundary-value elastoplasticproblem. The incremental problem in Mohr-Coulomb plasticity is combined with limit load analysis. Beside a conven-tional direct method of incremental limit analysis, a recent indirect one is introduced and its advantages are described. Thepaper contains 2D and 3D numerical experiments on slope stability with publicly available Matlab implementations.

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1 Introduction

A yield criterion defining a set of admissible stresses is an essential part of elastoplastic problems. In engineering practice,there is a broad class of yield criteria that are formulated in terms of principal stresses (i.e., in terms of eigenvalues ofstress tensors). We mention the Mohr-Coulomb, the Tresca, the Rankine, the Hoek-Brown or the unified strength ones[13, 17, 24, 25], for example. Such criteria usually have a multisurface representation leading to nonsmooth yield surfaceswith relatively complex structures of singular points. Therefore, the corresponding constitutive solution schemes (includingstress-strain relations and eventually also their derivatives) are technically very complicated and still challenging.

The main aim of this paper is to simplify the handling such criteria by using a specific form of the subdifferential of theeigenvalue yield function. A similar idea was introduced in the recent paper [34] for yield criteria containing one or twosingular points (apices) on the yield surface (e.g., the Drucker-Prager or the Menetrey-Willam ones). It led to simpler andmore correct implicit constitutive solution schemes, and it enabled a deeper analysis of the stress-strain operator. So thispaper approaches the subdifferential-based treatment to a broader class of yield functions and it also demonstrates otheradvantages of this technique.

Due to the technical complexity of implicit solution schemes for models with eigenvalue yield functions, we focus onlyon a particular but representative yield criterion: the Mohr-Coulomb one. This criterion is broadly exploited in soil and rockmechanics and its surface is a hexagonal pyramid aligned with the hydrostatic axis (see, e.g., [17]). We consider the modeldescribed in [17, Sect. 8.2], which can optionally contain a nonassociative flow rule and nonlinear isotropic hardening.The nonassociative flow rule enables to capture the dilatant behavior of a material. Further, due to the presence of thenonlinear hardening, one cannot find a constitutive solution in a closed form, and thus the problem is more challenging. Welet a hardening function be in an abstract form as in [17]. We refer to [5] for a particular example of nonlinear hardening insoil mechanics.

In literature, there are many various concepts of constitutive solution schemes for models containing yield criteria interms of principal stresses. We refer to the recent papers [13] and [22] for their detailed overview and historical develop-ment, respectively. It is worth mentioning that the solution schemes depend mainly on the formulation of the plastic flowrule, its discretization and eventual other approximations. Let us now briefly discuss these aspects.

The plastic flow rule is usually formulated by using the so-called Koiter rule in engineering practice. This rule wasintroduced for associative models with multisurface yield criteria in [23] and consequently extended to nonassociative

∗Corresponding author E-mail: [email protected]


2 S. Sysala, M. Cermak, and T. Ligursky: Subdifferential-based implicit return-mapping operators

models, see, e.g., [15]. It consists of several formulas that depend on a position of the unknown stress tensor on the yieldsurface. These formulas have a different number of plastic multipliers. Within the Mohr-Coulomb pyramid, one plasticmultiplier is used for the smooth portions, two multipliers at the edge points, and six multipliers at the apex. Therefore,the resulting solution schemes are different for each Koiter’s formula. However, only one of them usually leads to thecorrect stress tensor. Moreover, handling different numbers of plastic multipliers is not suitable for analyzing the stress-strain operator even if the solution can be found in a closed form. If the elastoplastic model contains a convex plasticpotential as the Mohr-Coulomb one, then it is possible to replace the Koiter rule with a subdifferential of the potential (see,e.g., [17]). Such a formulation is independent of the unknown stress position, contains just one plastic multiplier, and thusit is more convenient for mathematical analysis of the constitutive operators. In [34], it was shown that this formulation isalso convenient for solution of some constitutive problems. Finally, in some special cases, the constitutive problem can alsobe defined by using the principle of maximum plastic dissipation [17,19] or the theory of bipotentials [3,38] and solved bytechniques based on mathematical programming.

The (fully) implicit Euler discretization of the flow rule is frequently used in elastoplasticity. Then the solution issearched by the elastic predictor – plastic corrector method. Within the plastic correction, the so-called (implicit) return-mapping scheme is constructed. It is worth mentioning that plastic correction problems can be reduced to problems formu-lated only in terms of principal stresses [12, 13, 17]. Beside other Euler-type methods (see, e.g., [17, 28]), the cutting planemethods are also popular. We refer to [30] for a literature survey and recent development of these methods.

Many other approximative techniques have been suggested for simplifying solution schemes for multisurface yieldcriteria. These techniques are based on local or global smoothing of yield surfaces or plastic potentials. We refer to [13,Sect. 1.2] or [1, 4, 5] for literature surveys. However, such an approach is out of the scope of this paper.

From the discrete constitutive solution scheme, we obtain an implicit stress-strain operator for a fixed time step. Insertingthis operator into the balance equation and using a strain-displacement relation, we arrive at an incremental boundary-valueelastoplastic problem in terms of displacements. This problem is solved mostly by nonsmooth variants of the Newtonmethod [8, 18, 27, 31, 32] in each time step. Then, it is useful to construct the so-called consistent tangent operator repre-senting a (generalized) derivative of the stress-strain operator. Here, we use the framework based on the eigenprojectionsof symmetric second order tensors, see, e.g., [6, 17]. A similar approach is also used in the recent book [4] with a slightlydifferent terminology like the spectral directions or the spin of a tensor. Another approach is introduced, e.g., in [12,13,16],where the consistent tangent operator is determined by the tangent operator representing the relation between the stress andstrain rates.

Further, this paper is devoted to a limit load problem, which is frequently combined with the Mohr-Coulomb model.It is an additional problem to the elastoplastic one, where a load history is prescribed by a fixed external force, which ismultiplied by an enlarging load parameter with an unknown limit value. It is well known that the investigated body collapseswhen this critical value is exceeded [10,37,38]. Therefore, this value is an important safety parameter and no solution existsbeyond it. Strip-footing collapse or slope stability are traditional applications of this problem (see, e.g., [9, 17, 29]). Thesimplest computational technique is based on the so-called incremental limit analysis, where the load parameter is enlargedup to its limit value. Then, the boundary-value elastoplastic problem is solved for enlarging values of this parameter.Beside the conventional direct method of incremental limit analysis, we also introduce an indirect method and describe itsadvantages based on recent results from [7, 20, 21, 35, 36].

The rest of the paper is organized as follows. In Sect. 2, a particular form of the subdifferential of a specific eigenvaluefunction is derived. In Sect. 3, the constitutive initial value problem with the Mohr-Coulomb yield criterion is formulated byusing the subdifferential of the plastic potential and discretized by the implicit Euler method. In Sect. 4, the existence anduniqueness of a solution to the discretized problem is proven and an improved solution scheme is derived. In Sect. 5, thestress-strain and the consistent tangent operators are constructed. In Sect. 6, the direct and indirect method of incrementallimit analysis are introduced. Both methods are combined with the semismooth Newton method. In Sect. 7, 2D and 3Dnumerical experiments related to slope stability are described. In Sect. 8, some concluding remarks are mentioned. Thepaper also contains Appendix, where the solution scheme is simplified under the plane strain assumptions.

In this paper, the second order tensors, matrices, and vectors are denoted by bold letters. We also use the followingnotation: R+ := {z ∈ R; z ≥ 0} and R3×3

sym stands for the space of symmetric, second order tensors. The standardscalar product in R3 and the biscalar product in R3×3

sym are denoted by · and :, respectively. The symbol ⊗ means the tensorproduct. Further, the fourth order tensors are denoted by capital blackboard letters, e.g., C, and used for representationof linear mappings from R3×3

sym to R3×3sym . In particular, we consider 36 independent components Cijkl of C satisfying the

following symmetries:

Cijkl = Cijlk = Cjikl = Cjilk, i, j, k, l = 1, 2, 3. (1.1)Copyright line will be provided by the publisher

ZAMM header will be provided by the publisher 3

2 Subdifferential of an eigenvalue function

In this section, we derive a particular form of the subdifferential of a specific eigenvalue function. This auxiliary result willbe crucial for an efficient construction of the constitutive and consistent tangent operators in Mohr-Coulomb plasticity. Let

η =

3∑i=1

ηiei ⊗ ei, η1 ≥ η2 ≥ η3, (2.1)

be the spectral decomposition of a tensor η ∈ R3×3sym . Here, ηi ∈ R, ei ∈ R3, i = 1, 2, 3, denote the eigenvalues, and the

eigenvectors of η, respectively. The eigenvalues η1, η2, η3 can be computed by using the Haigh-Westargaard coordinates(see, e.g., [17, Appendix A]), and they are uniquely determined with respect to the prescribed ordering. Let ω1, ω2, ω3

denote the corresponding eigenvalue functions, i.e. ηi := ωi(η), i = 1, 2, 3. Further, we define the following set ofadmissible eigenvectors of η:

V (η) = {(e1, e2, e3) ∈ R3 × R3 × R3 | ei · ej = δij ; ηei = ηiei, i, j = 1, 2, 3; η1 ≥ η2 ≥ η3}.

The Mohr-Coulomb yield function or the plastic potential corresponds to the following eigenvalue function:

g(η) = aω1(η)− bω3(η), η ∈ R3×3sym, (2.2)

where the parameters a, b ≥ 0 are chosen appropriately. Notice that the convexity of the eigenvalue function g can bederived from:

ω1(η) = maxe∈R3

|e|=1

η : (e⊗ e) = maxe∈R3

|e|=1

(ηe) · e, ω3(η) = mine∈R3

|e|=1

η : (e⊗ e). (2.3)

A particular form of the subdifferential ∂g(η) can be found by using a framework introduced in [26, Chapter 2]. We deriveanother form of ∂g(η), which is more convenient for the purposes of this paper.

