1.
BASISTA Michał
- Micromechanical and Lattice Modeling of Brittle Damage.
- (Praca habilitacyjna).
- Warszawa 2001 s. 237.
- Prace
IPPT 3/2001.
2.
SZEMPLIŃSKA
- STUPNICKA Wanda, TYRKIEL Elżbieta
- Bifurkacje, chaos i fraktale w dynamice wahadła.
- Warszawa 2001 s. 32.
- Prace
IPPT 2/2001.
3.
ANTUNEZ
Horacio J.
- Bulk-metal
Forming Processes from Computational Modelling via Sensitivity Analysis to Tool
Shape Optimization.
- (Praca
habilitacyjna). - Warszawa
2001 s. 207.
- Prace IPPT
1/2001.
This thesis is a
multi-aspect study concerned with theoretical modeling of damage in brittle
solids with special emphasis on rocks and plain concrete. Motivated by the
complex and diverse nature of damage processes in these materials, an integrated
approach involving micromechanical, phenomenological and lattice modeling is
pursued. It is shown that these seemingly disparate classes of models turn to be
complementary in their objectives and utility. The bulk of the work is devoted
to the local description of damage problems in rock-like materials under
quasi-static mechanical loading (tensile or compressive) at isothermal
conditions. Slightly off the mainstream but well in tune with the
micromechanical damage modeling promoted in this thesis, a type of environmental
damage of concrete due to chemically aggressive ambient is also investigated.
Application of the methods of physics of critical phenomena to brittle damage
and fracture problems constitutes an important part of this study. It is
suggested that the percolation and other disorder models be used at large
microdefects densities where the traditional methods of micromechanical and
continuum damage mechanics cease to be valid. The individual chapters of the
thesis can be summarized as follows.
Chapter 1 presents in a concise way the basic definitions and assumptions
of the damage mechanics. Main differences (and similarities, if any) between
damage and fracture, damage and plasticity are pointed out. It is stressed that
the presently used name of damage mechanics (shortly DM) shall not be identified
with early phenomenological damage theories (CDM) which were chronologically the
first to deal with the damage problems in the framework of continuum mechanics.
The former is much a wider class comprising micromechanical damage models,
statistical damage models, statistical damage models, and the CDM models
themselves. In Chapter 1, these three classes of models are briefly
characterized and critically evaluated in the context of brittle deformation.
Chapter 2 provides experimental background for the analytical models
proposed in this thesis. Mechanisms of tensile and compressive microcracking in
rocks inducing inelastic macroscopic stress-strain behavior, microcrack related
anisotropy, positive dilatancy, pressure dependence, hysteresis, etc. are
discussed in considerable detail. Additionally, some basic facts about the
subcritical microcrack growth as well as the acoustic emission application to
rock microcracking and fracture are reviewed. It seems that acoustic emission
(AE) is gaining a steadily growing reputation as a reliable, non-destructive
technique capable of unraveling focal mechanisms and locating, both temporary
and spatially, the evolving microcracks within the material volume. The
important point is that AE allows for continuous screening of the microcrack
nucleation, growth and localization into a dominant fault.
Chapters 3, 4, 5, 6, 7, 8 constitute the original research part of the
thesis. Chapter 3 is concerned with a relatively straightforward case of damage
development in brittle materials subjected to displacement-controlled
(homogeneous) tensile loading. On the example of plain concrete and using a
general internal-variable thermodynamic framework, it is shown in a systematic
manner how a workable micromechanical damage model can be constructed starting
with the behavior of a single tensile microcrack in the unit cell. A new
microcrack growth condition (3.5)-(3.6) is proposed based on an experimentally
supported dependence of the fracture toughness on the microcrack length. The
model is formulated for the uniaxial tension and then generalized for
two-dimensional case. Typical configurations of interacting microcracks having
analytical solutions for the K 1
factors are incorporated into the model to illustrate how the stress
amplification effects might influence the overall s-e
behavior in tension.
Chapter 4 deals with the compressive damage in rocks. Of several possible
micromechanism of inelastic rock deformation in brittle regime, the sliding
microcrack model is selected for modeling purposes owing to its remarkable
versatility in replicating the macroscopically observed effects of rock response
in compression. The original contribution of this chapter includes application
of the Rice thermodynamic formalism with microstructural variables to the
sliding microcrack deformational mechanism, detailed analysis of all phases of
sliding microcrack behavior in loading and unloading, derivation of the ensuing
incremental stress-strain equations, their numerical implementation and
comparison of the computed results with the available test data on granite to
show the model at work.
Chapter 5 is devoted to estimation of the stress intensity factors for
interacting slits endowed with frictional and cohesive resistance and for
sliding microcracks with developed tension wings under overall compression. The
very effective analytical-numerical method of Kachanov (1987) is extended and
modified to account for friction and cohesion on the crack faces. Also, the case
of point-force loading on interacting cracks is considered. Using a developed
FORTRAN code, a number of test problems of crack interaction is solved and
compared with the corresponding „exact” BEM results.
