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Fatigue design of structures under thermomechanical
loadings
Eric Charkaluk, André Bignonnet, Andreï Constantinescu, Ky Dang Van
To cite this version:
Eric Charkaluk, André Bignonnet, Andreï Constantinescu, Ky Dang Van. Fatigue design of structures
under thermomechanical loadings. Fatigue and Fracture of Engineering Materials and Structures,
Wiley-Blackwell, 2002, 25 (12), pp.1199-1206. �10.1046/j.1460-2695.2002.00612.x�. �hal-00111365�
Fatigue design of structures under thermomechanical loadings
E. CHARKALUK
1
, A. BIGNONNET
2
, A. CONSTANTINESCU
3
and K. DANG VAN
3
1
Laboratoire de MeÂcanique de Lille, UMR CNRS 8107, boulevard Paul Langevin, 59655, Villeneuve d'Ascq Cedex, France,
2
P.S.A Peugeot Citroen ±
Direction de la Recherche et de l'Innovation Automobile, chemin de la Malmaison ± 91 570, Bievres, France,
3
Laboratoire de Mecanique des Solides ± Ecole
Polytechnique, 91 128 Palaiseau, France
ABSTRACT This paper presents a global approach to the design of structures that experience
thermomechanical fatigue loading, which has been applied successfully in the case of
cast-iron exhaust manifolds. After a presentation of the design context in the automotive
industry, the important hypotheses and choices of this approach, based on a thermal 3D
computation, an elastoviscoplastic constitutive law and the dissipated energy per cycle as
a damage indicator associated with a failure criterion, are first pointed out. Two particu-
lar aspects are described in more detail: the viscoplastic constitutive models, which
permit a finite element analysis of complex structures and the fatigue criterion based
on the dissipated energy per cycle. The FEM results associated with this damage
indicator permit the construction of a design curve independent of temperature; an
agreement is observed between the predicted durability and the results of isothermal as
well as non isothermal tests on specimens and thermomechanical fatigue tests on real
components on an engine bench. These results show that thermomechanical fatigue
design of complex structures can be performed in an industrial context.
Keywords dissipated energy; fatigue; thermomechanics; viscoplasticity.
Nomenclature
a hardening strain tensor
H
hardening modulus
E, E
elastic modulus and elastic compliance tensor
e
mechanical strain tensor
e
elastic
elastic strain tensor
e
p,v,vp
respectively plastic, viscous, viscoplastic strain tensor
g
plastic multiplier
Z
viscosity
J
2
second invariant of the stress tensor
K
p
, K
p
elastic modulus (tensor) of the plastic part
K
v
, K
v
elastic modulus (tensor) of the viscous part
m
exponant of the viscosity
N
1
number of cycles to the first visible crack
N
2
number of cycles to a through crack
n
Poisson's ratio
s
p,v,vp
respectively plastic, viscous, viscoplastic deviatoric stress tensor
s
p,v,vp
respectively plastic, viscous, viscoplastic stress tensor
s
y
yield stress
Dw
dissipated energy density per cycle
X
hardening stress tensor
Correspondence: E. Charkaluk, Laboratoire de Me
Â
canique de Lille, UMR
CNRS 8107, boulevard Paul Langevin, 59655, Villeneuve d'Ascq Cedex,
France. E-mail: eric.charkaluk@univ-lille1.fr
1
INTRODUCTION
One can distinguish two principal approaches in fatigue
research: the first one concerns the understanding of
physical phenomena at the material scale and is the
focus of a large part of the work; the second one tries
to model these phenomena to establish lifetime criteria at
the scale of the structure. This paper addresses the life-
time prediction of structures in the Thermomechanical
Fatigue (TMF) domain for direct industrial applications
with a design time reduction in mind.
In the HCF (High Cycle Fatigue) domain, a number
of methods permitting the fatigue lifetime assessment
of structures during the design process are now success-
fully used in an industrial context. A general review of
methods and criteria available today was recently written
by Socie and Marquis.