Lemma 2.1 Let g : R3×3sym → R be defined by (2.2). Then for any η ∈ R3×3

sym , it holds:

∂g(η) =

{ν =

3∑i=1

νiei ⊗ ei ∈ R3×3sym

∣∣∣ (e1, e2, e3) ∈ V (η); a ≥ ν1 ≥ ν2 ≥ ν3 ≥ −b;

3∑i=1

νi = a− b; (ν1 − a)[ω1(η)− ω2(η)] = 0; (ν3 + b)[ω2(η)− ω3(η)] = 0

}. (2.4)

P r o o f. Since g(0) = 0 and g(2η) = 2g(η), the standard definition of ∂g(η) is equivalent to:

∂g(η) = {ν ∈ R3×3sym | g(η) = ν : η; g(τ ) ≥ ν : τ ∀τ ∈ R3×3

sym}. (2.5)

First, we derive necessary and sufficient conditions on ν ∈ R3×3sym ensuring

g(τ ) ≥ ν : τ ∀τ ∈ R3×3sym. (2.6)

To this end, consider the following spectral decomposition of ν:

ν =

3∑i=1

νifi ⊗ fi, ν1 ≥ ν2 ≥ ν3, (f1, f2, f3) ∈ V (ν). (2.7)

Choose τ = ±I , where I is the unit tensor in R3×3sym . Then we have from (2.6), (2.7):

ν1 + ν2 + ν3 = a− b. (2.8)

Choose τ = f1 ⊗ f1 and τ = −f3 ⊗ f3. Then we derive from (2.6), (2.7), respectively:

ν1 ≤ a, ν3 ≥ −b. (2.9)Copyright line will be provided by the publisher


Conversely, let τ ∈ R3×3sym be arbitrarily chosen and denote ξi := τ : (fi ⊗ fi), i = 1, 2, 3, where f1, f2, f3 are from (2.7).

Then,

ξ1 + ξ2 + ξ3 = τ : I = ω1(τ ) + ω2(τ ) + ω3(τ ), ω1(τ ) ≥ ξi ≥ ω3(τ ), ∀i = 1, 2, 3, (2.10)

follow from I =∑3i=1 fi ⊗ fi and (2.3), respectively. Consequently,

ν : τ =

3∑i=1

νiξi = ξ1(ν1 − ν2) + (ξ1 + ξ2)(ν2 − ν3) + (ξ1 + ξ2 + ξ3)ν3

(2.10)= ξ1(ν1 − ν2) + (τ : I − ξ3)(ν2 − ν3) + ν3τ : I

(2.10)

≤ ω1(τ )(ν1 − ν2) + [τ : I − ω3(τ )](ν2 − ν3) + ν3τ : I =

3∑i=1

νiωi(τ )

= ν1[ω1(τ )− ω2(τ )] + (ν1 + ν2)[ω2(τ )− ω3(τ )] + (ν1 + ν2 + ν3)ω3(τ )

(2.8)= ν1[ω1(τ )− ω2(τ )] + (a− b− ν3)[ω2(τ )− ω3(τ )] + (a− b)ω3(τ )

(2.9)

≤ a[ω1(τ )− ω2(τ )] + a[ω2(τ )− ω3(τ )] + (a− b)ω3(τ )

= aω1(τ )− bω3(τ ) = g(τ ) ∀τ ∈ R3×3sym. (2.11)

Thus the conditions (2.7)–(2.9) are necessary and sufficient for (2.6).

Secondly, assume that ν belongs to ∂g(η). Then (2.7)–(2.9) hold. Since g(η)(2.5)= ν : η, equalities must hold for

τ = η within the derivation of (2.11), i.e., we have:

(ξ1 − ω1(η))(ν1 − ν2) = 0, (ξ3 − ω3(η))(ν2 − ν3) = 0, (2.12)

(ν1 − a)[ω1(η)− ω2(η)] = 0, (ν3 + b)[ω2(η)− ω3(η)] = 0. (2.13)

On the one hand, it is easy to show that if ξi = ωi(η) then fi is an eigenvector of η related to ωi(η) for i = 1 ori = 3. On the other hand, if νi = νi+1 for some i ∈ {1, 2} then the eigenvectors fi, fi+1 can be replaced with anyei, ei+1 ∈ span{fi, fi+1} satisfying |ei| = |ei+1| = 1 and ei · ei+1 = 0 in (2.7). From these facts, it is easy to see that theequalities in (2.12) imply:

∃(e1, e2, e3) ∈ V (η) : ν =

3∑i=1

νiei ⊗ ei, ν1 ≥ ν2 ≥ ν3. (2.14)

To summarize, any element ν ∈ ∂g(η) admits a spectral decomposition in the form of (2.14) with ν1, ν2, ν3 satisfying(2.8), (2.9) and (2.13). Therefore,

∂g(η) ⊂

{ν =

3∑i=1

νiei ⊗ ei ∈ R3×3sym

∣∣∣ (e1, e2, e3) ∈ V (η); a ≥ ν1 ≥ ν2 ≥ ν3 ≥ −b;

3∑i=1

νi = a− b; (ν1 − a)[ω1(η)− ω2(η)] = 0; (ν3 + b)[ω2(η)− ω3(η)] = 0

}. (2.15)

Conversely, using (2.5) and (2.11), one can easily check that any element from the set on the right hand side of (2.15)belongs to ∂g(η).

Remark 2.2 One can easily specify the eigenvalues ν1, ν2, and ν3 in (2.4) if all eigenvalues of η are not the same. Ifη1 > η2 > η3 then ν1 = a, ν2 = 0, and ν3 = −b. If η1 = η2 > η3 then a ≥ ν1 ≥ ν2 ≥ 0, ν1 + ν2 = a, and ν3 = −b. Ifη1 > η2 = η3 then ν1 = a, 0 ≥ ν2 ≥ ν3 ≥ −b, and ν2 + ν3 = −b.



3 The constitutive problem with the Mohr-Coulomb yield criterion

An initial value (evolution) constitutive problem studied at an arbitrary material point is an ingredient of the overall elasto-plastic problem. Nevertheless, the analysis of this auxiliary problem is nontrivial and crucial for the analysis of the overallelastoplastic problem. The solvability analysis of evolution constitutive problems can be found, e.g., in [14]. We referto [17, 19, 28] for a complete formulation of the initial boundary-value elastoplastic problem.

In this section, we introduce the infinitesimal constitutive initial value problem and its implicit Euler discretization for amodel containing the Mohr-Coulomb yield criterion, a nonassociative plastic flow rule, and nonlinear isotropic hardening[17].

3.1 The initial value constitutive problem

Unlike [17], we formulate the initial value constitutive problem by using the subdifferential of the plastic potential:

Given the history of the infinitesimal strain tensor ε = ε(t), t ∈ [0, tmax], and the initial values εp(0) = εp0, εp(0) = εp0 ≥

0, find (σ(t), εp(t), εp(t), λ(t)) such that

σ = De : (ε− εp), κ = H(εp),

εp ∈ λ∂g(σ), ˙εp = −λ∂f(σ,κ)∂κ ,

λ ≥ 0, f(σ, κ) ≤ 0, λf(σ, κ) = 0

(3.1)

hold for each instant t ∈ [0, tmax].

Here, σ, εp, εp, λ denote the Cauchy stress tensor, the plastic strain, the hardening variable, and the plastic multiplier,respectively. The dot symbol means the pseudo-time derivative of a quantity. The fourth order tensor De represents thelinear isotropic elastic law:

σ = De : εe =1

3(3K − 2G)(I : εe)I + 2Gεe, De =

1

3(3K − 2G)I ⊗ I + 2GI, (3.2)

where εe = ε − εp is the elastic part of the strain tensor, K,G > 0, 3K > 2G, denote the bulk, and shear moduli,respectively, and Iη = η for any η ∈ R3×3

sym , i.e., [I]ijkl = 12 (δikδjl + δilδjk), employing the symmetries (1.1). Further, we

let the function H represent the non-linear isotropic hardening in an abstract form and assume that it is a nondecreasing,continuous, and piecewise smooth function satisfying H(0) = 0. Finally, the functions f and g represent the yield functionand the plastic potential for the Mohr-Coulomb model, respectively. They are defined as follows:

f(σ, κ) = (1 + sinφ)ω1(σ)− (1− sinφ)ω3(σ)− 2(c0 + κ) cosφ, (3.3)g(σ) = (1 + sinψ)ω1(σ)− (1− sinψ)ω3(σ), (3.4)

where ω1 and ω3 are the maximal and minimal eigenvalue functions introduced in Sect. 2, and the material parametersc0 > 0, φ, ψ ∈ (0, π/2) represent the initial cohesion, the friction angle, and the dilatancy angle, respectively. Notice thatf, g are convex functions with respect to the stress variable. Recall that the function g is a special case of the functionconsidered in Sect. 2 for the choice

a := 1 + sinψ, b := 1− sinψ (3.5)

and thus ∂g(σ) is given by Lemma 2.1. Clearly, ∂f(σ, κ)/∂κ = −2 cosφ.It is worth mentioning that the value of tmax need not be always known, see Sect. 6.

3.2 The implicit discretization of the constitutive problem

Let 0 = t0 < t1 < . . . < tk < . . . < tN = tmax be a partition of the interval [0, tmax] and denote σk := σ(tk),εk := ε(tk), εpk := εp(tk), εpk := εp(tk), εp,trk := εp(tk−1), εtrk := ε(tk) − εp(tk−1), and σtrk := De : εtrk . Here, thesuperscript tr is the standard notation for the so-called trial variables (see, e.g., [17]), which are known. If it is clear thatthe step k is fixed then we shall omit the subscript k and write σ, ε, εp, εp, εp,tr, εtr, and σtr to simplify the notation. Thek-th step of the incremental constitutive problem discretized by the implicit Euler method reads as:

Given σtr and εp,tr, find σ, εp, and4λ satisfying:

σ = σtr −4λDe : ν, ν ∈ ∂g(σ),

εp = εp,tr +4λ(2 cosφ),

4λ ≥ 0, f(σ, H(εp)) ≤ 0, 4λf(σ, H(εp)) = 0.

(3.6)



Unlike problem (3.1), the unknown εp is not introduced in (3.6). It can simply be computed from the formula εp(tk) =ε(tk)− D−1e : σ(tk) and used as the input parameter for the next step.