In Chapter 6, a micromechanical model is formulated for the deformation
of hardened concrete exposed to chemical corrosion by sulphate ions migrating
into the concrete structure from ground water. The model is quite complex
involving several coupled physico-chemical processes such as non-steady
diffusion, heterogeneous chemical reactions, expansion of reaction products,
microcracking of heterogeneous matrix (hardened cement paste with ettringite
inclusion), and percolation. These processes are modeled on the microscale and
resulting equations are volume-averaged leading to the macroscopic expansions
which closely match the test data of Ouyang et al. (1988) on ASTM recommended
specimens.
Chapter 7 suggests a micromechanically-based phenomenological damage
model for brittle response of solids. For deformation processes dominated by
Mode I microcracking, the inelastic change of the compliance tensor is
identified as the flux, and the fourth-order tensor
Q = 1/2 (s Ä
s)
as the affinity. The conditions under which a damage potential exists are
indicated. Illustrative examples are worked out.
Chapter 8 is devoted to modeling brittle damage and fracture using
methods of statistical physics of the critical phenomena - an alternative
modeling methodology which is entirely different from the conventional
(micromechanical or phenomenological) damage mechanics models. In the search for
possible interrelations between damage mechanics and statistical lattice models,
two classes of disordered systems are given particular attention: percolation
model and central-force model. It is found that the percolation theory can be
quite useful when verifying the accuracy of the effective-continua methods in
predicting the effective elastic constants of a damaged solid. Furthermore,
results of the numerical simulations on central-force triangular in Hansen, et
al. (1989) are compared with the corresponding analytical estimates yielded by
the parallel bar model (widely used in damage mechanics). To this end, a number
of original results is obtained in this chapter including accuracy assessment of
the effective-media methods, selection of the secant effective compliance tensor
as the damage variable, microcrack interaction in tension, etc.
Opracowanie
materiału przedstawionego w tym zeszycie zostało poprzedzone bogatym doświadczeniem
dydaktycznym, studiami literatury naukowej na temat zjawisk drgań chaotycznych
w układach fizycznych oraz publikacjami serii oryginalnych prac naukowych w międzynarodowych
czasopismach naukowych. Dodatkowym, ale bardzo istotnym doświadczeniem były
seminaria, referaty lub krótkie serie wykładów, przedstawione zarówno w IPPT
PAN, jak i na wyższych uczelniach dla tych środowisk naukowych, których
zainteresowanie zjawiskami drgań chaotycznych w prostych deterministycznych układach
nie było poprzedzone systematycznymi studiami na ten temat. Ograniczony czas
seminarium lub referatu na konferencji naukowej dawał tu bodźce do przemyślenia,
jaką wybrać metodę referowania materiału tak, by trafił on do wyobraźni i
przekonania słuchaczy w sposób prosty, a zarazem pobudził ich zainteresowanie
i zachęcił do głębszego studiowania przedmiotu. Ten kierunek myślenia
doprowadził do spostrzeżenia, że w tej nieuchwytnie matematycznie dziedzinie
dobrą metodą jest przedstawienie zarówno zagadnień podstawowych, jak i
zaawansowanych, przy maksymalnym wykorzystaniu interpretacji geometrycznej.
Interpretacja ta posługuje się w dużej mierze rysunkami: zarówno wykresami
schematycznymi, jak i graficzną interpretacją wyników obliczeń
komputerowych.
Po wygłoszeniu referatów na temat własnych wyników
w dziedzinie drgań chaotycznych na konferencjach krajowych, często padało
pytanie o literaturę podstawowa na te tematy. Chodziło oczywiście o książkę
dostępną w Polsce, i to książkę nadającą się do wstępnego zapoznania się
z przedmiotem. Najczęściej odpowiadam wtedy, że najlepiej zacząć od książki
F. Moona pt. Chaotic vibrations, an introduction for applied scientists and
engineers [1], aczkolwiek zdawałam sobie sprawę, że książka ta nie jest
powszechnie dostępna w Polsce. Poza tym jest ona dość obszerna, a
przedstawiony materiał jest tak poszatkowany na dużą liczbę rozdziałów, że
przestudiowanie jej wcale nie jest łatwe. Istnieje jednak pierwsza wersja tej
książki o mniejszej objętości. Otóż, jak pisze prof. Moon we wstępie, bodźcem
do napisania tej książki było zaproszenie IPPT PAN w roku 1984 do wygłoszenia
8 godzin wykładów na temat drgań chaotycznych, i że książka ta jest właśnie
rozszerzeniem tematu tych wykładów. Tak więc, pierwszą, krótszą wersję
książki F. Moona można znaleźć w zeszycie IPPT 28/1985 pt. Chaos w
nieliniowej mechanice [2], zawierającym prace przygotowane na konferencję
szkoleniową pod tym samym tytułem, która odbyła się w Jabłonnie w dniach
12-17 sierpnia 1984 r.
