Table Of ContentLecture Notes in Applied
and Computational Mechanics
Volume 10
Series Editors
Prof. Dr.-lng. Friedrich Pfeiffer
Prof. Dr.-lng. Peter Wriggers
Springer
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Heidelberg
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London
ONLINE LIBRARY
Milan Engineering
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Tokyo http://www.springer.de/engine/
Deformation and Failure
in Metallic Materials
Kolumban Hutter
Herbert Baaser (Eds.)
, Springer
Professor Dr. KOLUMBAN HUTTER
DR. HERBERT BAASER
Technical University of Darmstadt
Department of Mechanics
Hochschulstr. 1
D-64289 Darmstadt
GERMANY
e-mail: [email protected]
e-mail: [email protected]
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ISBN 978-3-642-05649-9 ISBN 978-3-540-36564-8 (eBook)
DOI 10.1007/978-3-540-36564-8
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Preface
A "Sonderforschungsbereich" (SFB) is a programme of the "Deutsche For
schungsgemeinschaft" to financially support a concentrated research effort of
a number of scientists located principally at one university, research labora
tory or a number of these situated in close proximity to one another so that
active interaction among individual scientists is easily possible.
Such SFBs are devoted to a topic, in our case "Deformation and Fail
ure in Metallic and Granular Materials", and financing is based on a peer
reviewed proposal for three (now four) years with the intention of several
prolongations after evaluation of intermediate progress and continuation re
ports. An SFB is terminated in general by a formal workshop, in which the
state of the art of the achieved results is presented in oral or/and poster
communications to which also guests are invited with whom the individual
project investigators may have collaborated. Moreover, a research report in
book form is produced in which a number of articles from these lectures are
selected and collected, which present those research results that withstood a
rigorous reviewing process (with generally two or three referees).
The theme deformation and failure of materials is presented here in two
volumes of the Lecture Notes in Applied and Computational Mechanics by
Springer Verlag, and the present volume is devoted to metallic continua.
The complementary volume (Lecture Notes in Applied and Computational
Mechanics, vol. 11, Eds. K. HUTTER & N. KIRCHNER) is dedicated to the
Dynamic Response of Granular and Porous Materials under Large and Catas
trophic Deformations.
The SFB "Deformation and Failure in Metallic and Granular Materials"
lasted from October 1994 until December 2002, thus a total of slightly more
than eight years, and had an interdisciplinary focus: Teachers, researchers, in
cluding Ph.D. students from various University Departments were involved,
namely Mathematics, Mechanics, Material Sciences, Civil and Mechanical
Engineering. Many projects were headed by researchers from two different
departments. Each project had one - sometimes two - principal researchers
who either as Ph.D. students or postdoctoral assistants would perform the
actual research under the supervision of the proposers of the individual pro
posals.
This volume tries to summarise the obtained results not so much in a
form as it would appear in specialised peer reviewed periodicals, but such
that a broader community is able to follow the red lines of the arguments in
each article. This required that the authors were asked to include also items
that may be obvious to specialists. We hope that this endeavour has been
achieved.
VI
This book on the Deformation and Failure in Metallic Mater'ials is divided
into four major parts:
(I) Constitutive behaviour - modelling and numerics,
(II) Constitutive behaviour - experiments and verification,
(III) Constitutive behaviour - mathematical foundation and
(IV) Damage and fracture.
These four topics define the main classes into which the 16 different articles
may fit. They reflect a fairly objective cross section of the research that was
conducted during the eight years this SFB existed.
Part I. Constitutive Behaviour - Modelling and Numerics
This largest part of the book consists of five articles which all deal in one way
or an other with numerical integration of systems of differential equations,
mixed with partial and ordinary differential equations and in two specific
cases also containing algebraic equations. The equations derive from descrip
tions of material equations involving inelastic material behaviour in the spirit
of plasticity, viscoplasticity and elasto-viscoplasticity, generally involving a
yield surface. Numerical integration techniques of material equations of the
mentioned class has been a center of activity of the SFB.