4 Solution of the incremental constitutive problem

The aim of this section is to derive an improved solution scheme to problem (3.6). The solution scheme builds on thestandard elastic predictor – plastic corrector method and its improvement is based on the form of ∂g(σ) introduced inLemma 2.1. Within the elastic prediction, we assume4λ = 0. Then, it is readily seen that the triple

σ = σtr, εp = εp,tr, 4λ = 0 (4.1)

is a solution to (3.6) under the condition

f(σtr, H(εp,tr)) ≤ 0. (4.2)

The plastic correction happens when 4λ > 0. Then the unknown generalized stress (σ, H(εp)) lies on the yield surfaceand thus the corresponding plastic correction problem reads as:

Given σtr and εp,tr, find σ, εp, and4λ satisfying:

σ = σtr −4λDe : ν, ν ∈ ∂g(σ),


4λ > 0, f(σ, H(εp)) = 0.

(4.3)

A solution scheme to problem (4.3) is usually called an implicit return-mapping scheme. Since its derivation is technicallycomplicated, we divide the rest of this section into several subsections for easier orientation in the text. In Sect. 4.1, problem(4.3) is reduced and written in terms of principal stresses. In parallel Sects. 4.2–4.5, we derive the return mappings to:the smooth portion, the “left” edge, the “right” edge, and the apex of the pyramidal yield surface, respectively, and thecorresponding a priori decision criteria. In Sect. 4.6, we prove: the existence and uniqueness of solutions of problems (3.6)and (4.3), the continuous dependence of the solutions on the trial variables, and other useful results.

4.1 Plastic correction problem in terms of principal stresses

First, we reduce problem (4.3) by using the spectral decomposition of σ (see Sect. 2):

σ =3∑i=1

σiei ⊗ ei, σ1 ≥ σ2 ≥ σ3, (e1, e2, e3) ∈ V (σ), σi = ωi(σ), i = 1, 2, 3. (4.4)

From the definition of f introduced in Sect. 3, it is easy to see that (4.3)3 can be written only in terms of the principalstresses σ1, σ3 instead of the whole stress tensor σ. To re-formulate (4.3)1, we use Lemma 2.1 and (3.5): there exists(e1, e2, e3) ∈ V (σ) such that ν =

∑3i=1 νiei ⊗ ei, where

1 + sinψ ≥ ν1 ≥ ν2 ≥ ν3 ≥ −1 + sinψ, ν1 + ν2 + ν3 = 2 sinψ,

(ν1 − 1− sinψ)(σ1 − σ2) = 0, (ν3 + 1− sinψ)(σ2 − σ3) = 0.

}(4.5)

Since I =∑3i=1 ei ⊗ ei, (3.2) implies

De : ν =

3∑i=1

[2

3(3K − 2G) sinψ + 2Gνi

]ei ⊗ ei. (4.6)

Then one can substitute (4.4) and (4.6) into (4.3)1:

σtr = σ +4λDe : ν =

3∑i=1

σtri ei ⊗ ei, where σtri = σi +4λ[

2

3(3K − 2G) sinψ + 2Gνi

]. (4.7)

Notice that (4.7)1 defines the spectral decomposition of σtr. Since σ1 ≥ σ2 ≥ σ3 and ν1 ≥ ν2 ≥ ν3, we have:Copyright line will be provided by the publisher


(i) σtr1 ≥ σtr2 ≥ σtr3 ;

(ii) if σtri = σtrj then σi = σj , νi = νj .

From (i), it follows that the eigenvalues σtr1 , σtr2 , σ

tr3 are ordered and thus determined uniquely by using the eigenvalue

functions: σtri = ωi(σtr), i = 1, 2, 3. From (ii), we conclude that σ =

∑3i=1 σie

tri ⊗ etri , ν =

∑3i=1 νie

tri ⊗ etri for any

(etr1 , etr2 , e

tr3 ) ∈ V (σtr). The following lemma summarizes the proven results.

Lemma 4.1 Let σtr and εp,tr be given, satisfy f(σtr, H(εp,tr)) > 0 and σtr =∑3i=1 σ

tri etri ⊗ etri , σtr1 ≥ σtr2 ≥ σtr3 ,

(etr1 , etr2 , e

tr3 ) ∈ V (σtr), be the spectral decomposition of σtr. If (σ, εp,4λ) is a solution to (4.3) and σi are the ordered

eigenvalues of σ then (σ1, σ2, σ3, εp,4λ) is a solution to:

σi = σtri −4λ[23 (3K − 2G) sinψ + 2Gνi

], i = 1, 2, 3,


(1 + sinφ)σ1 − (1− sinφ)σ3 − 2(c0 +H(εp)) cosφ = 0,

σ1 ≥ σ2 ≥ σ3, 4λ > 0,

ν1, ν2, ν3 satisfy (4.5).

(4.8)

Conversely, if (σ1, σ2, σ3, εp,4λ) is a solution to (4.8) then (σ, εp,4λ), σ =

∑3i=1 σie

tri ⊗ etri , solves (4.3).

To be in accordance with problems (3.6) and (4.3), we do not include ν1, ν2, ν3 in the list of unknowns. From (4.5),it follows that the values of ν1, ν2, ν3 can be specified depending on the multiplicities of σ1, σ2, σ3 as in Remark 2.2.Therefore, we shall distinguish four types of the return mapping to the yield surface: the return mapping to the smoothportion (σ1 > σ2 > σ3), the return mapping to the left edge (σ1 = σ2 > σ3), the return mapping to the right edge(σ1 > σ2 = σ3), and the return mapping to the apex (σ1 = σ2 = σ3). These cases will be studied separately in parallelSects. 4.2–4.5. The terminology follows from [17], another one is used, e.g., in [24].

Within the notation introduced below, we shall use the subscripts s, l, r, a to distinguish the return type and the su-perscript “tr” to emphasize a known quantity depending only on trial variables. Further, we shall consider that σtr withσtr1 ≥ σtr2 ≥ σtr3 and εp,tr are given.

4.2 Return mapping to the smooth portion (σ1 > σ2 > σ3)

To derive the return mapping to the smooth portion of the yield surface, we introduce the auxiliary values

γtrs,l :=σtr1 − σtr2

2G(1 + sinψ)≥ 0, γtrs,r :=

σtr2 − σtr32G(1− sinψ)

≥ 0, (4.9)

and a function qtrs : R+ → R,

qtrs (γ) := (1 + sinφ)σtr1 − (1− sinφ)σtr3 − 2[c0 +H

(εp,tr + γ(2 cosφ)

)]cosφ

− γ[

4

3(3K − 2G) sinψ sinφ+ 4G(1 + sinψ sinφ)

]. (4.10)

One can observe that the function qtrs is continuous, piecewise smooth and decreasing in R+, making use of the propertiesof the function H . In addition, qtrs (0) = f(σtr, H(εp,tr)).

Theorem 4.2 Problem (4.8) has a solution (σ1, σ2, σ3, εp,4λ) satisfying σ1 > σ2 > σ3 if and only if the following

conditions hold:

f(σtr, H(εp,tr)) > 0,

qtrs (min{γtrs,l, γtrs,r}) < 0.

}(4.11)

If such a solution exists then its components are uniquely determined by the following equations:

qtrs (4λ) = 0, 4λ ∈ (0,min{γtrs,l, γtrs,r}), (4.12)

σ1 = σtr1 −4λ[

2

3(3K − 2G) sinψ + 2G(1 + sinψ)

], (4.13)

σ2 = σtr2 −4λ[

2

3(3K − 2G) sinψ

], (4.14)

σ3 = σtr3 −4λ[

2

3(3K − 2G) sinψ − 2G(1− sinψ)

], (4.15)

εp = εp,tr +4λ(2 cosφ). (4.16)Copyright line will be provided by the publisher


P r o o f. Assume that there exists a solution (σ1, σ2, σ3, εp,4λ) to (4.8) satisfying σ1 > σ2 > σ3. Then ν1 = 1+sinψ,

ν2 = 0, ν3 = −(1 − sinψ) as it follows from (4.5). Inserting ν1, ν2, ν3 into (4.8)1, we obtain (4.13)–(4.15). Inserting(4.13), (4.15), and (4.8)2 into (4.8)3, we find that4λ solves qtrs (4λ) = 0. From (4.13)–(4.15), one can also derive

σ1 − σ2 = σtr1 − σtr2 − 2G(1 + sinψ)4λ, (4.17)σ2 − σ3 = σtr2 − σtr3 − 2G(1− sinψ)4λ, (4.18)

which together with σ1 > σ2 > σ3 yields 4λ ∈ (0,min{γtrs,l, γtrs,r}). Since the function qtrs is continuous and decreasingin R+, the solution (σ1, σ2, σ3, ε

p,4λ) is uniquely determined by (4.12)–(4.16). Moreover, the conditions (4.11) musthold due to qtrs (0) = f(σtr, H(εp,tr)).

Conversely, assume that (4.11) holds. Then there exists a unique solution 4λ ∈ (0,min{γtrs,l, γtrs,r}) to the equationqtrs (4λ) = 0. Define σ1, σ2, σ3 and εp by (4.13)–(4.16), respectively. As in the first part of the proof, (4.17)–(4.18) hold.From (4.17), (4.18), and 4λ ∈ (0,min{γtrs,l, γtrs,r}), we have σ1 > σ2 > σ3. Finally, it is easy to see that the quintet(σ1, σ2, σ3, ε

p,4λ) is a solution to (4.8) for ν1 = 1 + sinψ, ν2 = 0, ν3 = −(1− sinψ).

Remark 4.3 It is important to note that (4.11) implies min{γtrs,l, γtrs,r} > 0 and consequently σtr1 > σtr2 > σtr3 .

4.3 Return mapping to the left edge (σ1 = σ2 > σ3)

To derive the return mapping to the left edge of the yield surface, we use γtrs,l from (4.9) and define another value

γtrl,a :=σtr1 + σtr2 − 2σtr3

2G(3− sinψ)(4.19)

and an auxiliary function qtrl : R+ → R,

qtrl (γ) :=1

2(1 + sinφ)(σtr1 + σtr2 )− (1− sinφ)σtr3 − 2

[c0 +H


)]cosφ

− γ[

4

3(3K − 2G) sinψ sinφ+G(1 + sinψ)(1 + sinφ) + 2G(1− sinψ)(1− sinφ)

]. (4.20)

Notice that the function qtrl is continuous, piecewise smooth and decreasing in R+. From the assumption σtr1 ≥ σtr2 ≥ σtr3 ,we have γtrl,a ≥ 0 and qtrl (0) ≤ f(σtr, H(εp,tr)).

