Preprint 2011
Efficient modeling of localized material failure by
means of a variationally consistent embedded strong
J. Mosler, L. Stanković and R. Radulović
Materials Mechanics
Helmholtz-Zentrum Geesthacht
discontinuity approach
This is a preprint of an article accepted by:
International Journal for Numerical Methods in Engineering (2011)
Efficient modeling of localized material failure by means of a
variationally consistent embedded strong discontinuity approach
J. Mosler & R. Radulović
L. Stanković
Materials Mechanics
Institute of Materials Research
Helmholtz-Zentrum Geesthacht
D-21502 Geesthacht, Germany
E-Mail: joern.mosler@hzg.de
Institute of Mechanics
Ruhr University Bochum
D-44780 Bochum, Germany
SUMMARY
This paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite
strains. Focus is on localized plastic deformation particularly relevant for slip bands. In contrast to already existing models,
the proposed implementation allows to consider several interacting discontinuities in each finite element. Based on a proper
re-formulation of the kinematics, an efficient parameterization of the deformation gradient is derived. It permits to compute
the strains explicitly which improves the performance significantly. However, the most important novel contribution of the
present paper is the advocated variational constitutive update. Within this framework, every aspect is naturally driven by energy
minimization, i.e., all unknown variables are jointly computed by minimizing the stress power. The proposed update relies
strongly on an extended principle of maximum dissipation. This framework provides enough flexibility for different failure
types and for a broad class of non-associative evolution equations. By discretizing the aforementioned continuous variational
principle, an efficient numerical implementation is obtained. It shows, in addition to its physical and mathematical elegance,
several practical advantages. For instance, the physical minimization principle itself specifies automatically and naturally the
set of active strong discontinuities.
1 INTRODUCTION
Traditionally, the modeling of cracks and shear bands is one of the most active research areas in (computational)
mechanics, see, e.g., [1, 2] and references cited therein for a comprehensive overview. Since the pioneering works
[3–5] have been published, particularly, so-called cohesive models have been frequently applied to the numerical
analysis of localized material failure. In contrast to standard stress-strain-based approaches, cohesive models
are governed by means of a traction-separation law. For instance, in case of brittle material failure, the normal
component of the traction vector is driven by the crack opening displacement (see [6]), while for ductile mode-II
or mode-III failure, a Schmid-type constitutive law connecting the resolved shear stress to the slip deformation is
often utilized, cf. [7].
Cohesive material models are usually based on a discontinuous approximation of the displacement field (strong
discontinuities) for capturing the material’s failure, although the separation can be modeled in a smeared fashion
by using an inelastic strain field as well, cf. [8]. Consequently, the discontinuities caused by cracks or shear bands
are explicitly and naturally included within this framework. In addition to this positive feature, such models show
numerous further advantages. For instance, they fulfill all criteria regarded as suitable for an efficient analysis of
failure in large-scale engineering structures as enumerated by Belytschko, Fish & Engelmann [9]. More precisely,
strong discontinuity approaches or cohesive models avoid the pathological mesh dependence in case of strainsoftening [10] and they account naturally for the multiscale character of material failure, i.e., the thickness of
the discontinuity is by definition zero and thus, it is infinitesimally smaller than the characteristic length scale of
the respective macroscopic structure. Accordingly, the idea is to incorporate the kinematics associated with the
small-scale (the softening zone) into a large-scale macroscopic material model. This is typical of most efficient
multiscale approaches, cf. [11–17], and permits to use relatively coarse macroscopic discretizations. This is a
significant difference compared to enhanced continuum models, such as non-local theories [18, 19] or gradient
enhanced models [20, 21] which require, at least, three to five finite elements across the thickness of the failure
zone resulting in high computational cost.
Clearly, for a realistic numerical analysis of material failure by means of strong discontinuity approaches the
kinematics of cracks or shear bands have to be approximated sufficiently accurately. For that purpose, two different
classes of finite element formulations can be utilized. The first of those falls into the range of interface elements,
cf. [22–24], where a jump in the deformation field is allowed to occur only at the boundaries between neighboring
1
2
J. Mosler, L. Stanković and R. Radulović
elements, while discontinuities can evolve arbitrarily within the second approach, i.e., displacement jumps are
also accounted for in the interior of finite elements, see [11, 14, 25–27]. Within the latter framework, a further
classification into models based on an element-wise approximation such as [11, 25, 26] (also known as Embedded
Strong Discontinuity Approaches), and methods relying on the eXtended Finite Element Method (X-FEM) or
Partition of Unity Finite Element Method (PU-FEM), cf. [14, 27] can be made. It bears emphasis that the X-FEM
[14, 27] and classical interface elements [22–24] are based on a continuous crack or shear band path representation.
More precisely, the field of displacement discontinuities is continuous at the element boundaries, e.g., the crack
width is approximated in a continuous fashion. By way of contrast, this is not the case for the Embedded Strong
Discontinuity Approach (ESDA), since it falls into the class of Enhanced Assumed Strain methods (EAS) and thus,
it relies on a local and incompatible enhancement, cf. [28]. Nevertheless, a continuous crack path is frequently
enforced within this framework as well, see [29, 30], though this is not required in general, cf. [31]. Further details
about the underlying kinematics and a comparison between the different models are omitted here. For a more
comprehensive overview, the reader is referred to [2, 32].
Within the present paper, focus is on a numerical implementation suitable for the analysis of numerous interacting and crossing cracks and shear bands in a fully three-dimensional setting at finite strains. Hence, several
simultaneously active discontinuities per finite element have to be taken into account. The aforementioned physical processes can be observed at almost every scale ranging from micro-defects to the macroscopic failure of
engineering structures. Consequently, the analysis of such phenomena is important for understanding the effect of
micro-defects on the resulting structural response (by using representative volume elements), but also for predicting the ultimate load of complex purely macroscopic mechanical systems. Though numerical models allowing for
a representation of an arbitrary crack path such as the X-FEM or the ESDA are very powerful for the simulation
of a restricted number of cracks or shear bands (also in the three-dimensional case), their application to problems
showing large numbers of simultaneously active discontinuities has not been demonstrated yet. More precisely,
an algorithm suitable for tracking each of those numerous discontinuities individually in a fully three-dimensional
setting has not been reported in the literature so far.
Multiple discontinuities can naturally be modeled by using classical interface elements. Unfortunately, such
formulations show a pronounced mesh bias, cf. [33], and thus, they have to be combined with adaptive strategies,
see [34–36]. Moreover, within interface formulations the field of displacement discontinuities is approximated in
a continuous fashion (as in the X-FEM) and hence, they require a modification of the global structure of the finite
element code resulting in higher numerical cost. The brief analysis given here points out the need for a novel strong
discontinuity approach for the analysis of numerous interacting and crossing cracks or slip bands. In the present
paper, an approach based on the ESDA is employed. For allowing interacting and crossing failure surfaces, several
discontinuities are considered in each finite element. Since this leads to increased numerical cost, efficiency of
the kinematics is of utmost importance. For that purpose, a novel parameterization of the deformation gradient is
elaborated which avoids the computation of the inverse of a 4th-order tensor and thus, it increases the numerical
performance significantly. A further boost of the efficiency is obtained by assuming the slip bands or the cracks
to be aligned with the facets of the underlying finite element discretization. Here, a strong link to the kinematics
of classical interface formulations exists. However, in sharp contrast, the topology of the mesh remains fixed, i.e.,
no duplication of nodes is required and the global finite element level is not affected. Evidently, this procedure
guarantees a continuous approximation of the crack path. The only shortcoming of this novel approach is that it
entails a certain mesh bias. Fortunately, this bias is not very pronounced as demonstrated by means of selected
numerical examples. Furthermore, if desired, it can be further reduced by using adaptive strategies, see [34–36],
or by other advanced methods which will also be discussed in the present paper.
The second requirement for the numerical analysis of material failure is a physically sound constitutive model.
Here, focus is on plasticity-based material laws. Such laws are particularly important for the modeling of slip
bands. A critical literature review reveals that most of the interface models are postulated in an ad-hoc manner and
not derived consistently from the principles of thermodynamics. For instance, the traction vector T is frequently
assumed to depend explicitly on the displacement discontinuity via T = T ([[u]]). Clearly, such a model is, at best,
Cauchy-elastic, cf. [37, 38]. Thus, it does neither guarantee a vanishing dissipation in case of elastic unloading nor
a truly positive dissipation in case of irreversible processes. For this reason, only thermodynamically consistent
models are considered in what follows. They are usually based on the Helmholtz energy of the interfaces (cracks
and shear bands) and evolution equations are derived by means of the second law of thermodynamics. Taking
large plastic deformations into account, such models can be found, for instance, in [2, 7, 39, 40]. Relatively recently, combinations with ductile damage evolution, rate effects and temperature effects were proposed in a series
of papers by Fagerström & Larsson, see [41–44]. Based on the cited publications, a class of non-associative plasticity models is elaborated in the present paper. It can naturally enforce certain failure modes (such as mode-II or
mode-III) and covers, amongst others, an anisotropic Drucker-Prager model. In sharp contrast to existing cohesive
models, the novel approach proposed here is fully variational. More precisely, every aspect is driven by energy
minimization, i.e., the unknown displacement discontinuity and the internal variables are jointly computed by min-
A variationally consistent embedded strong discontinuity approach
3
imizing the stress power. For providing enough flexibility to account even for non-associative evolution equations,
the flow rule and the evolution equations are enforced a priori by utilizing a convenient re-parameterization. The
dissipation functional can be chosen independently of those equations. The resulting variational principle can thus
be understood as a novel, extended principle of maximum dissipation. A similar concept was recently developed
in [45, 46] for classical plasticity models (stress-strain-based). By discretizing this variational principle, an efficient numerical implementation is derived. It can be applied to a broad range of, even non-associative, plasticity
models. Furthermore, it accounts for several discontinuities within each finite element. In contrast to implementations based on a modified return-mapping scheme, no artificial search strategy for determining the active internal
surfaces is required. This active set is naturally controlled by energy minimization. It will be shown that the advocated algorithmic formulation is formally identical to that for standard (continuous deformation) material models,
cf. [16, 45, 47, 48].
The paper is organized as follows: First, the kinematics associated with material failure is discussed in Section 2. For that purpose, the fundamentals of the Embedded Strong Discontinuity Approach are briefly summarized. Subsequently, an efficient parameterization of the deformation gradient is elaborated and finally, the
formulation is generalized to encompass several interacting discontinuities. The constitutive models are addressed
in Section 3. While a standard hyperelastic model for the bulk material is briefly described in Subsection 3.1,
the material response of slip bands and cracks is discussed in Subsection 3.2. Subsection 3.2 represents the main
part and contains the most important novel contribution of the present paper. Starting by summarizing some general principles for the derivation of interface models in Paragraph 3.2.1, a class of constitutive laws is developed
in Paragraph 3.2.2. In Paragraph 3.2.3, this class of prototype models is recast into a variationally consistent
framework. Within this novel variational method, everything is consistently driven by energy minimization, i.e.,
the internal variables and the deformation follow naturally from energy minimization. The respective numerical
implementation is elaborated in Section 4 and the performance of the resulting scheme is carefully analyzed in
Section 5.
2 KINEMATICS OF EMBEDDED STRONG DISCONTINUITIES
This section is concerned with the kinematics induced by strong discontinuities. More precisely, focus is on a
certain approximation of discontinuous displacement fields originally advocated in [11] (see also [49–51]). In
contrast to classical interface formulations [22] and eXtended Finite Element Methods (X-FEM, cf. [14, 52]), the
considered approximation of the deformation field is based on the Enhanced Assumed Strain concept (EAS) [28]
resulting in a globally incompatible displacement representation. As a result, this scheme is often referred to as
embedded strong discontinuity approach (ESDA).
First, the kinematics associated with a single discontinuity is presented in Subsection 2.1. While the fundamentals are addressed in Paragraph 2.1.1, a novel re-formulation of the discontinuous deformation mapping is
elaborated in Paragraph 2.1.2. Though it is equivalent to the original description, the new equation is numerically
more efficient. Finally, the extensions necessary for several possibly interacting and crossing discontinuities are
discussed in Subsection 2.2.
2.1 Single strong discontinuity
2.1.1 A concise review of the fundamentals
The fundamentals of the kinematics characterizing the ESDA (Embedded Strong Discontinuity Approach) are
briefly summarized in this paragraph. Further details and mathematical aspects can be found, e.g., in [2, 32].
Following [11, 49, 51], the discontinuous deformation mapping associated with a crack or a shear band ∂s Ω is
approximated by
u = û + [ u]] (Hs − ϕ), with û, ϕ ∈ C ∞ .
(1)
In Eq. (1), û represents the standard, continuous displacement field, [ u]] denotes the displacement discontinuity
at ∂s Ω, Hs is the Heaviside function with respect to the singular surface ∂s Ω and ϕ, not to be confused with the
deformation mapping ϕ, is a smooth ramp function necessary to prescribe the boundary conditions in terms of
û (see [51]). Clearly, considering a single finite element crossed by a discontinuity ∂s Ω, implies the partition of
its domain Ω into Ω = Ω− ∪ ∂s Ω ∪ Ω+ , see Fig. 1. Here, the normal vector N defining locally the topology of
∂s Ω points into Ω+ . With such notations, the ramp function ϕ of a certain finite element is designed by using the
standard interpolation functions Ni associated with node i as
n
ϕ=
Ω+
X
i=1
Ni
(2)
4
J. Mosler, L. Stanković and R. Radulović
∂s Ω
N Ω
Ω+
F
[ u]]
Ω−
F̄
F̃
J
Figure 1: Kinematics induced by strong discontinuities: The deformation can be locally multiplicatively decomposed into a (standard) continuous deformation F̄ and an additional mode related to the kinematics of material
failure F̃ , cf. [7].
where the summation in Eq. (2) is to be performed over all shape functions corresponding to nodes belonging to
the closure of Ω+ , cf. [32, 49].