W latach późniejszych ukazały się polskie tłumaczenia
niektórych książek opartych na materiale pełnych cykli wykładów, przeważnie
na studiach doktoranckich. Wymienię tu przede wszystkim książkę H.G.
Schustera pt. Chaos deterministyczny [3] oraz E. Otta pt. Chaos w układach
dynamicznych [4], obie ukierunkowane na studia fizyczne. Warta uwagi jest książka
J. Kudrewicza pt. Fraktale i chaos [5]. Z powszechnym zainteresowaniem spotkała
się książka popularno-naukowa I. Stewarta pod intrygującym tytułem Czy Bóg
gra w kości? [6].
Przedstawione rozważania na temat książek dostępnych
w Polsce zarówno na rynkach księgarskich jak i w bibliotekach naukowych, jak również
własne doświadczenia dydaktyczne doprowadziły do wniosku, że warto pokusić
się o upowszechnienie wiedzy na temat drgań chaotycznych w deterministycznych
prostych oscylatorach przez opracowanie publikacji ujmującej tematykę w zupełnie
inny sposób niż klasyczne ujęcie podręcznikowe. Ten inny sposób polega m.
in. na:
·
skierowanie uwagi czytelnika na jeden, a w dalszej kolejności na następne,
dobrze znany deterministyczny model dysypatywnego układu drgającego o jednym
stopniu swobody; model, który można sprowadzić do modelu fizycznego kulki
poruszającej się po wyznaczonym torze pod działaniem znanych i ciągłych w
opisie matematycznym sił. A ponieważ trudno o bardziej znany układ drgający
zbadany doświadczalnie niż wahadło matematyczne poddane działaniu zewnętrznego
periodycznego wymuszenia, przedstawiony zeszyt dotyczy właśnie tego układu;
·
przypomnieniu najpierw własności układu liniowego, a dalej słabo
nieliniowego, przez pryzmat wyników badań doświadczalnych i komputerowych,
bez stosowania wzorów i przekształceń matematycznych. Następnie, w miarę
zwiększania amplitudy wymuszenia i zbliżania się do zjawisk o charakterze
chaotycznym, wyjaśnieniu i interpretowaniu pojawienia się takich zjawisk jak
bifurkacje lokalne, granice obszarów przyciągania itd., również w
interpretacji geometrycznej. Nie odrywamy tu uwagi czytelnika pokazując np. pełną
klasyfikację różnych typów stateczności i niestateczności punktów równowagi
(osobliwych), czy pełnej listy różnorodnych typów bifurkacji. Czytelnik
obserwuje tylko te zjawiska, które się pojawiają w rozważanej dynamice wahadła;
·
oddzielenie od tekstu podstawowego tych fragmentów, które można ominąć przy
pierwszym czytaniu. Fragmenty te (pisane mniejszą czcionką) zawierają
rozszerzenie materiału, przedstawiając zarówno uwagi na temat tych problemów,
które występują w dynamice wahadła, jak i pewne dodatkowe uwagi teoretyczne,
odsyłając czytelnika do odnośnej literatury;
·
ujęciu w ten prosty sposób również zaawansowanych problemów i najnowszych
wyników dotyczących związku między teoretycznym pojęciem globalnej
bifurkacji a fraktalną strukturą granic obszarów przyciągania, zjawiskiem
chaosu przejściowego i wrażliwością na warunki początkowe;
·
połączeniu w jedną całość koncepcji drgań chaotycznych i fraktali,
poprzez pokazanie fraktalnej struktury dziwnego (chaotycznego) atraktora.
Część przedstawionych wyników została opublikowana
w czasopismach International Journal of Bifurcation and Chaos, Nonlinear
Dynamics oraz Computer Assisted Mechanics and Engineering Science w latach
1997-2001, a część została wykonana dla potrzeb niniejszego opracowania.
Wszystkie obliczenia komputerowe i graficzne opracowanie wyników wykonane zostały
przez dr Elżbietę Tyrkiel, współautorkę niniejszej publikacji.
Towards the end of
the sixties and the beginning of the seventies, electronic computers started to
be developed in a larger scale than up to then, and thus became available to
researches. And quite quickly a number of already available methods to solve
engineering problems began to be applied.
Until then such methods has required, even to obtain at least a roughly
meaningful result, a prohibitively heavy calculation works as they had to be
applied by hand. Basically this involved the solution of relatively large
systems of equations or the repetitive solution of rather simple problems.