Theorem 4.4 Problem (4.8) has a solution (σ1, σ2, σ3, εp,4λ) satisfying σ1 = σ2 > σ3 if and only if the following

conditions hold:


qtrl (γtrs,l) ≥ 0, qtrl (γtrl,a) < 0.

}(4.21)


qtrl (4λ) = 0, 4λ ∈ [γtrs,l, γtrl,a), (4.22)

σ1 = σ2 =1

2(σtr1 + σtr2 )−4λ

[2

3(3K − 2G) sinψ +G(1 + sinψ)

], (4.23)

σ3 = σtr3 −4λ[

2

3(3K − 2G) sinψ − 2G(1− sinψ)

], (4.24)

εp = εp,tr +4λ(2 cosφ). (4.25)

P r o o f. Assume that there exists a solution (σ1, σ2, σ3, εp,4λ) to (4.8) satisfying σ1 = σ2 > σ3. Then from (4.5), we

have

ν3 = −(1− sinψ), ν1 + ν2 = 1 + sinψ and 1 + sinψ ≥ ν1 ≥ ν2 ≥ 0. (4.26)Copyright line will be provided by the publisher


From here, (4.8)1 and σ1 = (σ1 + σ2)/2, we obtain (4.23) and (4.24). Inserting (4.23), (4.24), and (4.8)2 into (4.8)3, wefind that4λ solves qtrl (4λ) = 0. Further, from (4.8)1, (4.23) and (4.24), we obtain

σ1 − σ2 = σtr1 − σtr2 − 2G(ν1 − ν2)4λ, (4.27)

σ1 − σ3 =1

2(σtr1 + σtr2 )− σtr3 −G(3− sinψ)4λ. (4.28)

Equations (4.27) and (4.28) imply 4λ ∈ [γtrs,l, γtrl,a) making use of σ1 = σ2 > σ3 and ν1 − ν2 ≤ 1 + sinψ (see (4.26)3).

Since the function qtrl is continuous and decreasing in R+, the solution (σ1, σ2, σ3, εp,4λ) is uniquely determined by

(4.22)–(4.25). Moreover, the conditions (4.21) must hold as it follows from 4λ > 0 and f(σtr, H(εp,tr)) ≥ qtrl (0) >qtrl (4λ) = 0.

Conversely, assume that (4.21) holds. Then there exists a unique solution4λ ∈ [γtrs,l, γtrl,a) to the equation qtrl (4λ) = 0.

It holds that 4λ > 0. Indeed, if 4λ = γtrs,l = 0 then σtr1 = σtr2 and f(σtr, H(εp,tr)) = qtrl (0) = qtrl (4λ) = 0. Thiscontradicts (4.21)1. Further, define σ1 = σ2, σ3 and εp by (4.23)–(4.25), respectively. As in the first part of the proof,(4.28) holds. From4λ < γtrl,a and (4.28), we have σ1 = σ2 > σ3. The equations (4.23)–(4.24) coincide with (4.8)1 if andonly if ν3 = −(1− sinψ) and

ν1 =1

4G4λ(σtr1 − σtr2 ) +

1

2(1 + sinψ), ν2 = − 1

4G4λ(σtr1 − σtr2 ) +

1

2(1 + sinψ).

Hence, one can easily see that (4.26) and consequently (4.5) hold making use of4λ ≥ γtrs,l. Therefore, (σ1, σ2, σ3, εp,4λ)

is the solution to (4.8).

Remark 4.5 It is important to note that (4.21) implies 0 ≤ γtrs,l < γtrl,a and consequently σtr1 ≥ σtr2 > σtr3 . Further, thesolution components σ1 = σ2, σ3, εp, and4λ depend on σtr1 , σtr2 only through σtr1 + σtr2 .

4.4 Return mapping to the right edge (σ1 > σ2 = σ3)

To derive the return mapping to the right edge of the yield surface, we use γtrs,r from (4.9) and define another value

γtrr,a :=2σtr1 − σtr2 − σtr3

2G(3 + sinψ)(4.29)

and an auxiliary function qtrr : R+ → R,

qtrr (γ) := (1 + sinφ)σtr1 −1

2(1− sinφ)(σtr2 + σtr3 )− 2

[c0 +H


)]cosφ

− γ[

4

3(3K − 2G) sinψ sinφ+ 2G(1 + sinψ)(1 + sinφ) +G(1− sinψ)(1− sinφ)

]. (4.30)

Notice that the function qtrr is continuous, piecewise smooth and decreasing in R+. From the assumption σtr1 ≥ σtr2 ≥ σtr3 ,we have γtrr,a ≥ 0 and qtrr (0) ≤ f(σtr, H(εp,tr)).

Theorem 4.6 Problem (4.8) has a solution (σ1, σ2, σ3, εp,4λ) satisfying σ1 > σ2 = σ3 if and only if the following

conditions hold:


qtrr (γtrs,r) ≥ 0, qtrr (γtrr,a) < 0.

}(4.31)


qtrr (4λ) = 0, 4λ ∈ [γtrs,r, γtrr,a), (4.32)

σ1 = σtr1 −4λ[

2

3(3K − 2G) sinψ + 2G(1 + sinψ)

], (4.33)

σ3 =1

2(σtr2 + σtr3 )−4λ

[2

3(3K − 2G) sinψ −G(1− sinψ)

], (4.34)

εp = εp,tr +4λ(2 cosφ). (4.35)

For the sake of brevity, we skip the proof of Theorem 4.6 since it is quite analogous to the proof of Theorem 4.4.Remark 4.7 It is important to note that (4.31) implies 0 ≤ γtrs,r < γtrr,a and consequently σtr1 > σtr2 ≥ σtr3 . Further, the

solution components σ1, σ2 = σ3, εp, and4λ depend on σtr2 , σtr3 only through σtr2 + σtr3 .Copyright line will be provided by the publisher


4.5 Return mapping to the apex (σ1 = σ2 = σ3)

Define an auxiliary function qtra : R+ → R,

qtra (γ) :=2

3(σtr1 + σtr2 + σtr3 ) sinφ− 2

[c0 +H


)]cosφ− γ[4K sinψ sinφ], (4.36)

which is continuous, piecewise smooth, decreasing and unbounded from below in R+. From the assumption σtr1 ≥ σtr2 ≥σtr3 , we have qtra (0) ≤ f(σtr, H(εp,tr)). Further, we shall use the values γtrl,a ≥ 0 and γtrr,a ≥ 0 defined by (4.19) and(4.29), respectively.

Theorem 4.8 Problem (4.8) has a solution (σ1, σ2, σ3, εp,4λ) satisfying σ1 = σ2 = σ3 if and only if the following

conditions hold:


qtra (max{γtrl,a, γtrr,a}) ≥ 0.

}(4.37)


qtra (4λ) = 0, 4λ ≥ max{γtrl,a, γtrr,a}, (4.38)

σ1 = σ2 = σ3 =1

3(σtr1 + σtr2 + σtr3 )−4λ[2K sinψ], (4.39)

εp = εp,tr +4λ(2 cosφ). (4.40)

P r o o f. Assume that there exists a solution (σ1, σ2, σ3, εp,4λ) to (4.8) satisfying σ1 = σ2 = σ3. Then from (4.5), we

have

ν1 ≥ ν2 ≥ ν3, ν1 + ν2 + ν3 = 2 sinψ, 2ν1 − ν2 − ν3 ≤ 3 + sinψ, ν1 + ν2 − 2ν3 ≤ 3− sinψ. (4.41)

Since σ1 = (σ1 +σ2 +σ3)/3, we obtain (4.39) from (4.8)1 and (4.41)2. Inserting (4.39) and (4.8)2 into (4.8)3, we find that4λ solves qtra (4λ) = 0. Further, from (4.8)1, we obtain

0 = 2σ1 − σ2 − σ3 = 2σtr1 − σtr2 − σtr3 −4λ[2G(2ν1 − ν2 − ν3)], (4.42)

0 = σ1 + σ2 − 2σ3 = σtr1 + σtr2 − 2σtr3 −4λ[2G(ν1 + ν2 − 2ν3)]. (4.43)

This and (4.41) yield 4λ ≥ max{γtrl,a, γtrr,a}. Since the function qtra is continuous and decreasing in R+, the solution(σ1, σ2, σ3, ε

p,4λ) is uniquely determined by (4.38)–(4.40). Moreover, the conditions (4.37) must hold as it follows from4λ > 0 and f(σtr, H(εp,tr)) ≥ qtra (0) > qtra (4λ) = 0.

Conversely, assume that (4.37) holds. Owing to the fact that qtra is unbounded from below in R+, there exists a uniquesolution 4λ ≥ max{γtrl,a, γtrr,a} to the equation qtra (4λ) = 0. It holds that 4λ > 0. Indeed, if 4λ = γtrl,a = γtrr,a = 0

then σtr1 = σtr2 = σtr3 and f(σtr, H(εp,tr)) = qtra (0) = qtra (4λ) = 0. However, this contradicts (4.37)1. Further, defineσ1 = σ2 = σ3 and εp by (4.39) and (4.40), respectively. (4.39) coincides with (4.8)1 if and only if

νi =1

6G4λ(2σtri − σtrj − σtrk

)+

2

3sinψ, i, j, k = 1, 2, 3, i 6= j 6= k 6= i.

From here and 4λ ≥ max{γtrl,a, γtrr,a}, it is easy to see that ν1, ν2, ν3 satisfy (4.5). Therefore, (σ1, σ2, σ3, εp,4λ) is the

solution to (4.8).

Remark 4.9 It is important to note that the solution components σ1 = σ2 = σ3, εp, and 4λ depend on σtr1 , σtr2 , σtr3only through σtr1 + σtr2 + σtr3 when (4.37) holds.