By applying the generalized derivative to the Heaviside function, i.e., DHs = N δs (see [53, 54]), the deformation gradient is computed as
F =1+
∂ [ u]]
∂ϕ
∂ û
+
(Hs − ϕ) + [ u]] ⊗ N δs − [ u]] ⊗
∂X
∂X
∂X
(3)
with δs representing the Dirac-delta distribution. In what follows, a spatially constant approximation of the displacement discontinuity is chosen. Consequently, GRAD [ u]] = 0. More general interpolations can be found, for
instance, in [55]. With this assumption, the only spatially varying variable in Eq. (3) is the continuous displacement
field û. In line with the standard (continuous) finite element method, û is defined by means of shape functions,
i.e.,
nX
node
(4)
Ni ûei
û =
i=1
and continuity at the element boundaries is enforced by a standard assembling procedure. As a result, û is globally
conforming and continuous. In Eq. (4), ûei are the displacements at node i. Following [7, 56], Eq. (3) can be
rewritten into the equivalent multiplicative decomposition
F = F̄ · F̃ ,
F̄
F̃
with
=
=
1 + GRADû − [[u]] ⊗ GRADϕ
−1
1 + J ⊗ N δs ,
J := F̄ · [ u]] .
(5)
Here, the assumption GRAD [ u]] = 0 has been made again. According to Eq. (5), F̄ is the regularly distributed
part of F , while F̃ is associated with the singular distribution resulting from the displacement discontinuity, see
Fig. 1. In Eq. (5), J denotes the material counterpart of the displacement discontinuity, i.e., J is the pull-back of the
spatial vector [ u]] and belongs to the intermediate configuration implied by the multiplicative decomposition (5)1 .
Representation (5) is particularly convenient in case of plastic deformation. Further details are omitted here. They
can be found in [2, 32].
2.1.2 An equivalent, more efficient description
In what follows, the displacement discontinuity ([[u]] or J ) is assumed to be associated with purely irreversible
deformations. Furthermore and fully analogously to classical plasticity theory (continuous deformation), its evolution is postulated to be of the type J̇ = f (•). It bears emphasis, that such an evolution equation automatically
fulfills the principle of material objectivity, since J belongs to the intermediate configuration. Thus, the strains in
Ω± = Ω− ∪ Ω+ will be computed from
F̄ = 1 + GRADû − F̄ · J ⊗ GRADϕ.
(6)
As a result, for elastic elastic unloading (or for an elastic trial state), Eq. (6) reads
tr
tr
F̄ n+1 = F̂ n+1 − F̄ n+1 · J n ⊗ GRADϕ,
F̂ := 1 + GRADûn+1
(7)
A variationally consistent embedded strong discontinuity approach
5
tr
where the indices n and n + 1 correspond to pseudo times tn and tn+1 . Though being linear in F̄ n+1 , Eq. (7)
is implicit. According to [39], for a known continuous field û (which is the case in displacement-driven finite
element formulations), the straightforward solution of Eq. (7) is given by
tr
F̄ n+1 = A−1 : F̂ n+1 ,
Aikpq := Iikpq + [Iijpq Jj GRADϕk ]|tn .
(8)
As evident, the computation of the inverse of the 4th-order tensor A is numerically very costly. Since this inverse
has to be computed several times within the algorithm, the resulting performance of the scheme is relatively poor.
For this reason, Eq. (6) will be re-written into an equivalent, but numerically more efficient, counterpart.
For re-formulating Eq. (6), the deformation gradient F̄ is simply multiplied by the displacement discontinuity
J yielding
F̂ · J
F̄ · J = F̂ · J − F̄ · J (GRADϕ · J ) ⇒ F̄ · J =
.
(9)
1 + GRADϕ · J
Hence, Eq. (9)2 implies F̄ · J || F̂ · J , cf. [40]. Inserting Eq. (9)2 into Eq. (6), leads finally to
F̄ = F̂ −
1
F̂ · J ⊗ GRADϕ.
1 + GRADϕ · J
(10)
Eq. (10) is better suited than Eq. (6) for, at least, two reasons. First, in case of elastic loading, Eq. (10) reads
F̄ n+1 = F̂ n+1 −
1
F̂ n+1 · J n ⊗ GRADϕ.
1 + GRADϕ · J n
(11)
Consequently, the strains F̄ n+1 can be computed directly, without requiring the inverse of a 4th-order tensor.
Thus, the resulting algorithm is significantly more efficient than that previously proposed in [39]. Secondly, for
slip bands characterized by mode-II or mode-III failure, J · N = 0 holds. Hence,
F̄ · J = F̂ · J
and F̄ = F̂ − F̂ · J ⊗ GRADϕ.
(12)
While Eq. (12)2 is a further simplification of Eq. (11), Eq. (12)1 implies that the push-forwards of the material
displacement jump with respect to F̂ and F̄ are identical. A similar relation for the push-forward of a tangent
vector was derived in [40]. More precisely, the relation
F̂ · M
F̄ · M
=
,
||F̄ · M ||
||F̂ · M ||
M ·N =0
(13)
was derived in [40]. It bears emphasis that Eq. (12)1 encompasses Eq. (13) and thus, it is more general. Furthermore, since in displacement-based finite element formulations û and F̂ are usually known, Eq. (12)1 yields the
convenient relation
[ u]] = F̄ · J = F̂ · J .
(14)
Eq. (14) is directly related to the assumption GRAD [ u]] = 0.
Remark 1 According to [57], the embedded strong discontinuity approach shares some similarities with fracture
energy models based on a characteristic length lc . Conceptually, such a length scale smears the displacement
discontinuity over the respective finite element, i.e., it transforms a deformation jump into an equivalent inelastic
strain. In [8], a characteristic length has been proposed which can be re-written as lc = (N ·GRADϕ)−1 , cf. [57].
Thus, if the ramp function ϕ is not chosen properly, lc can be negative giving unphysical results. This is particular
true in a three-dimensional setting. As a result, some authors elaborated ideas for selecting the optimal ϕ. For
instance, the method presented in [31] is equivalent to minimizing the length scale lc . Physically speaking, this
strategy reduces artificial smearing of the localized deformation. In the present paper, the discontinuity is assumed
to be parallel to the facets of the considered finite element. Hence, lc = (N · N ||GRADϕ||)−1 = 1/||GRADϕ|| >
0. Thus, the aforementioned numerical problems cannot occur.
2.2 Multiple strong discontinuities
Conceptually, the modifications of the kinematics in case of multiple simultaneously active strong discontinuities
are relatively straightforward. For linearized kinematics, first ideas can be found in [58].
Starting from Eq. (6), a superposition of ns displacement jumps leads to a deformation gradient in Ω± of the
type
ns
X
F̄ = F̂ − F̄ ·
J (β) ⊗ GRADϕ(β) ,
F̂ = 1 + GRADû.
(15)
β=1
6
J. Mosler, L. Stanković and R. Radulović
Figure 2: Continuity of the singular surface ∂s Ω across finite elements: a) stochastically generated discontinuities;
b) discontinuities parallel to the element facets (edges in a two-dimensional setting)
Alternatively, extending Eq. (12)2 which holds for slip planes (J · N = 0) yields
F̄ = F̂ − F̂ ·
ns
X
β=1
J (β) ⊗ GRADϕ(β) .
(16)
Clearly, both superpositions (15) and (16) are physically reasonable. Unfortunately and in contrast to a single
discontinuity, Eqs. (15) and (16) are not equivalent, even if mode-II or mode-III failure is considered. A mathematical analysis of the difference between Eq. (15) and Eq. (16) could be performed by rewriting Eq. (15) into
F̄ = A−1 : F̂ and Eq. (16) into F̄ = Ã−1 : F̂ and finally checking ||A − Ã−1 ||. Such an analysis is beyond
the scope of the present paper. Instead, numerical experiments have been considered. More precisely, it turned
out within all computations that both approximations predict almost identical results, cf. [59]. For this reason, the
numerically more efficient formulation (16) will be applied in what follows (compare to Paragraph 2.1.2).
So far, the normal vector N defining the local topology of the cracks or the slip bands has been assumed as
known. Clearly, the position and orientation of the discontinuities depend on the considered material as well as on
the loading state. In case of a single discontinuity per element, different approaches can be found in the literature.
They can be subdivided into two classes: methods based on a continuous crack path such as [29, 30] and those
without enforcing crack path continuity, cf. [31]. For a comprehensive overview, the interested reader is referred
to [32]. Strictly speaking, a continuous crack path means that ∂s Ω is approximated in a continuous fashion. The
field of displacement discontinuities is in any case discontinuous at the element boundaries within the framework
of embedded strong discontinuity approaches.
In the present paper, a numerical method without explicitly enforcing crack path continuity is chosen for
two reasons. First, within each finite element up to four crossing displacement jumps are allowed to propagate
(tetrahedron element with four facets). For such a case, the design of a continuous crack path is very complex and
a subject of its own. Second and even more importantly, the novel approach proposed here should be applicable
to mechanical problems showing a large number of simultaneously active discontinuities such as in fragmentation
processes, see Subsections 5.1.3 and 5.2.2. To the best knowledge of the authors, such complex problems have
not been analyzed numerically yet by an embedded strong discontinuity approach with a continuous crack path
representation.
Even if the crack path is modeled in a discontinuous fashion, the computation of four different normal vectors
within a certain finite element is cumbersome. Usually, certain conditions such as orthogonality are enforced,
cf. [32]. Here, and in line with classical interface formulations it is assumed that the crack surfaces are aligned
with the facets or the edges of the respective finite element, i.e., the topology of possible discontinuities is a priori
known. In contrast to interface elements, the topology of the mesh remains, however, fixed, i.e., no duplication
of nodes is required. Clearly, this procedure guarantees a continuous approximation of the crack path, see Fig. 2.
Furthermore, an arbitrary single displacement jump can be approximated reasonably by a linear combination of
the facet modes. It turns out that the induced mesh bias is not very pronounced an can be effectively reduced
(see Subsection 5.2.1). Alternatively, adaptive strategies such as those in [34–36] can be applied. However, such
methods are beyond the scope of the present paper.
Summarizing the aforementioned discussion, the orientation of the normal vectors within a certain finite element is a priori assumed to be of the type
N (i) = GRADϕ(i) /||GRADϕ(i) ||.
(17)
Considering (linear) tetrahedron elements, two different failure modes are possible (two different types of ramp
functions ϕ(i) ): one node in Ω+ (or equivalently, three nodes), or two nodes in Ω+ . While the first of those is
associated with a facet, the latter corresponds to an edge.
A variationally consistent embedded strong discontinuity approach
7
In addition to reducing the numerical cost for computing the unknown orientation of N , the proposed method
shows two further advantages. First, it results in a symmetric tangent stiffness matrix which opens up the possibility
of deriving variationally consistent update schemes. Such a method will be developed in Subsections 3.2.3 and 4.2.
Second, as will be shown in Section 4, the numerical implementation avoids local snap-back problems as reported
in [60] (originally, the authors call it loss of uniqueness, cf. [58]).
3 CONSTITUTIVE EQUATIONS
This section is concerned with constitutive models describing the material response. While Subsection 3.1 is associated with points belonging to Ω± , i.e., the bulk material, Subsection 3.2 is related to traction-separation-laws.
First, the fundamentals of interface models are briefly summarized in Paragraph 3.2.1. Subsequently, a physically
sound class of constitutive models is presented in Paragraph 3.2.2. Finally, a novel variationally consistent approach is described in Paragraph 3.2.3. In line with so-called variational constitutive updates for classical stressstrain-based material models (see [16, 45, 47, 48]), all state variables, together with the unknown deformation
mapping, follow jointly from minimizing a suitable energy functional.
3.1 Constitutive model for the bulk material
Since the deformation gradient is regularly distributed in the bulk material (F = F̄ holds), standard stress-strainbased continuum models can be applied. For the sake of simplicity, an elastic response is assumed here. However,
other, more complex constitutive models can be easily incorporated as well.
Following the fundamentals of hyperelasticity and objectivity, a stored energy functional of the type
Ψreg = Ψreg (C̄),
C̄ := F̄
T
· F̄ = C
∀X ∈ Ω±
(18)
is postulated. By utilizing the standard Coleman & Noll procedure [61–63], the well-known stress response
τ = F̄ · S · F̄
T
S = 2 ∂C̄ Ψreg ,
and
(19)
is computed. Here, S denotes the second Piola-Kirchhoff stress tensor and τ are the Kirchhoff stresses. In the
numerical analyses presented in Section 5, a neo-Hooke-type energy functional Ψreg of the type
1
λ
J2 − 1
−
+ µ ln (J) + µ trC̄ − 3
(20)
Ψreg (C̄) = λ
4
2
2
is adopted, where J, tr, λ, µ are the determinant of the deformation gradient F̄ , the trace operation and the Lamé
constants.
3.2 Constitutive model for material interfaces
3.2.1 Fundamentals of traction-separation laws
In contrast to the bulk material, dissipative processes are taken into account within the interface ∂s Ω. More precisely, it is assumed that the localized deformations J are of purely irreversible, plastic nature. Clearly, other
mechanical processes such as those related to an elastic or damage-induced response can be modeled in a similar
fashion, cf. [13, 64, 65]. Assuming the interface model to be independent of the bulk material law, a stored-energy
functional of the type
Ψ(C̄, J , α) = Ψreg (C̄) + Ψsing (α) δs
(21)
is considered, cf. [39]. The localized nature of the deformation is reflected by the Dirac-delta distribution, see. [32,
51, 64, 66–68]. In Eq. (21), α is a displacement-like internal variable containing, among others, the displacement
discontinuity J .