1
We shall cite here a practical
approach presented by Dang Van et al.
2
which is ex-
tensively and successfully used in the design of all com-
ponents experiencing HCF by some automotive
industries.
3, 4
We should point out that even if the choice
of the criterion is important, the determination of the
limit conditions and also of an equivalent fatigue loading
obtained for example by the SSIA (Stress± strength Inter-
ference Analysis)
5
is essential for the design approach.
In the TMF design field, there are still a large number
of unresolved problems. Historically, TMF design
concerned principally the nuclear and the aeronautic
industries. In the first one, structures are usually
designed with lifetime criteria often issued for example
from the work of Coffin
6, 7
from General Electric and
used with high safety factors. A series of similar design
approaches have been formalized in different codes (R5
8
and RCC-MR,
9
). These rules are based on the periodic
monitoring of structures which permits the eventual
presence of cracks to be detected. The concept of crack
propagation then allows the lifetime of the structure to
be related to a critical crack size and provides the oppor-
tunity to assess the integrity of the structure during a
defined period.
In the aeronautic industry, for the design of turbine
blades, Chaboche
10
made great advances in the eighties.
These structures, which can include complex cooling
channels, are essentially subjected to centrifugal loads
with hold times at high temperature. For these applica-
tions, he developed complex constitutive and damage
laws to take into account the elastoviscoplasticity of the
material behavior and the creep±fatigue interaction in
the damage process and obtained interesting results.
In other industries, take for example the automotive
one, design paradigms are different. Components like
exhaust-manifolds, cylinder heads or cylinder blocks
have a complex geometry and the thermomechanical
loading includes different phases (binding, preloading
and thermal cycles) which induce 3D stress±strain distri-
butions. The main objective is not just safety, but also
lifetime prediction without intermediate monitoring.
Therefore engineers need robust computational methods
for the prediction of macroscopic fatigue crack initiation
which must guarantee the integrity of the structures
during the complete lifetime.
The first difficulty in such a design method comes from
the numerical evaluation of multiaxial strains and
stresses. This is due to combined thermal and mechanical
loadings with variable temperatures and a large range of
temperature inducing different constitutive behaviors.
The second difficulty arises from the fatigue criterion
itself as different damage mechanisms lead to the final
failure.
In the last decades we witnessed a series of major ad-
vances in the understanding of the underlying physical
phenomena and the numerical computations on struc-
tures. But to our knowledge, no global solution to this
problem has been proposed with results on actual indus-
trial structures (for a critical review and a list of unre-
solved problems see Chaboche, 10).
The aim of this study is to assess the lifetime of cast-iron
exhaust manifolds subjected to thermomechanical
loading. In this paper we propose a fatigue design method
based on elastoviscoplastic constitutive models for the
strain and stress computations and an energy criterion
for failure. In the first section we discuss the overall items
to be analyzed in a global design approach. The second
section presents an overview of the constitutive behavior
and of the Finite Element (FEM) computations. The ex-
perimental and computed fatigue results are discussed in
the final section. This global approach leads to a reason-
able lifetime prediction of actual structures and can easily
be extended to other materials and structures.
11
GLOBAL APPROACH IN FATIGUE PROBLEMS
The industrial design of parts generally passes through a
series of iterations based on design, testing and correc-
tion. In order to reduce the number of iterations (actually
about 3±5) and, implicitly, design costs and to obtain
better performance of the parts, engineers are obliged
to introduce predictive structural computations. Failure
should be directly assessed by computations during the
design. Ideally, prototype testing should not be a part of
the design cycle but only a final validation of predictions
in a real environment.
In a thermomechanical design process, we propose to
simulate the behavior of a prototype structure under vari-
ous loading conditions and to assess its lifetime. However,
not all loading conditions can generally be tested. The
choice of a unique representative test using the stress±
strength interference analysis (SSIA) is discussed for
2
example in Thomas et al.