4.6 Existence and uniqueness of a solution to the plastic correction problem

In parallel Sects. 4.2–4.5, we have derived separate decision criteria for each return type — the necessary and sufficientconditions (4.11), (4.21), (4.31), and (4.37). From Theorems 4.2, 4.4, 4.6, and 4.8, it follows that if one of these conditionsis satisfied for given trial variables σtr and εp,tr, then there exists a unique solution (σ1, σ2, σ3, ε

p,4λ) to problem (4.8)belonging to the corresponding part of the pyramidal yield surface. The aim of this section is to show that just one of



these conditions happens for any σtr, εp,tr, f(σtr, εp,tr) > 0, implying the existence and the uniqueness of solutions ofproblems (4.8), (4.3), and (3.6). We shall also discuss the continuous dependence of the solutions on σtr and εp,tr.

Recall that the auxiliary values γtrs,l, γtrs,r, γ

trl,a, γtrr,a have been defined by (4.9), (4.19), and (4.29) and that the intervals

Ctrs := (0,min{γtrs,l, γtrs,r}), Ctrl := [γtrs,l, γtrl,a), Ctrr := [γtrs,r, γ

trr,a), Ctra := [max{γtrl,a, γtrr,a},+∞)

have appeared in Theorems 4.2, 4.4, 4.6, and 4.8, respectively. Further, one can arrange (4.19) and (4.29) to the followingforms:

γtrl,a =1 + sinψ

3− sinψγtrs,l +

(1− 1 + sinψ

3− sinψ

)γtrs,r and γtrr,a =

3 + 3 sinψ

6 + 2 sinψγtrs,l +

[1− 3 + 3 sinψ

6 + 2 sinψ

]γtrl,a.

Since (1 + sinψ)/(3− sinψ) ∈ (0, 1), γtrl,a lies between γtrs,l and γtrs,r. Similarly, γtrr,a lies between γtrs,l and γtrl,a. Therefore,only three orderings are possible:

γtrs,l < γtrr,a < γtrl,a < γtrs,r, γtrs,r < γtrl,a < γtrr,a < γtrs,l or γtrs,l = γtrr,a = γtrl,a = γtrs,r. (4.44)

For these three cases, we obtain

Ctrs = (0, γtrs,l), Ctrl = [γtrs,l, γtrl,a), Ctrr = ∅, Ctra = [γtrl,a,+∞),

Ctrs = (0, γtrs,r), Ctrl = ∅, Ctrr = [γtrs,r, γtrr,a), Ctra = [γtrr,a,+∞),

Ctrs = (0, γtrs,l), Ctrl = ∅, Ctrr = ∅, Ctra = [γtrs,l,+∞),

respectively. Hence, we see that Ctrs , Ctrl , Ctrr , Ctra are mutually disjoint and their union is equal to (0,+∞) in all cases.Further, the functions qtrs , qtrl , qtrr , and qtra have been defined by (4.10), (4.20), (4.30), and (4.36), respectively. It holds thatqtrs (0) = f(σtr, H(εp,tr)) and

qtrs (γtrs,l) = qtrl (γtrs,l), qtrs (γtrs,r) = qtrr (γtrs,r), qtrl (γtrl,a) = qtra (γtrl,a), qtrr (γtrr,a) = qtra (γtrr,a). (4.45)

Therefore, we arrive at the following result.Lemma 4.10 There exists a unique function qtr : R+ → R satisfying:

qtr|Ctrs

= qtrs , qtr|Ctrl

= qtrl , qtr|Ctrr

= qtrr , qtr|Ctra

= qtra ,

qtr(0) = f(σtr, H(εp,tr)),

qtr is continuous, piecewise smooth, decreasing and unbounded from below in R+.

Using the function qtr, one can equivalently rewrite the criteria (4.11), (4.21), (4.31), and (4.37) as follows:

(smooth portion) qtr(0) > 0, qtr(min{γtrs,l, γtrs,r}) < 0,

(left edge) qtr(0) > 0, qtr(γtrs,l) ≥ 0, qtr(γtrl,a) < 0, γtrs,l < γtrs,r,

(right edge) qtr(0) > 0, qtr(γtrs,r) ≥ 0, qtr(γtrr,a) < 0, γtrs,r < γtrs,l,

(apex) qtr(0) > 0, qtr(max{γtrl,a, γtrr,a}) ≥ 0,

respectively. From the properties of qtr and (4.44), it follows that for any given data σtr and εp,tr, just one of these criteriais satisfied. Therefore, Theorems 4.2, 4.4, 4.6, 4.8, Lemma 4.1, and the elastic condition (4.2) yield the following solvabilityresults.

Theorem 4.11 Let qtr(0) = f(σtr, H(εp,tr)) > 0. Then problems (4.3) and (4.8) have unique solutions. In particular,if the criterion (4.11), (4.21), (4.31), or (4.37) holds, then the solution (σ1, σ2, σ3, ε

p,4λ) to problem (4.8) is uniquelydetermined by (4.12)–(4.16), (4.22)–(4.25), (4.32)–(4.35), or (4.38)–(4.40), respectively. Moreover, the solution component4λ > 0 is uniquely defined by the equation qtr(4λ) = 0.

Theorem 4.12 The discretized constitutive problem (3.6) has a unique solution.Remark 4.13 The existence and uniqueness of a solution to the constitutive problem is an expected result, which is

not usually discussed in literature. In addition, the criteria (4.11), (4.21), (4.31), and (4.37) have been derived for eachreturn type without the knowledge of the solution. They eliminate blind guesswork from the solution schemes introducedin parallel Sects. 4.2–4.5. Similar criteria were known only for a linear function H (see, e.g., [24]), for which the solutioncomponents can be found in closed forms.



It follows from Theorem 4.12 that there exists a unique function S : R3×3sym × R+ → R3×3

sym such that the stress tensor σsought in problem (3.6) satisfies

σ = S(σtr, εp,tr). (4.46)

The implicit function S can be constructed by using Lemma 4.1 and the return-mapping solution schemes introduced inTheorems 4.2, 4.4, 4.6, and 4.8.

Theorem 4.14 The function S is continuous in R3×3sym × R+.

P r o o f. For the sake of brevity, we only sketch the proof. The functions ωi : σtr 7→ σtri , i = 1, 2, 3, are continuous andpiecewise smooth in R3×3

sym [6]. From (4.10), (4.20), (4.30), (4.36), and (4.45), it follows that the function q(γ,σtr, εp,tr) :=

qtr(γ) is also continuous and piecewise smooth, hence locally Lipschitz continuous. Since it is easy to see that qtr hasalso negative one-sided derivatives, the implicit function theorem [11, Sect. 7.1] implies the continuity of the function(σtr, εp,tr) 7→ 4λ, where4λ is the unique solution of q(4λ,σtr, εp,tr) = 0. Further, it is possible to show the followingstatements:

• If4λ = 0 then formulas (4.13)–(4.15) yield σi = σtri , i = 1, 2, 3. This corresponds with the elastic solution (4.1).

• If4λ = γtrs,l then formulas (4.13)–(4.15) and (4.23)–(4.24) lead to the same principal stresses.

• If4λ = γtrs,r then formulas (4.13)–(4.15) and (4.33)–(4.34) lead to the same principal stresses.

• If4λ = γtrl,a then formulas (4.23)–(4.24) and (4.39) lead to the same principal stresses.

• If4λ = γtrr,a then formulas (4.33)–(4.34) and (4.39) lead to the same principal stresses.

Hence, the function (σtr, εp,tr) 7→ (σ1, σ2, σ3) is also continuous. Finally, we conclude that the function S is continuoussince the tensors σ and σtr are coaxial (see Lemma 4.1).

5 Stress-strain and consistent tangent operators

Recalling (4.46) and the relations εp,tr = εp(tk−1), εtr = ε(tk)−εp(tk−1), σtr = De : εtr from Sect. 3.2, one can definefunctions S : R3×3

sym × R+ → R3×3sym and T : R3×3

sym × R3×3sym × R+ → R3×3

sym as follows:

S(εtr, εp,tr

):= S

(De : εtr, εp,tr

), (5.1)

T (ε, εp(tk−1), εp(tk−1)) := S(ε− εp(tk−1), εp(tk−1)). (5.2)

Obviously, the function T represents a relation between the stress σ = σ(tk) and the strain ε = ε(tk). A (general-ized) derivative of T with respect to ε is known as the consistent tangent operator in literature. We use the notationT (ε, εp(tk−1), εp(tk−1)) for this operator. In order to derive T, we shall find the Frechet derivative of S with respect toεtr for εp,tr fixed at the points where it exists. This derivative will be denoted by DS (εtr, εp,tr). The symbol D will alsobe used for partial derivatives of other functions with respect to εtr.

According to (3.2), the tensors σtr and εtr have the same eigenvectors and their eigenvalues are related as follows:

σtri =1

3(3K − 2G)(εtr1 + εtr2 + εtr3 ) + 2Gεtri , i = 1, 2, 3. (5.3)

Hence, one can easily evaluate S (εtr, εp,tr) by using (5.1). The derivative DS (εtr, εp,tr) can be found in the followingopen sets:

M tre = {εtr ∈ R3×3

sym | qtrs (0) = f(σtr, H(εp,tr)

)< 0},

M trs = {εtr ∈ R3×3

sym | qtrs (0) > 0, qtrs (min{γtrs,l, γtrs,r}) < 0},M trl = {εtr ∈ R3×3

sym | qtrl (γtrs,l) > 0, qtrl (γtrl,a) < 0},M trr = {εtr ∈ R3×3

sym | qtrr (γtrs,r) > 0, qtrr (γtrr,a) < 0},M tra = {εtr ∈ R3×3

sym | qtra (max{γtrl,a, γtrr,a}) > 0},

It follows from Sect. 4 that the stress σ = S (εtr, εp,tr) lies in the elastic domain, at the smooth portion, on the left edge,on the right edge and at the apex of the yield surface if εtr ∈ M tr

e , εtr ∈ M trs , εtr ∈ M tr

l , εtr ∈ M trr , εtr ∈ M tr

a ,respectively. Therefore, these sets are mutually disjoint. One can also see that the union of their closures is equal to R3×3

sym .Copyright line will be provided by the publisher


If εtr ∈M tre then the elastic response happens and we simply arrive at

S(εtr, εp,tr

)= De : εtr, DS

(εtr, εp,tr

)= De. (5.4)

The remaining cases are studied separately in parallel Sects. 5.1–5.4. We shall use the eigenprojections of a secondorder tensor instead of the eigenvectors similarly as in [17]. The derivatives of the eigenprojections are derived in [6] andsummarized in [17]. For the sake of simplicity, we shall assume that H is differentiable at εp,trk +4λ(2 cosφ) and denoteH1 := H ′(εp,trk +4λ(2 cosφ)).