The coupling of the constitutive model for points belonging to Ω± and that corresponding to X ∈ ∂s Ω is
provided by the condition of traction continuity, i.e,
T ± − T s = 0,
∀X ∈ ∂s Ω.
(22)
Here, T ± denotes the limits of the traction vector T at the negative and the positive boundary of the discontinuity
and T s is the traction vector within the discontinuity. Since the identity T ± = P ± · N holds (P is the first
Piola-Kirchhoff stress tensor), T ± depends on the bulk’s material model, while T s results from the interface law.
Accordingly, Eq. (22) represents a physically sound coupling condition. It is noteworthy that due to the underlying
8
J. Mosler, L. Stanković and R. Radulović
Petrov-Galerkin discretization of the ESDA, this compatibility condition is not a priori included, but has to be
enforced explicitly, cf. [40, 69].
For developing constitutive models capturing irreversible mechanical processes, the constraints imposed by the
second law of thermodynamics have to be fulfilled. For that purpose, the dissipation D = P : Ḟ − Ψ̇ is computed.
After a straightforward calculation and after inserting the elastic response (19), the reduced dissipation inequality
is finally obtained as
h
i
D = T̄ · J̇ + q · α̇ δs ≥ 0,
q := −∂α Ψsing ,
T̄ := Σ · N ,
Σ := C̄ · S.
(23)
Here, Σ are the Mandel stresses, q is a stress-like internal variable conjugate to α, T̄ is the push-forward of the
traction vector T to the intermediate configuration and the superposed dot represents the material time derivative.
Further details, including the analogy to crystal plasticity theory, can be found in [39].
By exploiting the analogous structure between Eq. (23) and that of standard (continuous deformation) finite
strain plasticity theory, the space of admissible stresses
ET̄ := (T̄ , q) ∈ R3 × Rn | φ(T̄ , q) ≤ 0
(24)
is introduced. In Eq. (24), T̄ is the traction vector at the singular surface obtained from the bulk’s model, i.e., the
± sign has been dropped for the sake of conciseness. It bears emphasis that for the special choice, q = T̄ s and
φ(T̄ , q) = ||T̄ − q||, φ = 0 is equivalent to the condition of traction continuity. Consequently, the concept of
admissible stresses encompasses the condition of traction continuity leading to a more general formulation. For
this reason, it is adopted in what follows. More details can be found in [39].
Based on Ineq. (23) and Eq. (24), physically sound evolution equations can be derived. More precisely, applying the postulate of maximum dissipation
compute: min L,
with
T̄ ,q,λ
L(T̄ , q, λ) := −D + λ φ δs .
(25)
yields
J̇ = λ ∂T̄ φ,
α̇ = λ ∂q φ.
(26)
In line with standard plasticity theory, the plastic multiplier λ is computed from the consistency condition φ̇ = 0.
More general non-associative evolution equations can be derived by means of two additional convex potentials: a
plastic potential g = g(T̄ , q) and a hardening/softening potential h = h(T̄ , q), i.e.,
J̇ = λ ∂T̄ g,
α̇ = λ ∂q h.
(27)
3.2.2 A prototype model
In what follows, a class of constitutive models characterized by yield functions of the type
φ = T̄ eq (T̄ ) − qi (αi ) − q0eq
(28)
will be considered. Accordingly, φ depends on an equivalent stress measure T̄ eq which is assumed to be a positively
homogeneous function of degree one. For associative evolution equations, the dissipation corresponding to Eq. (28)
reads in this case
D = λ q0eq δs ≥ 0.
(29)
Hence, D is indeed positive and equally importantly, it can be computed in closed form. For several applications,
it is convenient to decompose the traction vector into a normal and a shear part, i.e.,
T̄N := T̄ · N ,
T̄ S := T̄ − T̄N N .
(30)
Based on this decomposition, the equivalent stress measure can be additively split in a similar fashion:
T̄ eq (T̄ ) = T̄Neq (T̄N ) + T̄Seq(T̄ S ).
(31)
Again, T̄Neq and T̄Seq are postulated to be homogeneous functions of degree one. If T̄Neq and T̄Seq are positively
homogeneous of degree n and m, a positively homogeneous equivalent stress T̄ eq of degree one can be defined as
q
q
(32)
T̄ eq (T̄ ) = n T̄Neq(T̄N ) + m T̄Seq(T̄ S )
A variationally consistent embedded strong discontinuity approach
9
cf. [45]. Therefore and without loss of generality, the order of T̄Neq and T̄Seq is assumed to be one. The aforementioned class of yield functions is very broad and contains many important constitutive models. For instance, by
setting
1 p
T̄ S · H · T̄ S
(33)
φN (T̄N ) = κ T̄N ,
φS (T̄ S ) =
2
an anisotropic Drucker-Prager model suitable for the analysis of slip bands is obtained. Here, H is a tensor
describing the material anisotropy. By way of contrast, the case TSeq = 0 corresponds to a Rankine-type yield
function representing an admissible choice for the modeling of cracks.
The class of prototype models described here is completed by suitable evolutions equations. While such equations can be derived in a relatively straightforward manner for mixed-mode failure, slip bands require further
attention, i.e., mode-II or mode-III failure has to be enforced explicitly in that case (J · N = 0). Clearly, this
requires a non-associative constitutive law. A physically sound method for deriving such a law is given by the
framework of generalized standard materials, cf. [70, 71]. Accordingly, physically sound evolution equations
(D ≥ 0) can be developed by using a convex plastic potential g and a convex hardening/softening potential h. Obviously, if g depends only on the shear components of the traction vector T̄ , mode-I failure is naturally excluded.
For this reason, a plastic potential of the type
g = h = T̄Seq(T̄ S ) − qi (αi ) − q0eq .
(34)
is suitable for the analysis of slip bands. Eq. (34) leads finally to the evolution equations
J̇ = λ
∂ T̄Seq
,
∂ T̄
α̇ = −λ
(35)
and the dissipation is computed as
φ=0
D = λ (T̄Seq − qi ) δs = λ (q0eq − T̄Neq) δs .
(36)
In contrast to their associative counterparts, non-associative evolution equations entail an additional non-constant
term in the dissipation inequality (compare to Eq. (29)).
Remark 2 In the present paper, only isotropic hardening/softening models are considered. However, it is noteworthy that linear kinematic hardening of Prager-Ziegler-type can easily be included as well. For a geometrically
linearized framework, further details can be found in [65].
Remark 3 For a function T̄ eq(T̄ ) being positively homogeneous of degree one, T̄Seq (T̄ ) = ∂T̄ g · T̄ is fulfilled (see
Eq. (34)). Differentiating this identity once again yields
∂T̄2 g · T̄ = 0.
(37)
This identity which represents a compatibility condition between the stress vector and the flow direction will be
utilized for checking consistency of the variational constitutive updates presented in the next section.
3.2.3 Variational principles for traction-separation laws
In what follows, the constitutive model presented in the previous section, is recast into a variationally consistent
framework. Within this framework all state variables, together with the unknown deformation mapping, follow
jointly from minimizing an incrementally defined energy potential. In case of classical (continuous) associative
finite strain plasticity, such methods have been proposed in [16, 45, 47], while for non-associative evolution equations first ideas can be found in [46]. Though variational constitutive updates show several advantages compared
to other (standard) approaches such as the existence of a natural distance or the possibility of applying state-ofthe-art optimization schemes, they have not been developed yet for problems showing strong discontinuities, i.e.,
for cohesive traction-separation laws. In this paragraph, such a variationally consistent framework is elaborated.
It can be applied to a broad range of different constitutive models. First, the associative case is considered. Subsequently, an enhanced postulate of maximum dissipation is advocated allowing to account for non-associative
evolution equations as well. Several simultaneously active and possibly intersecting discontinuities are considered
within all derivations.
Conceptually and analogously to [16, 45, 47], the novel variational framework is based on minimizing the
stress power Ẽ = P : Ḟ = Ψ̇ + D. However and in contrast to the continuous case, the Helmholtz energy
Ψ contains, in addition to the standard regularly distributed part, a Dirac-delta distribution. The same holds for
10
J. Mosler, L. Stanković and R. Radulović
the dissipation. Starting with the constitutive framework discussed in the previous paragraph and considering ns
possibly interacting and crossing strong discontinuities, the stress power is given by
Ẽ(ϕ̇, J̇
(1)
(1)
, α̇
(1)
, T̄
,q
(1)
, . . . , J̇
(ns )
(ns )
, α̇
, T̄
(ns )
,q
(ns )
+
) = Ψ̇reg (ϕ̇) + c
ns
X
i=1
with χ(i) = χ(i) (T̄
(i)
ns
X
(i)
Ψ̇sing (α̇(i) ) δs(i)
i=1
(i)
D (J̇
(i)
(i)
, α̇ , T̄
(i)
(i)
,q ) +
ns
X
χ
(i)
δs(i)
i=1
!
(38)
,
, q (i) ) being the characteristic function of the space of admissible stresses ET̄ (i) , i.e.,
χ(i) = χ(i) (T̄
(i)
, q (i) ) =
(
(i)
0 , if (T̄ , q (i) ) ∈ ET̄ (i)
(i)
∞ , if (T̄ , q (i) ) ∈
/ ET̄ (i) .
(39)
In Eq. (38), c is a constant parameter related to the relative position (or area) of the discontinuity within the
respective finite element. It can be considered as c = 1 at the moment. Its precise physical interpretation will
be explained later. According to Eq. (38), χ(i) vanishes for admissible stresses and Ẽ represents indeed the stress
power. Inadmissible states are penalized by χ(i) = ∞ and thus, they are a priori excluded, if minimization of Ẽ is
the overriding physical principle. In line with [16, 45, 47], the postulate of maximum dissipation (maximization of
Ẽ with respect to the stress-like variables) is applied next, yielding
E(ϕ̇, J̇
(1)
(1)
, α̇
, . . . , J̇
(ns )
(ns )
, α̇
) = Ψ̇reg (ϕ̇) + c
ns
X
i=1
(i)
Ψ̇sing (α̇(i) )
δs(i)
+
ns
X
χ
∗(i)
δs(i)
i=1
!
.
(40)
Here and henceforth, χ∗(i) is the Legendre transformation of χ(i) , cf. [72]. Physically speaking, χ∗(i) is the
dissipation, provided only admissible states and associative evolution equations are considered. Consequently, E is
still the stress power, but now as a function of displacement-like variables only. As shown in Paragraph 3.2.2 (see
Eq. (29)), for associative evolution equations based on a positively homogeneous yield function, χ∗(i) is obtained
eq(i)
in closed form as χ∗(i) = λ(i) q0
≥ 0. Though such an assumption is usually made, it is not mandatory, cf.
[16, 45, 47].
Similarly to [16, 45, 47], it can be shown that the minimizers of E are equivalent to the underlying associative
flow rule and the corresponding evolution equations. More precisely, they follow naturally from the locally defined
problem
(1)
(ns )
(1)
(ns )
(J̇ , α̇(1) , . . . , J̇
, α̇(ns ) ) = arg inf E(ϕ̇, J̇ , α̇(1) , . . . , J̇
, α̇(ns ) )
.
(41)
ϕ̇=const
Since variational principle (41) is not standard and furthermore, it depends on the parameter c, its consistency will
be explicitly proved. However, a more suitable re-parameterization of the functional E is given first.
Although the solution of Eq. (41) looks straightforward, it bears emphasis that several highly non-linear constraints related to the flow rule and the evolution equations have to be included. For instance, self-penetration
has to be avoided a priori. Since only isotropic hardening/softening models are considered in the present paper,
α(i) = α(i) is scalar-valued and thus, the respective constraint can be enforced trivially, i.e., α̇(i) = −λ(i) ≤ 0.
Consequently, the only non-trivial constraint is associated with the flow rule
J̇
(i)
= λ ∂T̄ (i) φ(i) ,
(42)
or more precisely, it is related to the flow direction. In what follows, two convenient parameterizations of the flow
direction will be proposed.
For elaborating suitable parameterizations of the flow direction, the underlying ideas are explained first by
means of a von Mises model defined by the yield function φ = ||T̄ S || − q0eq . Without loss of generality, hardening
or softening can be neglected, since those mechanisms do not affect the flow rule. The generalizations necessary
for other flow rules will be discussed subsequently. Evidently, in case of a von Mises-type yield function, the
resulting flow direction for discontinuity i reads
(i)
(i)
∂T̄ (i) φ
with
(i)
T̄ S
=
T̄ S
(i)
n
o
∈ M(i) := M (i) M (i) · N (i) = 0, M (i) : M (i) = 1
||T̄ S ||
(i)
:= 1 − N (i) ⊗ N (i) · T̄ .
(43)
A variationally consistent embedded strong discontinuity approach
11
Accordingly, a straightforward three-dimensional parameterization of the flow direction depending on a so-called
˜ (i) 6= T̄ (i) is given by
pseudo-stress vector T̄
q
˜ (i) = 3.