12
in the HCF context and can
be extended for TMF conditions but will not be pre-
sented here.
In this work we study cast-iron exhaust manifolds. The
material is a nodular graphite cast-iron with silicon and
molybdenum additives.
The thermomechanical loading can be represented
by heating and cooling cycles between 20 8C and
800 8C (approximately 1000±5000 thermal shocks de-
pending on whether the application is a gasoline or diesel
engine) combined with mechanical clamping imposed by
the bolting of the hot exhaust manifold onto the colder,
more rigid, cylinder head. This thermal cycle is an
equivalent loading in the sense of the SSIA and should
represent the same damage as the complete lifetime of
the structure in the customer's vehicle. The temperature
distribution is computed from the transient heat equa-
tion with boundary conditions, heat flux and heat ex-
change coefficients determined by a 3D transient gas
flow simulation. The spatial thermal distribution is then
checked by infrared thermography of a structure. More
details of this crucial point of the design approach are
presented by Lederer et al.
13
After the thermal loading determination, a global ap-
proach to thermomechanical fatigue design can be
divided into two steps:
iAStress±Strain Computation depending on the defin-
ition of the thermomechanical loading and the choice of
the elastoviscoplastic constitutive law, and
ii A Fatigue Criterion including a damage evolution law
and a failure criterion.
An important assumption is the coupling/uncoupling of
the behavior and the damage. Lemaitre and Chaboche
14
proposed interesting constitutive laws coupled with
damage for continuum mechanics. It might therefore
seem unusual to uncouple the stress±strain behavior
from the damage. We accepted this hypothesis and the
underlying approximations in order to be able to compute
a 3D structure ('10
5
degrees of freedom) in a reasonable
computational time (some hours) with existing computers.
Stress±Strain computation
It is obvious that the mechanical behavior should be
reliable at both minimum and maximum temperatures,
where cast iron has plastic and viscous behavior, respect-
ively. One can distinguish two classes of models: physical
ones representing precisely the microscopic phenomena
at a material scale and specifically related to a certain
type of macroscopical loading: creep, relaxation, . . .
and phenomenological ones representing a macroscopic
mechanical behavior. In order to keep the final objective
of a global simple representation of the material behavior
at the scale of the structure, we shall restrict our choices
to the second class of models.
Fatigue criterion
The second aspect concerns the fatigue criterion. Two
different ways could be followed to determine a damage
evolution law. The first one is the continuum damage
mechanics developed in particular by Lemaitre et al.
14
The second one considers parametric laws relating
mechanical fields and lifetime such as the Manson±
Coffin law, or the Strain±Range Partioning method.
15
Damage evolution laws based on continuum damage
mechanics induce two particular problems. On the one
hand, models are complex and calibration of the material
parameters is difficult. On the other hand the damage
evolution laws are often based on mean and maximal
stresses
14
which are not easily defined in a non isothermal
cycle. For example, the often proposed ratio (s/s
u
), where
s can be the von Mises equivalent stress or the mean stress
and s
u
the ultimate stress, has no physical or mechanical
interpretation in this complex loading context.
Parametric laws are based on the pioneering work of
Manson and Coffin
6, 16
with a proposed relation between
cumulated plastic strains and lifetime. However, this
cannot be simply extended for non isothermal multiaxial
loadings due to the same lack of mechanical and physical
meaning of plastic strain range in this particular con-
text.
17
This type of parametric law is interesting in an
industrial context because of its simplicity but it requires
an analysis of the more adapted mechanical fields in a
thermomechanical context.
The last point concerns the failure criterion, which
must guarantee the integrity of the structure during its
lifetime. In our particular context, the design objective
imposes that no macroscopic cracks should form during the
lifetime. Crack growth can be separated in two stages: an
initial one where the crack is confined to a small volume
and does not influence the macroscopic behavior of the
structure and a final one where the crack length directly
influences the load distribution in the structure. This
design criterion will impose the condition that the struc-
ture does not enter the final stage of cracking during its
lifetime.