5.1 Return operators to the smooth portion (εtr ∈M trs )

From Remark 4.3 and (5.3), we have εtr1 > εtr2 > εtr3 . This implies the differentiability of ω1, ω2, ω3 at εtr [6]. Moreover,one can introduce the eigenprojections Etr

i := Ei(εtr) of εtr, i = 1, 2, 3 [17]:

Etri :=

(εtr − εtrj I)(εtr − εtrk I)

(εtri − εtrj )(εtri − εtrk ). (5.5)

It holds:

Etri = etri ⊗ etri = Dωi(ε

tr), i = 1, 2, 3, (5.6)

and the derivatives Etri := DEi(εtr), i = 1, 2, 3, satisfy:

Etri =D((εtr)2)− (εtrj + εtrk )I− (2εtri − εtrj − εtrk )Etr

i ⊗Etri − (εtrj − εtrk )[Etr

j ⊗Etrj −E

trk ⊗E

trk ]

(εtri − εtrj )(εtri − εtrk ),

(5.7)

for any i = 1, 2, 3, i 6= j 6= k 6= i, where the components of the fourth order tensors D((εtr)2) and I satisfy [D((εtr)2)]ijkl =12 (δik[εtr]jl + δjl[ε

tr]ik + δjk[εtr]il + δil[εtr]jk) and [I]ijkl = 1

2 (δikδjl + δilδjk), respectively. From here and Lemma 4.1,we have

S(εtr, εp,tr

)=

3∑i=1

σiEtri , DS

(εtr, εp,tr

)=

3∑i=1

[σiEtri +Etr

i ⊗Dσi], (5.8)

where σ1, σ2, σ3 satisfy (4.13)–(4.15). From (4.10), (4.12)–(4.16), and (5.3), we derive

Dσ1 =1

3(3K − 2G)I + 2GEtr

1 −D(4λ)

[2

3(3K − 2G) sinψ + 2G(1 + sinψ)

],

Dσ2 =1

3(3K − 2G)I + 2GEtr

2 −D(4λ)

[2

3(3K − 2G) sinψ

],

Dσ3 =1

3(3K − 2G)I + 2GEtr

3 −D(4λ)

[2

3(3K − 2G) sinψ − 2G(1− sinψ)

],

D(4λ) =2G(1 + sinφ)Etr

1 − 2G(1− sinφ)Etr3 + 2

3 (3K − 2G) sinφI43 (3K − 2G) sinψ sinφ+ 4G(1 + sinψ sinφ) + 4H1 cos2 φ

.

Inserting Dσi, i = 1, 2, 3, into (5.8), we arrive at

DS(εtr, εp,tr

)=

3∑i=1

[σiEtri + 2GEtr

i ⊗Etri

]+

1

3(3K − 2G)I ⊗ I

−[2G(1 + sinψ)Etr

1 − 2G(1− sinψ)Etr3 +

2

3(3K − 2G) sinψI

]⊗D(4λ). (5.9)

This formula can be extended continuously to the boundary of M trs as it follows from [6, 17].



5.2 Return operators to the left edge (εtr ∈M trl )

From Remark 4.5 and (5.3), it follows that εtr1 ≥ εtr2 > εtr3 and σ1 = σ2, σ3, εp, and4λ depend on εtr1 , εtr2 only throughεtr1 + εtr2 . Therefore, it is sufficient to consider only the functions ω3 and ω12 := ω1 + ω2 at εtr and their derivatives.Notice that the derivatives of ω1 and ω2 need not be well-defined at εtr due to the possible equality εtr1 = εtr2 . From theidentity εtr1 + εtr2 + εtr3 = εtr : I , we obtain:

Dω3(εtr) = E3(εtr) =: Etr3 , Dω12(εtr) = I −E3(εtr) =: E12(εtr) =: Etr

12, (5.10)

where the function E3 is the same as in Sect. 5.1. Further, we use the equality

(εtr1 − εtr2 )(Etr1 ⊗E

tr1 −E

tr2 ⊗E

tr2 ) = (εtr − εtr3 E

tr3 )⊗Etr

12 +Etr12 ⊗ (εtr − εtr3 E

tr3 )− (εtr1 + εtr2 )Etr

12 ⊗Etr12

in order to continuously extend the function Etr3 := DE3(εtr) = −DE12(εtr) defined by (5.7) for i = 3 to εtr1 = εtr2 . Weobtain

Etr3 =D((εtr)2)− (εtr1 + εtr2 )I− [εtr ⊗Etr

12 +Etr12 ⊗ εtr] + (εtr1 + εtr2 )Etr

12 ⊗Etr12

(εtr3 − εtr1 )(εtr3 − εtr2 )

+(εtr1 + εtr2 − 2εtr3 )Etr

3 ⊗Etr3 + εtr3 [Etr

12 ⊗Etr3 +Etr

3 ⊗Etr12]

(εtr3 − εtr1 )(εtr3 − εtr2 ). (5.11)

Notice that if εtr1 = εtr2 > εtr3 then εtr has only two eigenprojections: Etr12 and Etr

3 , and εtr = εtr1 Etr12 + εtr3 E

tr3 .

Conversely, if εtr1 > εtr2 > εtr3 , then Etr12 = Etr

1 +Etr2 .

From the equality σ1 = σ2 and Lemma 4.1, we arrive at

S(εtr, εp,tr

)= σ1E

tr12 + σ3E

tr3 , DS

(εtr, εp,tr

)= (σ3 − σ1)Etr3 +Etr

12 ⊗Dσ1 +Etr3 ⊗Dσ3. (5.12)

From (4.20), (4.22)–(4.25), and (5.3), we derive

Dσ1 =1

3(3K − 2G)I +GEtr

12 −D(4λ)

[2

3(3K − 2G) sinψ +G(1 + sinψ)

],

Dσ3 =1

3(3K − 2G)I + 2GEtr

3 −D(4λ)

[2

3(3K − 2G) sinψ − 2G(1− sinψ)

],

D(4λ) =G(1 + sinφ)Etr

12 − 2G(1− sinφ)Etr3 + 2

3 (3K − 2G) sinφI43 (3K − 2G) sinψ sinφ+G(1 + sinψ)(1 + sinφ) + 2G(1− sinψ)(1− sinφ) + 4H1 cos2 φ

.

Hence,

DS(εtr, εp,tr

)= (σ3 − σ1)Etr3 +GEtr

12 ⊗Etr12 + 2GEtr

3 ⊗Etr3 +

1

3(3K − 2G)I ⊗ I

−[G(1 + sinψ)Etr

12 − 2G(1− sinψ)Etr3 +

2

3(3K − 2G) sinψI

]⊗D(4λ). (5.13)

This formula can be extended continuously to the boundary of M trl as it follows from [6, 17]. In particular, if εtr is such

that qtrl (γtrs,l) = 0 and qtrl (γtrl,a) < 0 then εtr1 ≥ εtr2 > εtr3 still holds (see Remark 4.5) and the form (5.13) remains valid.

5.3 Return operators to the right edge (εtr ∈M trr )

From Remark 4.7 and (5.3), it follows that εtr1 > εtr2 ≥ εtr3 and σ1, σ2 = σ3, εp, and4λ depend on εtr2 , εtr3 only throughεtr2 + εtr3 . Therefore, it is sufficient to consider only the functions ω1 and ω23 := ω2 + ω3 at εtr and their derivatives.Similarly as in Sect. 5.2, one can derive:

Dω1(εtr) = E1(εtr) =: Etr1 , Dω23(εtr) = I −E1(εtr) =: E23(εtr) =: Etr

23, (5.14)

Etr1 := DE1(εtr) = −DE23(εtr)

=D((εtr)2)− (εtr2 + εtr3 )I− [εtr ⊗Etr

23 +Etr23 ⊗ εtr] + (εtr2 + εtr3 )Etr

23 ⊗Etr23

(εtr1 − εtr2 )(εtr1 − εtr2 )

+(εtr2 + εtr3 − 2εtr1 )Etr

1 ⊗Etr1 + εtr1 [Etr

23 ⊗Etr1 +Etr

1 ⊗Etr23]

(εtr1 − εtr2 )(εtr1 − εtr3 ). (5.15)



Notice that if εtr1 > εtr2 = εtr3 then εtr has only two eigenprojections: Etr1 and Etr

23, and εtr = εtr1 Etr1 + εtr3 E

tr23.

Conversely, if εtr1 > εtr2 > εtr3 , then Etr23 = Etr

2 +Etr3 .

From the equality σ1 = σ2, we arrive at

S(εtr, εp,tr

)= σ1E

tr1 + σ3E

tr23, DS

(εtr, εp,tr

)= (σ1 − σ3)Etr1 +Etr

1 ⊗Dσ1 +Etr23 ⊗Dσ3. (5.16)

From (4.30), (4.32)–(4.35), and (5.3), we derive

Dσ1 =1

3(3K − 2G)I + 2GEtr

1 −D(4λ)

[2

3(3K − 2G) sinψ + 2G(1 + sinψ)

],

Dσ3 =1

3(3K − 2G)I +GEtr

23 −D(4λ)

[2

3(3K − 2G) sinψ −G(1− sinψ)

],

D(4λ) =2G(1 + sinφ)Etr

1 −G(1− sinφ)Etr23 + 2

3 (3K − 2G) sinφI43 (3K − 2G) sinψ sinφ+ 2G(1 + sinψ)(1 + sinφ) +G(1− sinψ)(1− sinφ) + 4H1 cos2 φ

.