• BUTTNER & SIMEON study the numerical properties of materials hav
ing constitutive relations of the elastoplastic and viscoplastic type with
yield surfaces and classify these equations in the context of differential
algebraic equations and thereby motivate the temporal discretillation by
implicit RUNGE-KuTTA schemes. They apply the methods to deduce ex
istence and uniqueness results for coersive methods and demonstrate that
algebraically stable methods preserve the contractivity of the elastoplas
tic flow. By combining the fully implicit RUNGE-KUTTA methods with
the two-level NEWTON method of computational plasticity they deduce a
return mapping technique that delivers improved accuracy and efficiency
of the numerical solution
• In their article on elasto-visco-plasticity SCHERF & SIMEON provide a
specific example that an initial boundary value problem of elasto-visco
plasticity leads to a system of differential-algebraic equations. They show
that the above mentioned integration technique can be improved by using
the second order time integration method BDF-2. They further propose
an alternative to the standard two-stage NEWTON process and use it in
simulation experiments to demonstrate the possible gain in accuracy and
efficiency.
• G RUTTMANN & EIDEL present a constitutive model for orthotropic elasto
plasticity at finite plastic strain and propose a scheme of its numerical
implementation. The essential ingredients of the model are the multi
plicative decomposition of the deformation tensor into elastic and plastic
VII
parts and an invariant setting of the elastic free energy and the yield
function by means of structure tensors reflecting the privileged directions
of the material. The exponential map approach is used to integrate the
associated flow rule, thus preserving plastic incompressibility. Implemen
tation of the constitutive relations into a brick-type shell element paired
with special interpolation techniques allows prediction of locking free de
formations, even for very thin structures. Typical examples illustrate the
suitability of the method.
• Of very similar intent is the article by TSAKMAKIS & WILLUWEIT de
voted to the time integration algorithms for finite deformation plasticity.
Because of the ordinary differential-algebraic-equation structure of the
classical plasticity theories, special elastic predictor-plastic corrector in
tegration schemes are proposed that preserve the plastic incompressibility
condition and employ the exponential map approach. The model equa
tions also incorporate isotropic and kinematic hardening.
• SANSOUR, KOLLMANN & BOCKO propose a non-linear model of finite
strain multiplicative inelasticity with an anisotropic elastic constitutive
law. Since complete proposals for anisotropic strain energy functions are
not known to date the authors motivate their own proposal and only af
terwards direct their attention to numerical integration which is consider
ably complicated by the anisotropy. The numerical details are presented
in full and then illustrated with applications to shells.
Part II. Constitutive Behaviour - Experiments and Verification
Experiments can be used in connection with constitutive models of continuous
systems in basically two different forms. First, one may verify a theoretical
model by comparing computational results for a typical boundary value prob
lem with experimental results for a realistic arrangement of the same problem.
Second, experiments may be used to validate a continuum mechanical model
by identifying its phenomenological parameters.
• The article by HARTH, LEHN & KOLLMANN is of this second type. The
material parameters of their inelastic constitutive model are identified by
minimization of the distance between the model response and the experi
mental data using an optimization algorithm. Because the amount of test
data is not sufficient for a statistical analysis, a method of stochastic si
mulation is used to generate artificial data with the same behaviour as the
experimental data. This procedure requires some care in the application
of the loading history of the experiments when the parameters are iden
tified, but the method is a suitable tool to enlarge data sets. Parameters
of a number of inelastic constitutive models are thus identified.
• DAFALIAS, SCHICK & TSAKMAKIS address the problem of induced aniso
tropy in large deformation plasticity. In metallic materials, deformation
induced anisotropy is reflected by the translation, rotation and distortion
of the yield surface. The authors present a thermodynamically consistent
VIII
model describing the evolving anisotropy and show its imprint in the
deformation of the yield surface. Various load histories, applied in an
experimental arrangement by ISHIKAWA, are analysed and it is shown hm\"
the yields surfaces obtained from the experiments and the computations
compare with one another.
• EMMEL, STIEFEL, GROSS & RODEL compare experimental and numeri
cal results in a problem of damage and failure of ceramic metal composites
due to temperature induced stresses. Two different structures, a metal
layer between two ceramic supports and an inter-penetrating network,
correlated to two different failure modes are studied. Cavitation is found
to occur as is interface debonding in conjunction with crack branching at
the interface. Experimental observations together with numerical calcu
lations lead to a theoretical description for the prediction of the failure
mode and the critical stresses in the composite.