˜ (i) , dim T̄
˜ (i) · (1 − N (i) ⊗ N (i) ) · T̄
˜ (i) , λ(i) = √1
T̄
λ(i) ∂T̄ (i) φ(i) = 1 − N (i) ⊗ N (i) · T̄
b
(44)
Alternatively, an equivalent two-dimensional parameterization is obtained by enforcing the orthogonality between
˜ (i) and N (i) a priori, i.e.,
T̄
q
1
(i)
(i)
(i)
(i)
(i)
(i)
˜ (i) · T̄
˜ (i) , dim T̄
˜ (i) = 2.
˜
˜
T̄
(45)
λ ∂T̄ (i) φ = T̄ , with T̄ · N = 0, λ = √
b
The coefficient b in Eqs. (44) and (45) is usually non-constant. It can be computed by calculating the norm of
Eqs. (44)1 or Eqs. (45)1 yielding
(46)
b = ∂T̄ (i) φ(i) · ∂T̄ (i) φ(i) .
Consequently, for a von Mises-type or an isotropic Drucker-Prager-type model b is constant. More precisely, in
case of von Mises plasticity, b = 1.
Clearly, Eqs. (44) and (45) are suitable parameterizations of the flow rule. However, in many cases, a decomposition into an amplitude and into a direction is more convenient. For that purpose, Eq. (44) is slightly modified
leading to

 ˜ (i) 
 1 − N (i) ⊗ N (i) · T̄
˜ (i) ∈ R3 .
T̄
(47)
∂T̄ (i) φ(i) ∈
˜ (i) || 
 || 1 − N (i) ⊗ N (i) · T̄
As a result, the flow rule (direction) can be parameterized as
o
n
˜ (i)
∂T̄ (i) φ(i) = ∂T̄ (i) g (i) ∈ ∂T̄ (i) g (i) ˜ (i) T̄
∈ R3 .
T̄
(48)
Interestingly, this parameterization of the flow direction is always meaningful. It even applies to non-associative
evolution equations or anisotropic plastic potentials. According to Eq. (48), the natural underlying idea is that
the space of admissible flow directions is obtained simply by evaluating the flow rule for all admissible stress
˜ (i) 6= T̄ (i) is referred to as the pseudo stress vector. Obviously, it
states. This interpretation explains that T̄
has to be enforced that the pseudo stress vector yields the same flow direction as its physical counterpart, i.e.,
∂T̄ (i) g (i) |T̄˜ (i) = ∂T̄ (i) g (i) |T̄ (i) . In what follows, only flow rules based on a plastic potential in the form of a
positively homogeneous function of degree one will be considered. In this case, the aforementioned compatibility
between the pseudo stresses and the physical stresses can be checked by analyzing condition (37). Interestingly,
the advocated novel parameterization of the flow rule is formally identical to that proposed in [45, 46] for standard
(continuous deformation) plasticity theory.
Inserting the novel parameterization (48) into the stress power (40) defines the alternative minimization problem
˜ (ns ) , λ(ns ) ).
˜ (1) , λ(1) , . . . , T̄
˜ (ns ) , λ(ns ) ) = arg
˜ (1) , λ(1) , . . . , , T̄
(49)
inf
E(ϕ̇, T̄
(ϕ̇, T̄
˜ (ns ) ,λ(ns )
˜ (1) ,λ(1) ,...,T̄
T̄
While for associative evolution equations both frameworks of energy minimization (compare Eq. (49) to Eq. (41))
lead, in principle, to identical results, only the novel variational problem (49) can be applied to non-associative
evolution equations. The reason is that the flow rule and the dissipation can be chosen independently. As a
consequence, Eq. (49) can be considered as an enhanced or extended principle of maximum dissipation. More
precisely, the stress power depending on a certain dissipation is minimized by simultaneously enforcing a suitable
flow rule. However, it bears emphasis that also for associative models minimization principle (49) shows an
important advantage: All highly nonlinear constraints related to the flow rule are already naturally included and
thus enforced.
Since variational principle (49) is non-standard and furthermore, it depends on the parameter c, consistency
will be shown next, i.e., it is proved that the minimizers of Eq. (49) are equivalent to the underlying evolution
equations and the yield function. According to Section 2, the kinematics associated with cracks and that related to
shear bands are slightly different. For this reason, consistency is proved for both cases separately.
Slip bands – mode-II and mode-III failure The functional to be minimized is the stress power. Thus, the strain
rate is required. By utilizing Eq. (12)2 , it is computed as
n1
ns
i=1
i=1
X
X ˙
(i)
˙
F̂ · J̇ ⊗ GRADϕ(i) .
F̂ · J (i) ⊗ GRADϕ(i) −
F̄˙ = F̂ −
(50)
12
J. Mosler, L. Stanković and R. Radulović
Hence, for a variation of Eq. (49) with respect to the plastic multiplier and the pseudo stresses, the continuous part
of the displacement field can be considered as constant, i.e.,
û = const
⇒
˙
F̂ = 0
⇒
F̄˙ = −
ns
X
i=1
F̂ · J̇
(i)
⊗ GRADϕ(i) .
(51)
T
By utilizing the Mandel stresses Σ = F̂ · ∂F̄ Ψreg and assuming slip planes parallel to the facets of the finite
elements (GRADϕ(i) ||N (i) ) (see Remark 7), the stress power reads
!
ns
ns
ns
X
X
X
(i)
(i)
(i) (i) (i)
(i)
(i)
(i)
.
(52)
D
q λ δs +
(λ ∂T̄ (i) g ⊗ N ||GRADϕ ||) + c
E = −Σ :
{z
}
|
{z
} |
i=1
i=1
i=1
(i)
GRADϕ(i)
J˙
In what follows, constitutive models falling into the range of the general framework as presented in Paragraph 3.2.2
will be utilized. Thus, the stress power (52) can be re-written as
ns
X
ns
X
ns
X
(i)
λ(i) (q0eq − (T̄Neq )(i) ) δs(i) .
|
{z
}
i=1
i=1
= D(i) , see Eq. (36)
(53)
Based on Eq. (53) consistency of the novel minimization principle can be analyzed. First, the stationarity condition
of E with respect to plastic multiplier λ(i) is computed. It results in
n
h
io
(i)
(i)
eq(i)
δλ E = −T̄S
||GRADϕ(i) || + δs(i) c q (i) + q0eq − (T̄Neq )(i)
δλ(i) = 0.
(54)
E = −Σ :
(λ(i) ∂T̄ (i) g (i) ⊗ N (i) ||GRADϕ(i) ||) + c
{z
}
|
{z
} |
i=1
(i)
GRADϕ(i)
J˙
q (i) λ(i) δs(i) +
Unfortunately, Eq. (54) cannot be interpreted easily, since it contains, in addition to a regular part, a singular
distribution. For that purpose, Eq. (54) is integrated over the domain of a finite element e. This leads to
Z
io
h
n
(i)
eq(i)
eq (i)
eq (i)
(i)
δλ(i) = 0.
(55)
c
q
+
q
−
(
T̄
)
||GRADϕ(i) || V e + A(i)
δλ E dV = −T̄S
s
0
N
Ωe
For the sake of simplicity, constant stresses and constant internal variables have been assumed. However, the
more general case follows identical lines and it is included in what follows. As shown in the appendix, for threedimensional tetrahedron elements based on a linear approximation of the undeformed configuration and a linear
ramp function ϕ defined by the shape functions of the corner nodes, the identity
||GRADϕ(i) || V e
(i)
As
= c(i)
geom
(56)
(i)
is fulfilled. The parameter cgeom is related to the relative position of the slip bands within the respective finite
element and does not depend on the size of the element. For instance, if the slip bands are chosen to be identical
(i)
to the facets of the finite elements, cgeom = 2. Hence, combining Eq. (55) with Eq. (56) leads to the natural choice
c = cgeom and thus, the integrated stationarity condition of E with respect to the plastic multiplier of slip band i is
finally obtained as
Z
(i)
Ωe
(i)
δλ E dV = −A(i)
δλ(i) = 0.
s cgeom φ
(57)
Consequently, stability of E with respect to the plastic multiplier δλ(i) is equivalent to fulfilling the yield function
at the discontinuity in a weak form (or continuity of the traction vector), i.e.,
Z
(i)
δλ E dV ≥ 0 ⇔ φ(i) ≤ 0.
(58)
Ωe
The derivation of the stationarity condition reveals the physical interpretation of the parameter c: While in classical
Embedded Strong Discontinuity Approaches (ESDA) the size of the slip band does not influence the results, cf.
[11, 50, 50, 51], this is not the case within the novel variational formulation. Only if c = cgeom is chosen, both
methods give identical results. Clearly, from a physical point of view, the area of the discontinuity should affect
the softening behavior. In this respect, the novel variational formulation is more realistic. However, in the present
A variationally consistent embedded strong discontinuity approach
13
paper, focus is on a variational re-formulation of the original ESDA. The influence of the slip band size on the
solution is beyond the scope of this work and will be analyzed in a forthcoming paper.
Fully analogously to the variation of the integrated stress power with respect to the plastic multiplier, station˜ (i) leads finally to
arity with respect to the pseudo stresses T̄
(i)
˜ (i) = 0
δT̄˜ (i) E = −λ(i) ||GRADϕ(i) || T̄ · ∂T̄2 (i) g (i) ˜ (i) · δ T̄
T̄
(59)
(i)
⇔ T̄ · ∂T̄2 (i) g (i) ˜ (i) = 0.
T̄
Hence, the compatibility condition between the pseudo stresses and their physical counterparts is fulfilled as well
(compare to Eq. (37)). As a result, the advocated novel minimization principle is indeed consistent.
The fundamentals of the advocated novel variational constitutive update are completed by the stress response.
By decomposing the minimization problem (49), into a local part inf E|û=const , the reduced potential
Ered = inf
˜ (1) ,λ(1) ,...,T̄
˜ (ns ) ,λ(ns )
T̄
˜ (1) , λ(1) , . . . , T̄
˜ (ns ) , λ(ns ) ).
E(ϕ̇, T̄
(60)
is defined from which the stresses can be derived, i.e.,
P = ∂F̄˙ Ered .
(61)
The general case including mixed-mode cracking Applying the same procedure as before to the general case
(J · N 6= 0) shows that variationally consistency cannot be guaranteed anymore. This is closely related to the
simplified kinematics (12) holding only for J · N = 0 and to the Petrov-Galerkin-type approximation of the
underlying finite element formulation (see Remark 7). However, this effect is only relevant for large deformations.
For this reason, focus is now on a geometrically linearized setting. In this case, the strain tensor reads
ε̄ = [GRADû]sym −
n1 X
[ u]]
(i)
i=1
⊗ GRADϕ(i)
sym
.
(62)
Denoting the stress tensor of the geometrically linearized theory as σ and the respective traction vector as T , the
stress power is computed as (compare to Eq. (52))
!
ns
ns
ns
X
X
X
(i)
(i)
(i) (i) (i)
(i)
(i) sym
(i)
D
q λ δs +
+c
(λ ∂T (i) g ⊗ N ||GRADϕ ||)
E = −σ :
i=1
i=1
i=1
(63)
ns n
io
h
X
(i)
eq(i)
λ(i) = 0.
−TS
||GRADϕ(i) || + δs(i) c q (i) + q0eq − (TNeq )(i)
=
i=1
Accordingly, the geometrically linearized framework leads to the same variational principle (compare Eq. (63) to
Eq. (54)). However and in sharp contrast to the finite strain kinematics, the strain field (62) holds now for any
material failure type. Consequently, mode-II and mode-III but also mode-I or mixed-mode material failure can be
accounted for. The remaining steps necessary for proving variational consistency are omitted here. They are fully
identical to those already presented in the previous paragraph.
Remark 4 The proposed variational constitutive update requires four degrees of freedom per discontinuity. Three
˜ (i) = 3), while an additional unknown describes
unknowns are necessary for defining the flow direction (dim T̄
(i)
the amplitude of inelastic deformations (dim λ = 1). Thus, from a computational point of view, it is more
efficient to parameterize the plastic deformations directly reducing the number of unknowns by one. The respective
modifications are straightforward and have already been explained for a von Mises prototype model. Furthermore,
(i)
in case of slip bands, J (i) · N (i) = 0 and consequently, J̇ · N (i) = 0. As a result, enforcing this constraint in
advance, leads to a further reduction of the number of unknowns. More precisely, the modeling of each slip band
requires then a parameterization depending on only two degrees of freedom, i.e.,
(i)
(i)
(i)
(i)
J̇ = λM1 M 1 + λM2 M 2 ,
(i)
(i)
(i)
Ṁ 1 = Ṁ 2 = 0 = 1 − ||M 1 || = 1 − ||M 2 ||,
(M 1 × M 2 ) · N = 1 (64)
(i)
Again, the unknowns λM1 and λM2 can be computed conveniently by minimizing the associated stress power.
Remark 5 A positively homogeneous plastic potential of degree one is considered within the present paper. It
bears emphasis that this property is crucial for employing the parameterization (48) in terms of pseudo stresses.
As stated in [46], the proposed decomposition of the flow rule into an amplitude and into a direction yields unique
results only in this case.
14
J. Mosler, L. Stanković and R. Radulović
Remark 6 By utilizing the chain rule, the identity
∂T̄ (i) g (i) ⊗ GRADϕ(i) = ∂Σ g̃ (i) ||GRADϕ(i) ||
(65)
can be shown in a straightforward manner. Here, g̃ (i) is the plastic potential, but now in terms of the Mandel
stresses, i.e.,
(i)
g̃ (i) (Σ) = (g (i) ◦ T̄ )(Σ).
(66)
Hence, by employing again the concept of pseudo stresses, the modified flow rule (65) can be parameterized as
o
n
(67)
∂T̄ (i) g (i) ⊗ GRADϕ(i) ∈ ∂Σ g̃ (i) ||GRADϕ(i) || Σ̃ ∈ R3×3 .