Hence,

DS(εtr, εp,tr

)= (σ1 − σ3)Etr1 + 2GEtr

1 ⊗Etr1 +GEtr

23 ⊗Etr23 +

1

3(3K − 2G)I ⊗ I

−[2G(1 + sinψ)Etr

1 −G(1− sinψ)Etr23 +

2

3(3K − 2G) sinψI

]⊗D(4λ). (5.17)

This formula can be extended continuously to the boundary of M trr as it follows from [6, 17]. In particular, if εtr is such

that qtrr (γtrs,r) = 0 and qtrr (γtrr,a) < 0 then εtr1 > εtr2 ≥ εtr3 still holds (see Remark 4.7) and the form (5.17) remains valid.

5.4 Return operators to the apex (εtr ∈M tra )

From Sect. 4.5, it follows that σ1 = σ2 = σ3 =: p. Moreover, p and4λ depend on εtr only through ptr := 13 (σtr1 +σtr2 +

σtr3 ) = K(εtr1 + εtr2 + εtr3 ). Hence,

S(εtr, εp,tr

)= pI, p

(4.39)= ptr − (2K sinψ)4λ, (5.18)

DS(εtr, εp,tr

)=

∂p

∂ptrKI ⊗ I =

(1− 2K sinψ

∂4λ∂ptr

)KI ⊗ I.

From the implicit equation qtra (4λ) = 0, we obtain

∂4λ∂ptr

(4.36)=

sinφ

2K sinψ sinφ+ 2H1 cos2 φ.

Hence,

DS(εtr, εp,tr

)= K

(1− K sinψ sinφ

K sinψ sinφ+H1 cos2 φ

)I ⊗ I. (5.19)

This formula is also well-defined on the boundary of M tra .

5.5 Comments

For each return type, we have derived just one innovative formula for DS without any other branching that depends onthe multiplicities of εtr1 , ε

tr2 , ε

tr3 . This has been achieved due to deeper analysis of dependencies within the constitutive

solution, see Remarks 4.3, 4.5, 4.7, and 4.9. An additional branching for DS is introduced, e.g., in [17, Appendix A]. Inmany other references, DS is correctly derived only under the assumption εtr1 > εtr2 > εtr3 . However, such formulas cancause significant rounding errors in vicinity of multiple eigenvalues of εtr.

It is readily seen that formulas (5.4), (5.9), (5.13), (5.17), and (5.19) for DS (εtr, εp,tr) are symmetric if ψ = φ, i.e., theplastic flow rule is associative. Moreover, it has been mentioned that these formulas can be extended continuously to theboundaries of the sets M tr

e , M trs , M tr

l , M trr , and M tr

a , respectively. This enables to define the consistent tangent operatorT (ε, εp(tk−1), εp(tk−1)) in the sense of the Clarke generalized derivative in R3×3

sym × R3×3sym × R+. Further, it follows

from Theorem 4.14 that the stress-strain operator T is a continuous function. One can also expect the semismoothnessof T (· , εp(tk−1), εp(tk−1)) in R3×3

sym on the basis of results from this section. One can use implicit function and inversetheorems for semismooth functions and the fact that the piecewise smooth functions are semismooth, for example. However,this investigation seems to be more involved and we shall not go into details here. The semismoothness of elastoplasticconstitutive operators has been analyzed, e.g., in [8, 18, 27, 31, 32, 34].



6 A direct and indirect method of incremental limit analysis

Up to now, we have studied only the constitutive elastoplastic problem and its implicit Euler discretization. To completethe elastoplastic problem, one must standardly add the balance equation, the initial and boundary conditions and the strain-displacement relation. In particular, inserting T

(ε(uk) , εp(tk−1), εp(tk−1)

), ε(uk) = 1

2 (∇uk+∇Tuk), into the principleof virtual work, we receive an incremental boundary-value elastoplastic problem formulated in terms of displacements[17,28,34]. In Mohr-Coulomb plasticity, an additional problem of limit analysis is often considered for a load that dependslinearly on t, see [9, 17, 29].

The presence of a limit value tlim > 0 of t, which can be viewed as a load parameter in this context, is a feature ofperfectly plastic problems. But it also appears when the hardening function H is bounded from above as in Sect. 7.1.The limit value tlim is an important safety parameter and no solution exists beyond it. It determines the collapse state ofthe elastoplastic body and can be defined by using a special convex minimization problem formulated either in terms ofdisplacements or stresses [10, 37, 38]. From the analysis of this problem, it follows that tlim is independent of the timediscretization and the elastic tensor De, but its dependence on a space discretization parameter h can be significant, mainlyfor the simplest finite elements.

In this section, we introduce a direct and indirect method of incremental limit analysis. For the sake of brevity, we focusonly on an algebraic formulation of the problem. To this end, we consider a finite element approximation of the incrementalboundary-value problem. We refer to [17, Appendix D] for the standard algebraic representation of the second and fourthorder tensors.

The vector of internal forces and the consistent tangent stiffness matrix at the k-th step are represented by functionsF k : Rn → Rn and Kk : Rn → Rn×n, respectively. It is worth mentioning that F k and Kk are assembled by using theoperators T and T at each integration point and depend on the solution from the previous step tk−1, see [34]. Further, weconsider a load vector in the form tkl at step k, where l ∈ Rn is fixed, and the corresponding problem reads as:

(Pk)t Given tk ∈ R+, find uk ∈ Rn : F k(uk) = tkl,

whereuk is the displacement vector. Within incremental limit load analysis, we adaptively construct an increasing sequence0 < t1 < t2 < . . . < tk < . . . < tlim depending on the solvability of (Pk)t in order to estimate the unknown limit valuetlim := tlim(h). In practice, the increment of tk decreases when a chosen numerical method does not converge at step k.Such a blind determination of tk is an evident drawback of this direct incremental method.

A more sophisticated adaptive strategy is based on a local and/or global material response of the body to the prescribedload history. To this end, we compute the values αk = bTuk, k = 1, 2, . . ., where uk is a solution to (Pk)t and b is suitablychosen so that the sequence {αk} is increasing and unbounded. There are many ways how to do it. One can detect a pointon the investigated body where it is expected that a selected displacement is the most sensitive to the applied forces, forexample. Then bTuk is the restriction of the displacement vector to the corresponding component. More universally, onecan also set b = l. This choice represents the work of the external forces, is meaningful even for continuous setting of theproblem and was analyzed for generalized Hencky’s plasticity in [7, 20, 21, 35, 36]. Clearly, if the increment αk − αk−1enlarges significantly with increasing k, then it is convenient to reduce the increment of t for the next step.

The knowledge of a suitable b also enables to introduce the indirect method of incremental limit analysis, where thesequences {tk} and {uk} are computed by using the following auxiliary problem:

(Pk)α Given (b, αk) ∈ Rn × R+, find (uk, tk) ∈ Rn × R+ :

{F k(uk) = tkl,

bTuk = αk.

Clearly, if (uk, tk) is a solution to (Pk)α, then uk also solves (Pk)t for tk and tk ≤ tlim. Usually, one can expect thatproblem (Pk)α has a solution for any αk. In such a case, αk can be chosen arbitrarily large and the indirect method doesnot include any blind guesswork unlike the direct one. This is the main advantage of the indirect method. One can expectthat tk → tlim as αk → +∞ for the associative Mohr-Coulomb model. This is proven for b = l and generalized Hencky’splasticity in [7, 21]. For the nonassociative Mohr-Coulomb model with ψ � φ, we can observe that tk ≈ tlim for somefinite k and the sequence {tk} is nonincreasing for k > k. In such a case, the material exhibits softening behavior and theindirect method is more convenient. It is also worth mentioning that the indirect method is similar to the arc-length methodintroduced, e.g., in [16, 17].

We solve problems (Pk)t and (Pk)α by the semismooth Newton method:Algorithm 1 (ALG-t)

1: initialization: u0k

2: for i = 0, 1, 2, . . . doCopyright line will be provided by the publisher


3: find δui ∈ Rn: Kk(uik)δui = tkl− F k(uik)

4: compute ui+1k = uik + δui

5: if ‖δui‖/(‖ui+1k ‖+ ‖uik‖) ≤ εNewton then stop

6: end for7: set uk = ui+1

k .

Algorithm 2 (ALG-α)

1: initialization: u0k, t0k

2: for i = 0, 1, 2, . . . do3: find vi, wi ∈ Rn: Kk(uik)vi = tikl− F k(uik), Kk(uik)wi = l

4: compute δti = [αk − bT (uik + vi)]/bTwi

5: compute δui = vi + δtiwi

6: set ui+1k = uik + δui, ti+1

k = tik + δti

7: if ‖δui‖/(‖ui+1k ‖+ ‖uik‖) ≤ εNewton then stop

8: end for9: set uk = ui+1

k , tk = ti+1k .

If T (· , εp(tk−1), εp(tk−1)) is semismooth in R3×3sym , then one can easily show that F k is semismooth in Rn. The

semismoothness is an essential assumption ensuring the local superlinear convergence of both algorithms above (see, e.g.,[7]). We use a linear extrapolation of the solutions from two previous steps for their initialization. In particular, we prescribethe following values of u0

k in ALG-t and u0k, t0k in ALG-α, k ≥ 2:

u0k = uk−1 +

αk − αk−1αk−1 − αk−2

(uk−1 − uk−2), t0k = tk−1 +αk − αk−1αk−1 − αk−2

(tk−1 − tk−2).

We have observed that this initialization is more convenient than u0k = uk−1, t0k = tk−1.

The direct and indirect method of incremental limit analysis are compared in Sect. 7.1, where specific heuristics for theconstruction of the sequences {tk} and {αk} for problems (Pk)t and (Pk)α are also described. Let us mention that anadaptive construction of {αk} is not necessary but it makes the computations more effective.

7 Numerical experiments – slope stability

We have implemented the direct and indirect method of incremental limit analysis for a 3D slope stability problem and itsplane strain reduction in Matlab. The experimental codes named SS-MC-NP-3D, SS-MC-NH, and SS-MC-NH-Acontrolare publicly available [33]. The codes are vectorized and include our improved return-mapping scheme for the Mohr-Coulomb model in combination with ALG-t or ALG-α. One can choose: a) several types of finite elements with appropriatenumerical quadratures; b) locally refined meshes with various densities.

Fig. 1 Cross section of the body with the coarsest mesh for quadrilateral elements (in meters).