The objective of this study is the interpretation of FEM
computations in order to obtain the critical damage zone
and the lifetime of the structure.
THERMOMECHANICAL FATIGUE ANALYSIS
FEM computations
The numerical evaluations depend essentially on the
choice of a constitutive law and an underlying integration
algorithm.
3
In order to take into account the plastic and viscous
behavior at low and high temperature, respectively, elas-
toviscoplastic constitutive laws have been chosen. Two
models have been finally selected, one classical unified
viscoplastic model
10
with a linear kinematic hardening:
e
elastic
1 n
E
n
E
trI
3
2
1
H
X
_
ee
vp
3
2
J
2
X
y
m
s X
J
2
X
;
_
_
ee
vp
E : e e
vp
1
and one involving an additive partition of stress into
viscous and plastic components
18
named two layer visco-
plastic model:
e
elastic
v
1 n
K
v
s
v
n
K
v
trs
v
I
e
elastic
p
1 n
K
p
p
n
K
p
trs
p
I
a
3
2
1
H
X
_
ee
v
3
2
J
2
s
v
m
s
v
J
2
s
v
_
ee
p
g
s
p
X
J
2
p
X
;
_
aa
_
ee
p
s
p
K
p
: e e
p
s
v
K
v
: e e
v
s
p
s
p
s
v
2
The main difference between the two models is the
coupling or uncoupling of the viscous and the plastic
dissipation mechanisms.
19
This could have an impact
on the final damage law if one wished to relate the
different constitutive quantities to the separate damage
variables. However, as already stated, both laws are
purely phenomenological and do not correspond to any
mechanical behavior at a microscopic scale.
The material parameters have been identified from iso-
thermal tension±compression±relaxation tests and are
supposed to vary linearly between the test temperatures.
It is obvious from this choice of constitutive models that
the transient cyclic behaviour of the material, hardening
or softening, has been neglected. This hypothesis has
later been verified on the LCF tests and has been
shown as reasonably correct.
Three numerical integration algorithms have been
tested under the Abaqus Standard finite element code
for the integration of the constitutive law: forward
Euler scheme (explicit), backward Euler scheme with a
first order series expansion (explicit), and a completely
implicit backward Euler scheme with radial return.
20
In
the anisothermal context with a changing yield limit it
turned out
19
that only the backward Euler scheme with a
radial return performed with reasonable convergence
rates and large time steps. The time performance
between the different schemes varied between 1 and 5.
The complete numerical computation for an exhaust
manifold, typically '10
5
degrees of freedom, took sev-
eral hours on a CRAY C-90 parallel computer.
Two particular aspects can be underlined. On one hand,
the comparison of the stress±strain loop obtained with
both models (see on Fig. 1) of a structure shows very
close mechanical responses which indicates a similar
modeling of the material behavior in the structure with
those different constitutive laws.
On the other hand, it is important to observe the multi-
axial aspects of the stress and strain tensors. Figure 2
shows the stress±strain loops obtained on an exhaust
manifold subjected to a thermomechanical loading
cycle, near the surface where s
zz
0.
One can remark that the other directions (xx and yy)
are both submitted to high stress and strain level. In
Figs 3 and 4, s
xx
and s
yy
are compared for a thermal
fatigue test specimen and for an exhaust manifold.
In the case of the specimen, the loading path is highly
non proportional and for the exhaust manifold, it is non
proportional but with approximately constant biaxiality
ratio. Those different examples show the multiaxial
mechanical responses of the different structures under
transient thermal loading. It will be important to deter-
mine a fatigue criterion compatible with this type of
− 0.05 − 0.04 − 0.03 − 0.02 − 0.01 0
− 400
− 300
− 200
− 100
0
100
200
300
400
500
Mechanical strain
Axial stress (MPa)
unified model
two layer model
Fig. 1 Comparison of both constitutive models on the mechanical
response obtained from the simulation of the thermal fatigue test.
4