By that time a good amount of theoretical knowledge, which allowed the
successful solution of many linear problems, was available. Mechanics, material
science, algebra and numerical analysis provided the necessary background for
these methods to be applied.
The first applications of numerical methods concerned those linear
problems. Among them, static problems of civil engineering were on top of the
list. And this to the extent that many concepts and names taken from this field
have remained in the new developing computer methods even when applied to other
problems. Typical examples of problems solved in this period are linear
elasticity and heat conduction.
Shortly afterwards, it became clear that nonlinear problems should also
be addressed, and this to satisfy countless practical requirements. Hence, the
development of a number of disciplines started. In material science, an
outstanding step was performed by including more complex constitutive equations,
which involved more state and history variables. On the other hand, more
powerful mathematical tools were developed to describe large displacement and
deformations. Boundary conditions of ongoing processes were more precisely
described including, among others, contact conditions and friction.
It would be hard to summarize all the developments of computational
mechanics in the recent decades. Just to mention the subjects they concerned,
effort was concentrated in material modelling, geometry and boundary conditions
description, numerical methods for discretization and for efficient solution of
the equation systems, and operations involving the mesh of finite elements
including optimization of the bandwidth, remeshing, mesh refinement and
coarsening, all aiming at minimizing an appropriate error norm. Practically all
the engineering problems of interest have been addressed, many of them involving
coupled problems.
Still further elaborations included the use of the solved system of
equations to perform sensitivity analysis by the so-called analytical methods.
These resulted to provide the sensitivity coefficients, and did that at a
residual cost as compared to the cost of solving the overall problem. The
inclusion of sensitivity analysis with respect to shape parameters finally
prepared the way to shape optimization since not only a given design functional,
but also its gradient in the design space were available to be used by an
optimization algorithm.
In this historical context, one of the fields which has attracted more
interest is that of metal forming simulations, probably for its interest for
industrial applications. Practically all the subjects quoted above for
computational mechanics in general are dealt with in this field.
A series of options are available for building a computational model for
metal forming simulation. Most frequently, each choice implies to reject other
possibilities, and some goals are obtained but some drawbacks arise. Hints about
the proper choice may be given by the type of process being simulated. This,
confirmed by the experience of many researches, supports the position that it is
more convenient to have several specialized programs, each of which effectively
analyzes a given class of problems, than to have a complex, much larger general
purpose program which, to given extent, is adapted to the characteristic
features of (almost?) every practical problem.
Within the preceding framework, the present work is a collection of
developments performed at the Institute of Fundamental Technological Research of
PAS. All of them are concerning one specific way of modelling metal forming
processes; this is especially suited for hot forming conditions, and has been
conceived for steady-state processes, although it can be applied in transient
ones as well.
The naturally suited application of this model is extrusion; however, it
can be used also in free forging, cutting and rolling (including seamless tube
rolling) for which results are shown as well.
Stationary and transient processes are separately presented here; not
only for the sake of clarity of the exposition and the for historical reasons,
but to show the analysis problem, sensitivity analysis and shape optimization as
successive steps within a logical line of thinking. Essentially this process was
followed in the computer simulation of metal forming, feeding into each step all
the results obtained in the preceding ones.
In this work, the steady state is presented first, starting by the flow
approach. Following, the discretization by finite elements is given. Here the
different features accounted for in the model are explained. Once the analysis
model is complete, the sensitivity is ready to be introduced. First, parametric
sensitivity is discussed, what is incidentally useful to show the available
methods for sensitivity analysis. Shape sensitivity is considered next, followed
by the optimization algorithm which finds, according to given criteria and
design restrictions, the optimum design.
Afterwards transient processes are considered. A full transient
formulation is considered and used to obtain an incremental method that makes of
linear elements due to a proper time-step splitting. Attention is focused on the
full explicit version. Further, sensitivity analysis within such model and
discretization is shown. In addition, the pseudo-concentration method is briefly
revisited and used in connection to Fourier series expansion of the problem of
seamless tube rolling.
Some additional -but significant- topics are discussed in the course of
the main presentation. The simulation of the almost perfect plasticity poses the
problem of uniqueness (or its lack), and this is discussed in the context of
cutting simulation. Pressure stabilization necessary to apply a friction model
based on a Coulomb-type law. In this context a bilinear interpolation is
introduced, which takes advantage of a method for similar pressure stabilization
used in fluid mechanics. Sensitivity analysis suggests that its results can have
an additional application in evaluating the effect on the solution of numerical
parameters needed in some models. This is the case of the upwind parameters in
coupled thermo-mechanical problems and the step in time integration of transient
processes. The introduction of shape sensitivity analysis of forming processes
gives the occasion to consider the extension to large displacements of the two
available methods. In the shape optimization part, the problem of shape
parameterization requires special attention. Two different techniques of
interpolating points in the discretized domain in terms of the design parameters
are proposed.