Σ̃
Since this parameterization can be applied to any of the active yield surfaces, the influence of all strong discontinuities on the deformation gradient can be described by the plastic multipliers and the pseudo stresses, i.e.,
ns
X
i
J̇
(i)
⊗ GRADϕ(i) =
ns
X
i
with
λ(i) ∂Σ φ(i) ǫp :=
ns
X
i=1
Σ̃
||GRADϕ(i) || = ǫ̇p (λ(1) , . . . , λ(ns ) , Σ̃)
J (i) ⊗ GRADϕ(i) .
(68)
(69)
This new parameterization leads to ns + dim Σ̃ unknowns. Consequently, in case of an elastically isotropic bulk
material ns + 6 degrees of freedom have to be computed. If pseudo stress vectors are utilized, ns + 3 ns variables
are required. Consequently, a pseudo stress tensor is computationally more efficient than its vector counterpart, if
more than one discontinuity is active.
Remark 7 The derivation of a variationally consistent strong discontinuity approach relies strongly on two terms.
First, the constitutive model has to be driven by a certain variational principle. In the present paper, this principle
is an extended principle of maximum dissipation in which non-associative evolution equations are enforced by
using a parameterization by means of pseudo stresses. Second, the finite element formulation has to conserve this
material symmetry (variational consistency). While this is automatically guaranteed for Bubnov-Galerkin-type
approaches such as the extended finite element method or classical interface elements, the considered embedded
strong discontinuity approach (ESDA) is based on a Petrov-Galerkin discretization. However, the ESDA is only
symmetric provided the cracks or shear bands are parallel to the facets or the edges of the underlying finite
element discretization. Thus, the assumption of cracks or shear bands being parallel to the elements’ facets and
edges would not be required in case of the extended finite element method. In summary, the proposed variational
constitutive update could also be incorporated into a broad variety of other finite element formulations based on
strong discontinuities including the extended finite element method or classical interface elements. If the respective
formulation was intrinsically variationally consistent (Bubnov-Galerkin-type), the resulting numerical approach
would be automatically symmetric, regardless of the orientation of the discontinuities and the failure mode. For
instance, the finite strain counterpart of the proposed mixed-mode model would be variationally consistent in this
case for any discontinuity. Details about this consistency for other finite element formulations are beyond the scope
of the present paper, but will be discussed in a forthcoming paper.
4 NUMERICAL IMPLEMENTATION
This section is concerned with the numerical implementation of the embedded strong discontinuity approach presented in the previous sections. While in Subsection 4.1 an algorithmic formulation based on a modified returnmapping scheme is briefly discussed, a novel variational constitutive is elaborated in Subsection 4.2. Since the
constitutive framework, including the variational principle, associated with slip bands and that related to cracks are
formally identical (cf. Subsection 3.2.3), the numerical implementations will only be discussed in detail for one of
such material failures. More precisely, focus is on slip bands.
4.1 Return-mapping algorithm
The numerical implementations for finite elements based the embedded strong discontinuity approach can be
classified into two groups, cf. [32]. The majority of the models relies strongly on the underlying Enhanced
Assumed Strain (EAS) concept, cf. [28]. More precisely, the degrees of freedom associated with the continuous
deformation û and those corresponding to the displacement discontinuities are computed simultaneously. For that
A variationally consistent embedded strong discontinuity approach
15
purpose, a static condensation technique is usually applied, see [11, 51]. Alternatively, some authors advocated a
numerical implementation based on a modified return-mapping scheme, cf. [39, 40, 57, 73]. By extensively using
the analogy between the strong discontinuity kinematics and that related to multiplicative plasticity theory, it can
be shown that the algorithms originally derived for standard (continuous) plasticity theory can also be applied to
the discontinuous case. This is precisely the framework which will be followed in the present paper.
According to Subsection 2.2, two different approximations exist in case of multiple strong discontinuities,
compare Eq. (15) to Eq. (16). Both have been implemented. However, since their algorithmic formulations are
similar, only one of those will be described in detail. Due to the efficiency of approximation (16), it is chosen as
prototype.
4.1.1 Elastic predictor
Assuming a purely elastic loading step, the material displacement jumps J (i) remain constant. It bears emphasis
that this is not true for their spatial counterparts, i.e.,
J̇
With assumption J̇
(i)
tr
F̄ n+1
(i)
=0
6⇒
[ u̇]]
(i)
=
i
d h
F̂ · J (i) = 0.
dt
(70)
= 0, the (elastic) trial deformation gradient at time tn+1 can be computed according to
= F̂ n+1 − F̂ n+1 ·
ns
X
β=1
(β)
J (β)
,
n ⊗ GRAD ϕ
F̂ n+1 = 1 + GRAD ûn+1
(71)
tr
which, in turn, defines the right Cauchy-Green trial tensor C̄ n+1 , the second Piola-Kirchhoff trial stresses S tr
n+1
and finally, the trial stress vector
(β) tr
tr
(β)
T̄ n+1 = C̄ n+1 · S tr
n+1 · N
(72)
at internal surface β. Subsequently, the discrete loading condition
(β) tr
(β) tr
φ(β) tr := φ(β) (T̄ n+1 , q n+1 ) > 0 with
(β) tr
q n+1 = q (β)
n .
(73)
can be checked. Clearly, if φ(β) tr ≤ 0 for all beta, the trial step represents already the physical solution. If one of
the yield functions is, however, active (φ(β) tr > 0), a plastic corrector step is required. In the next paragraph, such
a corrector step is described in detail. For the sake of simplicity, it is assumed that the set of active yield surfaces
Jact is known. Clearly, this is usually not the case. More precisely, the computation of this set is highly challenging
itself and thus, a subject of its own, cf. [74]. Some comments concerning the employed active set search can be
found in Paragraph 4.1.3.
4.1.2 Plastic corrector
(β)
(β)
For active yield surfaces, the assumption J̇
= 0 was obviously not correct. Hence, J n+1 6= J (β)
n . In line with
the classical return-mapping scheme for continuous deformations, cf. [75, 76], a backward Euler integration is
applied for approximating the flow rule and the evolution equations. More precisely,
(i)
(i)
J n+1 = J (i)
∂T̄ (i) g (i) n + ∆λ
n+1
(74)
(i)
(i)
(i) αn+1 = α(i)
+
∆λ
∂
g
.
q (i)
n
n+1
For solving the set of nonlinear equations (74), together with φ(i) = 0, ∀i ∈ Jact , Newton’s method is applied.
However, a direct implementation of those equations would lead to a singular Hessian, if several discontinuities
were active. This problem is fully analogous to that of crystal plasticity theory, cf. [74]. For that reason, the plastic
strain-type tensor
ns
X
p
ǫ :=
J (β) ⊗ GRADϕ(β)
(75)
β=1
is introduced and Eq. (74)1 is replaced by
ǫp n+1 = ǫp n +
X
β∈Jact
∆λ(β) ∂ T̄(β) g (β) |n+1 ⊗ GRAD ϕ(β) .
(76)
16
J. Mosler, L. Stanković and R. Radulović
Finally, the residuum R
is defined with
p
R ǫ = −ǫp n+1 + ǫp n +
and
R :=
X
∆λ(β) ∂ T̄(β) g (β) |n+1 ⊗ GRAD ϕ(β)
β∈Jact
p
Rǫ
Rα
,
(77)


(1)
(1)
∂ q(1) g (1) |n+1
 −αn+1 + α(1)

n + ∆λ
R α :=
,
...


nact
(nact )
(nact )
(nact )
−αn+1 + αn
+ ∆λ
∂ q(nact ) φ
|n+1
nact := dim Jact .
(78)
(79)
and the set of equations
R=0
∧
(β)
φn+1 = 0 , β = 1, . . . , nact .
(80)
is solved by means of Newton’s method. Within this scheme, the variables (ǫp , q (i) , ∆λ(i) ) have been chosen as
the independent unknowns. Although the linearizations necessary for an asymptotically quadratic convergence are
quite lengthy, they can be computed in a standard fashion. For a single discontinuity they can be found in [39],
while the case of multiple discontinuities is described in [59].
4.1.3 Numerical aspects
Choice of the unknown variables within the return-mapping scheme According to the previous paragraph,
the variables (ǫp , q(i) , ∆λ(i) ) have been chosen as independent unknowns within the return-mapping scheme.
(i)
As already mentioned, the replacement of the stress vectors T̄ by ǫp is motivated by an analogy to crystal
(i)
plasticity theory: If T̄ had been chosen, the resulting Hessian would have been singular in case of multiple active
discontinuities. However, the choice of ǫp as unknown variable is not the only choice. The Mandel stresses Σ could
have been utilized as well. As a matter of fact, the original implementation was based on this parameterization.
However, convergence problems due to the linearization of Σ had been observed. Such drawbacks are not present
within the current implementation.
Enforcing a positive plastic multiplier In several numerical examples, the converged return-mapping state
showed a negative plastic multiplier. Even by employing the standard active set search strategy as briefly summarized in the following paragraph, such problems could not be eliminated. For this reason, ∆λ ≥ 0 was explicitly
enforced by setting ∆λ = a2 .
Active set search The set of potentially active singular surfaces within the return-mapping scheme is not necessarily constant, i.e., Jn+1 := {i|φ(i) > 0} 6= const. This is a well known problem frequently reported in crystal
plasticity theory, cf. [74]. Usually, two different strategies are applied. For solving this problem, either the set
Jn+1 := {i|φ(i) > 0} is updated during the iterations, or it is modified at a converged step and an additional
Newton iteration is performed subsequently, cf. [75, 76]. Without going to much into detail, it is noted that both
schemes are purely heuristic. However, the latter guarantees, at least, a quadratic convergence. Consequently, this
approach is chosen. In the next section, a novel variational constitutive update will be presented. This method
naturally solves the aforementioned problem.
4.2 Variational constitutive updates
In contrast to the return-mapping scheme presented in the previous section, the novel variational constitutive update discussed here is directly based on the underlying extended principle of maximum dissipation, see Eq. (49)
and Eq. (60). Conceptually, the overriding idea is astonishingly simple: The continuous variational principle is
approximated by a consistent time integration yielding a consistent numerical implementation.
According to Paragraph 3.2.3, the aforementioned extended principle of maximum dissipation is the minimization of the stress power. Hence, application of this principle within a numerical formulation requires a proper time
discretization. In line with the return-mapping scheme presented in the previous subsection, a backward-Euler
integration is applied here. More precisely,
(i)
(i)
J n+1 = J n+1 + ∆λ(i) ∂T̄ (i) g (i) ˜ (i)
T̄ n+1
(81)
(i)
(i)
(i)
(i) αn+1 = αn + ∆λ
∂q (i) g .
n+1
A variationally consistent embedded strong discontinuity approach
17
Evidently, the only difference between Eq. (81) and Eq. (74) is the parameterization of the flow direction by means
˜ (i) . As pointed out in Remark 6, Eq. (81) can be replaced by an equivalent description
of pseudo stress vectors T̄
1
in terms of a pseudo stress tensor Σ̃, i.e.,
X
∆λ(i) ∂Σ̃ g̃ (i) ||GRADϕ(i) ||
(82)
ǫp n+1 = ǫp n +
i∈Jact
Σ̃n+1
with ǫp as defined in Eq. (69). According to Remark 6, this parameterization is more efficient than that depending
on pseudo stress vectors, if more than one discontinuity is active within the considered finite element. For that
reason, parameterization (82) is chosen in what follows. However, efficiency is not the only advantage of this
approach. More precisely, it will be shown that in this case, the resulting variational constitutive update is almost
identical to that previously derived for standard (stress-strain-based) material models, cf. [45, 46].
Based on time integrations (81)2 and (82) the integrated stress power can be computed. For that purpose,
Eq. (81)2 and Eq. (82) are inserted into the Helmholtz energy Ψ yielding
Ψ(tn+1 ) =: Ψn+1 = Ψreg |n+1 + c
ns
X
i=1
(i)
δs(i) Ψsing |n+1 .
(83)
The second term necessary for computing the integrated stress power is the dissipation. With Eq. (36), a backwardEuler integration yields
tZ
n+1
ns
X
(i)
.
(84)
∆λ(i) δs(i) q0eq − (T̄Neq)(i) D dt =
n+1
i=1
tn
Finally, by combining Eqs. (81)–(84), the integrated stress power
Iinc :=
tZ
n+1
E dt = Ψn+1 − Ψn + c
tn
tZ
n+1
D dt
(85)
tn
is completely defined. Since the time discretization (81) is consistent, Eq. (85) is a consistent first-order approximation of the underlying functional (49). Accordingly, the novel variational constitutive update reads
(∆λ(1) , . . . , ∆λ(ns ) , Σ̃) = arg inf Iinc (∆λ(1) , . . . , ∆λ(ns ) , Σ̃).
(86)
Consistency of the novel variational constitutive update (86) is not obvious. Consequently, it will be proved
here. For that purpose, the stationarity conditions are analyzed. After some relatively straightforward computations, the gradients of the energy Iinc are obtained as
eq (i) T
∂Iinc
eq (i)
eq (i)
(i) ∂(T̄ N )
(i)
(i)
(i)
e
− ∆λ
+ c δs
q0
− (T̄ N )
= − F̂ · ∂F̄ Ψ : ∂T̄ (i) g ⊗ GRADϕ
∂∆λ(i)
∂∆λ(i)
eq (i) eq (i)
eq (i)
eq (i)
(i) ∂(T̄ N )
(i)
(i)
− ∆λ
q0
− (T̄ N )
=
−(T̄ S ) ||GRADϕ ||
+ c δs
∂∆λ(i)
(87)
and
ns eq
X
T
∂(T̄ N )(i)
∂Iinc
2 (i)
g̃ |Σ̃ ||GRADϕ(i) || − c ∆λ(i) δs(i)
.