Part III. Constitutive Behaviour - Mathematical Foundation
It is well known that continuum mechanics paired with inelastic constitutive
models offers a wealth of nonlinear initial boundary value problems for ex
istence and uniqueness tests. The mathematicians in our SFB devoted their
activities to such questions, ideally by taking up models dealt with by the
engineers in a more applied context. This fruitful interplay has made the
models designed by engineers and material scientists more reliable and more
trustworthy.
• One central activity among the mathematicians within our SFn has been
with the existence theory of initial boundary value problems for the class
of monotone operators. The article by CHELMINSKI deals with such an
existence theory of global in time solutions to systems describing the in
elastic behaviour of metals. Geometrically linear deformations are studied
and the inelastic constitutive models are of the monotone type, leaving
nonmonotone constitutive relations to a few closing remarks.
• The initial boundary value problems that are presented by NEFF de
rive from models of finite elastoplasticity and apply the multiplicative
decomposition of the deformation gradient. For such a model, based on
the ESHELBY tensor, the behaviour of the systems at frozen plastic flow
is studied. Extending the analysis to viscoplastic flows and introducing
stringent elastic stability assumptions and a nonlocal extension in space
the author proves local existence in time of the corresponding initial value
problems. A new model is also introduced that is suitable for small elas
tic strains. Its key feature is an independent elastic rotation for which
a closure involving the elastic deformation tensor is proposed. For this
case, the equilibrium equations at frozen plastic flow are now linear ellip
tic leading straight forwardly to existence and uniqueness results without
additional stability assumptions.
• EBENFELD attacks in his analysis of initial boundary value problems in
continuum mechanics those material formulations which automatically
IX
lead to problems that admit smooth solutions. The models, constrained
this way yield equations of hyperbolic or parabolic type and the con
ditions constitute the mathematical counterpart to those conditions ob
tained by the second law of thermodynamics. The mathematical formu
lation of the problem is presented but no proofs are given.
• A challenging mathematical question in the context of homogenization is
the simultaneous proof of existence of solutions of the homogenized prob
lem at the macrolevel and a corresponding problem on the microlevel.
This problem is attacked by ALBER in his article. Within the context of
viscoplasticity (with monotone constitutive operators) it is shown that
the derived homogenized initial boundary value problem possesses a so
lution. From this, an asymptotic solution of the microproblem is con
structed with the property that the difference between the exact and
asymptotic solutions tends to zero if the length scale of the microscale
becomes vanishingly small.
Part IV. Damage and Fracture
Both are important notions in a balanced treatment of metallic and composite
materials. Failure occurs because of damage of the material and requires the
introduction of a model for its evolution. The four articles in this part address
such questions.
• The article by THOMAS et al. gives a review of the effect of the stress state
on deformation, roughness and damage in metal forming. The overview
presents elasto-plastic constitutive relations including anisotropy induced
during metal forming and by crack propagation. The arising material pa
rameters are identified by experiments. This leads to a modified G URSON
model. The article also discusses the feasibility of different experimental
techniques by which the local strain is determined.
• BAASER & GROSS realize 3D simulations of ductile damage and crack
propagation processes by the use of continuum damage mechanics such as
the models proposed by G URSON or ROUSSELIER. The numerical imple
mentation in the finite element method (FEM) is based on locking-free
20-noded brick elements in the finite strain regime. The main attention
is directed to the well-known disadvantage of modeling strain-softening
behaviour, the mesh-dependence occuring due to the loss of ellipticity of
the leading differential equations. The initially given well-posedness of
the system is checked permanently during the iteration process by evolv
ing the determinant of the acoustic tensor in a very efficient way. They
claim, that softening behaviour can be treated very well until the type
of diffential equations changes and the results become questionable due
to mesh-dependence. However, the point of loosing ellipticity can not be
computed a priori, the testing has to be exhibited in a steady manner.
• SCHMIDT, RICHTER & G ROSS discuss the modelling and simulation of
mode I crack growth in metal foames. Its initiation and subsequent re-