We consider the benchmark plane strain problem introduced in [17, Page 351] and its 3D extension. The 2D cross-section of the body with the coarsest mesh considered in [33, SS-MC-NH] is depicted in Fig. 1. The 3D geometry andthe corresponding hexahedral mesh arise from 2D by extruding. The height of the slope is 10 m and the inclination 45◦.We assume that the body is fixed at the bottom and zero normal displacements are prescribed on the lateral sides. Thebody is subjected to the self-weight. We set the specific weight ρg = 20 kN/m3 with ρ being the mass density and gthe gravitational acceleration. The resulting volume force is multiplied by the load factor t. In accordance with [17], thesequence {αk} is the settlement at the corner point of the top of the slope here. Further, we set E = 20 000 kPa andν = 0.49, hence, G = 67 114 kPa and K = 3 333 333 kPa. We consider only associative Mohr-Coulomb models withφ = ψ = 20◦. The hardening function H and the initial cohesion c0 will be introduced for particular experiments below.

We shall present one experiment for the plane strain (2D) problem and another one for the 3D problem. The primaryaim of these experiments is to illustrate numerically that the formulas derived in Sects. 4, 5, and Appendix work well. Thiscan be confirmed by observing superlinear convergence of ALG-t and ALG-α and their stability in vicinity of the limitload. We prescribe a high precision to these algorithms by setting εNewton = 10−12 in both experiments. Other aims willbe specified below.

7.1 Comparison of the direct and indirect method in 2D

We shall compare the direct method (code SS-MC-NH) and the indirect one (code SS-MC-NH-Acontrol) of incrementallimit analysis on the slope stability benchmark in 2D. We consider the associative Mohr-Coulomb model with c0 = 40 kPaand nonlinear isotropic hardening defined as in [34]:

H(εp) =

{Hεp − H2

4(c−c0) (εp)2 if εp ≤ 2(c− c0)/H,

c− c0 otherwise,

where c = 50 kPa and H = 10 000 kPa. Here, H represents the initial slope of H and the material response is perfectlyplastic for sufficiently large values of εp. The function H is smooth.

We use the eight-noded quadrilaterals (Q′2 elements) with the 3× 3 quadrature formula [2, Table 5.7] and a mesh with37 265 finite-element nodes (including the centers of the edges) and 110 592 integration points. The mesh has a similarscheme as the coarser one in Fig. 1. At each integration point with a plastic response, we solve the corresponding nonlinearequation qtr(4λ) = 0 by the Newton method with the a priori known lower bounds on4λ as the initial choice. Since theMatlab code is vectorized, we use a fixed number of 10 of these inner Newton’s iterations at these points. We have observedthat this choice is sufficient even in vicinity of the limit load since the nonlinear function H is almost constant there.

Recall that we solve problem (Pk)t with ALG-t in each step of the direct method. We set 4t0 = 0.5 (the initialload increment). If ALG-t converges within 50 iterations for step k ≥ 1 and if the computed increment of the settlementsatisfies αk − αk−1 < 0.5 m, then we set 4tk+1 = 4tk, where 4tk = tk − tk−1. Otherwise, the increment is dividedby two. Within the indirect method, where problems (Pk)α are solved by ALG-α, we set 4α0 = 0.0414 m as the initialincrement of the settlement to have comparable results with the direct method. If the computed load increment satisfies|tk − tk−1| > 5× 10−3, then we set4αk+1 = 4αk, otherwise,4αk+1 = 24αk,4αk = αk − αk−1. In both methods,the loading process is terminated when the computed settlement exceeds 4 meters.

Fig. 2 Load path of the direct method. Fig. 3 Load path of the indirect method.



Fig. 4 Number of iterations of ALG-t within the direct method. Fig. 5 Number of iterations of ALG-α within the indirectmethod.

Fig. 6 Convergence of ALG-t in selected steps. Fig. 7 Convergence of ALG-α in selected steps.

The direct and indirect method are compared in Figs. 2–7. In Figs. 2 and 3, successful load steps of the methods aredepicted by circular points. The obtained loading paths coincide practically and they are in accordance with [17, 20, 34].The computed limit value is equal to 4.057 in both cases, which is close to the estimate 4.045 from [9]. However, othercomparisons turn out that the indirect method behaves better than the direct one. First, both methods need 18 load steps,but the direct one requires 8 additional load steps without successful convergence, whereas convergence is achieved in eachstep of the indirect one. One can also see that the positions of the circular points are more convenient with respect to thecurvature of the loading path in Fig. 3 than in Fig. 2. Second, the rates of convergence of ALG-t and ALG-α are similaronly up to step 11 of the respective methods. The number of iterations between steps 12 and 18 is smaller within theindirect method, where the convergence is superlinear in each step (see Fig. 7). On the other hand, Fig. 6 shows that theconvergence is superlinear only up to a relative error of 10−10 in steps 11 and 16 of the direct method, and then the erroroscillates. This can also be observed in a few other steps (e.g., steps 14 and 15). Finally, the computational time of thedirect and indirect method was approximately 9 and 7 minutes, respectively, on a current laptop.

7.2 Associative perfectly plastic 3D problem

Within the 3D slope stability experiment (code SS-MC-NP-3D), we set c0 = 50 kPa and H = 0 kPa, which yields theperfectly plastic model. For this one, we shall compare loading paths for the Q1 and Q′2 hexahedral elements with 8 and20 nodes, respectively. We use the 2× 2× 2 and 3× 3× 3 noded quadrature formulas [2, Table 5.7] for these elements,respectively. Two hexahedral meshes have been prepared for this experiment. For the Q1 elements, the meshes contain5103 and 37 597 finite-element nodes, 34 560 and 276 480 integration points, respectively. For theQ′2 elements, the meshescontain 19 581 and 147 257 finite-element nodes, 116 640 and 933 120 integration points, respectively. We use the directmethod of incremental limit analysis terminated when the computed settlement exceeds 5 meters.



Fig. 8 Comparison of the loading paths for the Q1 and Q′2 elements.

The obtained loading paths are depicted in Fig. 8. One can observe that the estimated limit values of t are close to theexpected value 4.045 for the Q′2 elements but not for the Q1 elements. It would be necessary to use much finer meshes forestimating tlim with these elements. Figs. 9 and 10 illustrate failure at the end of the loading process for the Q′2 elementson the finer mesh.

Fig. 9 The total displacement and the deformed shape at theend of the loading process (in meters).

Fig. 10 The plastic multipliers at the end of the loading process(dimensionless).

8 Conclusion

This paper has extended the subdifferential-based constitutive solution technique from [34] to elastoplastic models con-taining the Mohr-Coulomb yield criterion. It has enabled a deeper analysis of the constitutive problem discretized by theimplicit Euler method and consequently has led to several improvements within solution schemes. For example, a prioridecision criteria characterizing each type of the return mapping have been derived even when the solution cannot be foundin a closed form. Construction of the consistent tangent operator has also been simplified.

The improved constitutive solution schemes have been implemented within slope stability problems in 2D and 3D. Tothis end, the direct and also the indirect method of incremental limit analysis have been used in combination with thesemismooth Newton method. Its local superlinear convergence has been observed within both methods. Further, it hasbeen illustrated that the indirect method leads to a more stable control of the loading process or that higher order finiteelements reduce strong dependence on the mesh.

Acknowledgements This work has been supported by The Ministry of Education, Youth and Sports (of the Czech Republic) from theNational Programme of Sustainability (NPU II), project “IT4Innovations excellence in science – LQ1602”. The authors would also liketo thank Pavel Marsalek for generating the quadrilateral meshes in 2D and 3D.



Appendix: Simplified constitutive handling of the plane strain problem

The results of Sects. 4 and 5 are, of course, also valid for the plane strain problem. Nevertheless, one can simplify theforms of the eigenprojections and their derivatives in this case since it suffices to consider only the subspace R3×3

ps of R3×3sym

containing trial tensors in the form

η =

η11 η12 0η12 η22 00 0 η33

.

We use the symbol D for the Frechet derivatives of functions defined in R3×3ps . Define the functions

ω1(η) :=1

2

[η11 + η22 +

√(η11 − η22)2 + 4η212

],

ω2(η) :=1

2

[η11 + η22 −

√(η11 − η22)2 + 4η212

],

ω3(η) := η33

in R3×3ps . Then ηi = ωi(η), i = 1, 2, 3, are the eigenvalues of η. These values are not ordered in general. We only know

that η1 ≥ η2. Further, define

η(η) :=

η11 η12 0η12 η22 0

0 0 0

, I :=

1 0 00 1 00 0 0

,

E1(η) :=

{η−η2Iη1−η2 if η1 > η2,

I if η1 = η2,E2(η) := I − E1(η), E3(η) :=

0 0 00 0 00 0 1

,

and

E1(η) :=

{1

η1−η2 [I− E1 ⊗ E1 − E2 ⊗ E2] if η1 > η2,

O if η1 = η2,E2(η) := −E1(η), E3(η) := O,

where O denotes the zero fourth order tensor and [I]ijkl = 12 (δikδjl + δilδjk) if i, j, k, l = 1, 2, [I]ijkl = 0 otherwise.

Clearly, Dω3(η) = E3(η). If η1 > η2 then

Dωi(η) = Ei(η), DEi(η) = Ei(η), i = 1, 2, in R3×3ps .

It is worth mentioning that these formulas need not hold in R3×3sym in general. Similar formulas are also introduced in [17,

Appendix A].Now, it is necessary to reorder the eigenvalues of η ∈ R3×3

ps . Denote the ordered eigenvalues as η1, η2, η3, i.e., η1 :=max{η1, η3} and η3 := min{η2, η3}. Correspondingly, define functions ωi, Ei, Ei, i = 1, 2, 3, as reordered functions ωi,Ei, Ei, i = 1, 2, 3. To complete the notation, one can easily set

E12(η) := E1(η) +E2(η), E23(η) := E2(η) +E3(η) ∀η ∈ R3×3ps .

Finally, one can choose η = εtr and straightforwardly use εtri := ωi(εtr), Etr

i := Ei(εtr), Etri := Ei(εtr), i = 1, 2, 3,

Etr12 := E12(εtr), and Etr

23 := E23(εtr) within Sect. 5 when the plane strain assumptions are considered.

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