(88)
F̂ · ∂F̄ Ψe : ∂Σ
=−
∂ Σ̃
∂ Σ̃
i=1
Alternatively, a parameterization in terms of pseudo stress vectors leads to
∂Iinc
˜ (i)
∂ T̄
=
=
−
i
h T
F̂ · ∂F̄ Ψe · GRADϕ(i) · ∂T̄2 g (i) |T̄˜ (i)
−T̄
(i)
eq
−
c ∆λ(i) δs(i)
∂(T̄ N )(i)
˜ (i)
∂ T̄
eq
· ∂T̄2 g (i) |T̄˜ (i) ||GRADϕ(i) ||
− c ∆λ(i) δs(i)
(89)
∂(T̄ N )(i)
.
˜ (i)
∂ T̄
Fully analogously to Paragraph 3.2.3, the singular Dirac-Delta distributions are eliminated by integrating the equations over the respective finite element. For example, for constant strain tetrahedron elements, Eq. (87) is re-written
as
Z
∂Iinc
eq (i)
eq (i)
eq
e
(i)
(i)
dV
=
−V
(
T̄
)
||GRADϕ
||
+
c
A
q
−
(
T̄
)
− rλ (∆λ(i) ),
(90)
S
s
N
0
∂∆λ(i)
Ωe
18
J. Mosler, L. Stanković and R. Radulović
while Eq. (89) yields
Z
Ωe
∂Iinc
(i)
dV = −T̄ · ∂T̄2 g (i) |T̄˜ (i) ||GRADϕ(i) || V e − rT (∆λ(i) ).
˜ (i)
∂ T̄
(91)
Here, rλ and r T are two residuals depending on ∆λ, i.e.,
eq
rλ (∆λ(i) )
(i)
= c A(i)
s ∆λ
∂(T̄ N )(i)
∂∆λ(i)
eq
(i)
rT (∆λ )
= c
A(i)
s
(i)
∆λ
∂(T̄ N )(i)
.
˜ (i)
∂ T̄
(92)
Hence, in the limiting case ∆t → 0, these residua converge to zero and as a result, stationarity conditions (90) and
(91) simplify to
Z
∂Iinc dV = 0 ⇔ φ(i) = 0
(93)
∂∆λ(i) Ωe
and
Z
Ωe
∆t→0
∂Iinc dV = 0
∂ Σ̃ ∆t→0
⇔
T̄
(i)
· ∂T̄2 g (i) |T̄˜ (i) = 0.
(94)
As evident from Eqs. (93)–(94), the numerical approximation leads to the same stationarity conditions as the continuous model, cf. Eqs. (58)–(59), and thus, the algorithmic formulation is consistent. In line with Paragraph 3.2.3,
the parameter c has been chosen as cgeom (see appendix). Clearly, strictly speaking, equivalences (93)–(94) hold
only, if the stresses are constant within each finite element. For more general discretizations, these stationarity
conditions represent the respective weak forms. It is noteworthy that Eqs. (93)–(94) are formally identical to those
corresponding to variational constitutive updates for continuous models (based on stress-strain relationships), cf.
[45, 46]. This analogy is even more pronounced, if a parameterization in terms of a pseudo stress tensor is employed.
R
For solving the highly nonlinear minimization problem inf Iinc (∆λ(1) , · · · , ∆λ(ns ) , Σ̃)dV standard optimization schemes can be applied.
In case of gradient-type approaches such as BFGS-algorithms or conjugate
R
gradient methods, the function Iinc dV itself and its gradients (see Eqs. (87) and (88)) are required, cf. [77, 78].
If other algorithms showing a higher convergence rate such as Newton’s method are to be applied, the second
derivatives of Iinc have to be computed as well. Although they are not given explicitly
in the present paper, they
R
can be derived in a straightforward fashion. Once, the minimization problem inf Iinc (∆λ(1) , · · · , ∆λ(ns ) , Σ̃)dV
is solved, the stresses can be computed in a standard manner.
5 NUMERICAL EXAMPLES
The numerical robustness, the performance as well as the accuracy of the novel variational constitutive update are
analyzed in this section by means of selected numerical examples. First, the propagation of shear bands is considered in Subsection 5.2. Subsequently, cracking in quasi-brittle structures is discussed in Subsection 5.2. Within all
computations, 4-node tetrahedron elements based on a linear approximation of the continuous displacement field
are used.
5.1 Modeling of slip bands
5.1.1 Single element test: variational constitutive update vs. return-mapping algorithm
For demonstrating the robustness and the advantages of the novel variational constitutive update (Subsection 4.2)
compared to a numerical implementation based on the return-mapping scheme (Subsection 4.1), a single element
test as shown in Fig. 3 is analyzed. While for the bulk material a classical isotropic neo-Hooke-type model is
utilized (Young’s modulus E, Poisson’s ratio ν), the slip bands parallel to the facets of the finite element are
approximated by means of a von Mises-type yield function and associative evolution equations (g = φ, T̄Neq = 0,
cf. Paragraph 3.2.2). Furthermore, the exponential softening law
α(β)
eq (i)
(β)
(β)
(95)
1 − exp −
q (α ) = q0
H
is considered.
A variationally consistent embedded strong discontinuity approach
z
y
19
F,u
x
L
E
ν
q0eq
H
u
L
=
=
=
=
=
=
0.5
1.0 · 107
0.2
500
0.0008
0.01
[m]
[kN/m2 ]
[kN/m2 ]
[m]
L
L
Figure 3: Single element shear test: geometry and material parameters
After localization occurs, a certain iteration step is singled out for a comparative analysis between the novel
variational constitutive update (Subsection 4.1) and the modified return-mapping scheme (Subsection 4.2). According to the respective trial state, the initial set of active singular surfaces Jtr
act includes two potential slip bands, i.e.,
one with the normal in x-direction (β = 1) and one with the normal in y-direction (β = 4). The computed energy
landscape Iinc as a function depending on the two unknown slip increments ∆λ(1) and ∆λ(4) is shown in Fig. 4.
Interestingly, the two numerical approaches lead to different results. While the return-mapping solution signals two
3 · 10−4
1
2
∆λ(4) [m]
0.0
−3 · 10−4
−3 · 10−4
0.0
3 · 10−4
∆λ(1) [m]
Figure 4: Single element shear test: two-dimensional contour plot of the potential Iinc as a function depending on
∆λ(1) and ∆λ(4) . Point 1 indicates the solution obtained from the return-mapping scheme, while point 2 defines
the result corresponding to the proposed variational constitutive update.
active surfaces showing the same amplitude of plastic slip (∆λ(1) , ∆λ(4) ) = (1.0386994·10−5, 1.0386994·10−5),
the variational algorithm yields only one active surface (∆λ(1) , ∆λ(4) = 2.15982 · 10−5 , 0). A careful analysis
reveals that only the energy associated with the variational approach corresponds to a minimum. The extremum as
predicted by the return-mapping scheme is a maximum and hence, it is not the correct physical solution. It bears
emphasis that if Jtr
act is modified such that only the first slip plane is active, both numerical implementations lead
to identical results.
This example highlights the importance of computing the correct set of active slip planes. However, since
such a method depends crucially on the underlying physics, it is not naturally included within a return-mapping
scheme. More precisely, the correct solution can only be identified by checking the respective energy Iinc . By way
of contrast, the variational constitutive update directly driven by that energy automatically chooses the correct set
of active slip bands.
5.1.2 Numerical analysis of a strip with a circular hole
Within the next example, the propagation of a single slip band in a steel made strip with a circular hole is numerically analyzed. The geometry, the boundary conditions as well as the material parameters are summarized
in Fig. 5. As in the previous subsection, a standard neo-Hooke-type model (with material constants E and ν) is
chosen for the elastic response, while a von Mises yield function with exponential softening is adopted for the
20
J. Mosler, L. Stanković and R. Radulović
u, F
0.3
0.4
E
ν
σy
H
=
=
=
=
20690 [kN/cm2 ]
0.29
45
[kN/cm2 ]
4000 [kN/cm3 ]
0.3
0.8
0.4
0.8
Figure 5: Numerical analysis of a strip with a circular hole: dimensions (in [cm]) and material parameters; thickness of the strip t = 0.1 cm
mesh I (1020 elements)
mesh II (2721 elements)
mesh III (5210 elements)
Figure 6: Numerical analysis of a strip with a circular hole: distribution of the internal variable associated with the
relative shear sliding displacement for different finite element triangulations
modeling of slip bands (see Eq. (95)). A similar problem has been investigated numerically by several authors,
see e.g., [75]. Although the structure could be modeled by using plane stress conditions, a fully three-dimensional
setting is considered.
Three different displacement-controlled numerical analyses were conducted (see Fig. 6). Within the computations, up to four slip bands parallel to the facets of the triangulation were considered in each finite element. Since
the stress state within this structure is highly inhomogeneous, an initial imperfection for triggering the localization
process is not required. The final deformation as predicted by the different finite element triangulations, together
with the distribution of the maximum relative shear sliding displacement, are shown in Fig. 6. As evident from
Fig. 6, all discretizations predict a shear band with the same orientation. The angle between the normal vector
of that band and the loading direction is about 45◦ . This orientation agrees well with that of the maximum shear
stresses.
The load-displacement diagrams as obtained from the three different finite element analyses are summarized in
Fig. 6. Accordingly, a maximum force which is almost independent with respect to the underlying triangulation of
about 3.6 kN is observed. The post peak response is also not significantly affected by the underlying finite element
triangulation. Hence, the numerical approach gives mesh objective results.
5.1.3 Modeling of micro-shear bands – representative volume elements
In this subsection, the robustness and efficiency of the proposed variational constitutive update in case of thousands
of simultaneously active and crossing discontinuities is demonstrated. For that purpose, a representative volume
element characterized by pre-existing micro slip bands is considered. The material parameters, together with the
geometry and the boundary conditions, are given in Fig. 8. Again, the bulk is modeled by means of an isotropic
neo-Hooke-type law (Young’s modulus E, Poisson’s ratio ν) and the slip bands are approximated by utilizing a von
Mises model. In order to simulate the initial state of the material realistically, the yield strength q0eq is element-wise
stochastically distributed. For the softening response, an exponential evolution of the type (95) is adopted.
Two different unstructured discretizations, mesh I with 3588 and mesh II with 5884 constant-strain tetrahedral
elements, are considered. Since the initial strength of the slip bands is stochastically distributed and furthermore,
the facets of the elements define the set of admissible slip bands, both triangulations lead to different microstructures (the topology of pre-existing shear bands) and consequently, both will lead to different mechanical
responses. However, since the discretizations are almost isotropic, i.e., no direction is preferred, and the distribution
of the initial strength q0eq is assumed as uniform, both representative volume elements are macroscopically similar.
A variationally consistent embedded strong discontinuity approach
21
4
F [kN]
3
2
mesh I (1020 elements)
mesh II (2721 elements)
mesh III (5210 elements)
1
0
0
0.005
0.01
0.015
0.02
u [cm]
Figure 7: Numerical analysis of a strip with a circular hole: predicted load-displacement response for different
finite element triangulations (see Fig. 6)
u,F
p
h
b
p
b
d
h
p
E
ν
q0eq
H
=
=
=
=
=
=
=
=
4
4
8
1
1500
0.31
4.0 – 6.0
0.3
[cm]
[cm]
[cm]
[kN/cm2 ]
[kN/cm2 ]
[kN/cm2 ]
[cm]
d
u,F
Figure 8: Triaxial compression test: geometry and material parameters
As a result, they should show the same mechanical response on average. More precisely, the homogenized solution
should be invariant with respect to the triangulation.
The distribution of the shear band displacements is shown in Fig. 9. Accordingly and as expected, both tri-
6.6 · 10−2
9.9 · 10−2
9.2 · 10−4
8.1 · 10−4
αmax [cm]
mesh I
mesh II
Figure 9: Triaxial compression test: distribution of the internal variable αmax
22
J. Mosler, L. Stanković and R. Radulović
angulations lead to different results at the micro-scale. For analyzing the homogenized solution, the respective
quantities have to be averaged. For that purpose, the load-displacement diagram is chosen. Since the reaction force
is proportional to the average stress and the top displacement agrees with the average strain field, this diagram can
be interpreted as the resulting macroscopic stress-strain law. As evident from Fig. 10, although both discretizations
250
Reaction force F [kN]
II
Fmax
200
I
Fmax
150
100
mesh I
mesh II
50
0
0
0.05
0.10
0.15
0.20
0.25
Displacement u [cm]
Figure 10: Triaxial compression test: load-displacement diagram
predict a different microscopic response, they yield identical results at the macro-scale. The estimated ultimate
load for mesh I is reached at a vertical displacement of u = 0.12565 cm and has the magnitude F = 211.77 kN.
For mesh II, the ultimate load is evaluated at F = 214.94 kN at a vertical displacement of u = 0.1339 cm. Accordingly, the difference between the two spatial discretizations is 1.5%.
The example presented here demonstrated that the proposed variational constitutive update can easily handle
more than thousands of active cracks or shear bands. Since usually numerous pre-existing defects can be observed
at the micro-scale, the novel method seems to be promising for analyzing their effect on the macro-scale.
