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We describe a methodology for solving the constitutive problem and evaluating the consistent tangent operator for isotropic elasto/visco-plastic models whose yield function incorporates the third stress invariant . The developments presented are

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng
2004;
60
:461–498 (DOI: 10.1002/nme.970)
A return map algorithm for general isotropic elasto
/
visco-plasticmaterials in principal space
Luciano Rosati
1
,
∗
,
†
and Nunziante Valoroso
2
1
Dipartimento di Scienza delle Costruzioni
,
Università di Napoli Federico II
,
via Claudio 21
,
Napoli 80125
,
Italy
2
Istituto per le Tecnologie della Costruzione
,
Consiglio Nazionale delle Ricerche
,
Viale Marx 15
,
Roma 00137
,
Italy
SUMMARYWe describe a methodology for solving the constitutive problem and evaluating the consistent tangentoperator for isotropic elasto
/
visco-plastic models whose yield function incorporates the third stressinvariant
J
3
. The developments presented are based upon original results, proved in the paper, con-cerning the derivatives of eigenvalues and eigenprojectors of symmetric second-order tensors withrespect to the tensor itself and upon an srcinal algebra of fourth-order tensors
A
obtained as secondderivatives of isotropic scalar functions of a symmetric tensor argument
A
. The analysis, initiallyreferred to the small-strain case, is then extended to a formulation for the large deformation regime;for both cases we provide a derivation of the consistent tangent tensor which shows the analogybetween the two formulations and the close relationship with the tangent tensors of the Lagrangiandescription of large-strain elastoplasticity. Copyright
2004 John Wiley & Sons, Ltd.
KEY WORDS
:
J
3
plasticity and viscoplasticity; principal space formulation; return map algorithm;consistent tangent
1. INTRODUCTIONComputationalplasticity and viscoplasticity have been characterized by signiﬁcant advancementsin the last two decades mainly after the work by Simo, whose fundamental contributionsin the ﬁeld have been collected in two recent monographs
[
1,2
]
. The general framework outlined in these books and in several other papers on the subject, see e.g. References
[
3–7
]
among others, has addressed isotropic elasto(visco-) plastic materials, i.e. materials whoseelastic and (visco)plastic behaviours are described through an isotropic stored energy functionand an isotropic yield function.
∗
Correspondence to: Luciano Rosati, Dipartimento di Scienza delle Costruzioni, Università di Napoli Federico II,via Claudio 21, 80125 Napoli, Italy.
†
E-mail: rosati@unina.itContract
/
grant sponsor: Italian National Research Council (CNR)Contract
/
grant sponsor: Italian Ministry of Education, University and Research (MIUR)
Received 2 October 2002 Revised 26 July 2003
Copyright
2004 John Wiley & Sons, Ltd.
Accepted 5 August 2003
462
L. ROSATI AND N. VALOROSO
As a matter of fact, the approaches outlined in References
[
1,2
]
for small and large-strainisotropic plasticity are quite different. Actually, in the former case the solution of the constitutiveproblem and the evaluation of the consistent tangent are usually addressed in terms of (intrinsic)tensor quantities. On the contrary, most of the solution strategies presented in the literature
[
8–11
]
for elastoplastic models in the large-strain regime, substantially derived from Reference
[
12
]
, heavily hinge on the isotropy of the involved functions. Indeed, the assumption of isotropyis used to establish that the return mapping algorithm takes place at ﬁxed principal axis sothat only the principal values of the state variables need to be iterated upon.In the present work, we show how a fully tensorial approach naturally supplies a uniﬁedframework for small- and large-strain isotropic plasticity which, in addition, allows for a deeperinsight in the formulation and ﬁnite element implementation of the solution strategy both atthe constitutive and structural level.This is obtained by illustrating the relationships existing between the ﬁrst and secondderivatives of isotropic scalar functions
of a symmetric rank-two tensor
A
, assigned ei-ther as function of the eigenvalues of
A
or of its invariants and by proving some basicresults concerning the differentiation of the eigenprojectors
A
k
of
A
with respect to the ten-sor
A
itself. Moreover, the principal space representation and inversion of rank-four positive-deﬁnite symmetric tensors
A
obtained as the second derivative of
with respect to
A
isdiscussed.The developments carried out in the paper provide proper evidence to three main issues. First,the yield function can be assigned either in terms of eigenvalues or of the invariants of the stresstensor without affecting the derivation of the tensor quantities entering the return mapping andthe tangent operator. Second, the usual procedure of expressing the material tangent operator inthe principal reference frame and then transforming it to the given reference frame via matrixmanipulations can be by-passed since the expression of the tangent operator in the globalco-ordinate system can be directly constructed. Third, the terms entering the ﬁnal expressionof the consistent tangent operator can be clearly identiﬁed.The results of tensor analysis alluded to above are employed to carry out a return mappingalgorithm for associative isotropic elasto
/
visco-plastic models whose yield function dependsupon the
J
3
stress invariant.Yield criteria of this type are representative of a wide class of engineering materials suchas concrete and geomaterials, see e.g. References
[
13–21
]
, for speciﬁc applications and
[
22
]
for a detailed account of failure criteria incorporating the
J
3
invariant. Recently, the use of
J
3
-dependent surfaces has also been advocated, among others, in References
[
23,24
]
, for describingphase transitions in shape-memory alloys within the superelastic range.The solution of structural problems endowed with
J
3
-dependent yield functions representsa considerable challenge, by far more severe than the one associated with the classical
J
2
(Von Mises) model. For this reason, several approaches have been proposed in the literature
[
25–30
]
; in some cases
[
31
]
, implicit procedures are abandoned in favour of explicit ones
[
32,33
]
because of their complicated algorithmic structure.Within the framework of closest-point projection algorithms, two additional solution strategiesfor
J
3
-dependent plasticity models have been recently proposed by the authors
[
34,35
]
byproviding an explicit representation formula for the inverse of the elasto(visco)plastic compliancetensor
G
entering the exact algorithm linearization. In order to allow a direct extension to aprincipal space formulation, the inversion of
G
is here carried out on the basis of the generalproperties of positive-deﬁnite rank-four tensors discussed in the paper.