5.2 Modeling of cracking in brittle structures
5.2.1 Analysis of the mesh bias: mode-I cracking under simple tension
Having presented the predictive capabilities of the novel variationally consistent strong discontinuity approach
for the modeling of slip bands, focus is now on material failure in quasi-brittle structures. Within the present
subsection, the mesh bias induced by assuming that the cracks and the shear bands are parallel to the facets and
edges of the underlying finite element discretization is analyzed. For that purpose, one of the simplest examples
is considered: uniaxial tension within a geometrically linearized setting. The elastic material response of the
1 cm×1 cm× 1 cm cube shown in Fig. 11 is approximated by an isotropic Hooke model (Lamé constants λ =
11074 kN/cm2 , µ = 8019 kN/cm2 ). Cracking is taken into account by means of a Rankine-type yield function.
Here, crack initiation is assumed at a critical tension stress of ft = 20 kN/cm2 , while crack propagation is governed
by the exponential softening law (95) showing a fracture energy of Gf = 0.1 KNm/m2 . Localization is triggered
by a slight imperfection in the tensile strength.
The results obtained by means of three different finite element discretizations are summarized in Fig. 11.
Within this figure, the distribution of the internal variable associated with crack opening is shown. The first row in
Fig. 11 is associated with the facet modes, i.e., cracks are enforced to be parallel to the facets of the finite elements.
As evident from this figure and as expected, the macroscopic crack is orthogonal to the maximum principal stress
direction. The respective load-displacement diagrams are given in Fig. 12 (left). Within these diagrams, the
analytical solution of the boundary value is also provided. According to Fig. 12 (left), differences between the
analytical solution and the finite element analyses in which the cracks are enforced to be parallel to the elements’
facets are now evident. More precisely, although the numerical solutions seem to be independent of the underlying
discretization (mesh I-III), the respective ultimate loads are significantly higher than that of the analytical solution.
However, this overprediction is not very surprising: Cracking is governed by the normal stresses acting at the
elements’ facets. This stress is only identical to the externally applied stress, if the normal of the considered facet
is identical to the maximum principal stress direction. Otherwise, the stress at that facet is always lower and thus,
cracking occurs later.
Although from the authors’ experiences, the aforementioned mesh bias is less pronounced in more complex
boundary value problems, it is indeed important to develop advanced numerical approaches further reducing this
A variationally consistent embedded strong discontinuity approach
mesh I
(629 elements)
FE implementation
23
mesh II
(1236 elements)
mesh III
(2005 elements)
Only facet modes
Facet and edge modes
Facet modes and stress
projection
Figure 11: Numerical analysis of the simple tension test: distribution of the internal variable associated with crack
opening for different finite element triangulations and numerical implementations
Only facet modes
analytical
mesh I
mesh II
mesh III
20
15
10
5
0
analytical
mesh I
mesh II
mesh III
25
F [kN]
20
Facet modes and stress projection
20
15
10
5
0
0.01
0.02
u [cm]
0.03
0
analytical
mesh I
mesh II
mesh III
25
F [kN]
25
F [kN]
Facet and edge modes
15
10
5
0
0.01
0.02
u [cm]
0.03
0
0
0.01
0.02
0.03
u [cm]
Figure 12: Numerical analysis of the simple tension test: predicted load-displacement response for different finite
element triangulations and numerical implementations (see Fig. 11)
24
J. Mosler, L. Stanković and R. Radulović
artificial bias. As already mentioned with the introduction of the present paper, adaptive strategies could be used
for that purpose, see, e.g., [34–36]. However, two alternative methods are briefly discussed here. The first of those,
is straightforward: In addition to the facet modes, the edge modes are also allowed to be active. By doing so, the
angle between the maximum principal stress direction and the normal vector of an active crack can be reduced. The
solutions predicted by this approach are shown in Fig. 11 (second row) and Fig. 12 (middle). According to Fig. 11,
the topology of the crack still agrees reasonably with that of the analytical solution. However and as evident in
Fig. 12, the ultimate load has been improved significantly. Consequently, although the method shows indeed an
intrinsic mesh bias, this bias is not very pronounced anymore.
Alternatively, the mesh bias induced by enforcing the cracks to be parallel to the elements’ facets can be
reduced by applying a stress projection techniques. The underlying idea of such a projection technique is that
crack initiation at the critical facet should occur at the same time as for the analytical solution. One way of
achieving this is to multiply the crack initiation stress by the cosine of the angle between the facet’s normal N F
and the maximum principle stress direction N σ , i.e.,
q̃0eq = cos(N F , N σ ) q0eq .
(96)
For enforcing also an equivalence of the resulting macroscopic fracture energies, the original fracture energy at
the respective facet has to be reduced by the cosine as well. It bears emphasis that the aforementioned projection
technique is not restricted to any specific failure type. For instance, if slip bands are to be modeled, the vector
N σ has to be chosen as the maximum principal shear direction. For guaranteeing consistency of the softening
behavior, the fracture energy has again to be modified as explained for the Rankine-type model. Further details are
omitted here, but will be presented in a forthcoming paper.
The results obtained from the strong discontinuity approach based on facet modes combined with the aforementioned stress projection technique are summarized in Fig. 11 (third row) and Fig. 12 (right). Again, the orientation
of the predicted macroscopic crack is in good agreement with that of the analytical solution (see Fig. 12). Furthermore, the corresponding load-displacement diagram agrees now also excellently with the analytical solution
The mesh bias induced by enforcing the cracks to be parallel to the elements’ facets has been carefully investigated here. Three conclusion can be drawn from the respective analyses:
• Although the results obtained by the strong discontinuity approach based on facet modes do not strongly
depend on the finite element discretization, they can be still different compared to the analytical solution.
• Therefore, advanced solution techniques are required. The simplest of those is to include also the edge
modes. Alternatively, the aforementioned projection technique can be used. An additional method which
has not been discussed in detail here is mesh adaption.
• Although the difference in predicted mechanical response between the strong discontinuity approach based
on facet modes and that including also edge modes is significant for the simple tension test, it is less pronounced in case of more complex boundary value problems. As a matter of fact, most of the examples
presented here have been originally computed by using both approaches. However, since the respective
results were almost identical, they will not be discussed in detail here.
5.2.2 Modeling of micro-cracks – representative volume elements
Similar to Subsection 5.1.3, the robustness and efficiency of the proposed variational constitutive update in case of
thousands of simultaneously active and crossing discontinuities is demonstrated here. However, focus is now on the
evolution of micro-cracks. The material parameters, together with the geometry and the boundary conditions, are
given in Fig. 13. According to Fig. 13, the representative volume elements consist of ceramic particles embedded
within a polymer matrix. Two different stochastically generated volumes with the same ratio polymer/ceramics are
considered. As evident in Fig. 13, mesh I has been discretized by using a relatively coarse triangulation compared
to mesh II.
The elements crossed by cracks and the distribution of the resulting macroscopic crack (crack width) are shown
in Fig. 14. Accordingly, thousands of elements are enhanced by a discontinuous displacement field. Furthermore,
depending on the underlying finite element discretization and microstructure, several spots of cracking at the
micro scale can be seen. However, only one final macroscopic crack within the polymer matrix being orthogonal
to the external loading direction will eventually form. Due to the variational consistency of the elaborated strong
discontinuity approach, the resulting crack path can be interpreted as the one which is energetically favorable.
The load-displacement diagrams are depicted in Fig. 15. Although both discretizations predict a different
microscopic response, they yield identical results at the macro-scale. The estimated ultimate load is reached at a
vertical displacement of 0.22 cm and has the magnitude 0.65 kN.
A variationally consistent embedded strong discontinuity approach
25
mesh II
mesh I
Ceramics
µ
9842 kN/cm2
λ
11554 kN/cm2
Fracture energy
0.32 kNm/m2
Crack initiation stress 60 kN/cm2
Polymer
µ
λ
Fracture energy
Crack initiation stress
112 kN/cm2
219 kN/cm2
0.12 kNm/m2
4 kN/cm2
Figure 13: Numerical analysis of crack initiation and propagation in a ceramic-polymer composite of size
0.3cm×0.4cm×2.0cm: geometry of the microstructures and material parameters
mesh I
Elements crossed
by cracks
mesh II
Distribution of
crack width
Elements crossed
by cracks
Distribution of
crack width
Figure 14: Numerical analysis of crack initiation and propagation in a ceramic-polymer composite: elements
crosses by micro cracks and the evolution of the equivalent crack width (max || [[u]] ||) for two different discretizations
6 CONCLUSIONS
In the present paper, a novel embedded strong discontinuity approach has been proposed. In contrast to already
existing models, the presented implementation allows to consider several interacting and crossing slip bands or
cracks in each finite element. For deriving an efficient formulation of the kinematics, the effect of a single discontinuity was critically analyzed first. Based on this analysis a novel parameterization of the deformation gradient
F was elaborated. It permits to compute the regularly distributed part of F in an explicit manner and therefore, it
increases the numerical efficiency significantly compared to other approaches. For a single discontinuity in each
finite element, this parameterization is mathematically equivalent to its original counterpart, while for several slip
bands this does not hold anymore. However, numerical experiments indicated that the difference between the two
formulations can be neglected even in this case.
26
J. Mosler, L. Stanković and R. Radulović
0.7
mesh I
mesh II
F [kN]
0.525
0.35
0.175
0
0
0.01
0.02
0.03
0.04
u [cm]
Figure 15: Numerical analysis of crack initiation and propagation in a ceramic-polymer composite: loaddisplacement response as predicted by the discretizations shown in Fig. 14
The other novelty advocated in the present paper is the proposed variational constitutive update. In line with
previously published works on variational updates for standard (stress-strain-based) material models, this method
allows to compute the displacement discontinuities, together with the internal variables, jointly by minimizing a
certain energy functional. This functional turns out to be the stress power. For providing enough flexibility to
account even for non-associative evolution equations, the flow rule and the evolution equations were enforced
a priori by utilizing a convenient re-parameterization and the dissipation functional was chosen independently
of those equations. The resulting variational principle can be understood as an extended principle of maximum
dissipation, since the flow rule and the dissipation can be chosen independently. This variational principle was
discretized yielding an efficient numerical implementation inheriting all physical properties of the underlying continuous counterpart. The advocated modeling framework covers a broad range of different, also non-associative,
plasticity models and accounts for several discontinuities within each finite element. In contrast to implementations based on a modified return-mapping scheme, no artificial active search strategies for determining the active
discontinuities are required. The physical minimization principle itself specifies automatically and naturally this
set. Interestingly, the proposed algorithmic formulation is formally identical to that of standard (continuous deformation) material models. This could be seen particularly when the flow rule was parameterized in terms of a
pseudo stress tensor.
For deriving the proposed variational constitutive update a geometry parameter c had to be introduced. It is
related to the position of the discontinuity within the considered finite element. By choosing c = cgeom , the
variational update gives identical results as the original embedded strong discontinuity approach. Therefore, this
choice was made. However, the position of the discontinuity, or more precisely, the area of the slip band or the
crack, does influence the softening behavior. Though this effect has not been analyzed in detail here, it is naturally
included within the proposed variational update scheme and will be analyzed in a forthcoming paper.
7 APPENDIX
In this appendix, equation
||GRADϕ|| V !
= cgeom = const
(97)
A
is proved. For that purpose, an affine approximation of the undeformed configuration X ∈ Ω by means of natural
coordinates ξ of the type
X = Jξ · ξ + X0
(98)
is considered. Here, J ξ is the Jacobian of that approximation. Hence, the volume V , the area A and the gradient
of the ramp function ϕ can be computed using the well known equations
V = det Jξ Vξ ,
GRADϕ = ∂ξ ϕ · J −1
ξ ,
A = det Jξ Vξ ||Aξ · J ξ−1 ||.
(99)
Inserting Eqs. (99) into the left hand side of Eq. (97), yields the identity
||∂ξ ϕ · J −1
||GRADϕ|| V
ξ ||
=
.
−1
A
||Aξ · J ξ ||
(100)
A variationally consistent embedded strong discontinuity approach
Mode
∂ξ Ni
Ai
X e1 ∈ Ω+
e1
1/cgeom e1
X e2 ∈ Ω+
e2
1/cgeom e2
X e3 ∈ Ω+
e3
1/cgeom e2
27
X e4 ∈ Ω+
−e1 − e2 − e3
1/cgeom (−e1 − e2 − e3 )
Table 1: Analysis of Eq. (100) for all different failure modes
For analyzing Eq. (100), all possible failure modes are considered in Tab. 1. Here, cgeom is a geometry parameter
which depends only on the relative position of the area Ai within the finite element. If the surface Ai is chosen
to be identical to one of the facets, cgeom = 2. If the area approaches the opposite corner node, cgeom → 0. In
between, a linear transition applies. Inserting Tab. 1 into Eq. (100) and Eq. (97), the proof is completed, i.e.,
||GRADϕ|| V
= cgeom .
A
(101)
Acknowledgement
This work was completed under the financial support of the Deutsche Forschungsgemeinschaft (DFG) through
project BR 580/30 and the Integrated Materials Systems cluster within the Hamburg Excellence Initiative. The
authors wish to express their sincere gratitude to this support.
References
[1] I. Milne, R.O. Ritchie, and B.L. Karihaloo. Comprehensive Structural Integrity, volume 3: Numerical and
Computational Methods. Elsevier Science, 2004.
[2] J. Mosler. On the numerical modeling of localized material failure at finite strains by means of variational
mesh adaption and cohesive elements. Habilitation, Ruhr University Bochum, Germany, 2007.
[3] D.S. Dugdale. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids,
8:100–108, 1960.
[4] G.I. Barenblatt. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech., 7:55–
129, 1962.
[5] Griffith A. The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London, CCXXI-A:163–198,
1920.
[6] A. Hillerborg, M. Modeer, and P.E. Petersson. Analysis of crack formation and crack growth in concrete by
means of fracture mechanics and finite elements. Cement and Concrete Research, 6:773–782, 1976.
[7] F. Armero and K. Garikipati. An analysis of strong discontinuities in multiplicative finite strain plasticity and
their relation with the numerical simulation of strain localization in solids. International Journal for Solids
and Structures, 33:2863–2885, 1996.
[8] J. Oliver. A consistent characteristic length for smeared cracking models. International Journal for Numerical
Methods in Engineering, 28:461–474, 1989.
[9] T. Belytschko, J. Fish, and B.E. Engelmann. A finite element with embedded localization zones. Computer
Methods in Applied Mechanics and Engineering, 70:59–89, 1988.
[10] R. De Borst. Non-linear analysis of frictional materials. PhD thesis, Technical University Delft, 1986.
[11] J.C. Simo, J. Oliver, and F. Armero. An analysis of strong discontinuities induced by strain softening in
rate-independent inelastic solids. Computational Mechanics, 12:277–296, 1993.
[12] K. Garikipati and T.J.R. Hughes. A study of strain localization in a multiple scale framework - the onedimensional problem. Computer Methods in Applied Mechanics and Engineering, 159(3-4):193–222, 1998.
[13] F. Armero. Large-scale modeling of localized dissipative mechanisms in a local continuum: applications to
the numerical simulation of strain localization in rate-dependent inelastic solids. Mechanics of CohesiveFrictional Materials, 4:101–131, 1999.
28
J. Mosler, L. Stanković and R. Radulović
[14] N. Moës, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing.
International Journal for Numerical Methods in Engineering, 46:131–150, 1999.
[15] M. Ortiz and E.A. Repetto. Nonconvex energy minimisation and dislocation in ductile single crystals. J.
Mech. Phys. Solids, 47:397–462, 1999.
[16] C. Carstensen, K. Hackl, and A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity.
Proc. R. Soc. Lond. A, 458:299–317, 2002.
[17] C. Miehe and M. Lambrecht. Analysis of micro-structure development in shearbands by energy relaxation
of incremental stress potentials: Large-strain theory for standard dissipative materials. International Journal
for Numerical Methods in Engineering, 58:1–41, 2003.
[18] G. Pijaudier-Cabot and Z.P. Bažant. Nonlocal damage theory. Journal of Engineering Mechanics (ASCE),
113:1512–1533, 1987.
[19] Z.P. Bažant and G. Pijaudier-Cabot. Nonlocal damage, localization, instability and convergence. Journal of
Applied Mechanics, 55:287–293, 1988.
[20] H.B. Mühlhaus and E.C. Aifantis. A variational principle for gradient plasticity. International Journal for
Solids and Structures, 28:845–857, 1991.
[21] R. De Borst and H.B. Mühlhaus. Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering, 35:521–539, 1992.
[22] A. Needleman. An analysis of decohesion along an imperfect interface. International Journal of Fracture,
42:21–40, 1990.
[23] G.T. Camacho and M. Ortiz. Computational modelling of impact damage in brittle materials. International
Journal for Solids and Structures, 30(20-22):2899–2938, 1996.
[24] M. Ortiz and A. Pandolfi. Finite-deformation irreversible cohesive elements for three-dimensional crackpropagation analysis. International Journal for Numerical Methods in Engineering, 44(9):1267–1282, 1999.
[25] E.N. Dvorkin, A. M. Cuitiño, and G. Gioia. Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. International Journal for Numerical Methods in
Engineering, 30:541–564, 1990.
[26] M. Klisinski, K. Runesson, and S.. Sture. Finite element with inner softening band. Journal of Engineering
Mechanics (ASCE), 117(3):575–587, 1991.
[27] N. Sukumar, N. Moës, B. Moran, and T. Belytschko. Extended finite element method for three-dimensional
crack modelling. International Journal for Numerical Methods in Engineering, 48:1549–1570, 2000.
[28] J.C. Simo and S. Rifai. A class mixed assumed strain methods and the method of incompatible modes.
International Journal for Numerical Methods in Engineering, 29:1595–1638, 1990.
[29] J. Oliver, A.E. Huespe, E. Samaniego, and E.W.V. Chaves. On Strategies for Tracking Strong Discontinuities
in Computational Failure Mechanics. In Fifth World Congress on Computational Mechanics, 2002.
[30] Th.C. Gasser and G.A. Holzapfel. 3d crack propagation in unreinfoced concrete. A two-step algorithm for
tracking 3d crack paths. Computer Methods in Applied Mechanics and Engineering, 195(37-40):5198–5219,
2006.
[31] J.M. Sancho1, J. Planas, A.M. Fathy, J.C. Gálvez, and D.A. Cendón. Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity. International Journal
for Numerical and Analytical Methods in Geomechanics, 31:173–187, 2007.
[32] J. Mosler. On the modeling of highly localized deformations induced by material failure: The strong discontinuity approach. Archives of Computational Methods in Engineering, 11(4):389–446, 2004.
[33] K. Papoulia, S. Vavasis, and P. Ganguly. Spatial convergence of crack nucleation using a cohesive element
model on a pinwheel-based mesh. International Journal for Numerical Methods in Engineering, 67(1):1–16,
2006.
[34] P. Thoutireddy and M. Ortiz. A variational r-adaption and shape-optimization method for finite-deformation
elasticity. International Journal for Numerical Methods in Engineering, 61:1–21, 2004.
A variationally consistent embedded strong discontinuity approach
29
[35] J. Mosler and M. Ortiz. Variational h-adaption in finite deformation elasticity and plasticity. International
Journal for Numerical Methods in Engineering, 72(5):505–523, 2007.
[36] C. Miehe and E. Gürses. A robust algorithm for configurational-force-driven brittle crack propagation with
r-adaptive mesh alignment. International Journal for Numerical Methods in Engineering, 72:127–155, 2007.
[37] C. Truesdell and R. Toupin. The classical field theories. In S. Flügge, editor, Handbuch der Physik, volume 3.
Springer-Verlag, Berlin, 1960.
[38] C. Truesdell and W. Noll. The nonlinear field theories. In S. Flügge, editor, Handbuch der Physik, volume 3.
Springer-Verlag, Berlin, 1965.
[39] J. Mosler. Modeling strong discontinuities at finite strains - a novel numerical implementation. Computer
Methods in Applied Mechanics and Engineering, 195(33-36):4396–4419, 2006.
[40] R.I. Borja. Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov-Galerkin multiscale formulation. Computer Methods in Applied Mechanics and
Engineering, 191:2949–2978, 2002.
[41] M. Fagerström and R. Larsson. A framework for fracture modelling based on the material forces concept with
xfem kinematics. International Journal for Numerical Methods in Engineering, 62(13):1763–1788, 2005.
[42] M. Fagerström and R. Larsson. Theory and numerics for finite deformation fracture modelling using strong
discontinuities. International Journal for Numerical Methods in Engineering, 66(6):911–948, 2006.
[43] M. Fagerström and R. Larsson. Approaches to dynamic fracture modelling at finite deformations. Journal of
the Mechanics and Physics of Solids, 56(2):613–639, 2008.
[44] M. Fagerström and R. Larsson. A thermo-mechanical cohesive zone formulation for ductile fracture. Journal
of the Mechanics and Physics of Solids, 56(10):3037–3058, 2008.
[45] J. Mosler and O.T. Bruhns. On the implementation of rate-independent standard dissipative solids at finite strain – Variational constitutive updates. Computer Methods in Applied Mechanics and Engineering,
199:417–429, 2010.
[46] J. Mosler and O.T. Bruhns. Towards variational constitutive updates for non-associative plasticity models at
finite strain: models based on a volumetric-deviatoric split. International Journal for Solids and Structures,
46(7-8):1676–1684, 2009.
[47] M. Ortiz and L. Stainier. The variational formulation of viscoplastic constitutive updates. Computer Methods
in Applied Mechanics and Engineering, 171:419–444, 1999.
[48] C. Miehe. Strain-driven homogenization of inelastic microstructures and composites based on an incremental
variational formulation. International Journal for Numerical Methods in Engineering, 55:1285–1322, 2002.
[49] J.C. Simo and J. Oliver. A new approach to the analysis and simulation of strain softening in solids. In Z.P.
Bažant, Z. Bittnar, M. Jirásek, and J. Mazars, editors, Fracture and Damage in Quasibrittle Structures, pages
25–39. E. &F.N. Spon, London, 1994.
[50] J. Oliver and J.C. Simo. Modelling strong discontinuities in solid mechanics by means of strain softening
constitutive equations. In H. Mang, N. Bićanić, and R. de Borst, editors, Computational Modelling of concrete
structures, pages 363–372. Pineridge press, 1994.
[51] J. Oliver. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations part
1: Fundamentals. part 2: Numerical simulations. International Journal for Numerical Methods in Engineering, 39:3575–3623, 1996.
[52] G.N. Wells and L.J. Sluys. A new method for modelling cohesive cracks using finite elements. International
Journal for Numerical Methods in Engineering, 50:2667–2682, 2001.
[53] I. Stakgold. Boundary value problems of mathematical physics, volume I. Macmillien Series in Advanced
Mathematics and theoretical physics, 1967.
[54] I. Stakgold. Green’s functions and boundary value problems. Wiley, 1998.
[55] C. Linder and F. Armero. Finite elements with embedded strong discontinuities for the modeling of failure
in solids. International Journal for Numerical Methods in Engineering, 72(12):1391–1433, 2007.
30
J. Mosler, L. Stanković and R. Radulović
[56] K. Garikipati. On strong discontinuities in inelastic solids and their numerical simulation. PhD thesis,
Stanford University, 1996.
[57] J. Mosler and G. Meschke. Embedded cracks vs. smeared crack models: A comparison of elementwise
discontinuous crack path approaches with emphasis on mesh bias. Computer Methods in Applied Mechanics
and Engineering, 193(30-32):3351–3375, 2004.
[58] J. Mosler. On advanced solution strategies to overcome locking effects in strong discontinuity approaches.
International Journal for Numerical Methods in Engineering, 63(9):1313–1341, 2005.
[59] L. Stanković. Describing multiple surface localised failure by means of strong discontinuities at finite strains.
PhD thesis, Ruhr University Bochum, 2008.
[60] M. Jirásek and T. Zimmermann. Embedded crack model: Part I: Basic formulation, Part II: Combination
with smeared cracks. International Journal for Numerical Methods in Engineering, 50:1269–1305, 2001.
[61] B.D. Coleman and W. Noll. The thermodynamics of elastic materials with heat conduction and viscosity.
Arch. Rational Mech. Anal., 13:167178, 1963.
[62] B.D. Coleman. Thermodynamics of materials with memory. Arch. Rational Mech. Anal., 17:1–45, 1964.
[63] B.D. Coleman and M.E. Gurtin. Thermodynamics with internal state variables. J. Chem. Phys, 47:597–613,
1967.
[64] C. Miehe and J. Schröder. Post-critical discontinuous analysis of small-strain softening elastoplastic solids.
Archive of Applied Mechanics, 64:267–285, 1994.
[65] J. Mosler and O.T. Bruhns. A 3D anisotropic elastoplastic-damage model using discontinuous displacement
fields. International Journal for Numerical Methods in Engineering, 60:923–948, 2004.
[66] M. F. Snyman, W. W. Bird, and J. B. Martin. A simple formulation of a dilatant joint element governed by
Coulomb friction. Engineering Computations, 8:215–229, 1991.
[67] F. Armero. Localized anisotropic damage of brittle materials. In D.R.J. Owen, E. Oñate, and E. Hinton,
editors, Computational Plasticity, volume 1, pages 635–640, 1997.
[68] J. Oliver, A.E. Huespe, M.D.G. Pulido, and E. Samaniego. On the strong discontinuity approach in finite
deformation settings. International Journal for Numerical Methods in Engineering, 56:1051–1082, 2003.
[69] J. Mosler and G. Meschke. 3D modeling of strong discontinuities in elastoplastic solids: Fixed and rotating localization formulations. International Journal for Numerical Methods in Engineering, 57:1533–1576,
2003.
[70] J. Mandel. Plasticité Classique et Viscoplasticité. Cours and Lectures au CISM No. 97. International Center
for Mechanical Sciences, Springer-Verlag, New York, 1972.
[71] J. Lemaitre. A continuous damage mechanics model for ductile fracture. J. Eng. Mat. Techn., 107:83–89,
1985.
[72] R.T. Rockafellar. Convex analysis. Princeton University Press, 1970.
[73] R.I. Borja. A finite element model for strain localization analysis of strongly discontinuous fields based on
standard galerkin approximation. Computer Methods in Applied Mechanics and Engineering, 190:1529–
1549, 2000.
[74] M. Schmidt-Baldassari. Numerical concepts for rate-independent single crystal plasticity. Computer Methods
in Applied Mechanics and Engineering, 192(11-12):1261–1280, 2003.
[75] J.C. Simo and T.J.R. Hughes. Computational inelasticity. Springer, New York, 1998.
[76] J.C. Simo. Numerical analysis of classical plasticity. In P.G. Ciarlet and J.J. Lions, editors, Handbook for
numerical analysis, volume IV. Elsevier, Amsterdam, 1998.
[77] C. Geiger and C. Kanzow.
Springer, 1999.
Numerische Verfahren zur Lösung unrestringierter Optimierungsaufgaben.
[78] C. Geiger and C. Kanzow. Theorie und Numerik restringierter Optimierungsaufgaben. Springer, 2002.