Copyright
2004 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2004;
60
:461–498
ISOTROPIC ELASTO
/
VISCO-PLASTIC MATERIALS
463It is shown in particular that only the
dyadic part of
G
, expressed as linear combinationof dyadic tensor products
S
⊗
ij
=
S
i
⊗
S
j
between the eigenprojectors
S
k
of the stress deviator
S
, is strictly needed in the computation of the return map solution since it is associated withthe algorithm linearization at constant eigenvectors. On the contrary, the
non-dyadic part
of
G
, which is expressed as linear combination of the square tensor products
S
ij
=
S
i
S
j
between the eigenprojectors of
S
, is associated with the differentiation of the eigenprojectorswith respect to the stress tensor and plays no active role in the local Newton iteration scheme.These circumstances suggest to express the return mapping algorithm directly in terms of principal values by making reference, at least formally, to a yield function assigned in termsof stress eigenvalues.Nonetheless, irrespective of the expression originally assumed for the yield function, alltensorial quantities entering the principal return mapping and the expression of the consistenttangent tensor are directly expressed in the global co-ordinate system.This approach carries over in a natural way to the case of isotropic hyperelastoplasticity atﬁnite strains based on the multiplicative decomposition since, in this case, the stored energyfunction is usually given in terms of principal stretches and the solution of the constitutiveproblem can be computed in terms of principal values
[
12
]
. With reference to this last case,an original derivation of the consistent tangent tensor, allowed by the results provided inthe paper, is presented. It highlights the perfect analogy of the relevant expression with theones pertaining to the continuum and algorithmic tangent tensors obtained for the Lagrangiandescription of large-strain elastoplasticity
[
36
]
and the close relationships with the expressionof the small-strain elastoplastic tangent.Finally, a numerical example shows the performances of the proposed implementation.2. ELEMENTS OF TENSOR ALGEBRAWe provide a brief review of the basic properties of linear transformations (tensors) on athree-dimensional inner product space
V
over the real ﬁeld.Denoting by Lin
(
L
in
)
the space of all second- (fourth-)order tensors on
V
, we shallalso illustrate the properties of tensor products between elements of Lin. Unless differentlystated, we shall use boldface normal (capital) letters to indicate elements of
V
(
Lin
)
and
BOLDBLACKBOARD
symbols to denote fourth-order tensors.Given
a
,
b
,
c
,
d
∈
V
and
A
∈
Lin the following relations can be proved:
(
a
⊗
b
)
·
(
c
⊗
d
)
=
(
a
·
c
)(
b
·
d
)(
a
⊗
b
)(
c
⊗
d
)
=
(
b
·
c
)(
a
⊗
d
)
A
(
a
⊗
b
)
=
Aa
⊗
b
(
a
⊗
b
)
A
=
a
⊗
A
T
b
(1)with the superscript
T
standing for transpose.Given
A
,
B
∈
Lin the tensor product
A
⊗
B
, usually termed dyadic product, is the elementof
L
in such that
(
A
⊗
B
)
C
=
(
B
·
C
)
A
=
tr
(
B
T
C
)
A
∀
C
∈
Lin (2)where the symbol tr
(
·
)
denotes the trace operator.
Copyright
2004 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2004;
60
:461–498
464
L. ROSATI AND N. VALOROSO
More recently Del Piero
[
37
]
has introduced an additional tensor product
A
B
betweensecond-order tensors deﬁned by
(
A
B
)
C
=
ACB
T
∀
C
∈
Lin (3)it will be referred to in the sequel as the square tensor product on account of the symbolsrcinally proposed by Del Piero.The previous product allows one to represent the identity tensor
I
in
L
in as
I
=
1
1
(4)where
1
is the identity tensor in Lin.The following composition rules can be shown to hold:
(
A
B
)(
C
D
)
=
(
AC
)
(
BD
)(
A
B
)(
C
⊗
D
)
=
(
ACB
T
)
⊗
D
(
A
⊗
B
)(
C
D
)
=
A
⊗
(
C
T
BD
)
(5)for every
A
,
B
,
C
,
D
∈
Lin. Furthermore, if
A
and
B
are invertible, one can show that such istheir square product:
(
A
B
)
−
1
=
A
−
1
B
−
1
Composition formulas between elements of
L
in and Lin follow immediately from the previousdeﬁnitions and properties:
(
a
⊗
b
)
(
c
⊗
d
)
=
(
a
⊗
c
)
⊗
(
b
⊗
d
)(
A
⊗
B
)(
c
⊗
d
)
= [
B
·
(
c
⊗
d
)
]
A
=
(
B
T
c
·
d
)
A
(
A
B
)(
c
⊗
d
)
=
A
(
c
⊗
d
)
B
T
=
Ac
⊗
Bd
(6)for every
A
,
B
∈
Lin and
a
,
b
,
c
,
d
∈
V
.Analogous to the standard deﬁnition adopted for second-order tensors, we remind that thetranspose of a fourth-order tensor is deﬁned by
A
B
·
C
=
B
·
A
T
C
∀
B
,
C
∈
Linso that
A
is symmetric whenever
A
=
A
T
. For the translation of the previous tensor formalisminto the matrix one the reader is referred to Appendix A.
2.1. Eigenprojectors of second-order symmetric tensors
Let
A
be a second-order symmetric tensor, i.e. an element of the subspace Sym
⊆
Lin.According to the spectral theorem
[
38
]
there is at least one orthonormal basis for
V
consistingof eigenvectors of
A
; hence, denoting by
ˆ
a
i
,
{
i
=
1
,
2
,
3
}
, the eigenvalues of
A
, supposed tobe distinct, and by
a
i
the associated unit eigenvectors, it turns out to be
A
=
3
i
=
1
ˆ
a
i
a
i
⊗
a
i
=
3
i
=
1
ˆ
a
i
A
i
(7)
Copyright
2004 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2004;
60
:461–498
ISOTROPIC ELASTO
/
VISCO-PLASTIC MATERIALS
465where
A
i
is the eigenprojector corresponding to
ˆ
a
i
, i.e. the orthogonal projection operator ontothe null space of
A
− ˆ
a
i
1
. In particular formula
(
1
)
1
yields
A
i
·
A
j
=
1 if
i
=
j
0 otherwise(8)Additional properties of the eigenprojectors that will be extensively used in the sequel are
A
i
A
j
=
A
j
A
i
=
A
i
if
i
=
j
0
otherwise(9)and
A
1
+
A
2
+
A
3
=
1
(10)Immediate consequence of (7) and (9) is that
A
i
A
=
AA
i
= ˆ
a
i
A
i
(11)
Remark 2.1
The eigenvalues of
A
are not necessarily distinct. Hence, if
A
has exactly two distinct eigen-values
ˆ
a
1
and
ˆ
a
2
and
a
1
is a unit vector belonging to the characteristic space of
ˆ
a
1
, the spectralrepresentation of
A
reads
A
= ˆ
a
1
a
1
⊗
a
1
+ ˆ
a
2
(
1
−
a
1
⊗
a
1
)
(12)while, for three coincident eigenvalues, one has
A
= ˆ
a
1
being
ˆ
a
the unique eigenvalue of
A
.The presence of coalescent eigenvalues does not represent a real problem from the com-putational standpoint since, even in this situation, it is possible to deﬁne three orthogonaleigenprojectors; as an alternative, one can use a perturbation of the possibly repeated eigen-values; see e.g. References
[
2,39,40
]
.
A basic result
[
39,41
]
setting in correspondence the generic eigenvalue with the associatedeigenprojector is contained in the following:
Lemma 1
Let
ˆ
a
i
be an eigenvalue of
A
∈
Sym. The derivative of
ˆ
a
i
with respect to
A
is provided by
d
A
ˆ
a
i
=
a
i
⊗
a
i
=
A
i
(13)
Proof
By deﬁnition of eigenvalue it turns out to be
Aa
i
= ˆ
a
i
a
i
(14)so that
ˆ
a
i
=
Aa
i
·
a
i
=
A
·
(
a
i
⊗
a
i
)
(15)since the eigenvectors
a
i
have been assumed of unit length.
Copyright
2004 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng
2004;
60
:461–498

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